Changeset 10368 for NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex

NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/abstract_foreword.tex

-                      r9393
+                      r10368
 {\small
+The ocean engine of NEMO (Nucleus for European Modelling of the Ocean) is a primitive
+equation model adapted to regional and global ocean circulation problems. It is intended to
+be a flexible tool for studying the ocean and its interactions with the others components of
+the earth climate system over a wide range of space and time scales.
+Prognostic variables are the three-dimensional velocity field, a non-linear sea surface height,
+the \textit{Conservative} Temperature and the \textit{Absolute} Salinity.
+In the horizontal direction, the model uses a curvilinear orthogonal grid and in the vertical direction,
+a full or partial step $z$-coordinate, or $s$-coordinate, or a mixture of the two.
+The distribution of variables is a three-dimensional Arakawa C-type grid.
+Various physical choices are available to describe ocean physics, including TKE, and GLS vertical physics.
+Within NEMO, the ocean is interfaced with a sea-ice model (LIM or CICE), passive tracer and
+biogeochemical models (TOP) and, via the OASIS coupler, with several atmospheric general circulation models.
+It also support two-way grid embedding via the AGRIF software.
+  The ocean engine of NEMO (Nucleus for European Modelling of the Ocean) is a primitive equation model adapted to
+  regional and global ocean circulation problems.
+  It is intended to be a flexible tool for studying the ocean and its interactions with
+  the others components of the earth climate system over a wide range of space and time scales.
+  Prognostic variables are the three-dimensional velocity field, a non-linear sea surface height,
+  the \textit{Conservative} Temperature and the \textit{Absolute} Salinity.
+  In the horizontal direction, the model uses a curvilinear orthogonal grid and in the vertical direction,
+  a full or partial step $z$-coordinate, or $s$-coordinate, or a mixture of the two.
+  The distribution of variables is a three-dimensional Arakawa C-type grid.
+  Various physical choices are available to describe ocean physics, including TKE, and GLS vertical physics.
+  Within NEMO, the ocean is interfaced with a sea-ice model (LIM or CICE),
+  passive tracer and biogeochemical models (TOP) and,
+  via the OASIS coupler, with several atmospheric general circulation models.
+  It also support two-way grid embedding via the AGRIF software.
+}
 …
 \chapter*{Disclaimer}
+Like all components of NEMO, the ocean component is developed under the \href{http://www.cecill.info/}{CECILL license},
+which is a French adaptation of the GNU GPL (General Public License). Anyone may use it
+freely for research purposes, and is encouraged to communicate back to the NEMO team
+its own developments and improvements. The model and the present document have been
+made available as a service to the community. We cannot certify that the code and its manual
+are free of errors. Bugs are inevitable and some have undoubtedly survived the testing phase.
+Users are encouraged to bring them to our attention. The author assumes no responsibility
+for problems, errors, or incorrect usage of NEMO.
+Like all components of NEMO,
+the ocean component is developed under the \href{http://www.cecill.info/}{CECILL license},
+which is a French adaptation of the GNU GPL (General Public License).
+Anyone may use it freely for research purposes,
+and is encouraged to communicate back to the NEMO team its own developments and improvements.
+The model and the present document have been made available as a service to the community.
+We cannot certify that the code and its manual are free of errors.
+Bugs are inevitable and some have undoubtedly survived the testing phase.
+Users are encouraged to bring them to our attention.
+The author assumes no responsibility for problems, errors, or incorrect usage of NEMO.
 \vspace{1cm}

NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/annex_A.tex

-                      r9414
+                      r10368
 In order to establish the set of Primitive Equation in curvilinear $s$-coordinates
+($i.e.$ an orthogonal curvilinear coordinate in the horizontal and an Arbitrary Lagrangian
+Eulerian (ALE) coordinate in the vertical), we start from the set of equations established
+in \autoref{subsec:PE_zco_Eq} for the special case $k = z$ and thus $e_3 = 1$, and we introduce
+an arbitrary vertical coordinate $a = a(i,j,z,t)$. Let us define a new vertical scale factor by
+$e_3 = \partial z / \partial s$ (which now depends on $(i,j,z,t)$) and the horizontal
+slope of $s-$surfaces by :
+($i.e.$ an orthogonal curvilinear coordinate in the horizontal and
+an Arbitrary Lagrangian Eulerian (ALE) coordinate in the vertical),
+we start from the set of equations established in \autoref{subsec:PE_zco_Eq} for
+the special case $k = z$ and thus $e_3 = 1$,
+and we introduce an arbitrary vertical coordinate $a = a(i,j,z,t)$.
+Let us define a new vertical scale factor by $e_3 = \partial z / \partial s$ (which now depends on $(i,j,z,t)$) and
+the horizontal slope of $s-$surfaces by:
 \begin{equation} \label{apdx:A_s_slope}
 \sigma _1 =\frac{1}{e_1 }\;\left. {\frac{\partial z}{\partial i}} \right|_s
 …
 \end{equation}
+The chain rule to establish the model equations in the curvilinear $s-$coordinate
+system is:
+The chain rule to establish the model equations in the curvilinear $s-$coordinate system is:
 \begin{equation} \label{apdx:A_s_chain_rule}
 \begin{aligned}
 …
 \end{equation}
 In particular applying the time derivative chain rule to $z$ provides the expression
 for $w_s$,  the vertical velocity of the $s-$surfaces referenced to a fix z-coordinate:
+In particular applying the time derivative chain rule to $z$ provides the expression for $w_s$,
+the vertical velocity of the $s-$surfaces referenced to a fix z-coordinate:
 \begin{equation} \label{apdx:A_w_in_s}
 w_s   =  \left.   \frac{\partial z }{\partial t}   \right|_s
 …
 \label{sec:A_continuity}
 Using (\autoref{apdx:A_s_chain_rule}) and the fact that the horizontal scale factors
+$e_1$ and $e_2$ do not depend on the vertical coordinate, the divergence of
 the velocity relative to the ($i$,$j$,$z$) coordinate system is transformed as follows
 in order to obtain its expression in the curvilinear $s-$coordinate system:
+Using (\autoref{apdx:A_s_chain_rule}) and
+the fact that the horizontal scale factors $e_1$ and $e_2$ do not depend on the vertical coordinate,
+the divergence of the velocity relative to the ($i$,$j$,$z$) coordinate system is transformed as follows in order to
+obtain its expression in the curvilinear $s-$coordinate system:
 \begin{subequations}
 …
 \end{subequations}
 Here, $w$ is the vertical velocity relative to the $z-$coordinate system.
 Introducing the dia-surface velocity component, $\omega $, defined as
 the volume flux across the moving $s$-surfaces per unit horizontal area:
+Here, $w$ is the vertical velocity relative to the $z-$coordinate system.
+Introducing the dia-surface velocity component,
+$\omega $, defined as the volume flux across the moving $s$-surfaces per unit horizontal area:
 \begin{equation} \label{apdx:A_w_s}
 \omega  = w - w_s - \sigma _1 \,u - \sigma _2 \,v    \\
 \end{equation}
 with $w_s$ given by \autoref{apdx:A_w_in_s}, we obtain the expression for
 the divergence of the velocity in the curvilinear $s-$coordinate system:
+with $w_s$ given by \autoref{apdx:A_w_in_s},
+we obtain the expression for the divergence of the velocity in the curvilinear $s-$coordinate system:
 \begin{subequations}
 \begin{align*} {\begin{array}{*{20}l}
 …
 \end{subequations}
+As a result, the continuity equation \autoref{eq:PE_continuity} in the
+$s-$coordinates is:
+As a result, the continuity equation \autoref{eq:PE_continuity} in the $s-$coordinates is:
 \begin{equation} \label{apdx:A_sco_Continuity}
 \frac{1}{e_3 } \frac{\partial e_3}{\partial t}
 …
  +\frac{1}{e_3 }\frac{\partial \omega }{\partial s} = 0
 \end{equation}
 A additional term has appeared that take into account the contribution of the time variation
 of the vertical coordinate to the volume budget.
+A additional term has appeared that take into account
+the contribution of the time variation of the vertical coordinate to the volume budget.
 …
 \label{sec:A_momentum}
 Here we only consider the first component of the momentum equation,
+Here we only consider the first component of the momentum equation,
 the generalization to the second one being straightforward.
 …
 $\bullet$ \textbf{Total derivative in vector invariant form}
 Let us consider \autoref{eq:PE_dyn_vect}, the first component of the momentum
+equation in the vector invariant form. Its total $z-$coordinate time derivative,
 $\left. \frac{D u}{D t} \right|_z$ can be transformed as follows in order to obtain
+Let us consider \autoref{eq:PE_dyn_vect}, the first component of the momentum equation in the vector invariant form.
+Its total $z-$coordinate time derivative,
+$\left. \frac{D u}{D t} \right|_z$ can be transformed as follows in order to obtain
 its expression in the curvilinear $s-$coordinate system:
 …
 \end{subequations}
+%
+Applying the time derivative chain rule (first equation of (\autoref{apdx:A_s_chain_rule}))
+to $u$ and using (\autoref{apdx:A_w_in_s}) provides the expression of the last term
+of the right hand side,
+Applying the time derivative chain rule (first equation of (\autoref{apdx:A_s_chain_rule})) to $u$ and
+using (\autoref{apdx:A_w_in_s}) provides the expression of the last term of the right hand side,
 \begin{equation*} {\begin{array}{*{20}l}
 w_s  \;\frac{\partial u}{\partial s}
 …
 \end{array} }
 \end{equation*}
 leads to the $s-$coordinate formulation of the total $z-$coordinate time derivative,
+leads to the $s-$coordinate formulation of the total $z-$coordinate time derivative,
 $i.e.$ the total $s-$coordinate time derivative :
 \begin{align} \label{apdx:A_sco_Dt_vect}
 …
   + \frac{1}{e_3 } \omega \;\frac{\partial u}{\partial s}
 \end{align}
 Therefore, the vector invariant form of the total time derivative has exactly the same
+mathematical form in $z-$ and $s-$coordinates. This is not the case for the flux form
 as shown in next paragraph.
+Therefore, the vector invariant form of the total time derivative has exactly the same mathematical form in
+$z-$ and $s-$coordinates.
+This is not the case for the flux form as shown in next paragraph.
 $\ $\newline    % force a new ligne
 …
 $\bullet$ \textbf{Total derivative in flux form}
+Let us start from the total time derivative in the curvilinear $s-$coordinate system
+we have just establish. Following the procedure used to establish (\autoref{eq:PE_flux_form}),
+it can be transformed into :
+Let us start from the total time derivative in the curvilinear $s-$coordinate system we have just establish.
+Following the procedure used to establish (\autoref{eq:PE_flux_form}), it can be transformed into :
 %\begin{subequations}
 \begin{align*} {\begin{array}{*{20}l}
 …
 \end{subequations}
 which leads to the $s-$coordinate flux formulation of the total $s-$coordinate time derivative,
 $i.e.$ the total $s-$coordinate time derivative in flux form :
+$i.e.$ the total $s-$coordinate time derivative in flux form:
 \begin{flalign}\label{apdx:A_sco_Dt_flux}
 \left. \frac{D u}{D t} \right|_s   = \frac{1}{e_3}  \left. \frac{\partial ( e_3\,u)}{\partial t} \right|_s
 …
 \end{flalign}
 which is the total time derivative expressed in the curvilinear $s-$coordinate system.
+It has the same form as in the $z-$coordinate but for the vertical scale factor
+that has appeared inside the time derivative which comes from the modification
+of (\autoref{apdx:A_sco_Continuity}), the continuity equation.
+It has the same form as in the $z-$coordinate but for
+the vertical scale factor that has appeared inside the time derivative which
+comes from the modification of (\autoref{apdx:A_sco_Continuity}),
+the continuity equation.
 $\ $\newline    % force a new ligne
 …
 \end{split}
 \end{equation*}
 Applying similar manipulation to the second component and replacing
 $\sigma _1$ and $\sigma _2$ by their expression \autoref{apdx:A_s_slope}, it comes:
+Applying similar manipulation to the second component and
+replacing $\sigma _1$ and $\sigma _2$ by their expression \autoref{apdx:A_s_slope}, it comes:
 \begin{equation} \label{apdx:A_grad_p_1}
 \begin{split}
 …
 \end{equation}
+An additional term appears in (\autoref{apdx:A_grad_p_1}) which accounts for the
+tilt of $s-$surfaces with respect to geopotential $z-$surfaces.
+As in $z$-coordinate, the horizontal pressure gradient can be split in two parts
+following \citet{Marsaleix_al_OM08}. Let defined a density anomaly, $d$, by $d=(\rho - \rho_o)/ \rho_o$,
+and a hydrostatic pressure anomaly, $p_h'$, by $p_h'= g \; \int_z^\eta d \; e_3 \; dk$.
+An additional term appears in (\autoref{apdx:A_grad_p_1}) which accounts for
+the tilt of $s-$surfaces with respect to geopotential $z-$surfaces.
+As in $z$-coordinate,
+the horizontal pressure gradient can be split in two parts following \citet{Marsaleix_al_OM08}.
+Let defined a density anomaly, $d$, by $d=(\rho - \rho_o)/ \rho_o$,
+and a hydrostatic pressure anomaly, $p_h'$, by $p_h'= g \; \int_z^\eta d \; e_3 \; dk$.
 The pressure is then given by:
 \begin{equation*}
 …
 \end{equation*}
 Substituing \autoref{apdx:A_pressure} in \autoref{apdx:A_grad_p_1} and using the definition of
 the density anomaly it comes the expression in two parts:
+Substituing \autoref{apdx:A_pressure} in \autoref{apdx:A_grad_p_1} and
+using the definition of the density anomaly it comes the expression in two parts:
 \begin{equation} \label{apdx:A_grad_p_2}
 \begin{split}
 …
 \end{split}
 \end{equation}
 This formulation of the pressure gradient is characterised by the appearance of a term depending on the
+the sea surface height only (last term on the right hand side of expression \autoref{apdx:A_grad_p_2}).
+This term will be loosely termed \textit{surface pressure gradient}
+whereas the first term will be termed the
+\textit{hydrostatic pressure gradient} by analogy to the $z$-coordinate formulation.
+In fact, the the true surface pressure gradient is $1/\rho_o \nabla (\rho \eta)$, and
+$\eta$ is implicitly included in the computation of $p_h'$ through the upper bound of
 the vertical integration.
+This formulation of the pressure gradient is characterised by the appearance of
+a term depending on the sea surface height only
+(last term on the right hand side of expression \autoref{apdx:A_grad_p_2}).
+This term will be loosely termed \textit{surface pressure gradient} whereas
+the first term will be termed the \textit{hydrostatic pressure gradient} by analogy to
+the $z$-coordinate formulation.
+In fact, the true surface pressure gradient is $1/\rho_o \nabla (\rho \eta)$,
+and $\eta$ is implicitly included in the computation of $p_h'$ through the upper bound of the vertical integration.
 $\ $\newline    % force a new ligne
 …
 $\bullet$ \textbf{The other terms of the momentum equation}
 The coriolis and forcing terms as well as the the vertical physics remain unchanged
+as they involve neither time nor space derivatives. The form of the lateral physics is
 discussed in \autoref{apdx:B}.
+The coriolis and forcing terms as well as the the vertical physics remain unchanged as
+they involve neither time nor space derivatives.
+The form of the lateral physics is discussed in \autoref{apdx:B}.
 …
 $\bullet$ \textbf{Full momentum equation}
 To sum up, in a curvilinear $s$-coordinate system, the vector invariant momentum equation
+solved by the model has the same mathematical expression as the one in a curvilinear
 $z-$coordinate, except for the pressure gradient term :
+To sum up, in a curvilinear $s$-coordinate system,
+the vector invariant momentum equation solved by the model has the same mathematical expression as
+the one in a curvilinear $z-$coordinate, except for the pressure gradient term:
 \begin{subequations} \label{apdx:A_dyn_vect}
 \begin{multline} \label{apdx:A_PE_dyn_vect_u}
 …
 \end{multline}
 \end{subequations}
 whereas the flux form momentum equation differ from it by the formulation of both
 the time derivative and the pressure gradient term  :
+whereas the flux form momentum equation differs from it by
+the formulation of both the time derivative and the pressure gradient term:
 \begin{subequations} \label{apdx:A_dyn_flux}
 \begin{multline} \label{apdx:A_PE_dyn_flux_u}
 …
 \end{equation}
+It is important to realize that the change in coordinate system has only concerned
+the position on the vertical. It has not affected (\textbf{i},\textbf{j},\textbf{k}), the
+orthogonal curvilinear set of unit vectors. ($u$,$v$) are always horizontal velocities
+so that their evolution is driven by \emph{horizontal} forces, in particular
+the pressure gradient. By contrast, $\omega$ is not $w$, the third component of the velocity,
+but the dia-surface velocity component, $i.e.$ the volume flux across the moving
+$s$-surfaces per unit horizontal area.
+It is important to realize that the change in coordinate system has only concerned the position on the vertical.
+It has not affected (\textbf{i},\textbf{j},\textbf{k}), the orthogonal curvilinear set of unit vectors.
+($u$,$v$) are always horizontal velocities so that their evolution is driven by \emph{horizontal} forces,
+in particular the pressure gradient.
+By contrast, $\omega$ is not $w$, the third component of the velocity, but the dia-surface velocity component,
+$i.e.$ the volume flux across the moving $s$-surfaces per unit horizontal area.
 …
 \label{sec:A_tracer}
 The tracer equation is obtained using the same calculation as for the continuity
 equation and then regrouping the time derivative terms in the left hand side :
+The tracer equation is obtained using the same calculation as for the continuity equation and then
+regrouping the time derivative terms in the left hand side :
 \begin{multline} \label{apdx:A_tracer}
 …
 The expression for the advection term is a straight consequence of (A.4), the
 expression of the 3D divergence in the $s-$coordinates established above.
+The expression for the advection term is a straight consequence of (A.4),
+the expression of the 3D divergence in the $s-$coordinates established above.
 \end{document}

NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/annex_B.tex

-                      r9407
+                      r10368
 \subsubsection*{In z-coordinates}
+In $z$-coordinates, the horizontal/vertical second order tracer diffusion operator
+is given by:
+In $z$-coordinates, the horizontal/vertical second order tracer diffusion operator is given by:
 \begin{eqnarray} \label{apdx:B1}
  &D^T = \frac{1}{e_1 \, e_2}      \left[
 …
 \subsubsection*{In generalized vertical coordinates}
 In $s$-coordinates, we defined the slopes of $s$-surfaces, $\sigma_1$ and
+$\sigma_2$ by \autoref{apdx:A_s_slope} and the vertical/horizontal ratio of diffusion
 coefficient by $\epsilon = A^{vT} / A^{lT}$. The diffusion operator is given by:
+In $s$-coordinates, we defined the slopes of $s$-surfaces, $\sigma_1$ and $\sigma_2$ by \autoref{apdx:A_s_slope} and
+the vertical/horizontal ratio of diffusion coefficient by $\epsilon = A^{vT} / A^{lT}$.
+The diffusion operator is given by:
 \begin{equation} \label{apdx:B2}
 …
 \end{subequations}
 Equation \autoref{apdx:B2} is obtained from \autoref{apdx:B1} without any
 additional assumption. Indeed, for the special case $k=z$ and thus $e_3 =1$,
 we introduce an arbitrary vertical coordinate $s = s (i,j,z)$ as in \autoref{apdx:A}
 and use \autoref{apdx:A_s_slope} and \autoref{apdx:A_s_chain_rule}.
 Since no cross horizontal derivative $\partial _i \partial _j $ appears in
 \autoref{apdx:B1}, the ($i$,$z$) and ($j$,$z$) planes are independent.
 The derivation can then be demonstrated for the ($i$,$z$)~$\to$~($j$,$s$)
 transformation without any loss of generality:
+Equation \autoref{apdx:B2} is obtained from \autoref{apdx:B1} without any additional assumption.
+Indeed, for the special case $k=z$ and thus $e_3 =1$,
+we introduce an arbitrary vertical coordinate $s = s (i,j,z)$ as in \autoref{apdx:A} and
+use \autoref{apdx:A_s_slope} and \autoref{apdx:A_s_chain_rule}.
+Since no cross horizontal derivative $\partial _i \partial _j $ appears in \autoref{apdx:B1},
+the ($i$,$z$) and ($j$,$z$) planes are independent.
+The derivation can then be demonstrated for the ($i$,$z$)~$\to$~($j$,$s$) transformation without
+any loss of generality:
 \begin{subequations}
 …
 \subsubsection*{In z-coordinates}
+The iso/diapycnal diffusive tensor $\textbf {A}_{\textbf I}$ expressed in the ($i$,$j$,$k$)
+curvilinear coordinate system in which the equations of the ocean circulation model are
+formulated, takes the following form \citep{Redi_JPO82}:
+The iso/diapycnal diffusive tensor $\textbf {A}_{\textbf I}$ expressed in
+the ($i$,$j$,$k$) curvilinear coordinate system in which
+the equations of the ocean circulation model are formulated,
+takes the following form \citep{Redi_JPO82}:
 \begin{equation} \label{apdx:B3}
 …
 \end{array} }} \right]
 \end{equation}
+where ($a_1$, $a_2$) are the isopycnal slopes in ($\textbf{i}$,
+$\textbf{j}$) directions, relative to geopotentials:
+where ($a_1$, $a_2$) are the isopycnal slopes in ($\textbf{i}$, $\textbf{j}$) directions, relative to geopotentials:
 \begin{equation*}
 a_1 =\frac{e_3 }{e_1 }\left( {\frac{\partial \rho }{\partial i}} \right)\left( {\frac{\partial \rho }{\partial k}} \right)^{-1}
 …
 \end{equation*}
 In practice, isopycnal slopes are generally less than $10^{-2}$ in the ocean, so
 $\textbf {A}_{\textbf I}$ can be simplified appreciably \citep{Cox1987}:
+In practice, isopycnal slopes are generally less than $10^{-2}$ in the ocean,
+so $\textbf {A}_{\textbf I}$ can be simplified appreciably \citep{Cox1987}:
 \begin{subequations} \label{apdx:B4}
 \begin{equation} \label{apdx:B4a}
 …
+Physically, the full tensor \autoref{apdx:B3}
+represents strong isoneutral diffusion on a plane parallel to the isoneutral
+surface and weak dianeutral diffusion perpendicular to this plane.
+However, the approximate `weak-slope' tensor \autoref{apdx:B4a} represents strong
+diffusion along the isoneutral surface, with weak
+\emph{vertical}  diffusion -- the principal axes of the tensor are no
+longer orthogonal. This simplification also decouples
+the ($i$,$z$) and ($j$,$z$) planes of the tensor. The weak-slope operator therefore takes the same
+form, \autoref{apdx:B4}, as \autoref{apdx:B2}, the diffusion operator for geopotential
+diffusion written in non-orthogonal $i,j,s$-coordinates. Written out
+explicitly,
+Physically, the full tensor \autoref{apdx:B3} represents strong isoneutral diffusion on a plane parallel to
+the isoneutral surface and weak dianeutral diffusion perpendicular to this plane.
+However,
+the approximate `weak-slope' tensor \autoref{apdx:B4a} represents strong diffusion along the isoneutral surface,
+with weak \emph{vertical} diffusion -- the principal axes of the tensor are no longer orthogonal.
+This simplification also decouples the ($i$,$z$) and ($j$,$z$) planes of the tensor.
+The weak-slope operator therefore takes the same form, \autoref{apdx:B4}, as \autoref{apdx:B2},
+the diffusion operator for geopotential diffusion written in non-orthogonal $i,j,s$-coordinates.
+Written out explicitly,
 \begin{multline} \label{apdx:B_ldfiso}
 …
 The isopycnal diffusion operator \autoref{apdx:B4},
 \autoref{apdx:B_ldfiso} conserves tracer quantity and dissipates its
+square. The demonstration of the first property is trivial as \autoref{apdx:B4} is the divergence
 of fluxes. Let us demonstrate the second one:
+\autoref{apdx:B_ldfiso} conserves tracer quantity and dissipates its square.
+The demonstration of the first property is trivial as \autoref{apdx:B4} is the divergence of fluxes.
+Let us demonstrate the second one:
 \begin{equation*}
 \iiint\limits_D T\;\nabla .\left( {\textbf{A}}_{\textbf{I}} \nabla T \right)\,dv
 …
 \end{subequations}
 \addtocounter{equation}{-1}
  the property becomes obvious.
+the property becomes obvious.
 \subsubsection*{In generalized vertical coordinates}
+Because the weak-slope operator \autoref{apdx:B4}, \autoref{apdx:B_ldfiso} is decoupled
+in the ($i$,$z$) and ($j$,$z$) planes, it may be transformed into
+generalized $s$-coordinates in the same way as \autoref{sec:B_1} was transformed into
+\autoref{sec:B_2}. The resulting operator then takes the simple form
+Because the weak-slope operator \autoref{apdx:B4},
+\autoref{apdx:B_ldfiso} is decoupled in the ($i$,$z$) and ($j$,$z$) planes,
+it may be transformed into generalized $s$-coordinates in the same way as
+\autoref{sec:B_1} was transformed into \autoref{sec:B_2}.
+The resulting operator then takes the simple form
 \begin{equation} \label{apdx:B_ldfiso_s}
 …
 \end{equation}
 where ($r_1$, $r_2$) are the isopycnal slopes in ($\textbf{i}$,
 $\textbf{j}$) directions, relative to $s$-coordinate surfaces:
+where ($r_1$, $r_2$) are the isopycnal slopes in ($\textbf{i}$, $\textbf{j}$) directions,
+relative to $s$-coordinate surfaces:
 \begin{equation*}
 r_1 =\frac{e_3 }{e_1 }\left( {\frac{\partial \rho }{\partial i}} \right)\left( {\frac{\partial \rho }{\partial s}} \right)^{-1}
 …
 \end{equation*}
+To prove  \autoref{apdx:B5}  by direct re-expression of \autoref{apdx:B_ldfiso} is
+straightforward, but laborious. An easier way is first to note (by reversing the
+derivation of \autoref{sec:B_2} from \autoref{sec:B_1} ) that the
+weak-slope operator may be \emph{exactly} reexpressed in
+non-orthogonal $i,j,\rho$-coordinates as
+To prove \autoref{apdx:B5} by direct re-expression of \autoref{apdx:B_ldfiso} is straightforward, but laborious.
+An easier way is first to note (by reversing the derivation of \autoref{sec:B_2} from \autoref{sec:B_1} ) that
+the weak-slope operator may be \emph{exactly} reexpressed in non-orthogonal $i,j,\rho$-coordinates as
 \begin{equation} \label{apdx:B5}
 …
 \end{array} }} \right).
 \end{equation}
+Then direct transformation from $i,j,\rho$-coordinates to
+$i,j,s$-coordinates gives \autoref{apdx:B_ldfiso_s} immediately.
+Note that the weak-slope approximation is only made in
+transforming from the (rotated,orthogonal) isoneutral axes to the
+non-orthogonal $i,j,\rho$-coordinates. The further transformation
+into $i,j,s$-coordinates is exact, whatever the steepness of
+the  $s$-surfaces, in the same way as the transformation of
+horizontal/vertical Laplacian diffusion in $z$-coordinates,
+Then direct transformation from $i,j,\rho$-coordinates to $i,j,s$-coordinates gives
+\autoref{apdx:B_ldfiso_s} immediately.
+Note that the weak-slope approximation is only made in transforming from
+the (rotated,orthogonal) isoneutral axes to the non-orthogonal $i,j,\rho$-coordinates.
+The further transformation into $i,j,s$-coordinates is exact, whatever the steepness of the $s$-surfaces,
+in the same way as the transformation of horizontal/vertical Laplacian diffusion in $z$-coordinates,
 \autoref{sec:B_1} onto $s$-coordinates is exact, however steep the $s$-surfaces.
 …
 \label{sec:B_3}
 The second order momentum diffusion operator (Laplacian) in the $z$-coordinate
+is found by applying \autoref{eq:PE_lap_vector}, the expression for the Laplacian
 of a vector,  to the horizontal velocity vector :
+The second order momentum diffusion operator (Laplacian) in the $z$-coordinate is found by
+applying \autoref{eq:PE_lap_vector}, the expression for the Laplacian of a vector,
+to the horizontal velocity vector:
 \begin{align*}
 \Delta {\textbf{U}}_h
 …
 \end{array} }} \right)
 \end{align*}
 Using \autoref{eq:PE_div}, the definition of the horizontal divergence, the third
 componant of the second vector is obviously zero and thus :
+Using \autoref{eq:PE_div}, the definition of the horizontal divergence,
+the third componant of the second vector is obviously zero and thus :
 \begin{equation*}
 \Delta {\textbf{U}}_h = \nabla _h \left( \chi \right) - \nabla _h \times \left( \zeta \right) + \frac {1}{e_3 } \frac {\partial }{\partial k} \left( {\frac {1}{e_3 } \frac{\partial {\textbf{ U}}_h }{\partial k}} \right)
 \end{equation*}
 Note that this operator ensures a full separation between the vorticity and horizontal
+divergence fields (see \autoref{apdx:C}). It is only equal to a Laplacian
 applied to each component in Cartesian coordinates, not on the sphere.
 The horizontal/vertical second order (Laplacian type) operator used to diffuse
 horizontal momentum in the $z$-coordinate therefore takes the following form :
+Note that this operator ensures a full separation between
+the vorticity and horizontal divergence fields (see \autoref{apdx:C}).
+It is only equal to a Laplacian applied to each component in Cartesian coordinates, not on the sphere.
+The horizontal/vertical second order (Laplacian type) operator used to diffuse horizontal momentum in
+the $z$-coordinate therefore takes the following form:
 \begin{equation} \label{apdx:B_Lap_U}
  {\textbf{D}}^{\textbf{U}} =
 …
 \end{align*}
 Note Bene: introducing a rotation in \autoref{apdx:B_Lap_U} does not lead to a
 useful expression for the iso/diapycnal Laplacian operator in the $z$-coordinate.
 Similarly, we did not found an expression of practical use for the geopotential
+horizontal/vertical Laplacian operator in the $s$-coordinate. Generally,
+\autoref{apdx:B_Lap_U} is used in both $z$- and $s$-coordinate systems, that is
 a Laplacian diffusion is applied on momentum along the coordinate directions.
+Note Bene: introducing a rotation in \autoref{apdx:B_Lap_U} does not lead to
+a useful expression for the iso/diapycnal Laplacian operator in the $z$-coordinate.
+Similarly, we did not found an expression of practical use for
+the geopotential horizontal/vertical Laplacian operator in the $s$-coordinate.
+Generally, \autoref{apdx:B_Lap_U} is used in both $z$- and $s$-coordinate systems,
+that is a Laplacian diffusion is applied on momentum along the coordinate directions.
 \end{document}

NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/annex_C.tex

-                      r9414
+                      r10368
 \label{sec:C.0}
 Notation used in this appendix in the demonstations :
+Notation used in this appendix in the demonstations:
 fluxes at the faces of a $T$-box:
 …
 $dv=e_1\,e_2\,e_3 \,di\,dj\,dk$  is the volume element, with only $e_3$ that depends on time.
 $D$ and $S$ are the ocean domain volume and surface, respectively.
 No wetting/drying is allow ($i.e.$ $\frac{\partial S}{\partial t} = 0$)
 Let $k_s$ and $k_b$ be the ocean surface and bottom, resp.
+$D$ and $S$ are the ocean domain volume and surface, respectively.
+No wetting/drying is allow ($i.e.$ $\frac{\partial S}{\partial t} = 0$).
+Let $k_s$ and $k_b$ be the ocean surface and bottom, resp.
 ($i.e.$ $s(k_s) = \eta$ and $s(k_b)=-H$, where $H$ is the bottom depth).
 \begin{flalign*}
 …
                                                        =  \int_D { \frac{1}{e_3} \partial_t \left( e_3 \, Q \right) dv } =0
 \end{equation*}
+equation of evolution of $Q$ written as the time evolution of the vertical content of $Q$
+like for tracers, or momentum in flux form, the quadratic quantity $\frac{1}{2}Q^2$ is conserved when :
+equation of evolution of $Q$ written as
+the time evolution of the vertical content of $Q$ like for tracers, or momentum in flux form,
+the quadratic quantity $\frac{1}{2}Q^2$ is conserved when:
 \begin{flalign*}
 \partial_t \left(   \int_D{ \frac{1}{2} \,Q^2\;dv }   \right)
 …
   -   \frac{1}{2} \int_D {  \frac{Q^2}{e_3} \partial_t (e_3) \;dv }
 \end{flalign}
 equation of evolution of $Q$ written as the time evolution of $Q$
 like for momentum in vector invariant form, the quadratic quantity $\frac{1}{2}Q^2$ is conserved when :
+equation of evolution of $Q$ written as the time evolution of $Q$ like for momentum in vector invariant form,
+the quadratic quantity $\frac{1}{2}Q^2$ is conserved when:
 \begin{flalign*}
 \partial_t \left( \int_D {\frac{1}{2} Q^2\;dv} \right)
 …
 +  \int_D { \frac{1}{2} Q^2 \, \partial_t e_3  \;e_1e_2\;di\,dj\,dk } \\
 \end{flalign*}
 that is in a more compact form :
+that is in a more compact form:
 \begin{flalign} \label{eq:Q2_vect}
 \partial_t \left( \int_D {\frac{1}{2} Q^2\;dv} \right)
 …
+The discretization of pimitive equation in $s$-coordinate ($i.e.$ time and space varying
+vertical coordinate) must be chosen so that the discrete equation of the model satisfy
+integral constrains on energy and enstrophy.
+The discretization of pimitive equation in $s$-coordinate ($i.e.$ time and space varying vertical coordinate)
+must be chosen so that the discrete equation of the model satisfy integral constrains on energy and enstrophy.
 Let us first establish those constraint in the continuous world.
 The total energy ($i.e.$ kinetic plus potential energies) is conserved :
+The total energy ($i.e.$ kinetic plus potential energies) is conserved:
 \begin{flalign} \label{eq:Tot_Energy}
   \partial_t \left( \int_D \left( \frac{1}{2} {\textbf{U}_h}^2 +  \rho \, g \, z \right) \;dv \right)  = & 0
 \end{flalign}
+under the following assumptions: no dissipation, no forcing
+(wind, buoyancy flux, atmospheric pressure variations), mass
+conservation, and closed domain.
+This equation can be transformed to obtain several sub-equalities.
+The transformation for the advection term depends on whether
+the vector invariant form or the flux form is used for the momentum equation.
+Using \autoref{eq:Q2_vect} and introducing \autoref{apdx:A_dyn_vect} in \autoref{eq:Tot_Energy}
+for the former form and
+Using \autoref{eq:Q2_flux} and introducing \autoref{apdx:A_dyn_flux} in \autoref{eq:Tot_Energy}
+for the latter form  leads to:
+under the following assumptions: no dissipation, no forcing (wind, buoyancy flux, atmospheric pressure variations),
+mass conservation, and closed domain.
+This equation can be transformed to obtain several sub-equalities.
+The transformation for the advection term depends on whether the vector invariant form or
+the flux form is used for the momentum equation.
+Using \autoref{eq:Q2_vect} and introducing \autoref{apdx:A_dyn_vect} in
+\autoref{eq:Tot_Energy} for the former form and
+using \autoref{eq:Q2_flux} and introducing \autoref{apdx:A_dyn_flux} in
+\autoref{eq:Tot_Energy} for the latter form leads to:
 \begin{subequations} \label{eq:E_tot}
 …
 Substituting the discrete expression of the time derivative of the velocity either in vector invariant,
 leads to the discrete equivalent of the four equations \autoref{eq:E_tot_flux}.
+leads to the discrete equivalent of the four equations \autoref{eq:E_tot_flux}.
 % -------------------------------------------------------------------------------------------------------------
 …
 \label{subsec:C_vor}
+Let $q$, located at $f$-points, be either the relative ($q=\zeta / e_{3f}$), or
+the planetary ($q=f/e_{3f}$), or the total potential vorticity ($q=(\zeta +f) /e_{3f}$).
+Two discretisation of the vorticity term (ENE and EEN) allows the conservation of
+the kinetic energy.
+Let $q$, located at $f$-points, be either the relative ($q=\zeta / e_{3f}$),
+or the planetary ($q=f/e_{3f}$), or the total potential vorticity ($q=(\zeta +f) /e_{3f}$).
+Two discretisation of the vorticity term (ENE and EEN) allows the conservation of the kinetic energy.
 % -------------------------------------------------------------------------------------------------------------
 %       Vorticity Term with ENE scheme
 …
 \label{subsec:C_vorENE}
 For the ENE scheme, the two components of the vorticity term are given by :
+For the ENE scheme, the two components of the vorticity term are given by:
 \begin{equation*}
 - e_3 \, q \;{\textbf{k}}\times {\textbf {U}}_h    \equiv
 …
 \end{equation*}
+This formulation does not conserve the enstrophy but it does conserve the
+total kinetic energy. Indeed, the kinetic energy tendency associated to the
+vorticity term and averaged over the ocean domain can be transformed as
+follows:
+This formulation does not conserve the enstrophy but it does conserve the total kinetic energy.
+Indeed, the kinetic energy tendency associated to the vorticity term and
+averaged over the ocean domain can be transformed as follows:
 \begin{flalign*}
 &\int\limits_D -  \left(  e_3 \, q \;\textbf{k} \times \textbf{U}_h  \right) \cdot \textbf{U}_h  \;  dv &&  \\
 …
 \end{aligned}   } \right.
 \end{equation}
+where the indices $i_p$ and $j_p$ take the following value:
+$i_p = -1/2$ or $1/2$ and $j_p = -1/2$ or $1/2$,
+where the indices $i_p$ and $j_p$ take the following value: $i_p = -1/2$ or $1/2$ and $j_p = -1/2$ or $1/2$,
 and the vorticity triads, ${^i_j}\mathbb{Q}^{i_p}_{j_p}$, defined at $T$-point, are given by:
 \begin{equation} \tag{\ref{eq:Q_triads}}
 …
 \end{equation}
+This formulation does conserve the total kinetic energy. Indeed,
+This formulation does conserve the total kinetic energy.
+Indeed,
 \begin{flalign*}
 &\int\limits_D - \textbf{U}_h \cdot   \left(  \zeta \;\textbf{k} \times \textbf{U}_h  \right)  \;  dv &&  \\
 …
 \label{subsec:C_zad}
+The change of Kinetic Energy (KE) due to the vertical advection is exactly
+balanced by the change of KE due to the horizontal gradient of KE~:
+The change of Kinetic Energy (KE) due to the vertical advection is exactly balanced by the change of KE due to the horizontal gradient of KE~:
 \begin{equation*}
     \int_D \textbf{U}_h \cdot \frac{1}{e_3 } \omega \partial_k \textbf{U}_h \;dv
 …
    +   \frac{1}{2} \int_D {  \frac{{\textbf{U}_h}^2}{e_3} \partial_t ( e_3) \;dv }  \\
 \end{equation*}
+Indeed, using successively \autoref{eq:DOM_di_adj} ($i.e.$ the skew symmetry
+property of the $\delta$ operator) and the continuity equation, then
+\autoref{eq:DOM_di_adj} again, then the commutativity of operators
+$\overline {\,\cdot \,}$ and $\delta$, and finally \autoref{eq:DOM_mi_adj}
+($i.e.$ the symmetry property of the $\overline {\,\cdot \,}$ operator)
+Indeed, using successively \autoref{eq:DOM_di_adj} ($i.e.$ the skew symmetry property of the $\delta$ operator)
+and the continuity equation, then \autoref{eq:DOM_di_adj} again,
+then the commutativity of operators $\overline {\,\cdot \,}$ and $\delta$, and finally \autoref{eq:DOM_mi_adj}
+($i.e.$ the symmetry property of the $\overline {\,\cdot \,}$ operator)
 applied in the horizontal and vertical directions, it becomes:
 \begin{flalign*}
 …
 \end{flalign*}
+There is two main points here. First, the satisfaction of this property links the choice of
+the discrete formulation of the vertical advection and of the horizontal gradient
+of KE. Choosing one imposes the other. For example KE can also be discretized
+as $1/2\,({\overline u^{\,i}}^2 + {\overline v^{\,j}}^2)$. This leads to the following
+expression for the vertical advection:
+There is two main points here.
+First, the satisfaction of this property links the choice of the discrete formulation of the vertical advection and
+of the horizontal gradient of KE.
+Choosing one imposes the other.
+For example KE can also be discretized as $1/2\,({\overline u^{\,i}}^2 + {\overline v^{\,j}}^2)$.
+This leads to the following expression for the vertical advection:
 \begin{equation*}
 \frac{1} {e_3 }\; \omega\; \partial_k \textbf{U}_h
 …
 \end{array}} } \right)
 \end{equation*}
 a formulation that requires an additional horizontal mean in contrast with
+the one used in NEMO. Nine velocity points have to be used instead of 3.
+a formulation that requires an additional horizontal mean in contrast with the one used in NEMO.
+Nine velocity points have to be used instead of 3.
 This is the reason why it has not been chosen.
 Second, as soon as the chosen $s$-coordinate depends on time, an extra constraint
 arises on the time derivative of the volume at $u$- and $v$-points:
+Second, as soon as the chosen $s$-coordinate depends on time,
+an extra constraint arises on the time derivative of the volume at $u$- and $v$-points:
 \begin{flalign*}
 e_{1u}\,e_{2u}\,\partial_t (e_{3u}) =\overline{ e_{1t}\,e_{2t}\;\partial_t (e_{3t}) }^{\,i+1/2}    \\
 …
 \gmcomment{
+A pressure gradient has no contribution to the evolution of the vorticity as the
+curl of a gradient is zero. In the $z$-coordinate, this property is satisfied locally
 on a C-grid with 2nd order finite differences (property \autoref{eq:DOM_curl_grad}).
+  A pressure gradient has no contribution to the evolution of the vorticity as the curl of a gradient is zero.
+  In the $z$-coordinate, this property is satisfied locally on a C-grid with 2nd order finite differences
+  (property \autoref{eq:DOM_curl_grad}).
+}
 When the equation of state is linear ($i.e.$ when an advection-diffusion equation
+for density can be derived from those of temperature and salinity) the change of
+KE due to the work of pressure forces is balanced by the change of potential
 energy due to buoyancy forces:
+When the equation of state is linear
+($i.e.$ when an advection-diffusion equation for density can be derived from those of temperature and salinity)
+the change of KE due to the work of pressure forces is balanced by
+the change of potential energy due to buoyancy forces:
 \begin{equation*}
 - \int_D  \left. \nabla p \right|_z \cdot \textbf{U}_h \;dv
 …
 \end{equation*}
 This property can be satisfied in a discrete sense for both $z$- and $s$-coordinates.
 Indeed, defining the depth of a $T$-point, $z_t$, as the sum of the vertical scale
+factors at $w$-points starting from the surface, the work of pressure forces can be
 written as:
+This property can be satisfied in a discrete sense for both $z$- and $s$-coordinates.
+Indeed, defining the depth of a $T$-point, $z_t$,
+as the sum of the vertical scale factors at $w$-points starting from the surface,
+the work of pressure forces can be written as:
 \begin{flalign*}
 &- \int_D  \left. \nabla p \right|_z \cdot \textbf{U}_h \;dv
 …
 \end{flalign*}
 The first term is exactly the first term of the right-hand-side of \autoref{eq:KE+PE_vect_discrete}.
+It remains to demonstrate that the last term, which is obviously a discrete analogue of
+$\int_D \frac{p}{e_3} \partial_t (e_3)\;dv$ is equal to the last term of \autoref{eq:KE+PE_vect_discrete}.
+It remains to demonstrate that the last term,
+which is obviously a discrete analogue of $\int_D \frac{p}{e_3} \partial_t (e_3)\;dv$ is equal to
+the last term of \autoref{eq:KE+PE_vect_discrete}.
 In other words, the following property must be satisfied:
 \begin{flalign*}
 …
 \end{flalign*}
 Let introduce $p_w$ the pressure at $w$-point such that $\delta_k [p_w] = - \rho \,g\,e_{3t}$.
+Let introduce $p_w$ the pressure at $w$-point such that $\delta_k [p_w] = - \rho \,g\,e_{3t}$.
 The right-hand-side of the above equation can be transformed as follows:
 …
+Note that this property strongly constrains the discrete expression of both
+the depth of $T-$points and of the term added to the pressure gradient in the
+$s$-coordinate. Nevertheless, it is almost never satisfied since a linear equation
+of state is rarely used.
+Note that this property strongly constrains the discrete expression of both the depth of $T-$points and
+of the term added to the pressure gradient in the $s$-coordinate.
+Nevertheless, it is almost never satisfied since a linear equation of state is rarely used.
 …
 \end{flalign*}
 Substituting the discrete expression of the time derivative of the velocity either in vector invariant or in flux form,
+leads to the discrete equivalent of the
+Substituting the discrete expression of the time derivative of the velocity either in
+vector invariant or in flux form, leads to the discrete equivalent of the ????
 …
 \label{subsec:C.3.3}
+In flux from the vorticity term reduces to a Coriolis term in which the Coriolis
+parameter has been modified to account for the ``metric'' term. This altered
+Coriolis parameter is discretised at an f-point. It is given by:
+In flux from the vorticity term reduces to a Coriolis term in which
+the Coriolis parameter has been modified to account for the ``metric'' term.
+This altered Coriolis parameter is discretised at an f-point.
+It is given by:
 \begin{equation*}
 f+\frac{1} {e_1 e_2 } \left( v \frac{\partial e_2 } {\partial i} - u \frac{\partial e_1 } {\partial j}\right)\;
 …
 \end{equation*}
 Either the ENE or EEN scheme is then applied to obtain the vorticity term in flux form.
 It therefore conserves the total KE. The derivation is the same as for the
 vorticity term in the vector invariant form (\autoref{subsec:C_vor}).
+Either the ENE or EEN scheme is then applied to obtain the vorticity term in flux form.
+It therefore conserves the total KE.
+The derivation is the same as for the vorticity term in the vector invariant form (\autoref{subsec:C_vor}).
 % -------------------------------------------------------------------------------------------------------------
 …
 \label{subsec:C.3.4}
+The flux form operator of the momentum advection is evaluated using a
+centered second order finite difference scheme. Because of the flux form,
+the discrete operator does not contribute to the global budget of linear
+momentum. Because of the centered second order scheme, it conserves
+the horizontal kinetic energy, that is :
+The flux form operator of the momentum advection is evaluated using
+a centered second order finite difference scheme.
+Because of the flux form, the discrete operator does not contribute to the global budget of linear momentum.
+Because of the centered second order scheme, it conserves the horizontal kinetic energy, that is:
 \begin{equation} \label{eq:C_ADV_KE_flux}
 …
 \end{equation}
 Let us first consider the first term of the scalar product ($i.e.$ just the the terms
 associated with the i-component of the advection) :
+Let us first consider the first term of the scalar product
+($i.e.$ just the the terms associated with the i-component of the advection):
 \begin{flalign*}
 &  - \int_D u \cdot \nabla \cdot \left(   \textbf{U}\,u   \right) \; dv   \\
 …
    \biggl\{     \left(   \frac{1}{e_{3t}} \frac{\partial e_{3t}}{\partial t}   \right) \; b_t     \biggr\}    &&& \\
 \end{flalign*}
+Applying similar manipulation applied to the second term of the scalar product
+leads to :
+Applying similar manipulation applied to the second term of the scalar product leads to:
 \begin{equation*}
  -  \int_D \textbf{U}_h \cdot     \left(                 {{\begin{array} {*{20}c}
 …
    \biggl\{     \left(   \frac{1}{e_{3t}} \frac{\partial e_{3t}}{\partial t}   \right) \; b_t     \biggr\}
 \end{equation*}
+which is the discrete form of
+$ \frac{1}{2} \int_D u \cdot \nabla \cdot \left(   \textbf{U}\,u   \right) \; dv $.
+which is the discrete form of $ \frac{1}{2} \int_D u \cdot \nabla \cdot \left(   \textbf{U}\,u   \right) \; dv $.
 \autoref{eq:C_ADV_KE_flux} is thus satisfied.
+When the UBS scheme is used to evaluate the flux form momentum advection,
+the discrete operator does not contribute to the global budget of linear momentum
+(flux form). The horizontal kinetic energy is not conserved, but forced to decay
+($i.e.$ the scheme is diffusive).
+When the UBS scheme is used to evaluate the flux form momentum advection,
+the discrete operator does not contribute to the global budget of linear momentum (flux form).
+The horizontal kinetic energy is not conserved, but forced to decay ($i.e.$ the scheme is diffusive).
 …
 \end{equation}
+The scheme does not allow but the conservation of the total kinetic energy but the conservation
+of $q^2$, the potential enstrophy for a horizontally non-divergent flow ($i.e.$ when $\chi$=$0$).
+Indeed, using the symmetry or skew symmetry properties of the operators ( \autoref{eq:DOM_mi_adj}
+and \autoref{eq:DOM_di_adj}), it can be shown that:
+The scheme does not allow but the conservation of the total kinetic energy but the conservation of $q^2$,
+the potential enstrophy for a horizontally non-divergent flow ($i.e.$ when $\chi$=$0$).
+Indeed, using the symmetry or skew symmetry properties of the operators
+( \autoref{eq:DOM_mi_adj} and \autoref{eq:DOM_di_adj}),
+it can be shown that:
 \begin{equation} \label{eq:C_1.1}
 \int_D {q\,\;{\textbf{k}}\cdot \frac{1} {e_3} \nabla \times \left( {e_3 \, q \;{\textbf{k}}\times {\textbf{U}}_h } \right)\;dv} \equiv 0
 \end{equation}
 where $dv=e_1\,e_2\,e_3 \; di\,dj\,dk$ is the volume element. Indeed, using
+\autoref{eq:dynvor_ens}, the discrete form of the right hand side of \autoref{eq:C_1.1}
 can be transformed as follow:
+where $dv=e_1\,e_2\,e_3 \; di\,dj\,dk$ is the volume element.
+Indeed, using \autoref{eq:dynvor_ens},
+the discrete form of the right hand side of \autoref{eq:C_1.1} can be transformed as follow:
 \begin{flalign*}
 &\int_D q \,\; \textbf{k} \cdot \frac{1} {e_3 } \nabla \times
 …
 \end{aligned}   } \right.
 \end{equation}
 where the indices $i_p$ and $k_p$ take the following value:
+where the indices $i_p$ and $k_p$ take the following values:
 $i_p = -1/2$ or $1/2$ and $j_p = -1/2$ or $1/2$,
 and the vorticity triads, ${^i_j}\mathbb{Q}^{i_p}_{j_p}$, defined at $T$-point, are given by:
 …
 This formulation does conserve the potential enstrophy for a horizontally non-divergent flow ($i.e.$ $\chi=0$).
+Let consider one of the vorticity triad, for example ${^{i}_j}\mathbb{Q}^{+1/2}_{+1/2} $,
+similar manipulation can be done for the 3 others. The discrete form of the right hand
+side of \autoref{eq:C_1.1} applied to this triad only can be transformed as follow:
+Let consider one of the vorticity triad, for example ${^{i}_j}\mathbb{Q}^{+1/2}_{+1/2} $,
+similar manipulation can be done for the 3 others.
+The discrete form of the right hand side of \autoref{eq:C_1.1} applied to
+this triad only can be transformed as follow:
 \begin{flalign*}
 …
+All the numerical schemes used in NEMO are written such that the tracer content
+is conserved by the internal dynamics and physics (equations in flux form).
+For advection, only the CEN2 scheme ($i.e.$ $2^{nd}$ order finite different scheme)
+conserves the global variance of tracer. Nevertheless the other schemes ensure
+that the global variance decreases ($i.e.$ they are at least slightly diffusive).
+For diffusion, all the schemes ensure the decrease of the total tracer variance,
+except the iso-neutral operator. There is generally no strict conservation of mass,
+as the equation of state is non linear with respect to $T$ and $S$. In practice,
+the mass is conserved to a very high accuracy.
+All the numerical schemes used in NEMO are written such that the tracer content is conserved by
+the internal dynamics and physics (equations in flux form).
+For advection,
+only the CEN2 scheme ($i.e.$ $2^{nd}$ order finite different scheme) conserves the global variance of tracer.
+Nevertheless the other schemes ensure that the global variance decreases
+($i.e.$ they are at least slightly diffusive).
+For diffusion, all the schemes ensure the decrease of the total tracer variance, except the iso-neutral operator.
+There is generally no strict conservation of mass,
+as the equation of state is non linear with respect to $T$ and $S$.
+In practice, the mass is conserved to a very high accuracy.
 % -------------------------------------------------------------------------------------------------------------
 %       Advection Term
 …
+Whatever the advection scheme considered it conserves of the tracer content as all
+the scheme are written in flux form. Indeed,  let $T$ be the tracer and $\tau_u$, $\tau_v$,
+and $\tau_w$ its interpolated values at velocity point (whatever the interpolation is),
+Whatever the advection scheme considered it conserves of the tracer content as
+all the scheme are written in flux form.
+Indeed, let $T$ be the tracer and its $\tau_u$, $\tau_v$, and $\tau_w$ interpolated values at velocity point
+(whatever the interpolation is),
 the conservation of the tracer content due to the advection tendency is obtained as follows:
 \begin{flalign*}
 …
 \end{flalign*}
+The conservation of the variance of tracer due to the advection tendency
+can be achieved only with the CEN2 scheme, $i.e.$ when
+$\tau_u= \overline T^{\,i+1/2}$, $\tau_v= \overline T^{\,j+1/2}$, and $\tau_w= \overline T^{\,k+1/2}$.
+The conservation of the variance of tracer due to the advection tendency can be achieved only with the CEN2 scheme,
+$i.e.$ when $\tau_u= \overline T^{\,i+1/2}$, $\tau_v= \overline T^{\,j+1/2}$, and $\tau_w= \overline T^{\,k+1/2}$.
 It can be demonstarted as follows:
 \begin{flalign*}
 …
+The discrete formulation of the horizontal diffusion of momentum ensures the
+conservation of potential vorticity and the horizontal divergence, and the
+dissipation of the square of these quantities ($i.e.$ enstrophy and the
+variance of the horizontal divergence) as well as the dissipation of the
+horizontal kinetic energy. In particular, when the eddy coefficients are
+horizontally uniform, it ensures a complete separation of vorticity and
+horizontal divergence fields, so that diffusion (dissipation) of vorticity
+(enstrophy) does not generate horizontal divergence (variance of the
+horizontal divergence) and \textit{vice versa}.
+These properties of the horizontal diffusion operator are a direct consequence
+of properties \autoref{eq:DOM_curl_grad} and \autoref{eq:DOM_div_curl}.
+When the vertical curl of the horizontal diffusion of momentum (discrete sense)
+is taken, the term associated with the horizontal gradient of the divergence is
+locally zero.
+The discrete formulation of the horizontal diffusion of momentum ensures
+the conservation of potential vorticity and the horizontal divergence,
+and the dissipation of the square of these quantities
+($i.e.$ enstrophy and the variance of the horizontal divergence) as well as
+the dissipation of the horizontal kinetic energy.
+In particular, when the eddy coefficients are horizontally uniform,
+it ensures a complete separation of vorticity and horizontal divergence fields,
+so that diffusion (dissipation) of vorticity (enstrophy) does not generate horizontal divergence
+(variance of the horizontal divergence) and \textit{vice versa}.
+These properties of the horizontal diffusion operator are a direct consequence of
+properties \autoref{eq:DOM_curl_grad} and \autoref{eq:DOM_div_curl}.
+When the vertical curl of the horizontal diffusion of momentum (discrete sense) is taken,
+the term associated with the horizontal gradient of the divergence is locally zero.
 % -------------------------------------------------------------------------------------------------------------
 …
 \label{subsec:C.6.1}
 The lateral momentum diffusion term conserves the potential vorticity :
+The lateral momentum diffusion term conserves the potential vorticity:
 \begin{flalign*}
 &\int \limits_D \frac{1} {e_3 } \textbf{k} \cdot \nabla \times
 …
 \label{subsec:C.6.3}
+The lateral momentum diffusion term dissipates the enstrophy when the eddy
+coefficients are horizontally uniform:
+The lateral momentum diffusion term dissipates the enstrophy when the eddy coefficients are horizontally uniform:
 \begin{flalign*}
 &\int\limits_D  \zeta \; \textbf{k} \cdot \nabla \times
 …
 \label{subsec:C.6.4}
+When the horizontal divergence of the horizontal diffusion of momentum
+(discrete sense) is taken, the term associated with the vertical curl of the
+vorticity is zero locally, due to \autoref{eq:DOM_div_curl}.
+The resulting term conserves the $\chi$ and dissipates $\chi^2$
+when the eddy coefficients are horizontally uniform.
+When the horizontal divergence of the horizontal diffusion of momentum (discrete sense) is taken,
+the term associated with the vertical curl of the vorticity is zero locally, due to \autoref{eq:DOM_div_curl}.
+The resulting term conserves the $\chi$ and dissipates $\chi^2$ when the eddy coefficients are horizontally uniform.
 \begin{flalign*}
 & \int\limits_D  \nabla_h \cdot
 …
 \label{sec:C.7}
 As for the lateral momentum physics, the continuous form of the vertical diffusion
+of momentum satisfies several integral constraints. The first two are associated
 with the conservation of momentum and the dissipation of horizontal kinetic energy:
+As for the lateral momentum physics,
+the continuous form of the vertical diffusion of momentum satisfies several integral constraints.
+The first two are associated with the conservation of momentum and the dissipation of horizontal kinetic energy:
 \begin{align*}
  \int\limits_D   \frac{1} {e_3 }\; \frac{\partial } {\partial k}
 …
 \end{align*}
+The first property is obvious. The second results from:
+The first property is obvious.
+The second results from:
 \begin{flalign*}
 \int\limits_D
 …
 \end{flalign*}
+The vorticity is also conserved. Indeed:
+The vorticity is also conserved.
+Indeed:
 \begin{flalign*}
 \int \limits_D
 …
 \end{flalign*}
+If the vertical diffusion coefficient is uniform over the whole domain, the
+enstrophy is dissipated, $i.e.$
+If the vertical diffusion coefficient is uniform over the whole domain, the enstrophy is dissipated, $i.e.$
 \begin{flalign*}
 \int\limits_D \zeta \, \textbf{k} \cdot \nabla \times
 …
       &\left[  \frac{A_u^{\,vm}} {e_{3uw}} \delta_{k+1/2} \left[ \delta_{j+1/2} \left[ e_{1u}\,u \right] \right]  \right]  \biggr\}  &&\\
 \end{flalign*}
+Using the fact that the vertical diffusion coefficients are uniform, and that in
+$z$-coordinate, the vertical scale factors do not depend on $i$ and $j$ so
+that: $e_{3f} =e_{3u} =e_{3v} =e_{3t} $ and $e_{3w} =e_{3uw} =e_{3vw} $,
+it follows:
+Using the fact that the vertical diffusion coefficients are uniform,
+and that in $z$-coordinate, the vertical scale factors do not depend on $i$ and $j$ so that:
+$e_{3f} =e_{3u} =e_{3v} =e_{3t} $ and $e_{3w} =e_{3uw} =e_{3vw} $, it follows:
 \begin{flalign*}
 \equiv A^{\,vm} \sum\limits_{i,j,k} \zeta \;\delta_k
 …
       \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} \right) \right)\; dv = 0    &&&\\
 \end{flalign*}
+and the square of the horizontal divergence decreases ($i.e.$ the horizontal
+divergence is dissipated) if the vertical diffusion coefficient is uniform over the
+whole domain:
+and the square of the horizontal divergence decreases ($i.e.$ the horizontal divergence is dissipated) if
+the vertical diffusion coefficient is uniform over the whole domain:
 \begin{flalign*}
 …
 \label{sec:C.8}
 The numerical schemes used for tracer subgridscale physics are written such
 that the heat and salt contents are conserved (equations in flux form).
 Since a flux form is used to compute the temperature and salinity,
 the quadratic form of these quantities ($i.e.$ their variance) globally tends to diminish.
+The numerical schemes used for tracer subgridscale physics are written such that
+the heat and salt contents are conserved (equations in flux form).
+Since a flux form is used to compute the temperature and salinity,
+the quadratic form of these quantities ($i.e.$ their variance) globally tends to diminish.
 As for the advection term, there is conservation of mass only if the Equation Of Seawater is linear.

NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/annex_D.tex

-                      r9407
+                      r10368
 A "model life" is more than ten years. Its software, composed of a few
+hundred modules, is used by many people who are scientists or students
+and do not necessarily know every aspect of computing very well.
 Moreover, a well thought-out program is easier to read and understand,
+less difficult to modify, produces fewer bugs and is easier to maintain.
 Therefore, it is essential that the model development follows some rules :
+A "model life" is more than ten years.
+Its software, composed of a few hundred modules, is used by many people who are scientists or students and
+do not necessarily know every aspect of computing very well.
+Moreover, a well thought-out program is easier to read and understand, less difficult to modify,
+produces fewer bugs and is easier to maintain.
+Therefore, it is essential that the model development follows some rules:
 - well planned and designed
 …
 - flexible.
 To satisfy part of these aims, \NEMO is written with a coding standard which
 is close to the ECMWF rules, named DOCTOR \citep{Gibson_TR86}.
 These rules present some advantages like :
+To satisfy part of these aims, \NEMO is written with a coding standard which is close to the ECMWF rules,
+named DOCTOR \citep{Gibson_TR86}.
+These rules present some advantages like:
 - to provide a well presented program
 - to use rules for variable names which allow recognition of their type
+- to use rules for variable names which allow recognition of their type
 (integer, real, parameter, local or shared variables, etc. ).
 …
 \label{sec:D_structure}
 Each program begins with a set of headline comments containing :
+Each program begins with a set of headline comments containing:
 - the program title
 …
 - the author name(s), the date of creation and any updates.
 - Each program is split into several well separated sections and
+- Each program is split into several well separated sections and
 sub-sections with an underlined title and specific labelled statements.
 - A program has not more than 200 to 300 lines.
 A template of a module style can be found on the NEMO depository
 in the following file : NEMO/OPA\_SRC/module\_example.
+A template of a module style can be found on the NEMO depository in the following file:
+NEMO/OPA\_SRC/module\_example.
 % ================================================================
 % Coding conventions
 …
 \label{sec:D_coding}
+- Use of the universal language \textsc{Fortran} 90, and try to avoid obsolescent
+features like statement functions, do not use GO TO and EQUIVALENCE statements.
+- A continuation line begins with the character {\&} indented by three spaces
+compared to the previous line, while the previous line ended with the character {\&}.
+- All the variables must be declared. The code is usually compiled with implicit none.
+- Use of the universal language \textsc{Fortran} 90, and try to avoid obsolescent features like statement functions,
+do not use GO TO and EQUIVALENCE statements.
+- A continuation line begins with the character {\&} indented by three spaces compared to the previous line,
+while the previous line ended with the character {\&}.
+- All the variables must be declared.
+The code is usually compiled with implicit none.
+- Never use continuation lines in the declaration of a variable. When searching a
+variable in the code through a \textit{grep} command, the declaration line will be found.
+- In the declaration of a PUBLIC variable, the comment part at the end of the line
+should start with the two characters "\verb?!:?". the following UNIX command, \\
+- Never use continuation lines in the declaration of a variable.
+When searching a variable in the code through a \textit{grep} command, the declaration line will be found.
+- In the declaration of a PUBLIC variable, the comment part at the end of the line should start with
+the two characters "\verb?!:?".
+The following UNIX command, \\
 \verb?grep var_name *90 \ grep \!: ? \\
 will display the module name and the line where the var\_name declaration is.
+- Always use a three spaces indentation in DO loop, CASE, or IF-ELSEIF-ELSE-ENDIF
+statements.
+- Always use a three spaces indentation in DO loop, CASE, or IF-ELSEIF-ELSE-ENDIF statements.
 - use a space after a comma, except when it appears to separate the indices of an array.
 …
 \label{sec:D_naming}
+The purpose of the naming conventions is to use prefix letters to classify
+model variables. These conventions allow the variable type to be easily
+known and rapidly identified. The naming conventions are summarised
+in the Table below:
+The purpose of the naming conventions is to use prefix letters to classify model variables.
+These conventions allow the variable type to be easily known and rapidly identified.
+The naming conventions are summarised in the Table below:
 …
 %--------------------------------------------------------------------------------------------------------------
 N.B.   Parameter here, in not only parameter in the \textsc{Fortran} acceptation, it is also used for code variables
 that are read in namelist and should never been modified during a simulation.
+N.B. Parameter here, in not only parameter in the \textsc{Fortran} acceptation,
+it is also used for code variables that are read in namelist and should never been modified during a simulation.
 It is the case, for example, for the size of a domain (jpi,jpj,jpk).

NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/annex_E.tex

-                      r9407
+                      r10368
 \newpage
 $\ $\newline    % force a new ligne
+ This appendix some on going consideration on algorithms used or planned to be used
+in \NEMO.
+This appendix some on going consideration on algorithms used or planned to be used in \NEMO.
 $\ $\newline    % force a new ligne
 …
 \label{sec:TRA_adv_ubs}
 The UBS advection scheme is an upstream biased third order scheme based on
 an upstream-biased parabolic interpolation. It is also known as Cell Averaged
+QUICK scheme (Quadratic Upstream Interpolation for Convective
 Kinematics). For example, in the $i$-direction :
+The UBS advection scheme is an upstream biased third order scheme based on
+an upstream-biased parabolic interpolation.
+It is also known as Cell Averaged QUICK scheme (Quadratic Upstream Interpolation for Convective Kinematics).
+For example, in the $i$-direction:
 \begin{equation} \label{eq:tra_adv_ubs2}
 \tau _u^{ubs} = \left\{  \begin{aligned}
 …
 - \frac{1}{2}\, |U|_{i+1/2} \;\frac{1}{6} \;\delta_{i+1/2}[\tau"_i]
 \end{equation}
 where $U_{i+1/2} = e_{1u}\,e_{3u}\,u_{i+1/2}$ and
 $\tau "_i =\delta _i \left[ {\delta _{i+1/2} \left[ \tau \right]} \right]$.
 By choosing this expression for $\tau "$ we consider a fourth order approximation
+of $\partial_i^2$ with a constant i-grid spacing ($\Delta i=1$).
+where $U_{i+1/2} = e_{1u}\,e_{3u}\,u_{i+1/2}$ and
+$\tau "_i =\delta _i \left[ {\delta _{i+1/2} \left[ \tau \right]} \right]$.
+By choosing this expression for $\tau "$ we consider a fourth order approximation of $\partial_i^2$ with
+a constant i-grid spacing ($\Delta i=1$).
 Alternative choice: introduce the scale factors:
 …
+This results in a dissipatively dominant (i.e. hyper-diffusive) truncation
+error \citep{Shchepetkin_McWilliams_OM05}. The overall performance of the
+advection scheme is similar to that reported in \cite{Farrow1995}.
+It is a relatively good compromise between accuracy and smoothness. It is
+not a \emph{positive} scheme meaning false extrema are permitted but the
+amplitude of such are significantly reduced over the centred second order
+method. Nevertheless it is not recommended to apply it to a passive tracer
+that requires positivity.
+The intrinsic diffusion of UBS makes its use risky in the vertical direction
+where the control of artificial diapycnal fluxes is of paramount importance.
+It has therefore been preferred to evaluate the vertical flux using the TVD
+scheme when \np{ln\_traadv\_ubs}\forcode{ = .true.}.
+For stability reasons, in \autoref{eq:tra_adv_ubs}, the first term which corresponds
+to a second order centred scheme is evaluated using the \textit{now} velocity
+(centred in time) while the second term which is the diffusive part of the scheme,
+is evaluated using the \textit{before} velocity (forward in time. This is discussed
+by \citet{Webb_al_JAOT98} in the context of the Quick advection scheme. UBS and QUICK
+schemes only differ by one coefficient. Substituting 1/6 with 1/8 in
+(\autoref{eq:tra_adv_ubs}) leads to the QUICK advection scheme \citep{Webb_al_JAOT98}.
+This option is not available through a namelist parameter, since the 1/6
+coefficient is hard coded. Nevertheless it is quite easy to make the
+substitution in \mdl{traadv\_ubs} module and obtain a QUICK scheme
+NB 1: When a high vertical resolution $O(1m)$ is used, the model stability can
+be controlled by vertical advection (not vertical diffusion which is usually
+solved using an implicit scheme). Computer time can be saved by using a
+time-splitting technique on vertical advection. This possibility have been
+implemented and validated in ORCA05-L301. It is not currently offered in the
+current reference version.
+NB 2 : In a forthcoming release four options will be proposed for the
+vertical component used in the UBS scheme. $\tau _w^{ubs}$ will be
+evaluated using either \textit{(a)} a centred $2^{nd}$ order scheme ,
+or  \textit{(b)} a TVD scheme, or  \textit{(c)} an interpolation based on conservative
+parabolic splines following \citet{Shchepetkin_McWilliams_OM05} implementation of UBS in ROMS,
+or  \textit{(d)} an UBS. The $3^{rd}$ case has dispersion properties similar to an
+eight-order accurate conventional scheme.
+NB 3 : It is straight forward to rewrite \autoref{eq:tra_adv_ubs} as follows:
+This results in a dissipatively dominant (i.e. hyper-diffusive) truncation error
+\citep{Shchepetkin_McWilliams_OM05}.
+The overall performance of the advection scheme is similar to that reported in \cite{Farrow1995}.
+It is a relatively good compromise between accuracy and smoothness.
+It is not a \emph{positive} scheme meaning false extrema are permitted but
+the amplitude of such are significantly reduced over the centred second order method.
+Nevertheless it is not recommended to apply it to a passive tracer that requires positivity.
+The intrinsic diffusion of UBS makes its use risky in the vertical direction where
+the control of artificial diapycnal fluxes is of paramount importance.
+It has therefore been preferred to evaluate the vertical flux using the TVD scheme when
+\np{ln\_traadv\_ubs}\forcode{ = .true.}.
+For stability reasons, in \autoref{eq:tra_adv_ubs}, the first term which corresponds to
+a second order centred scheme is evaluated using the \textit{now} velocity (centred in time) while
+the second term which is the diffusive part of the scheme, is evaluated using the \textit{before} velocity
+(forward in time).
+This is discussed by \citet{Webb_al_JAOT98} in the context of the Quick advection scheme.
+UBS and QUICK schemes only differ by one coefficient.
+Substituting 1/6 with 1/8 in (\autoref{eq:tra_adv_ubs}) leads to the QUICK advection scheme \citep{Webb_al_JAOT98}.
+This option is not available through a namelist parameter, since the 1/6 coefficient is hard coded.
+Nevertheless it is quite easy to make the substitution in \mdl{traadv\_ubs} module and obtain a QUICK scheme.
+NB 1: When a high vertical resolution $O(1m)$ is used, the model stability can be controlled by vertical advection
+(not vertical diffusion which is usually solved using an implicit scheme).
+Computer time can be saved by using a time-splitting technique on vertical advection.
+This possibility have been implemented and validated in ORCA05-L301.
+It is not currently offered in the current reference version.
+NB 2: In a forthcoming release four options will be proposed for the vertical component used in the UBS scheme.
+$\tau _w^{ubs}$ will be evaluated using either \textit{(a)} a centered $2^{nd}$ order scheme,
+or \textit{(b)} a TVD scheme, or \textit{(c)} an interpolation based on conservative parabolic splines following
+\citet{Shchepetkin_McWilliams_OM05} implementation of UBS in ROMS, or \textit{(d)} an UBS.
+The $3^{rd}$ case has dispersion properties similar to an eight-order accurate conventional scheme.
+NB 3: It is straight forward to rewrite \autoref{eq:tra_adv_ubs} as follows:
 \begin{equation} \label{eq:tra_adv_ubs2}
 \tau _u^{ubs} = \left\{  \begin{aligned}
 …
 \end{split}
 \end{equation}
 \autoref{eq:tra_adv_ubs2} has several advantages. First it clearly evidence that
+the UBS scheme is based on the fourth order scheme to which is added an
+upstream biased diffusive term. Second, this emphasises that the $4^{th}$ order
+part have to be evaluated at \emph{now} time step, not only the $2^{th}$ order
+part as stated above using \autoref{eq:tra_adv_ubs}. Third, the diffusive term is
+in fact a biharmonic operator with a eddy coefficient with is simply proportional
 to the velocity.
+\autoref{eq:tra_adv_ubs2} has several advantages.
+First it clearly evidences that the UBS scheme is based on the fourth order scheme to which
+is added an upstream biased diffusive term.
+Second, this emphasises that the $4^{th}$ order part have to be evaluated at \emph{now} time step,
+not only the $2^{th}$ order part as stated above using \autoref{eq:tra_adv_ubs}.
+Third, the diffusive term is in fact a biharmonic operator with a eddy coefficient which
+is simply proportional to the velocity.
 laplacian diffusion:
 …
 with ${A_u^{lT}}^2 = \frac{1}{12} {e_{1u}}^3\ |u|$,
 $i.e.$ $A_u^{lT} = \frac{1}{\sqrt{12}} \,e_{1u}\ \sqrt{ e_{1u}\,|u|\,}$
 it comes :
+it comes:
 \begin{equation} \label{eq:tra_ldf_lap}
 \begin{split}
 …
 \end{split}
 \end{equation}
+if the velocity is uniform ($i.e.$ $|u|=cst$) and choosing $\tau "_i =\frac{e_{1T}}{e_{2T}\,e_{3T}}\delta _i \left[ \frac{e_{2u} e_{3u} }{e_{1u} } \delta _{i+1/2}[\tau] \right]$
+if the velocity is uniform ($i.e.$ $|u|=cst$) and
+choosing $\tau "_i =\frac{e_{1T}}{e_{2T}\,e_{3T}}\delta _i \left[ \frac{e_{2u} e_{3u} }{e_{1u} } \delta _{i+1/2}[\tau] \right]$
 sol 1 coefficient at T-point ( add $e_{1u}$ and $e_{1T}$ on both side of first $\delta$):
 …
 \label{sec:LF}
+We adopt the following semi-discrete notation for time derivative. Given the values of a variable $q$ at successive time step, the time derivation and averaging operators at the mid time step are:
+We adopt the following semi-discrete notation for time derivative.
+Given the values of a variable $q$ at successive time step,
+the time derivation and averaging operators at the mid time step are:
 \begin{subequations} \label{eq:dt_mt}
 \begin{align}
 …
 \end{align}
 \end{subequations}
 As for space operator, the adjoint of the derivation and averaging time operators are
+$\delta_t^*=\delta_{t+\rdt/2}$ and $\overline{\cdot}^{\,t\,*}= \overline{\cdot}^{\,t+\Delta/2}$
+, respectively.
+As for space operator,
+the adjoint of the derivation and averaging time operators are $\delta_t^*=\delta_{t+\rdt/2}$ and
+$\overline{\cdot}^{\,t\,*}= \overline{\cdot}^{\,t+\Delta/2}$, respectively.
 The Leap-frog time stepping given by \autoref{eq:DOM_nxt} can be defined as:
 …
       =         \frac{q^{t+\rdt}-q^{t-\rdt}}{2\rdt}
 \end{equation}
 Note that \autoref{chap:LF} shows that the leapfrog time step is $\rdt$, not $2\rdt$
+as it can be found sometime in literature.
 The leap-Frog time stepping is a second order centered scheme. As such it respects
 the quadratic invariant in integral forms, $i.e.$ the following continuous property,
+Note that \autoref{chap:LF} shows that the leapfrog time step is $\rdt$,
+not $2\rdt$ as it can be found sometimes in literature.
+The leap-Frog time stepping is a second order centered scheme.
+As such it respects the quadratic invariant in integral forms, $i.e.$ the following continuous property,
 \begin{equation} \label{eq:Energy}
 \int_{t_0}^{t_1} {q\, \frac{\partial q}{\partial t} \;dt}
 …
    =  \frac{1}{2} \left( {q_{t_1}}^2 - {q_{t_0}}^2 \right) ,
 \end{equation}
+is satisfied in discrete form. Indeed,
+is satisfied in discrete form.
+Indeed,
 \begin{equation} \begin{split}
 \int_{t_0}^{t_1} {q\, \frac{\partial q}{\partial t} \;dt}
 …
       \equiv \frac{1}{2} \left( {q_{t_1}}^2 - {q_{t_0}}^2 \right)
 \end{split} \end{equation}
 NB here pb of boundary condition when applying the adjoin! In space, setting to 0
 the quantity in land area is sufficient to get rid of the boundary condition
 (equivalently of the boundary value of the integration by part). In time this boundary
 condition is not physical and \textbf{add something here!!!}
+NB here pb of boundary condition when applying the adjoint!
+In space, setting to 0 the quantity in land area is sufficient to get rid of the boundary condition
+(equivalently of the boundary value of the integration by part).
+In time this boundary condition is not physical and \textbf{add something here!!!}
 …
 \subsection{Griffies iso-neutral diffusion operator}
+Let try to define a scheme that get its inspiration from the \citet{Griffies_al_JPO98}
+scheme, but is formulated within the \NEMO framework ($i.e.$ using scale
+factors rather than grid-size and having a position of $T$-points that is not
+necessary in the middle of vertical velocity points, see \autoref{fig:zgr_e3}).
+In the formulation \autoref{eq:tra_ldf_iso} introduced in 1995 in OPA, the ancestor of \NEMO,
+the off-diagonal terms of the small angle diffusion tensor contain several double
+spatial averages of a gradient, for example $\overline{\overline{\delta_k \cdot}}^{\,i,k}$.
+It is apparent that the combination of a $k$ average and a $k$ derivative of the tracer
+allows for the presence of grid point oscillation structures that will be invisible
+to the operator. These structures are \textit{computational modes}. They
+will not be damped by the iso-neutral operator, and even possibly amplified by it.
+In other word, the operator applied to a tracer does not warranties the decrease of
+its global average variance. To circumvent this, we have introduced a smoothing of
+the slopes of the iso-neutral surfaces (see \autoref{chap:LDF}). Nevertheless, this technique
+works fine for $T$ and $S$ as they are active tracers ($i.e.$ they enter the computation
+of density), but it does not work for a passive tracer.   \citep{Griffies_al_JPO98} introduce
+a different way to discretise the off-diagonal terms that nicely solve the problem.
+The idea is to get rid of combinations of an averaged in one direction combined
+with a derivative in the same direction by considering triads. For example in the
+(\textbf{i},\textbf{k}) plane, the four triads are defined at the $(i,k)$ $T$-point as follows:
+Let try to define a scheme that get its inspiration from the \citet{Griffies_al_JPO98} scheme,
+but is formulated within the \NEMO framework
+($i.e.$ using scale factors rather than grid-size and having a position of $T$-points that
+is not necessary in the middle of vertical velocity points, see \autoref{fig:zgr_e3}).
+In the formulation \autoref{eq:tra_ldf_iso} introduced in 1995 in OPA, the ancestor of \NEMO,
+the off-diagonal terms of the small angle diffusion tensor contain several double spatial averages of a gradient,
+for example $\overline{\overline{\delta_k \cdot}}^{\,i,k}$.
+It is apparent that the combination of a $k$ average and a $k$ derivative of the tracer allows for
+the presence of grid point oscillation structures that will be invisible to the operator.
+These structures are \textit{computational modes}.
+They will not be damped by the iso-neutral operator, and even possibly amplified by it.
+In other word, the operator applied to a tracer does not warranties the decrease of its global average variance.
+To circumvent this, we have introduced a smoothing of the slopes of the iso-neutral surfaces
+(see \autoref{chap:LDF}).
+Nevertheless, this technique works fine for $T$ and $S$ as they are active tracers
+($i.e.$ they enter the computation of density), but it does not work for a passive tracer.
+\citep{Griffies_al_JPO98} introduce a different way to discretise the off-diagonal terms that
+nicely solve the problem.
+The idea is to get rid of combinations of an averaged in one direction combined with
+a derivative in the same direction by considering triads.
+For example in the (\textbf{i},\textbf{k}) plane, the four triads are defined at the $(i,k)$ $T$-point as follows:
 \begin{equation} \label{eq:Gf_triads}
 _i^k \mathbb{T}_{i_p}^{k_p} (T)
 …
                              \right)
 \end{equation}
 where the indices $i_p$ and $k_p$ define the four triads and take the following value:
 $i_p = -1/2$ or $1/2$ and $k_p = -1/2$ or $1/2$,
 $b_u= e_{1u}\,e_{2u}\,e_{3u}$ is the volume of $u$-cells,
+where the indices $i_p$ and $k_p$ define the four triads and take the following value:
+$i_p = -1/2$ or $1/2$ and $k_p = -1/2$ or $1/2$,
+$b_u= e_{1u}\,e_{2u}\,e_{3u}$ is the volume of $u$-cells,
 $A_i^k$ is the lateral eddy diffusivity coefficient defined at $T$-point,
 and $_i^k \mathbb{R}_{i_p}^{k_p}$ is the slope associated with each triad :
+and $_i^k \mathbb{R}_{i_p}^{k_p}$ is the slope associated with each triad:
 \begin{equation} \label{eq:Gf_slopes}
 _i^k \mathbb{R}_{i_p}^{k_p}
 …
 {\left(\alpha / \beta \right)_i^k  \ \delta_{k+k_p}[T^i ] - \delta_{k+k_p}[S^i ] }
 \end{equation}
+Note that in \autoref{eq:Gf_slopes} we use the ratio $\alpha / \beta$ instead of
+multiplying the temperature derivative by $\alpha$ and the salinity derivative
+by $\beta$. This is more efficient as the ratio $\alpha / \beta$ can to be
+evaluated directly.
+Note that in \autoref{eq:Gf_triads}, we chose to use ${b_u}_{\,i+i_p}^{\,k}$ instead of
+${b_{uw}}_{\,i+i_p}^{\,k+k_p}$. This choice has been motivated by the decrease
+of tracer variance and the presence of partial cell at the ocean bottom
+(see \autoref{apdx:Gf_operator}).
+Note that in \autoref{eq:Gf_slopes} we use the ratio $\alpha / \beta$ instead of
+multiplying the temperature derivative by $\alpha$ and the salinity derivative by $\beta$.
+This is more efficient as the ratio $\alpha / \beta$ can to be evaluated directly.
+Note that in \autoref{eq:Gf_triads}, we chose to use ${b_u}_{\,i+i_p}^{\,k}$ instead of ${b_{uw}}_{\,i+i_p}^{\,k+k_p}$.
+This choice has been motivated by the decrease of tracer variance and
+the presence of partial cell at the ocean bottom (see \autoref{apdx:Gf_operator}).
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 \begin{figure}[!ht] \begin{center}
 \includegraphics[width=0.70\textwidth]{Fig_ISO_triad}
 \caption{  \protect\label{fig:ISO_triad}
+Triads used in the Griffies's like iso-neutral diffision scheme for
 $u$-component (upper panel) and $w$-component (lower panel).}
+\caption{  \protect\label{fig:ISO_triad}
+  Triads used in the Griffies's like iso-neutral diffision scheme for
+  $u$-component (upper panel) and $w$-component (lower panel).}
 \end{center}
 \end{figure}
 …
 The four iso-neutral fluxes associated with the triads are defined at $T$-point.
 They take the following expression :
+They take the following expression:
 \begin{flalign} \label{eq:Gf_fluxes}
 \begin{split}
 …
 \end{flalign}
 The resulting iso-neutral fluxes at $u$- and $w$-points are then given by the
 sum of the fluxes that cross the $u$- and $w$-face (\autoref{fig:ISO_triad}):
+The resulting iso-neutral fluxes at $u$- and $w$-points are then given by
+the sum of the fluxes that cross the $u$- and $w$-face (\autoref{fig:ISO_triad}):
 \begin{flalign} \label{eq:iso_flux}
 \textbf{F}_{iso}(T)
 …
 %    \end{pmatrix}
 \end{flalign}
 resulting in a iso-neutral diffusion tendency on temperature given by the divergence
 of the sum of all the four triad fluxes :
+resulting in a iso-neutral diffusion tendency on temperature given by
+the divergence of the sum of all the four triad fluxes:
 \begin{equation} \label{eq:Gf_operator}
 D_l^T = \frac{1}{b_T}  \sum_{\substack{i_p,\,k_p}} \left\{
 …
 where $b_T= e_{1T}\,e_{2T}\,e_{3T}$ is the volume of $T$-cells.
+This expression of the iso-neutral diffusion has been chosen in order to satisfy
+the following six properties:
+This expression of the iso-neutral diffusion has been chosen in order to satisfy the following six properties:
 \begin{description}
+\item[$\bullet$ horizontal diffusion] The discretization of the diffusion operator
+recovers the traditional five-point Laplacian in the limit of flat iso-neutral direction :
+\item[$\bullet$ horizontal diffusion]
+  The discretization of the diffusion operator recovers the traditional five-point Laplacian in
+  the limit of flat iso-neutral direction:
 \begin{equation} \label{eq:Gf_property1a}
 D_l^T = \frac{1}{b_T}  \ \delta_{i}
 …
 \end{equation}
 \item[$\bullet$ implicit treatment in the vertical]  In the diagonal term associated
+with the vertical divergence of the iso-neutral fluxes (i.e. the term associated
+with a second order vertical derivative) appears only tracer values associated
+with a single water column. This is of paramount importance since it means
+that the implicit in time algorithm for solving the vertical diffusion equation can
+be used to evaluate this term. It is a necessity since the vertical eddy diffusivity
 associated with this term,
+\item[$\bullet$ implicit treatment in the vertical]
+  In the diagonal term associated with the vertical divergence of the iso-neutral fluxes
+  (i.e. the term associated with a second order vertical derivative)
+  appears only tracer values associated with a single water column.
+  This is of paramount importance since it means that
+  the implicit in time algorithm for solving the vertical diffusion equation can be used to evaluate this term.
+  It is a necessity since the vertical eddy diffusivity associated with this term,
 \begin{equation}
     \sum_{\substack{i_p, \,k_p}} \left\{
 …
 can be quite large.
 \item[$\bullet$ pure iso-neutral operator]  The iso-neutral flux of locally referenced
 potential density is zero, $i.e.$
+\item[$\bullet$ pure iso-neutral operator]
+  The iso-neutral flux of locally referenced potential density is zero, $i.e.$
 \begin{align} \label{eq:Gf_property2}
 \begin{matrix}
 …
 \end{matrix}
 \end{align}
 This result is trivially obtained using the \autoref{eq:Gf_triads} applied to $T$ and $S$
 and the definition of the triads' slopes \autoref{eq:Gf_slopes}.
 \item[$\bullet$ conservation of tracer] The iso-neutral diffusion term conserve the
 total tracer content, $i.e.$
+This result is trivially obtained using the \autoref{eq:Gf_triads} applied to $T$ and $S$ and
+the definition of the triads' slopes \autoref{eq:Gf_slopes}.
+\item[$\bullet$ conservation of tracer]
+  The iso-neutral diffusion term conserve the total tracer content, $i.e.$
 \begin{equation} \label{eq:Gf_property1}
 \sum_{i,j,k} \left\{ D_l^T \ b_T \right\} = 0
 \end{equation}
+This property is trivially satisfied since the iso-neutral diffusive operator
+is written in flux form.
+\item[$\bullet$ decrease of tracer variance] The iso-neutral diffusion term does
+not increase the total tracer variance, $i.e.$
+This property is trivially satisfied since the iso-neutral diffusive operator is written in flux form.
+\item[$\bullet$ decrease of tracer variance]
+  The iso-neutral diffusion term does not increase the total tracer variance, $i.e.$
 \begin{equation} \label{eq:Gf_property1}
 \sum_{i,j,k} \left\{ T \ D_l^T \ b_T \right\} \leq 0
 \end{equation}
+The property is demonstrated in the \autoref{apdx:Gf_operator}. It is a
+key property for a diffusion term. It means that the operator is also a dissipation
+term, $i.e.$ it is a sink term for the square of the quantity on which it is applied.
+It therfore ensure that, when the diffusivity coefficient is large enough, the field
+on which it is applied become free of grid-point noise.
+\item[$\bullet$ self-adjoint operator] The iso-neutral diffusion operator is self-adjoint,
+$i.e.$
+The property is demonstrated in the \autoref{apdx:Gf_operator}.
+It is a key property for a diffusion term.
+It means that the operator is also a dissipation term,
+$i.e.$ it is a sink term for the square of the quantity on which it is applied.
+It therfore ensures that, when the diffusivity coefficient is large enough,
+the field on which it is applied become free of grid-point noise.
+\item[$\bullet$ self-adjoint operator]
+  The iso-neutral diffusion operator is self-adjoint, $i.e.$
 \begin{equation} \label{eq:Gf_property1}
 \sum_{i,j,k} \left\{ S \ D_l^T \ b_T \right\} = \sum_{i,j,k} \left\{ D_l^S \ T \ b_T \right\}
 \end{equation}
 In other word, there is no needs to develop a specific routine from the adjoint of this
+operator. We just have to apply the same routine. This properties can be demonstrated
+quite easily in a similar way the "non increase of tracer variance" property has been
 proved (see \autoref{apdx:Gf_operator}).
+In other word, there is no needs to develop a specific routine from the adjoint of this operator.
+We just have to apply the same routine.
+This properties can be demonstrated quite easily in a similar way the "non increase of tracer variance" property
+has been proved (see \autoref{apdx:Gf_operator}).
 \end{description}
 …
 \subsection{Eddy induced velocity and skew flux formulation}
 When Gent and McWilliams [1990] diffusion is used (\key{traldf\_eiv} defined),
 an additional advection term is added. The associated velocity is the so called
+eddy induced velocity, the formulation of which depends on the slopes of iso-
+neutral surfaces. Contrary to the case of iso-neutral mixing, the slopes used
+here are referenced to the geopotential surfaces, $i.e.$ \autoref{eq:ldfslp_geo}
+is used in $z$-coordinate, and the sum \autoref{eq:ldfslp_geo}
 + \autoref{eq:ldfslp_iso} in $z^*$ or $s$-coordinates.
+When Gent and McWilliams [1990] diffusion is used (\key{traldf\_eiv} defined),
+an additional advection term is added.
+The associated velocity is the so called eddy induced velocity,
+the formulation of which depends on the slopes of iso-neutral surfaces.
+Contrary to the case of iso-neutral mixing, the slopes used here are referenced to the geopotential surfaces,
+$i.e.$ \autoref{eq:ldfslp_geo} is used in $z$-coordinate,
+and the sum \autoref{eq:ldfslp_geo} + \autoref{eq:ldfslp_iso} in $z^*$ or $s$-coordinates.
 The eddy induced velocity is given by:
 …
 \end{split}
 \end{equation}
 where $A_{e}$ is the eddy induced velocity coefficient, and $r_i$ and $r_j$ the
 slopes between the iso-neutral and the geopotential surfaces.
+where $A_{e}$ is the eddy induced velocity coefficient,
+and $r_i$ and $r_j$ the slopes between the iso-neutral and the geopotential surfaces.
 %%gm wrong: to be modified with 2 2D streamfunctions
+ In other words,
+the eddy induced velocity can be derived from a vector streamfuntion, $\phi$, which
+is given by $\phi = A_e\,\textbf{r}$ as $\textbf{U}^*  = \textbf{k} \times \nabla \phi$
+In other words, the eddy induced velocity can be derived from a vector streamfuntion, $\phi$,
+which is given by $\phi = A_e\,\textbf{r}$ as $\textbf{U}^*  = \textbf{k} \times \nabla \phi$.
 %%end gm
+A traditional way to implement this additional advection is to add it to the eulerian
+velocity prior to compute the tracer advection. This allows us to take advantage of
+all the advection schemes offered for the tracers (see \autoref{sec:TRA_adv}) and not just
+a $2^{nd}$ order advection scheme. This is particularly useful for passive tracers
+where \emph{positivity} of the advection scheme is of paramount importance.
+A traditional way to implement this additional advection is to add it to the eulerian velocity prior to
+compute the tracer advection.
+This allows us to take advantage of all the advection schemes offered for the tracers
+(see \autoref{sec:TRA_adv}) and not just a $2^{nd}$ order advection scheme.
+This is particularly useful for passive tracers where
+\emph{positivity} of the advection scheme is of paramount importance.
 % give here the expression using the triads. It is different from the one given in \autoref{eq:ldfeiv}
 % see just below a copy of this equation:
 …
 \end{equation}
+\citep{Griffies_JPO98} introduces another way to implement the eddy induced advection,
+the so-called skew form. It is based on a transformation of the advective fluxes
+using the non-divergent nature of the eddy induced velocity.
+For example in the (\textbf{i},\textbf{k}) plane, the tracer advective fluxes can be
+transformed as follows:
+\citep{Griffies_JPO98} introduces another way to implement the eddy induced advection, the so-called skew form.
+It is based on a transformation of the advective fluxes using the non-divergent nature of the eddy induced velocity.
+For example in the (\textbf{i},\textbf{k}) plane, the tracer advective fluxes can be transformed as follows:
 \begin{flalign*}
 \begin{split}
 …
 \end{split}
 \end{flalign*}
 and since the eddy induces velocity field is no-divergent, we end up with the skew
 form of the eddy induced advective fluxes:
+and since the eddy induces velocity field is no-divergent,
+we end up with the skew form of the eddy induced advective fluxes:
 \begin{equation} \label{eq:eiv_skew_continuous}
 \textbf{F}_{eiv}^T = \begin{pmatrix}
 …
                                  \end{pmatrix}
 \end{equation}
 The tendency associated with eddy induced velocity is then simply the divergence
+of the \autoref{eq:eiv_skew_continuous} fluxes. It naturally conserves the tracer
+content, as it is expressed in flux form and, as the advective form, it preserve the
+tracer variance. Another interesting property of \autoref{eq:eiv_skew_continuous}
+form is that when $A=A_e$, a simplification occurs in the sum of the iso-neutral
 diffusion and eddy induced velocity terms:
+The tendency associated with eddy induced velocity is then simply the divergence of
+the \autoref{eq:eiv_skew_continuous} fluxes.
+It naturally conserves the tracer content, as it is expressed in flux form and,
+as the advective form, it preserves the tracer variance.
+Another interesting property of \autoref{eq:eiv_skew_continuous} form is that when $A=A_e$,
+a simplification occurs in the sum of the iso-neutral diffusion and eddy induced velocity terms:
 \begin{flalign} \label{eq:eiv_skew+eiv_continuous}
 \textbf{F}_{iso}^T + \textbf{F}_{eiv}^T &=
 …
 \end{pmatrix}
 \end{flalign}
 The horizontal component reduces to the one use for an horizontal laplacian
+operator and the vertical one keep the same complexity, but not more. This property
+has been used to reduce the computational time \citep{Griffies_JPO98}, but it is
+not of practical use as usually $A \neq A_e$. Nevertheless this property can be used to
+choose a discret form of  \autoref{eq:eiv_skew_continuous} which is consistent with the
 iso-neutral operator \autoref{eq:Gf_operator}. Using the slopes \autoref{eq:Gf_slopes}
 and defining $A_e$ at $T$-point($i.e.$ as $A$, the eddy diffusivity coefficient),
 the resulting discret form is given by:
+The horizontal component reduces to the one use for an horizontal laplacian operator and
+the vertical one keeps the same complexity, but not more.
+This property has been used to reduce the computational time \citep{Griffies_JPO98},
+but it is not of practical use as usually $A \neq A_e$.
+Nevertheless this property can be used to choose a discret form of \autoref{eq:eiv_skew_continuous} which
+is consistent with the iso-neutral operator \autoref{eq:Gf_operator}.
+Using the slopes \autoref{eq:Gf_slopes} and defining $A_e$ at $T$-point($i.e.$ as $A$,
+the eddy diffusivity coefficient), the resulting discret form is given by:
 \begin{equation} \label{eq:eiv_skew}
 \textbf{F}_{eiv}^T   \equiv   \frac{1}{4} \left( \begin{aligned}
 …
 \end{equation}
 Note that \autoref{eq:eiv_skew} is valid in $z$-coordinate with or without partial cells.
 In $z^*$ or $s$-coordinate, the slope between the level and the geopotential surfaces
 must be added to $\mathbb{R}$ for the discret form to be exact.
 Such a choice of discretisation is consistent with the iso-neutral operator as it uses the
+same definition for the slopes. It also ensures the conservation of the tracer variance
+(see Appendix \autoref{apdx:eiv_skew}), $i.e.$ it does not include a diffusive component
 but is a "pure" advection term.
+In $z^*$ or $s$-coordinate, the slope between the level and the geopotential surfaces must be added to
+$\mathbb{R}$ for the discret form to be exact.
+Such a choice of discretisation is consistent with the iso-neutral operator as
+it uses the same definition for the slopes.
+It also ensures the conservation of the tracer variance (see Appendix \autoref{apdx:eiv_skew}),
+$i.e.$ it does not include a diffusive component but is a "pure" advection term.
 …
 This part will be moved in an Appendix.
 The continuous property to be demonstrated is :
+The continuous property to be demonstrated is:
 \begin{align*}
 \int_D  D_l^T \; T \;dv   \leq 0
 …
+%
 \allowdisplaybreaks
+\intertext{The summation is done over all $i$ and $k$ indices, it is therefore possible to introduce a shift of $-1$ either in $i$ or $k$ direction in order to regroup all the terms of the summation by triad at a ($i$,$k$) point. In other words, we regroup all the terms in the neighbourhood  that contain a triad at the same ($i$,$k$) indices. It becomes: }
+  \intertext{The summation is done over all $i$ and $k$ indices,
+  it is therefore possible to introduce a shift of $-1$ either in $i$ or $k$ direction in order to
+  regroup all the terms of the summation by triad at a ($i$,$k$) point.
+  In other words, we regroup all the terms in the neighbourhood that contain a triad at the same ($i$,$k$) indices.
+  It becomes: }
+%
 &\equiv -\sum_{i,k}
 …
+%
 \allowdisplaybreaks
+\intertext{Then outing in factor the triad in each of the four terms of the summation and substituting the triads by their expression given in \autoref{eq:Gf_triads}. It becomes: }
+  \intertext{Then outing in factor the triad in each of the four terms of the summation and
+  substituting the triads by their expression given in \autoref{eq:Gf_triads}.
+  It becomes: }
+%
 &\equiv -\sum_{i,k}
 …
 The last inequality is obviously obtained as we succeed in obtaining a negative summation of square quantities.
+Note that, if instead of multiplying $D_l^T$ by $T$, we were using another tracer field, let say $S$, then the previous demonstration would have let to:
+Note that, if instead of multiplying $D_l^T$ by $T$, we were using another tracer field, let say $S$,
+then the previous demonstration would have let to:
 \begin{align*}
 \int_D  S \; D_l^T  \;dv &\equiv  \sum_{i,k} \left\{ S \ D_l^T \ b_T \right\}    \\
 …
 &\equiv  \sum_{i,k} \left\{ D_l^S \ T \ b_T \right\}
 \end{align*}
+This means that the iso-neutral operator is self-adjoint. There is no need to develop a specific to obtain it.
+This means that the iso-neutral operator is self-adjoint.
+There is no need to develop a specific to obtain it.
 …
 This have to be moved in an Appendix.
 The continuous property to be demonstrated is :
+The continuous property to be demonstrated is:
 \begin{align*}
 \int_D \nabla \cdot \textbf{F}_{eiv}(T) \; T \;dv  \equiv 0
 …
 \end{matrix}
 \end{align*}
 The two terms associated with the triad ${_i^k \mathbb{R}_{+1/2}^{+1/2}}$ are the
 same but of opposite signs, they cancel out.
 Exactly the same thing occurs for the triad ${_i^k \mathbb{R}_{-1/2}^{-1/2}}$.
 The two terms associated with the triad ${_i^k \mathbb{R}_{+1/2}^{-1/2}}$ are the
 same but both of opposite signs and shifted by 1 in $k$ direction. When summing over $k$
+they cancel out with the neighbouring grid points.
 Exactly the same thing occurs for the triad ${_i^k \mathbb{R}_{-1/2}^{+1/2}}$ in the
+$i$ direction. Therefore the sum over the domain is zero, $i.e.$ the variance of the
 tracer is preserved by the discretisation of the skew fluxes.
+The two terms associated with the triad ${_i^k \mathbb{R}_{+1/2}^{+1/2}}$ are the same but of opposite signs,
+they cancel out.
+Exactly the same thing occurs for the triad ${_i^k \mathbb{R}_{-1/2}^{-1/2}}$.
+The two terms associated with the triad ${_i^k \mathbb{R}_{+1/2}^{-1/2}}$ are the same but both of opposite signs and
+shifted by 1 in $k$ direction.
+When summing over $k$ they cancel out with the neighbouring grid points.
+Exactly the same thing occurs for the triad ${_i^k \mathbb{R}_{-1/2}^{+1/2}}$ in the $i$ direction.
+Therefore the sum over the domain is zero,
+$i.e.$ the variance of the tracer is preserved by the discretisation of the skew fluxes.
 \end{document}

NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/annex_iso.tex

-                      r10146
+                      r10368
 %---------------------------------------------------------------------------------------------------------
 Two scheme are available to perform the iso-neutral diffusion.
 If the namelist logical \np{ln\_traldf\_triad} is set true,
 \NEMO updates both active and passive tracers using the Griffies triad representation
+of iso-neutral diffusion and the eddy-induced advective skew (GM) fluxes.
 If the namelist logical \np{ln\_traldf\_iso} is set true,
 the filtered version of Cox's original scheme (the Standard scheme) is employed (\autoref{sec:LDF_slp}).
 In the present implementation of the Griffies scheme,
+Two scheme are available to perform the iso-neutral diffusion.
+If the namelist logical \np{ln\_traldf\_triad} is set true,
+\NEMO updates both active and passive tracers using the Griffies triad representation of iso-neutral diffusion and
+the eddy-induced advective skew (GM) fluxes.
+If the namelist logical \np{ln\_traldf\_iso} is set true,
+the filtered version of Cox's original scheme (the Standard scheme) is employed (\autoref{sec:LDF_slp}).
+In the present implementation of the Griffies scheme,
 the advective skew fluxes are implemented even if \np{ln\_traldf\_eiv} is false.
+Values of iso-neutral diffusivity and GM coefficient are set as
+described in \autoref{sec:LDF_coef}. Note that when GM fluxes are used,
+the eddy-advective (GM) velocities are output for diagnostic purposes using xIOS,
+Values of iso-neutral diffusivity and GM coefficient are set as described in \autoref{sec:LDF_coef}.
+Note that when GM fluxes are used, the eddy-advective (GM) velocities are output for diagnostic purposes using xIOS,
 even though the eddy advection is accomplished by means of the skew fluxes.
 …
 The options specific to the Griffies scheme include:
 \begin{description}[font=\normalfont]
+\item[\np{ln\_triad\_iso}] See \autoref{sec:taper}. If this is set false (the default), then
+  `iso-neutral' mixing is accomplished within the surface mixed-layer
+  along slopes linearly decreasing with depth from the value immediately below
+  the mixed-layer to zero (flat) at the surface (\autoref{sec:lintaper}).
+  This is the same treatment as used in the default implementation \autoref{subsec:LDF_slp_iso}; \autoref{fig:eiv_slp}.
+  Where \np{ln\_triad\_iso} is set true, the vertical skew flux is further reduced
+  to ensure no vertical buoyancy flux, giving an almost pure
+  horizontal diffusive tracer flux within the mixed layer. This is similar to
+  the tapering suggested by \citet{Gerdes1991}. See \autoref{subsec:Gerdes-taper}
+\item[\np{ln\_botmix\_triad}] See \autoref{sec:iso_bdry}.
+\item[\np{ln\_triad\_iso}]
+  See \autoref{sec:taper}.
+  If this is set false (the default),
+  then `iso-neutral' mixing is accomplished within the surface mixed-layer along slopes linearly decreasing with
+  depth from the value immediately below the mixed-layer to zero (flat) at the surface (\autoref{sec:lintaper}).
+  This is the same treatment as used in the default implementation
+  \autoref{subsec:LDF_slp_iso}; \autoref{fig:eiv_slp}.
+  Where \np{ln\_triad\_iso} is set true,
+  the vertical skew flux is further reduced to ensure no vertical buoyancy flux,
+  giving an almost pure horizontal diffusive tracer flux within the mixed layer.
+  This is similar to the tapering suggested by \citet{Gerdes1991}. See \autoref{subsec:Gerdes-taper}
+\item[\np{ln\_botmix\_triad}]
+  See \autoref{sec:iso_bdry}.
   If this is set false (the default) then the lateral diffusive fluxes
   associated with triads partly masked by topography are neglected.
   If it is set true, however, then these lateral diffusive fluxes are applied,
   giving smoother bottom tracer fields at the cost of introducing diapycnal mixing.
+\item[\np{rn\_sw\_triad}]  blah blah to be added....
+\item[\np{rn\_sw\_triad}]
+  blah blah to be added....
 \end{description}
 The options shared with the Standard scheme include:
 …
 \section{Triad formulation of iso-neutral diffusion}
 \label{sec:iso}
 We have implemented into \NEMO a scheme inspired by \citet{Griffies_al_JPO98},
+We have implemented into \NEMO a scheme inspired by \citet{Griffies_al_JPO98},
 but formulated within the \NEMO framework, using scale factors rather than grid-sizes.
 \subsection{Iso-neutral diffusion operator}
+The iso-neutral second order tracer diffusive operator for small
+angles between iso-neutral surfaces and geopotentials is given by
+\autoref{eq:iso_tensor_1}:
+The iso-neutral second order tracer diffusive operator for small angles between
+iso-neutral surfaces and geopotentials is given by \autoref{eq:iso_tensor_1}:
 \begin{subequations} \label{eq:iso_tensor_1}
   \begin{equation}
 …
 %  {-r_1 } \hfill & {-r_2 } \hfill & {r_1 ^2+r_2 ^2} \hfill \\
 % \end{array} }} \right)
  Here \autoref{eq:PE_iso_slopes}
+Here \autoref{eq:PE_iso_slopes}
 \begin{align*}
   r_1 &=-\frac{e_3 }{e_1 } \left( \frac{\partial \rho }{\partial i}
 …
     }{\partial k} \right)^{-1}
 \end{align*}
+is the $i$-component of the slope of the iso-neutral surface relative to the computational
+surface, and $r_2$ is the $j$-component.
+We will find it useful to consider the fluxes per unit area in $i,j,k$
+space; we write
+is the $i$-component of the slope of the iso-neutral surface relative to the computational surface,
+and $r_2$ is the $j$-component.
+We will find it useful to consider the fluxes per unit area in $i,j,k$ space; we write
 \begin{equation}
   \label{eq:Fijk}
   \vect{F}_{\mathrm{iso}}=\left(f_1^{lT}e_2e_3, f_2^{lT}e_1e_3, f_3^{lT}e_1e_2\right).
 \end{equation}
+Additionally, we will sometimes write the contributions towards the
+fluxes $\vect{f}$ and $\vect{F}_{\mathrm{iso}}$ from the component
+$R_{ij}$ of $\Re$ as $f_{ij}$, $F_{\mathrm{iso}\: ij}$, with
+$f_{ij}=R_{ij}e_i^{-1}\partial T/\partial x_i$ (no summation) etc.
+Additionally, we will sometimes write the contributions towards the fluxes $\vect{f}$ and
+$\vect{F}_{\mathrm{iso}}$ from the component $R_{ij}$ of $\Re$ as $f_{ij}$, $F_{\mathrm{iso}\: ij}$,
+with $f_{ij}=R_{ij}e_i^{-1}\partial T/\partial x_i$ (no summation) etc.
 The off-diagonal terms of the small angle diffusion tensor
 \autoref{eq:iso_tensor_1}, \autoref{eq:iso_tensor_2} produce skew-fluxes along the
 $i$- and $j$-directions resulting from the vertical tracer gradient:
+\autoref{eq:iso_tensor_1}, \autoref{eq:iso_tensor_2} produce skew-fluxes along
+the $i$- and $j$-directions resulting from the vertical tracer gradient:
 \begin{align}
   \label{eq:i13c}
 …
 \end{align}
+The vertical diffusive flux associated with the $_{33}$
+component of the small angle diffusion tensor is
+The vertical diffusive flux associated with the $_{33}$ component of the small angle diffusion tensor is
 \begin{equation}
   \label{eq:i33c}
 …
 \end{equation}
+Since there are no cross terms involving $r_1$ and $r_2$ in the above, we can
+consider the iso-neutral diffusive fluxes separately in the $i$-$k$ and $j$-$k$
+planes, just adding together the vertical components from each
+plane. The following description will describe the fluxes on the $i$-$k$
+plane.
+There is no natural discretization for the $i$-component of the
+skew-flux, \autoref{eq:i13c}, as
+although it must be evaluated at $u$-points, it involves vertical
+gradients (both for the tracer and the slope $r_1$), defined at
+$w$-points. Similarly, the vertical skew flux, \autoref{eq:i31c}, is evaluated at
+$w$-points but involves horizontal gradients defined at $u$-points.
+Since there are no cross terms involving $r_1$ and $r_2$ in the above,
+we can consider the iso-neutral diffusive fluxes separately in the $i$-$k$ and $j$-$k$ planes,
+just adding together the vertical components from each plane.
+The following description will describe the fluxes on the $i$-$k$ plane.
+There is no natural discretization for the $i$-component of the skew-flux, \autoref{eq:i13c},
+as although it must be evaluated at $u$-points,
+it involves vertical gradients (both for the tracer and the slope $r_1$), defined at $w$-points.
+Similarly, the vertical skew flux, \autoref{eq:i31c},
+is evaluated at $w$-points but involves horizontal gradients defined at $u$-points.
 \subsection{Standard discretization}
 The straightforward approach to discretize the lateral skew flux
+\autoref{eq:i13c} from tracer cell $i,k$ to $i+1,k$, introduced in 1995
+into OPA, \autoref{eq:tra_ldf_iso}, is to calculate a mean vertical
+gradient at the $u$-point from the average of the four surrounding
+vertical tracer gradients, and multiply this by a mean slope at the
+$u$-point, calculated from the averaged surrounding vertical density
+gradients. The total area-integrated skew-flux (flux per unit area in
+$ijk$ space) from tracer cell $i,k$
+to $i+1,k$, noting that the $e_{{3}_{i+1/2}^k}$ in the area
+$e{_{3}}_{i+1/2}^k{e_{2}}_{i+1/2}i^k$ at the $u$-point cancels out with
+the $1/{e_{3}}_{i+1/2}^k$ associated with the vertical tracer
+gradient, is then \autoref{eq:tra_ldf_iso}
+\autoref{eq:i13c} from tracer cell $i,k$ to $i+1,k$, introduced in 1995 into OPA,
+\autoref{eq:tra_ldf_iso}, is to calculate a mean vertical gradient at the $u$-point from
+the average of the four surrounding vertical tracer gradients, and multiply this by a mean slope at the $u$-point,
+calculated from the averaged surrounding vertical density gradients.
+The total area-integrated skew-flux (flux per unit area in $ijk$ space) from tracer cell $i,k$ to $i+1,k$,
+noting that the $e_{{3}_{i+1/2}^k}$ in the area $e{_{3}}_{i+1/2}^k{e_{2}}_{i+1/2}i^k$ at the $u$-point cancels out with
+the $1/{e_{3}}_{i+1/2}^k$ associated with the vertical tracer gradient, is then \autoref{eq:tra_ldf_iso}
 \begin{equation*}
   \left(F_u^{13} \right)_{i+\hhalf}^k = \Alts_{i+\hhalf}^k
 …
   \frac{\delta_{i+1/2} [\rho]}{\overline{\overline{\delta_k \rho}}^{\,i,k}},
 \end{equation*}
+and here and in the following we drop the $^{lT}$ superscript from
+$\Alt$ for simplicity.
+Unfortunately the resulting combination $\overline{\overline{\delta_k
+    \bullet}}^{\,i,k}$ of a $k$ average and a $k$ difference %of the tracer
+reduces to $\bullet_{k+1}-\bullet_{k-1}$, so two-grid-point oscillations are
+invisible to this discretization of the iso-neutral operator. These
+\emph{computational modes} will not be damped by this operator, and
+may even possibly be amplified by it.  Consequently, applying this
+operator to a tracer does not guarantee the decrease of its
+global-average variance. To correct this, we introduced a smoothing of
+the slopes of the iso-neutral surfaces (see \autoref{chap:LDF}). This
+technique works for $T$ and $S$ in so far as they are active tracers
+($i.e.$ they enter the computation of density), but it does not work
+for a passive tracer.
+and here and in the following we drop the $^{lT}$ superscript from $\Alt$ for simplicity.
+Unfortunately the resulting combination $\overline{\overline{\delta_k\bullet}}^{\,i,k}$ of a $k$ average and
+a $k$ difference of the tracer reduces to $\bullet_{k+1}-\bullet_{k-1}$,
+so two-grid-point oscillations are invisible to this discretization of the iso-neutral operator.
+These \emph{computational modes} will not be damped by this operator, and may even possibly be amplified by it.
+Consequently, applying this operator to a tracer does not guarantee the decrease of its global-average variance.
+To correct this, we introduced a smoothing of the slopes of the iso-neutral surfaces (see \autoref{chap:LDF}).
+This technique works for $T$ and $S$ in so far as they are active tracers
+($i.e.$ they enter the computation of density), but it does not work for a passive tracer.
 \subsection{Expression of the skew-flux in terms of triad slopes}
 \citep{Griffies_al_JPO98} introduce a different discretization of the
 off-diagonal terms that nicely solves the problem.
+\citep{Griffies_al_JPO98} introduce a different discretization of the off-diagonal terms that
+nicely solves the problem.
 % Instead of multiplying the mean slope calculated at the $u$-point by
 % the mean vertical gradient at the $u$-point,
 …
  \end{center} \end{figure}
 % >>>>>>>>>>>>>>>>>>>>>>>>>>>>
+They get the skew flux from the products of the vertical gradients at
+each $w$-point surrounding the $u$-point with the corresponding `triad'
+slope calculated from the lateral density gradient across the $u$-point
+divided by the vertical density gradient at the same $w$-point as the
+tracer gradient. See \autoref{fig:ISO_triad}a, where the thick lines
+denote the tracer gradients, and the thin lines the corresponding
+triads, with slopes $s_1, \dotsc s_4$. The total area-integrated
+skew-flux from tracer cell $i,k$ to $i+1,k$
+They get the skew flux from the products of the vertical gradients at each $w$-point surrounding the $u$-point with
+the corresponding `triad' slope calculated from the lateral density gradient across the $u$-point divided by
+the vertical density gradient at the same $w$-point as the tracer gradient.
+See \autoref{fig:ISO_triad}a, where the thick lines denote the tracer gradients,
+and the thin lines the corresponding triads, with slopes $s_1, \dotsc s_4$.
+The total area-integrated skew-flux from tracer cell $i,k$ to $i+1,k$
 \begin{multline}
   \label{eq:i13}
 …
   _{k-\frac{1}{2}} \left[ T^i \right]/e_{{3w}_{i+1}}^{k+\frac{1}{2}},
 \end{multline}
+where the contributions of the triad fluxes are weighted by areas
+$a_1, \dotsc a_4$, and $\Alts$ is now defined at the tracer points
+rather than the $u$-points. This discretization gives a much closer
+stencil, and disallows the two-point computational modes.
+ The vertical skew flux \autoref{eq:i31c} from tracer cell $i,k$ to $i,k+1$ at the
+$w$-point $i,k+\hhalf$ is constructed similarly (\autoref{fig:ISO_triad}b)
+by multiplying lateral tracer gradients from each of the four
+surrounding $u$-points by the appropriate triad slope:
+where the contributions of the triad fluxes are weighted by areas $a_1, \dotsc a_4$,
+and $\Alts$ is now defined at the tracer points rather than the $u$-points.
+This discretization gives a much closer stencil, and disallows the two-point computational modes.
+The vertical skew flux \autoref{eq:i31c} from tracer cell $i,k$ to $i,k+1$ at
+the $w$-point $i,k+\hhalf$ is constructed similarly (\autoref{fig:ISO_triad}b) by
+multiplying lateral tracer gradients from each of the four surrounding $u$-points by the appropriate triad slope:
 \begin{multline}
   \label{eq:i31}
 …
 We notate the triad slopes $s_i$ and $s'_i$ in terms of the `anchor point' $i,k$
 (appearing in both the vertical and lateral gradient), and the $u$- and
+$w$-points $(i+i_p,k)$, $(i,k+k_p)$ at the centres of the `arms' of the
 triad as follows (see also \autoref{fig:ISO_triad}):
+(appearing in both the vertical and lateral gradient),
+and the $u$- and $w$-points $(i+i_p,k)$, $(i,k+k_p)$ at the centres of the `arms' of the triad as follows
+(see also \autoref{fig:ISO_triad}):
 \begin{equation}
   \label{eq:R}
 …
   { \alpha_i^k  \ \delta_{k+k_p}[T^i] - \beta_i^k \ \delta_{k+k_p}[S^i] }.
 \end{equation}
 In calculating the slopes of the local neutral surfaces,
 the expansion coefficients $\alpha$ and $\beta$ are evaluated at the anchor points of the triad,
+In calculating the slopes of the local neutral surfaces,
+the expansion coefficients $\alpha$ and $\beta$ are evaluated at the anchor points of the triad,
 while the metrics are calculated at the $u$- and $w$-points on the arms.
 …
     \includegraphics[width=0.80\textwidth]{Fig_GRIFF_qcells}
     \caption{   \protect\label{fig:qcells}
     Triad notation for quarter cells. $T$-cells are inside
       boxes, while the  $i+\half,k$ $u$-cell is shaded in green and the
       $i,k+\half$ $w$-cell is shaded in pink.}
+      Triad notation for quarter cells. $T$-cells are inside boxes,
+      while the  $i+\half,k$ $u$-cell is shaded in green and
+      the $i,k+\half$ $w$-cell is shaded in pink.}
   \end{center} \end{figure}
 % >>>>>>>>>>>>>>>>>>>>>>>>>>>>
 Each triad $\{_i^{k}\:_{i_p}^{k_p}\}$ is associated (\autoref{fig:qcells}) with the quarter
+cell that is the intersection of the $i,k$ $T$-cell, the $i+i_p,k$ $u$-cell and the $i,k+k_p$ $w$-cell.
 Expressing the slopes $s_i$ and $s'_i$ in \autoref{eq:i13} and \autoref{eq:i31} in this notation,
 we have $e.g.$ \ $s_1=s'_1={\:}_i^k \mathbb{R}_{1/2}^{1/2}$.
 Each triad slope $_i^k\mathbb{R}_{i_p}^{k_p}$ is used once (as an $s$)
+to calculate the lateral flux along its $u$-arm, at $(i+i_p,k)$,
 and then again as an $s'$ to calculate the vertical flux along its $w$-arm at $(i,k+k_p)$.
 Each vertical area $a_i$ used to calculate the lateral flux and horizontal area $a'_i$ used
+to calculate the vertical flux can also be identified as the area across the $u$- and $w$-arms
+of a unique triad, and we notate these areas, similarly to the triad slopes,
 as $_i^k{\mathbb{A}_u}_{i_p}^{k_p}$, $_i^k{\mathbb{A}_w}_{i_p}^{k_p}$,
 where $e.g.$ in \autoref{eq:i13} $a_{1}={\:}_i^k{\mathbb{A}_u}_{1/2}^{1/2}$,
+Each triad $\{_i^{k}\:_{i_p}^{k_p}\}$ is associated (\autoref{fig:qcells}) with the quarter cell that is
+the intersection of the $i,k$ $T$-cell, the $i+i_p,k$ $u$-cell and the $i,k+k_p$ $w$-cell.
+Expressing the slopes $s_i$ and $s'_i$ in \autoref{eq:i13} and \autoref{eq:i31} in this notation,
+we have $e.g.$ \ $s_1=s'_1={\:}_i^k \mathbb{R}_{1/2}^{1/2}$.
+Each triad slope $_i^k\mathbb{R}_{i_p}^{k_p}$ is used once (as an $s$) to
+calculate the lateral flux along its $u$-arm, at $(i+i_p,k)$,
+and then again as an $s'$ to calculate the vertical flux along its $w$-arm at $(i,k+k_p)$.
+Each vertical area $a_i$ used to calculate the lateral flux and horizontal area $a'_i$ used to
+calculate the vertical flux can also be identified as the area across the $u$- and $w$-arms of a unique triad,
+and we notate these areas, similarly to the triad slopes,
+as $_i^k{\mathbb{A}_u}_{i_p}^{k_p}$, $_i^k{\mathbb{A}_w}_{i_p}^{k_p}$,
+where $e.g.$ in \autoref{eq:i13} $a_{1}={\:}_i^k{\mathbb{A}_u}_{1/2}^{1/2}$,
 and in \autoref{eq:i31} $a'_{1}={\:}_i^k{\mathbb{A}_w}_{1/2}^{1/2}$.
 \subsection{Full triad fluxes}
+A key property of iso-neutral diffusion is that it should not affect
+the (locally referenced) density. In particular there should be no
+lateral or vertical density flux. The lateral density flux disappears so long as the
+area-integrated lateral diffusive flux from tracer cell $i,k$ to
+$i+1,k$ coming from the $_{11}$ term of the diffusion tensor takes the
+form
+A key property of iso-neutral diffusion is that it should not affect the (locally referenced) density.
+In particular there should be no lateral or vertical density flux.
+The lateral density flux disappears so long as the area-integrated lateral diffusive flux from
+tracer cell $i,k$ to $i+1,k$ coming from the $_{11}$ term of the diffusion tensor takes the form
 \begin{equation}
   \label{eq:i11}
 …
   \frac{\delta _{i+1/2} \left[ T^k\right]}{{e_{1u}}_{\,i+1/2}^{\,k}},
 \end{equation}
+where the areas $a_i$ are as in \autoref{eq:i13}. In this case,
+separating the total lateral flux, the sum of \autoref{eq:i13} and
+\autoref{eq:i11}, into triad components, a lateral tracer
+flux
+where the areas $a_i$ are as in \autoref{eq:i13}.
+In this case, separating the total lateral flux, the sum of \autoref{eq:i13} and \autoref{eq:i11},
+into triad components, a lateral tracer flux
 \begin{equation}
   \label{eq:latflux-triad}
 …
   \right)
 \end{equation}
+can be identified with each triad. Then, because the
+same metric factors ${e_{3w}}_{\,i}^{\,k+k_p}$ and
+${e_{1u}}_{\,i+i_p}^{\,k}$ are employed for both the density gradients
+in $ _i^k \mathbb{R}_{i_p}^{k_p}$ and the tracer gradients, the lateral
+density flux associated with each triad separately disappears.
+can be identified with each triad.
+Then, because the same metric factors ${e_{3w}}_{\,i}^{\,k+k_p}$ and ${e_{1u}}_{\,i+i_p}^{\,k}$ are employed for both
+the density gradients in $ _i^k \mathbb{R}_{i_p}^{k_p}$ and the tracer gradients,
+the lateral density flux associated with each triad separately disappears.
 \begin{equation}
   \label{eq:latflux-rho}
   {\mathbb{F}_u}_{i_p}^{k_p} (\rho)=-\alpha _i^k {\:}_i^k {\mathbb{F}_u}_{i_p}^{k_p} (T) + \beta_i^k {\:}_i^k {\mathbb{F}_u}_{i_p}^{k_p} (S)=0
 \end{equation}
+Thus the total flux $\left( F_u^{31} \right) ^i _{i,k+\frac{1}{2}} +
+\left( F_u^{11} \right) ^i _{i,k+\frac{1}{2}}$ from tracer cell $i,k$
+to $i+1,k$ must also vanish since it is a sum of four such triad fluxes.
+The squared slope $r_1^2$ in the expression \autoref{eq:i33c} for the
+$_{33}$ component is also expressed in terms of area-weighted
+squared triad slopes, so the area-integrated vertical flux from tracer
+cell $i,k$ to $i,k+1$ resulting from the $r_1^2$ term is
+Thus the total flux $\left( F_u^{31} \right) ^i _{i,k+\frac{1}{2}} + \left( F_u^{11} \right) ^i _{i,k+\frac{1}{2}}$ from
+tracer cell $i,k$ to $i+1,k$ must also vanish since it is a sum of four such triad fluxes.
+The squared slope $r_1^2$ in the expression \autoref{eq:i33c} for the $_{33}$ component is also expressed in
+terms of area-weighted squared triad slopes,
+so the area-integrated vertical flux from tracer cell $i,k$ to $i,k+1$ resulting from the $r_1^2$ term is
 \begin{equation}
   \label{eq:i33}
 …
     + \Alts_i^k a_{4}' s_{4}'^2 \right)\delta_{k+\frac{1}{2}} \left[ T^{i+1} \right],
 \end{equation}
+where the areas $a'$ and slopes $s'$ are the same as in
+\autoref{eq:i31}.
+Then, separating the total vertical flux, the sum of \autoref{eq:i31} and
+\autoref{eq:i33}, into triad components,  a vertical flux
+where the areas $a'$ and slopes $s'$ are the same as in \autoref{eq:i31}.
+Then, separating the total vertical flux, the sum of \autoref{eq:i31} and \autoref{eq:i33},
+into triad components, a vertical flux
 \begin{align}
   \label{eq:vertflux-triad}
 …
    {_i^k\mathbb{R}_{i_p}^{k_p}}{\: }_i^k{\mathbb{F}_u}_{i_p}^{k_p} (T) \label{eq:vertflux-triad2}
 \end{align}
+may be associated with each triad. Each vertical density flux $_i^k {\mathbb{F}_w}_{i_p}^{k_p} (\rho)$
+associated with a triad then separately disappears (because the
+lateral flux $_i^k{\mathbb{F}_u}_{i_p}^{k_p} (\rho)$
+disappears). Consequently the total vertical density flux $\left( F_w^{31} \right)_i ^{k+\frac{1}{2}} +
+\left( F_w^{33} \right)_i^{k+\frac{1}{2}}$ from tracer cell $i,k$
+to $i,k+1$ must also vanish since it is a sum of four such triad
+fluxes.
+We can explicitly identify (\autoref{fig:qcells}) the triads associated with the $s_i$, $a_i$, and $s'_i$, $a'_i$ used in the definition of
+the $u$-fluxes and $w$-fluxes in
+\autoref{eq:i31}, \autoref{eq:i13}, \autoref{eq:i11} \autoref{eq:i33} and
+\autoref{fig:ISO_triad} to  write out the iso-neutral fluxes at $u$- and
+$w$-points as sums of the triad fluxes that cross the $u$- and $w$-faces:
+may be associated with each triad.
+Each vertical density flux $_i^k {\mathbb{F}_w}_{i_p}^{k_p} (\rho)$ associated with a triad then
+separately disappears (because the lateral flux $_i^k{\mathbb{F}_u}_{i_p}^{k_p} (\rho)$ disappears).
+Consequently the total vertical density flux
+$\left( F_w^{31} \right)_i ^{k+\frac{1}{2}} + \left( F_w^{33} \right)_i^{k+\frac{1}{2}}$ from
+tracer cell $i,k$ to $i,k+1$ must also vanish since it is a sum of four such triad fluxes.
+We can explicitly identify (\autoref{fig:qcells}) the triads associated with the $s_i$, $a_i$,
+and $s'_i$, $a'_i$ used in the definition of the $u$-fluxes and $w$-fluxes in \autoref{eq:i31},
+\autoref{eq:i13}, \autoref{eq:i11} \autoref{eq:i33} and \autoref{fig:ISO_triad} to write out
+the iso-neutral fluxes at $u$- and $w$-points as sums of the triad fluxes that cross the $u$- and $w$-faces:
 %(\autoref{fig:ISO_triad}):
 \begin{flalign} \label{eq:iso_flux} \vect{F}_{\mathrm{iso}}(T) &\equiv
 …
 \label{subsec:variance}
+We now require that this operator should not increase the
+globally-integrated tracer variance.
+We now require that this operator should not increase the globally-integrated tracer variance.
 %This changes according to
 % \begin{align*}
 …
 %              + {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p}}  \ \delta_{k+1/2} [T]   \right\}      \\
 % \end{align*}
+Each triad slope $_i^k\mathbb{R}_{i_p}^{k_p}$ drives a lateral flux
+$_i^k{\mathbb{F}_u}_{i_p}^{k_p} (T)$ across the $u$-point $i+i_p,k$ and
+a vertical flux $_i^k{\mathbb{F}_w}_{i_p}^{k_p} (T)$ across the
+$w$-point $i,k+k_p$.  The lateral flux drives a net rate of change of
+variance, summed over the two $T$-points $i+i_p-\half,k$ and $i+i_p+\half,k$, of
+Each triad slope $_i^k\mathbb{R}_{i_p}^{k_p}$ drives a lateral flux $_i^k{\mathbb{F}_u}_{i_p}^{k_p} (T)$ across
+the $u$-point $i+i_p,k$ and a vertical flux $_i^k{\mathbb{F}_w}_{i_p}^{k_p} (T)$ across the $w$-point $i,k+k_p$.
+The lateral flux drives a net rate of change of variance,
+summed over the two $T$-points $i+i_p-\half,k$ and $i+i_p+\half,k$, of
 \begin{multline}
   {b_T}_{i+i_p-1/2}^k\left(\frac{\partial T}{\partial t}T\right)_{i+i_p-1/2}^k+
 …
  \end{aligned}
 \end{multline}
+while the vertical flux similarly drives a net rate of change of
+variance summed over the $T$-points $i,k+k_p-\half$ (above) and
+$i,k+k_p+\half$ (below) of
+while the vertical flux similarly drives a net rate of change of variance summed over
+the $T$-points $i,k+k_p-\half$ (above) and $i,k+k_p+\half$ (below) of
 \begin{equation}
 \label{eq:dvar_iso_k}
   _i^k{\mathbb{F}_w}_{i_p}^{k_p} (T) \,\delta_{k+ k_p}[T^i].
 \end{equation}
+The total variance tendency driven by the triad is the sum of these
+two. Expanding $_i^k{\mathbb{F}_u}_{i_p}^{k_p} (T)$ and
+$_i^k{\mathbb{F}_w}_{i_p}^{k_p} (T)$ with \autoref{eq:latflux-triad} and
+\autoref{eq:vertflux-triad}, it is
+The total variance tendency driven by the triad is the sum of these two.
+Expanding $_i^k{\mathbb{F}_u}_{i_p}^{k_p} (T)$ and $_i^k{\mathbb{F}_w}_{i_p}^{k_p} (T)$ with
+\autoref{eq:latflux-triad} and \autoref{eq:vertflux-triad}, it is
 \begin{multline*}
 -\Alts_i^k\left \{
 …
 \right \}.
 \end{multline*}
+The key point is then that if we require
+$_i^k{\mathbb{A}_u}_{i_p}^{k_p}$ and $_i^k{\mathbb{A}_w}_{i_p}^{k_p}$
+to be related to a triad volume $_i^k\mathbb{V}_{i_p}^{k_p}$ by
+The key point is then that if we require $_i^k{\mathbb{A}_u}_{i_p}^{k_p}$ and $_i^k{\mathbb{A}_w}_{i_p}^{k_p}$ to
+be related to a triad volume $_i^k\mathbb{V}_{i_p}^{k_p}$ by
 \begin{equation}
   \label{eq:V-A}
 …
   \right)^2\leq 0.
 \end{equation}
+Thus, the constraint \autoref{eq:V-A} ensures that the fluxes (\autoref{eq:latflux-triad}, \autoref{eq:vertflux-triad}) associated
+with a given slope triad $_i^k\mathbb{R}_{i_p}^{k_p}$ do not increase
+the net variance. Since the total fluxes are sums of such fluxes from
+the various triads, this constraint, applied to all triads, is
+sufficient to ensure that the globally integrated variance does not
+increase.
+The expression \autoref{eq:V-A} can be interpreted as a discretization
+of the global integral
+Thus, the constraint \autoref{eq:V-A} ensures that the fluxes
+(\autoref{eq:latflux-triad}, \autoref{eq:vertflux-triad}) associated with
+a given slope triad $_i^k\mathbb{R}_{i_p}^{k_p}$ do not increase the net variance.
+Since the total fluxes are sums of such fluxes from the various triads, this constraint, applied to all triads,
+is sufficient to ensure that the globally integrated variance does not increase.
+The expression \autoref{eq:V-A} can be interpreted as a discretization of the global integral
 \begin{equation}
   \label{eq:cts-var}
 …
   \int\!\mathbf{F}\cdot\nabla T\, dV,
 \end{equation}
+where, within each triad volume $_i^k\mathbb{V}_{i_p}^{k_p}$, the
+lateral and vertical fluxes/unit area
+where, within each triad volume $_i^k\mathbb{V}_{i_p}^{k_p}$, the lateral and vertical fluxes/unit area
 \[
 \mathbf{F}=\left(
 …
 \subsection{Triad volumes in Griffes's scheme and in \NEMO}
+To complete the discretization we now need only specify the triad
+volumes $_i^k\mathbb{V}_{i_p}^{k_p}$. \citet{Griffies_al_JPO98} identify
+these $_i^k\mathbb{V}_{i_p}^{k_p}$ as the volumes of the quarter
+cells, defined in terms of the distances between $T$, $u$,$f$ and
+$w$-points. This is the natural discretization of
+\autoref{eq:cts-var}. The \NEMO model, however, operates with scale
+factors instead of grid sizes, and scale factors for the quarter
+cells are not defined. Instead, therefore we simply choose
+To complete the discretization we now need only specify the triad volumes $_i^k\mathbb{V}_{i_p}^{k_p}$.
+\citet{Griffies_al_JPO98} identifies these $_i^k\mathbb{V}_{i_p}^{k_p}$ as the volumes of the quarter cells,
+defined in terms of the distances between $T$, $u$,$f$ and $w$-points.
+This is the natural discretization of \autoref{eq:cts-var}.
+The \NEMO model, however, operates with scale factors instead of grid sizes,
+and scale factors for the quarter cells are not defined.
+Instead, therefore we simply choose
 \begin{equation}
   \label{eq:V-NEMO}
   _i^k\mathbb{V}_{i_p}^{k_p}=\quarter {b_u}_{i+i_p}^k,
 \end{equation}
+as a quarter of the volume of the $u$-cell inside which the triad
+quarter-cell lies. This has the nice property that when the slopes
+$\mathbb{R}$ vanish, the lateral flux from tracer cell $i,k$ to
+$i+1,k$ reduces to the classical form
+as a quarter of the volume of the $u$-cell inside which the triad quarter-cell lies.
+This has the nice property that when the slopes $\mathbb{R}$ vanish,
+the lateral flux from tracer cell $i,k$ to $i+1,k$ reduces to the classical form
 \begin{equation}
   \label{eq:lat-normal}
 …
  = -\overline\Alts_{\,i+1/2}^k\;\frac{{e_{1w}}_{\,i + 1/2}^{\,k}\:{e_{1v}}_{\,i + 1/2}^{\,k}\;\delta_{i+ 1/2}[T^k]}{{e_{1u}}_{\,i + 1/2}^{\,k}}.
 \end{equation}
 In fact if the diffusive coefficient is defined at $u$-points, so that
+we employ $\Alts_{i+i_p}^k$ instead of  $\Alts_i^k$ in the definitions of the
 triad fluxes \autoref{eq:latflux-triad} and \autoref{eq:vertflux-triad},
+In fact if the diffusive coefficient is defined at $u$-points,
+so that we employ $\Alts_{i+i_p}^k$ instead of  $\Alts_i^k$ in the definitions of the triad fluxes
+\autoref{eq:latflux-triad} and \autoref{eq:vertflux-triad},
 we can replace $\overline{A}_{\,i+1/2}^k$ by $A_{i+1/2}^k$ in the above.
 \subsection{Summary of the scheme}
+The iso-neutral fluxes at $u$- and
+$w$-points are the sums of the triad fluxes that cross the $u$- and
+$w$-faces \autoref{eq:iso_flux}:
+The iso-neutral fluxes at $u$- and $w$-points are the sums of the triad fluxes that
+cross the $u$- and $w$-faces \autoref{eq:iso_flux}:
 \begin{subequations}\label{eq:alltriadflux}
   \begin{flalign}\label{eq:vect_isoflux}
 …
 \end{subequations}
  The divergence of the expression \autoref{eq:iso_flux} for the fluxes gives the iso-neutral diffusion tendency at
+The divergence of the expression \autoref{eq:iso_flux} for the fluxes gives the iso-neutral diffusion tendency at
 each tracer point:
 \begin{equation} \label{eq:iso_operator} D_l^T = \frac{1}{b_T}
 …
 The diffusion scheme satisfies the following six properties:
 \begin{description}
 \item[$\bullet$ horizontal diffusion] The discretization of the
   diffusion operator recovers \autoref{eq:lat-normal} the traditional five-point Laplacian in
   the limit of flat iso-neutral direction :
+\item[$\bullet$ horizontal diffusion]
+  The discretization of the diffusion operator recovers the traditional five-point Laplacian
+  \autoref{eq:lat-normal} in the limit of flat iso-neutral direction:
   \begin{equation} \label{eq:iso_property0} D_l^T = \frac{1}{b_T} \
     \delta_{i} \left[ \frac{e_{2u}\,e_{3u}}{e_{1u}} \;
 …
   \end{equation}
+\item[$\bullet$ implicit treatment in the vertical] Only tracer values
+  associated with a single water column appear in the expression
+  \autoref{eq:i33} for the $_{33}$ fluxes, vertical fluxes driven by
+  vertical gradients. This is of paramount importance since it means
+  that a time-implicit algorithm can be used to solve the vertical
+  diffusion equation. This is necessary
+ since the vertical eddy
+  diffusivity associated with this term,
+\item[$\bullet$ implicit treatment in the vertical]
+  Only tracer values associated with a single water column appear in the expression \autoref{eq:i33} for
+  the $_{33}$ fluxes, vertical fluxes driven by vertical gradients.
+  This is of paramount importance since it means that a time-implicit algorithm can be used to
+  solve the vertical diffusion equation.
+  This is necessary since the vertical eddy diffusivity associated with this term,
   \begin{equation}
     \frac{1}{b_w}\sum_{\substack{i_p, \,k_p}} \left\{
 …
       {b_u}_{i+i_p}^k\: \Alts_i^k \: \left(_i^k \mathbb{R}_{i_p}^{k_p}\right)^2
     \right\},
  \end{equation}
+  \end{equation}
   (where $b_w= e_{1w}\,e_{2w}\,e_{3w}$ is the volume of $w$-cells) can be quite large.
 \item[$\bullet$ pure iso-neutral operator] The iso-neutral flux of
   locally referenced potential density is zero. See
   \autoref{eq:latflux-rho} and \autoref{eq:vertflux-triad2}.
 \item[$\bullet$ conservation of tracer] The iso-neutral diffusion
   conserves tracer content, $i.e.$
+\item[$\bullet$ pure iso-neutral operator]
+  The iso-neutral flux of locally referenced potential density is zero.
+  See \autoref{eq:latflux-rho} and \autoref{eq:vertflux-triad2}.
+\item[$\bullet$ conservation of tracer]
+  The iso-neutral diffusion conserves tracer content, $i.e.$
   \begin{equation} \label{eq:iso_property1} \sum_{i,j,k} \left\{ D_l^T \
       b_T \right\} = 0
   \end{equation}
+  This property is trivially satisfied since the iso-neutral diffusive
+  operator is written in flux form.
+\item[$\bullet$ no increase of tracer variance] The iso-neutral diffusion
+  does not increase the tracer variance, $i.e.$
+  This property is trivially satisfied since the iso-neutral diffusive operator is written in flux form.
+\item[$\bullet$ no increase of tracer variance]
+  The iso-neutral diffusion does not increase the tracer variance, $i.e.$
   \begin{equation} \label{eq:iso_property2} \sum_{i,j,k} \left\{ T \ D_l^T
       \ b_T \right\} \leq 0
   \end{equation}
+  The property is demonstrated in
+  \autoref{subsec:variance} above. It is a key property for a diffusion
+  term. It means that it is also a dissipation term, $i.e.$ it
+  dissipates the square of the quantity on which it is applied.  It
+  therefore ensures that, when the diffusivity coefficient is large
+  enough, the field on which it is applied becomes free of grid-point
+  noise.
+\item[$\bullet$ self-adjoint operator] The iso-neutral diffusion
+  operator is self-adjoint, $i.e.$
+  The property is demonstrated in \autoref{subsec:variance} above.
+  It is a key property for a diffusion term.
+  It means that it is also a dissipation term,
+  $i.e.$ it dissipates the square of the quantity on which it is applied.
+  It therefore ensures that, when the diffusivity coefficient is large enough,
+  the field on which it is applied becomes free of grid-point noise.
+\item[$\bullet$ self-adjoint operator]
+  The iso-neutral diffusion operator is self-adjoint, $i.e.$
   \begin{equation} \label{eq:iso_property3} \sum_{i,j,k} \left\{ S \ D_l^T
       \ b_T \right\} = \sum_{i,j,k} \left\{ D_l^S \ T \ b_T \right\}
   \end{equation}
+  In other word, there is no need to develop a specific routine from
+  the adjoint of this operator. We just have to apply the same
+  routine. This property can be demonstrated similarly to the proof of
+  the `no increase of tracer variance' property. The contribution by a
+  single triad towards the left hand side of \autoref{eq:iso_property3}, can
+  be found by replacing $\delta[T]$ by $\delta[S]$ in \autoref{eq:dvar_iso_i}
+  and \autoref{eq:dvar_iso_k}. This results in a term similar to
+  \autoref{eq:perfect-square},
+  In other word, there is no need to develop a specific routine from the adjoint of this operator.
+  We just have to apply the same routine.
+  This property can be demonstrated similarly to the proof of the `no increase of tracer variance' property.
+  The contribution by a single triad towards the left hand side of \autoref{eq:iso_property3},
+  can be found by replacing $\delta[T]$ by $\delta[S]$ in \autoref{eq:dvar_iso_i} and \autoref{eq:dvar_iso_k}.
+  This results in a term similar to \autoref{eq:perfect-square},
 \begin{equation}
   \label{eq:TScovar}
 …
   \right).
 \end{equation}
+This is symmetrical in $T $ and $S$, so exactly the same term arises
+from the discretization of this triad's contribution towards the
+RHS of \autoref{eq:iso_property3}.
+This is symmetrical in $T $ and $S$, so exactly the same term arises from
+the discretization of this triad's contribution towards the RHS of \autoref{eq:iso_property3}.
 \end{description}
 \subsection{Treatment of the triads at the boundaries}\label{sec:iso_bdry}
+The triad slope can only be defined where both the grid boxes centred at
+the end of the arms exist. Triads that would poke up
+through the upper ocean surface into the atmosphere, or down into the
+ocean floor, must be masked out. See \autoref{fig:bdry_triads}. Surface layer triads
+$\triad{i}{1}{R}{1/2}{-1/2}$ (magenta) and
+$\triad{i+1}{1}{R}{-1/2}{-1/2}$ (blue) that require density to be
+specified above the ocean surface are masked (\autoref{fig:bdry_triads}a): this ensures that lateral
+tracer gradients produce no flux through the ocean surface. However,
+to prevent surface noise, it is customary to retain the $_{11}$ contributions towards
+the lateral triad fluxes $\triad[u]{i}{1}{F}{1/2}{-1/2}$ and
+$\triad[u]{i+1}{1}{F}{-1/2}{-1/2}$; this drives diapycnal tracer
+fluxes. Similar comments apply to triads that would intersect the
+ocean floor (\autoref{fig:bdry_triads}b). Note that both near bottom
+triad slopes $\triad{i}{k}{R}{1/2}{1/2}$ and
+$\triad{i+1}{k}{R}{-1/2}{1/2}$ are masked when either of the $i,k+1$
+or $i+1,k+1$ tracer points is masked, i.e.\ the $i,k+1$ $u$-point is
+masked. The associated lateral fluxes (grey-black dashed line) are
+masked if \np{ln\_botmix\_triad}\forcode{ = .false.}, but left unmasked,
+giving bottom mixing, if \np{ln\_botmix\_triad}\forcode{ = .true.}.
+The default option \np{ln\_botmix\_triad}\forcode{ = .false.} is suitable when the
+bbl mixing option is enabled (\key{trabbl}, with \np{nn\_bbl\_ldf}\forcode{ = 1}),
+or  for simple idealized  problems. For setups with topography without
+bbl mixing, \np{ln\_botmix\_triad}\forcode{ = .true.} may be necessary.
+The triad slope can only be defined where both the grid boxes centred at the end of the arms exist.
+Triads that would poke up through the upper ocean surface into the atmosphere,
+or down into the ocean floor, must be masked out.
+See \autoref{fig:bdry_triads}.
+Surface layer triads $\triad{i}{1}{R}{1/2}{-1/2}$ (magenta) and $\triad{i+1}{1}{R}{-1/2}{-1/2}$ (blue) that
+require density to be specified above the ocean surface are masked (\autoref{fig:bdry_triads}a):
+this ensures that lateral tracer gradients produce no flux through the ocean surface.
+However, to prevent surface noise, it is customary to retain the $_{11}$ contributions towards
+the lateral triad fluxes $\triad[u]{i}{1}{F}{1/2}{-1/2}$ and $\triad[u]{i+1}{1}{F}{-1/2}{-1/2}$;
+this drives diapycnal tracer fluxes.
+Similar comments apply to triads that would intersect the ocean floor (\autoref{fig:bdry_triads}b).
+Note that both near bottom triad slopes $\triad{i}{k}{R}{1/2}{1/2}$ and
+$\triad{i+1}{k}{R}{-1/2}{1/2}$ are masked when either of the $i,k+1$ or $i+1,k+1$ tracer points is masked,
+i.e.\ the $i,k+1$ $u$-point is masked.
+The associated lateral fluxes (grey-black dashed line) are masked if \np{ln\_botmix\_triad}\forcode{ = .false.},
+but left unmasked, giving bottom mixing, if \np{ln\_botmix\_triad}\forcode{ = .true.}.
+The default option \np{ln\_botmix\_triad}\forcode{ = .false.} is suitable when the bbl mixing option is enabled
+(\key{trabbl}, with \np{nn\_bbl\_ldf}\forcode{ = 1}), or for simple idealized problems.
+For setups with topography without bbl mixing, \np{ln\_botmix\_triad}\forcode{ = .true.} may be necessary.
 % >>>>>>>>>>>>>>>>>>>>>>>>>>>>
 \begin{figure}[h] \begin{center}
     \includegraphics[width=0.60\textwidth]{Fig_GRIFF_bdry_triads}
     \caption{  \protect\label{fig:bdry_triads}
+      (a) Uppermost model layer $k=1$ with $i,1$ and $i+1,1$ tracer
+      points (black dots), and $i+1/2,1$ $u$-point (blue square). Triad
+      slopes $\triad{i}{1}{R}{1/2}{-1/2}$ (magenta) and $\triad{i+1}{1}{R}{-1/2}{-1/2}$
+      (blue) poking through the ocean surface are masked (faded in
+      figure). However, the lateral $_{11}$ contributions towards
+      $\triad[u]{i}{1}{F}{1/2}{-1/2}$ and $\triad[u]{i+1}{1}{F}{-1/2}{-1/2}$
+      (yellow line) are still applied, giving diapycnal diffusive
+      fluxes.\newline
+      (a) Uppermost model layer $k=1$ with $i,1$ and $i+1,1$ tracer points (black dots),
+      and $i+1/2,1$ $u$-point (blue square).
+      Triad slopes $\triad{i}{1}{R}{1/2}{-1/2}$ (magenta) and $\triad{i+1}{1}{R}{-1/2}{-1/2}$ (blue) poking through
+      the ocean surface are masked (faded in figure).
+      However, the lateral $_{11}$ contributions towards $\triad[u]{i}{1}{F}{1/2}{-1/2}$ and
+      $\triad[u]{i+1}{1}{F}{-1/2}{-1/2}$ (yellow line) are still applied,
+      giving diapycnal diffusive fluxes.\newline
       (b) Both near bottom triad slopes $\triad{i}{k}{R}{1/2}{1/2}$ and
       $\triad{i+1}{k}{R}{-1/2}{1/2}$ are masked when either of the $i,k+1$
       or $i+1,k+1$ tracer points is masked, i.e.\ the $i,k+1$ $u$-point
       is masked. The associated lateral fluxes (grey-black dashed
       line) are masked if \protect\np{botmix\_triad}\forcode{ = .false.}, but left
       unmasked, giving bottom mixing, if \protect\np{botmix\_triad}\forcode{ = .true.}}
+      $\triad{i+1}{k}{R}{-1/2}{1/2}$ are masked when either of the $i,k+1$ or $i+1,k+1$ tracer points is masked,
+      i.e.\ the $i,k+1$ $u$-point is masked.
+      The associated lateral fluxes (grey-black dashed line) are masked if
+      \protect\np{botmix\_triad}\forcode{ = .false.}, but left unmasked,
+      giving bottom mixing, if \protect\np{botmix\_triad}\forcode{ = .true.}}
  \end{center} \end{figure}
 % >>>>>>>>>>>>>>>>>>>>>>>>>>>>
 \subsection{ Limiting of the slopes within the interior}\label{sec:limit}
 As discussed in \autoref{subsec:LDF_slp_iso}, iso-neutral slopes relative to
+geopotentials must be bounded everywhere, both for consistency with the small-slope
+approximation and for numerical stability \citep{Cox1987,
+  Griffies_Bk04}. The bound chosen in \NEMO is applied to each
 component of the slope separately and has a value of $1/100$ in the ocean interior.
+As discussed in \autoref{subsec:LDF_slp_iso},
+iso-neutral slopes relative to geopotentials must be bounded everywhere,
+both for consistency with the small-slope approximation and for numerical stability \citep{Cox1987, Griffies_Bk04}.
+The bound chosen in \NEMO is applied to each component of the slope separately and
+has a value of $1/100$ in the ocean interior.
 %, ramping linearly down above 70~m depth to zero at the surface
+It is of course relevant to the iso-neutral slopes $\tilde{r}_i=r_i+\sigma_i$ relative to
+geopotentials (here the $\sigma_i$ are the slopes of the coordinate surfaces relative to
+geopotentials) \autoref{eq:PE_slopes_eiv} rather than the slope $r_i$ relative to coordinate
+surfaces, so we require
+It is of course relevant to the iso-neutral slopes $\tilde{r}_i=r_i+\sigma_i$ relative to geopotentials
+(here the $\sigma_i$ are the slopes of the coordinate surfaces relative to geopotentials)
+\autoref{eq:PE_slopes_eiv} rather than the slope $r_i$ relative to coordinate surfaces, so we require
 \begin{equation*}
   |\tilde{r}_i|\leq \tilde{r}_\mathrm{max}=0.01.
 …
 and then recalculate the slopes $r_i$ relative to coordinates.
 Each individual triad slope
+ \begin{equation}
+   \label{eq:Rtilde}
+_i^k\tilde{\mathbb{R}}_{i_p}^{k_p} = {}_i^k\mathbb{R}_{i_p}^{k_p}  + \frac{\delta_{i+i_p}[z_T^k]}{{e_{1u}}_{\,i + i_p}^{\,k}}
+ \end{equation}
+is limited like this and then the corresponding
+$_i^k\mathbb{R}_{i_p}^{k_p} $ are recalculated and combined to form the fluxes.
+Note that where the slopes have been limited, there is now a non-zero
+iso-neutral density flux that drives dianeutral mixing.  In particular this iso-neutral density flux
+is always downwards, and so acts to reduce gravitational potential energy.
+\begin{equation}
+  \label{eq:Rtilde}
+  _i^k\tilde{\mathbb{R}}_{i_p}^{k_p} = {}_i^k\mathbb{R}_{i_p}^{k_p}  + \frac{\delta_{i+i_p}[z_T^k]}{{e_{1u}}_{\,i + i_p}^{\,k}}
+\end{equation}
+is limited like this and then the corresponding $_i^k\mathbb{R}_{i_p}^{k_p} $ are recalculated and
+combined to form the fluxes.
+Note that where the slopes have been limited, there is now a non-zero iso-neutral density flux that
+drives dianeutral mixing.
+In particular this iso-neutral density flux is always downwards,
+and so acts to reduce gravitational potential energy.
 \subsection{Tapering within the surface mixed layer}\label{sec:taper}
+Additional tapering of the iso-neutral fluxes is necessary within the
+surface mixed layer. When the Griffies triads are used, we offer two
+options for this.
+Additional tapering of the iso-neutral fluxes is necessary within the surface mixed layer.
+When the Griffies triads are used, we offer two options for this.
 \subsubsection{Linear slope tapering within the surface mixed layer}\label{sec:lintaper}
+This is the option activated by the default choice
+\np{ln\_triad\_iso}\forcode{ = .false.}. Slopes $\tilde{r}_i$ relative to
+geopotentials are tapered linearly from their value immediately below the mixed layer to zero at the
+surface, as described in option (c) of \autoref{fig:eiv_slp}, to values
+This is the option activated by the default choice \np{ln\_triad\_iso}\forcode{ = .false.}.
+Slopes $\tilde{r}_i$ relative to geopotentials are tapered linearly from their value immediately below
+the mixed layer to zero at the surface, as described in option (c) of \autoref{fig:eiv_slp}, to values
 \begin{subequations}
   \begin{equation}
    \label{eq:rmtilde}
      \rMLt =
   -\frac{z}{h}\left.\tilde{r}_i\right|_{z=-h}\quad \text{ for  } z>-h,
+    \label{eq:rmtilde}
+    \rMLt =
+    -\frac{z}{h}\left.\tilde{r}_i\right|_{z=-h}\quad \text{ for  } z>-h,
   \end{equation}
+and then the $r_i$ relative to vertical coordinate surfaces are appropriately
+adjusted to
+  and then the $r_i$ relative to vertical coordinate surfaces are appropriately adjusted to
   \begin{equation}
    \label{eq:rm}
  \rML =\rMLt -\sigma_i \quad \text{ for  } z>-h.
+    \label{eq:rm}
+    \rML =\rMLt -\sigma_i \quad \text{ for  } z>-h.
   \end{equation}
 \end{subequations}
 …
 \end{equation}
+This slope tapering gives a natural connection between tracer in the
+mixed-layer and in isopycnal layers immediately below, in the
+thermocline. It is consistent with the way the $\tilde{r}_i$ are
+tapered within the mixed layer (see \autoref{sec:taperskew} below)
+so as to ensure a uniform GM eddy-induced velocity throughout the
+mixed layer. However, it gives a downwards density flux and so acts so
+as to reduce potential energy in the same way as does the slope
+limiting discussed above in \autoref{sec:limit}.
+This slope tapering gives a natural connection between tracer in the mixed-layer and
+in isopycnal layers immediately below, in the thermocline.
+It is consistent with the way the $\tilde{r}_i$ are tapered within the mixed layer
+(see \autoref{sec:taperskew} below) so as to ensure a uniform GM eddy-induced velocity throughout the mixed layer.
+However, it gives a downwards density flux and so acts so as to reduce potential energy in the same way as
+does the slope limiting discussed above in \autoref{sec:limit}.
+As in \autoref{sec:limit} above, the tapering
+\autoref{eq:rmtilde} is applied separately to each triad
+$_i^k\tilde{\mathbb{R}}_{i_p}^{k_p}$, and the
+$_i^k\mathbb{R}_{i_p}^{k_p}$ adjusted. For clarity, we assume
+$z$-coordinates in the following; the conversion from
+$\mathbb{R}$ to $\tilde{\mathbb{R}}$ and back to $\mathbb{R}$ follows exactly as described
+above by \autoref{eq:Rtilde}.
+As in \autoref{sec:limit} above, the tapering \autoref{eq:rmtilde} is applied separately to
+each triad $_i^k\tilde{\mathbb{R}}_{i_p}^{k_p}$, and the $_i^k\mathbb{R}_{i_p}^{k_p}$ adjusted.
+For clarity, we assume $z$-coordinates in the following;
+the conversion from $\mathbb{R}$ to $\tilde{\mathbb{R}}$ and back to $\mathbb{R}$ follows exactly as
+described above by \autoref{eq:Rtilde}.
 \begin{enumerate}
+\item Mixed-layer depth is defined so as to avoid including regions of weak
+vertical stratification in the slope definition.
+ At each $i,j$ (simplified to $i$ in
+\autoref{fig:MLB_triad}), we define the mixed-layer by setting
+the vertical index of the tracer point immediately below the mixed
+layer, $k_{\mathrm{ML}}$, as the maximum $k$ (shallowest tracer point)
+such that the potential density
+${\rho_0}_{i,k}>{\rho_0}_{i,k_{10}}+\Delta\rho_c$, where $i,k_{10}$ is
+the tracer gridbox within which the depth reaches 10~m. See the left
+side of \autoref{fig:MLB_triad}. We use the $k_{10}$-gridbox
+instead of the surface gridbox to avoid problems e.g.\ with thin
+daytime mixed-layers. Currently we use the same
+$\Delta\rho_c=0.01\;\mathrm{kg\:m^{-3}}$ for ML triad tapering as is
+used to output the diagnosed mixed-layer depth
+$h_{\mathrm{ML}}=|z_{W}|_{k_{\mathrm{ML}}+1/2}$, the depth of the $w$-point
+above the $i,k_{\mathrm{ML}}$ tracer point.
+\item We define `basal' triad slopes
+${\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p}$ as the slopes
+of those triads whose vertical `arms' go down from the
+$i,k_{\mathrm{ML}}$ tracer point to the $i,k_{\mathrm{ML}}-1$ tracer point
+below. This is to ensure that the vertical density gradients
+associated with these basal triad slopes
+${\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p}$ are
+representative of the thermocline. The four basal triads defined in the bottom part
+of \autoref{fig:MLB_triad} are then
+\item
+  Mixed-layer depth is defined so as to avoid including regions of weak vertical stratification in
+  the slope definition.
+  At each $i,j$ (simplified to $i$ in \autoref{fig:MLB_triad}),
+  we define the mixed-layer by setting the vertical index of the tracer point immediately below the mixed layer,
+  $k_{\mathrm{ML}}$, as the maximum $k$ (shallowest tracer point) such that
+  the potential density ${\rho_0}_{i,k}>{\rho_0}_{i,k_{10}}+\Delta\rho_c$,
+  where $i,k_{10}$ is the tracer gridbox within which the depth reaches 10~m.
+  See the left side of \autoref{fig:MLB_triad}.
+  We use the $k_{10}$-gridbox instead of the surface gridbox to avoid problems e.g.\ with thin daytime mixed-layers.
+  Currently we use the same $\Delta\rho_c=0.01\;\mathrm{kg\:m^{-3}}$ for ML triad tapering as is used to
+  output the diagnosed mixed-layer depth $h_{\mathrm{ML}}=|z_{W}|_{k_{\mathrm{ML}}+1/2}$,
+  the depth of the $w$-point above the $i,k_{\mathrm{ML}}$ tracer point.
+\item
+  We define `basal' triad slopes ${\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p}$ as
+  the slopes of those triads whose vertical `arms' go down from the $i,k_{\mathrm{ML}}$ tracer point to
+  the $i,k_{\mathrm{ML}}-1$ tracer point below.
+  This is to ensure that the vertical density gradients associated with
+  these basal triad slopes ${\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p}$ are representative of the thermocline.
+  The four basal triads defined in the bottom part of \autoref{fig:MLB_triad} are then
 \begin{align}
   {\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p} &=
 …
 {\:}^{k_{\mathrm{ML}}}_i{\mathbb{R}_{\mathrm{base}}}_{\,1/2}^{-1/2}. \notag
 \end{align}
+The vertical flux associated with each of these triads passes through the $w$-point
+$i,k_{\mathrm{ML}}-1/2$ lying \emph{below} the $i,k_{\mathrm{ML}}$ tracer point,
+so it is this depth
+The vertical flux associated with each of these triads passes through
+the $w$-point $i,k_{\mathrm{ML}}-1/2$ lying \emph{below} the $i,k_{\mathrm{ML}}$ tracer point, so it is this depth
 \begin{equation}
   \label{eq:zbase}
   {z_\mathrm{base}}_{\,i}={z_{w}}_{k_\mathrm{ML}-1/2}
 \end{equation}
+(one gridbox deeper than the
+diagnosed ML depth $z_{\mathrm{ML}})$ that sets the $h$ used to taper
+the slopes in \autoref{eq:rmtilde}.
+\item Finally, we calculate the adjusted triads
+${\:}_i^k{\mathbb{R}_{\mathrm{ML}}}_{\,i_p}^{k_p}$ within the mixed
+layer, by multiplying the appropriate
+${\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p}$ by the ratio of
+the depth of the $w$-point ${z_w}_{k+k_p}$ to ${z_{\mathrm{base}}}_{\,i}$. For
+instance the green triad centred on $i,k$
+one gridbox deeper than the diagnosed ML depth $z_{\mathrm{ML}})$ that sets the $h$ used to taper the slopes in
+\autoref{eq:rmtilde}.
+\item
+  Finally, we calculate the adjusted triads ${\:}_i^k{\mathbb{R}_{\mathrm{ML}}}_{\,i_p}^{k_p}$ within
+  the mixed layer, by multiplying the appropriate ${\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p}$ by
+  the ratio of the depth of the $w$-point ${z_w}_{k+k_p}$ to ${z_{\mathrm{base}}}_{\,i}$.
+  For instance the green triad centred on $i,k$
 \begin{align}
   {\:}_i^k{\mathbb{R}_{\mathrm{ML}}}_{\,1/2}^{-1/2} &=
 …
 \begin{figure}[h]
 %  \fcapside {
+    \caption{\protect\label{fig:MLB_triad} Definition of
+      mixed-layer depth and calculation of linearly tapered
+      triads. The figure shows a water column at a given $i,j$
+      (simplified to $i$), with the ocean surface at the top. Tracer points are denoted by
+      bullets, and black lines the edges of the tracer cells; $k$
+      increases upwards. \newline
+      \hspace{5 em}We define the mixed-layer by setting the vertical index
+      of the tracer point immediately below the mixed layer,
+      $k_{\mathrm{ML}}$, as the maximum $k$ (shallowest tracer point)
+      such that ${\rho_0}_{i,k}>{\rho_0}_{i,k_{10}}+\Delta\rho_c$,
+      where $i,k_{10}$ is the tracer gridbox within which the depth
+      reaches 10~m. We calculate the triad slopes within the mixed
+      layer by linearly tapering them from zero (at the surface) to
+      the `basal' slopes, the slopes of the four triads passing through the
+      $w$-point $i,k_{\mathrm{ML}}-1/2$ (blue square),
+      ${\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p}$. Triads with
+    different $i_p,k_p$, denoted by different colours, (e.g. the green
+    triad $i_p=1/2,k_p=-1/2$) are tapered to the appropriate basal triad.}
+  \caption{\protect\label{fig:MLB_triad}
+    Definition of mixed-layer depth and calculation of linearly tapered triads.
+    The figure shows a water column at a given $i,j$ (simplified to $i$), with the ocean surface at the top.
+    Tracer points are denoted by bullets, and black lines the edges of the tracer cells;
+    $k$ increases upwards. \newline
+    \hspace{5 em}
+    We define the mixed-layer by setting the vertical index of the tracer point immediately below the mixed layer,
+    $k_{\mathrm{ML}}$, as the maximum $k$ (shallowest tracer point) such that
+    ${\rho_0}_{i,k}>{\rho_0}_{i,k_{10}}+\Delta\rho_c$,
+    where $i,k_{10}$ is the tracer gridbox within which the depth reaches 10~m.
+    We calculate the triad slopes within the mixed layer by linearly tapering them from zero
+    (at the surface) to the `basal' slopes,
+    the slopes of the four triads passing through the $w$-point $i,k_{\mathrm{ML}}-1/2$ (blue square),
+    ${\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p}$.
+    Triads with different $i_p,k_p$, denoted by different colours,
+    (e.g. the green triad $i_p=1/2,k_p=-1/2$) are tapered to the appropriate basal triad.}
 %}
   {\includegraphics[width=0.60\textwidth]{Fig_GRIFF_MLB_triads}}
 …
 \subsubsection{Additional truncation of skew iso-neutral flux components}
 \label{subsec:Gerdes-taper}
+The alternative option is activated by setting \np{ln\_triad\_iso} =
+  true. This retains the same tapered slope $\rML$  described above for the
+calculation of the $_{33}$ term of the iso-neutral diffusion tensor (the
+vertical tracer flux driven by vertical tracer gradients), but
+replaces the $\rML$ in the skew term by
+The alternative option is activated by setting \np{ln\_triad\_iso} = true.
+This retains the same tapered slope $\rML$  described above for the calculation of the $_{33}$ term of
+the iso-neutral diffusion tensor (the vertical tracer flux driven by vertical tracer gradients),
+but replaces the $\rML$ in the skew term by
 \begin{equation}
   \label{eq:rm*}
 …
 \end{equation}
 This operator
 \footnote{To ensure good behaviour where horizontal density
   gradients are weak, we in fact follow \citet{Gerdes1991} and set
+$\rML^*=\mathrm{sgn}(\tilde{r}_i)\min(|\rMLt^2/\tilde{r}_i|,|\tilde{r}_i|)-\sigma_i$.}
+then has the property it gives no vertical density flux, and so does
 not change the potential energy.
 This approach is similar to multiplying the iso-neutral  diffusion
+coefficient by $\tilde{r}_{\mathrm{max}}^{-2}\tilde{r}_i^{-2}$ for steep
 slopes, as suggested by \citet{Gerdes1991} (see also \citet{Griffies_Bk04}).
+\footnote{
+  To ensure good behaviour where horizontal density gradients are weak,
+  we in fact follow \citet{Gerdes1991} and
+  set $\rML^*=\mathrm{sgn}(\tilde{r}_i)\min(|\rMLt^2/\tilde{r}_i|,|\tilde{r}_i|)-\sigma_i$.}
+then has the property it gives no vertical density flux, and so does not change the potential energy.
+This approach is similar to multiplying the iso-neutral diffusion coefficient by
+$\tilde{r}_{\mathrm{max}}^{-2}\tilde{r}_i^{-2}$ for steep slopes,
+as suggested by \citet{Gerdes1991} (see also \citet{Griffies_Bk04}).
 Again it is applied separately to each triad $_i^k\mathbb{R}_{i_p}^{k_p}$
+In practice, this approach gives weak vertical tracer fluxes through
+the mixed-layer, as well as vanishing density fluxes. While it is
+theoretically advantageous that it does not change the potential
+energy, it may give a discontinuity between the
+fluxes within the mixed-layer (purely horizontal) and just below (along
+iso-neutral surfaces).
+In practice, this approach gives weak vertical tracer fluxes through the mixed-layer,
+as well as vanishing density fluxes.
+While it is theoretically advantageous that it does not change the potential energy,
+it may give a discontinuity between the fluxes within the mixed-layer (purely horizontal) and
+just below (along iso-neutral surfaces).
 % This may give strange looking results,
 % particularly where the mixed-layer depth varies strongly laterally.
 …
 \subsection{Continuous skew flux formulation}\label{sec:continuous-skew-flux}
+ When Gent and McWilliams's [1990] diffusion is used,
+an additional advection term is added. The associated velocity is the so called
+eddy induced velocity, the formulation of which depends on the slopes of iso-
+neutral surfaces. Contrary to the case of iso-neutral mixing, the slopes used
+here are referenced to the geopotential surfaces, $i.e.$ \autoref{eq:ldfslp_geo}
+is used in $z$-coordinate, and the sum \autoref{eq:ldfslp_geo}
++ \autoref{eq:ldfslp_iso} in $z^*$ or $s$-coordinates.
+When Gent and McWilliams's [1990] diffusion is used, an additional advection term is added.
+The associated velocity is the so called eddy induced velocity,
+the formulation of which depends on the slopes of iso-neutral surfaces.
+Contrary to the case of iso-neutral mixing, the slopes used here are referenced to the geopotential surfaces,
+$i.e.$ \autoref{eq:ldfslp_geo} is used in $z$-coordinate,
+and the sum \autoref{eq:ldfslp_geo} + \autoref{eq:ldfslp_iso} in $z^*$ or $s$-coordinates.
 The eddy induced velocity is given by:
 …
 \end{equation}
 \end{subequations}
+with $A_{e}$ the eddy induced velocity coefficient, and $\tilde{r}_1$ and $\tilde{r}_2$ the slopes between the iso-neutral and the geopotential surfaces.
+The traditional way to implement this additional advection is to add
+it to the Eulerian velocity prior to computing the tracer
+advection. This is implemented if \key{traldf\_eiv} is set in the
+default implementation, where \np{ln\_traldf\_triad} is set
+false. This allows us to take advantage of all the advection schemes
+offered for the tracers (see \autoref{sec:TRA_adv}) and not just a $2^{nd}$
+order advection scheme. This is particularly useful for passive
+tracers where \emph{positivity} of the advection scheme is of
+paramount importance.
+However, when \np{ln\_traldf\_triad} is set true, \NEMO instead
+implements eddy induced advection according to the so-called skew form
+\citep{Griffies_JPO98}. It is based on a transformation of the advective fluxes
+using the non-divergent nature of the eddy induced velocity.
+For example in the (\textbf{i},\textbf{k}) plane, the tracer advective
+fluxes per unit area in $ijk$ space can be
+transformed as follows:
+with $A_{e}$ the eddy induced velocity coefficient,
+and $\tilde{r}_1$ and $\tilde{r}_2$ the slopes between the iso-neutral and the geopotential surfaces.
+The traditional way to implement this additional advection is to add it to the Eulerian velocity prior to
+computing the tracer advection.
+This is implemented if \key{traldf\_eiv} is set in the default implementation,
+where \np{ln\_traldf\_triad} is set false.
+This allows us to take advantage of all the advection schemes offered for the tracers
+(see \autoref{sec:TRA_adv}) and not just a $2^{nd}$ order advection scheme.
+This is particularly useful for passive tracers where
+\emph{positivity} of the advection scheme is of paramount importance.
+However, when \np{ln\_traldf\_triad} is set true,
+\NEMO instead implements eddy induced advection according to the so-called skew form \citep{Griffies_JPO98}.
+It is based on a transformation of the advective fluxes using the non-divergent nature of the eddy induced velocity.
+For example in the (\textbf{i},\textbf{k}) plane,
+the tracer advective fluxes per unit area in $ijk$ space can be transformed as follows:
 \begin{flalign*}
 \begin{split}
 …
 \end{split}
 \end{flalign*}
 and since the eddy induced velocity field is non-divergent, we end up with the skew
 form of the eddy induced advective fluxes per unit area in $ijk$ space:
+and since the eddy induced velocity field is non-divergent,
+we end up with the skew form of the eddy induced advective fluxes per unit area in $ijk$ space:
 \begin{equation} \label{eq:eiv_skew_ijk}
 \textbf{F}_\mathrm{eiv}^T = \begin{pmatrix}
 …
 \end{split}
 \end{equation}
 Note that  \autoref{eq:eiv_skew_physical} takes the same form whatever the
+vertical coordinate, though of course the slopes
 $\tilde{r}_i$ which define the $\psi_i$ in \autoref{eq:eiv_psi} are relative to geopotentials.
 The tendency associated with eddy induced velocity is then simply the convergence
 of the fluxes (\autoref{eq:eiv_skew_ijk}, \autoref{eq:eiv_skew_physical}), so
+Note that \autoref{eq:eiv_skew_physical} takes the same form whatever the vertical coordinate,
+though of course the slopes $\tilde{r}_i$ which define the $\psi_i$ in \autoref{eq:eiv_psi} are relative to
+geopotentials.
+The tendency associated with eddy induced velocity is then simply the convergence of the fluxes
+(\autoref{eq:eiv_skew_ijk}, \autoref{eq:eiv_skew_physical}), so
 \begin{equation} \label{eq:skew_eiv_conv}
 \frac{\partial T}{\partial t}= -\frac{1}{e_1 \, e_2 \, e_3 }      \left[
 …
    + e_{1} \psi_2 \partial_j T \right)  \right]
 \end{equation}
+ It naturally conserves the tracer content, as it is expressed in flux
+ form. Since it has the same divergence as the advective form it also
+ preserves the tracer variance.
+It naturally conserves the tracer content, as it is expressed in flux form.
+Since it has the same divergence as the advective form it also preserves the tracer variance.
 \subsection{Discrete skew flux formulation}
 The skew fluxes in (\autoref{eq:eiv_skew_physical}, \autoref{eq:eiv_skew_ijk}), like the off-diagonal terms
+(\autoref{eq:i13c}, \autoref{eq:i31c}) of the small angle diffusion tensor, are best
+expressed in terms of the triad slopes, as in \autoref{fig:ISO_triad}
+and (\autoref{eq:i13}, \autoref{eq:i31}); but now in terms of the triad slopes
+$\tilde{\mathbb{R}}$ relative to geopotentials instead of the
+$\mathbb{R}$ relative to coordinate surfaces. The discrete form of
 \autoref{eq:eiv_skew_ijk} using the slopes \autoref{eq:R} and
+The skew fluxes in (\autoref{eq:eiv_skew_physical}, \autoref{eq:eiv_skew_ijk}),
+like the off-diagonal terms (\autoref{eq:i13c}, \autoref{eq:i31c}) of the small angle diffusion tensor,
+are best expressed in terms of the triad slopes, as in \autoref{fig:ISO_triad} and
+(\autoref{eq:i13}, \autoref{eq:i31});
+but now in terms of the triad slopes $\tilde{\mathbb{R}}$ relative to geopotentials instead of
+the $\mathbb{R}$ relative to coordinate surfaces.
+The discrete form of \autoref{eq:eiv_skew_ijk} using the slopes \autoref{eq:R} and
 defining $A_e$ at $T$-points is then given by:
 …
     \end{pmatrix},
   \end{flalign}
   where the skew flux in the $i$-direction associated with a given
   triad is (\autoref{eq:latflux-triad}, \autoref{eq:triadfluxu}):
+  where the skew flux in the $i$-direction associated with a given triad is (\autoref{eq:latflux-triad},
+  \autoref{eq:triadfluxu}):
   \begin{align}
     \label{eq:skewfluxu}
 …
 \end{subequations}
+Such a discretisation is consistent with the iso-neutral
+operator as it uses the same definition for the slopes.  It also
+ensures the following two key properties.
+Such a discretisation is consistent with the iso-neutral operator as it uses the same definition for the slopes.
+It also ensures the following two key properties.
 \subsubsection{No change in tracer variance}
+The discretization conserves tracer variance, $i.e.$ it does not
+include a diffusive component but is a `pure' advection term. This can
+be seen
+%either from Appendix \autoref{apdx:eiv_skew} or
+by considering the
+fluxes associated with a given triad slope
+$_i^k{\mathbb{R}}_{i_p}^{k_p} (T)$. For, following
+\autoref{subsec:variance} and \autoref{eq:dvar_iso_i}, the
+associated horizontal skew-flux $_i^k{\mathbb{S}_u}_{i_p}^{k_p} (T)$
+drives a net rate of change of variance, summed over the two
+$T$-points $i+i_p-\half,k$ and $i+i_p+\half,k$, of
+The discretization conserves tracer variance, $i.e.$ it does not include a diffusive component but is a `pure' advection term.
+This can be seen %either from Appendix \autoref{apdx:eiv_skew} or
+by considering the fluxes associated with a given triad slope $_i^k{\mathbb{R}}_{i_p}^{k_p} (T)$.
+For, following \autoref{subsec:variance} and \autoref{eq:dvar_iso_i},
+the associated horizontal skew-flux $_i^k{\mathbb{S}_u}_{i_p}^{k_p} (T)$ drives a net rate of change of variance,
+summed over the two $T$-points $i+i_p-\half,k$ and $i+i_p+\half,k$, of
 \begin{equation}
 \label{eq:dvar_eiv_i}
   _i^k{\mathbb{S}_u}_{i_p}^{k_p} (T)\,\delta_{i+ i_p}[T^k],
 \end{equation}
 while the associated vertical skew-flux gives a variance change summed over the
 $T$-points $i,k+k_p-\half$ (above) and $i,k+k_p+\half$ (below) of
+while the associated vertical skew-flux gives a variance change summed over
+the $T$-points $i,k+k_p-\half$ (above) and $i,k+k_p+\half$ (below) of
 \begin{equation}
 \label{eq:dvar_eiv_k}
   _i^k{\mathbb{S}_w}_{i_p}^{k_p} (T) \,\delta_{k+ k_p}[T^i].
 \end{equation}
+Inspection of the definitions (\autoref{eq:skewfluxu}, \autoref{eq:skewfluxw})
+shows that these two variance changes (\autoref{eq:dvar_eiv_i}, \autoref{eq:dvar_eiv_k})
+sum to zero. Hence the two fluxes associated with each triad make no
+net contribution to the variance budget.
+Inspection of the definitions (\autoref{eq:skewfluxu}, \autoref{eq:skewfluxw}) shows that
+these two variance changes (\autoref{eq:dvar_eiv_i}, \autoref{eq:dvar_eiv_k}) sum to zero.
+Hence the two fluxes associated with each triad make no net contribution to the variance budget.
 \subsubsection{Reduction in gravitational PE}
+The vertical density flux associated with the vertical skew-flux
+always has the same sign as the vertical density gradient; thus, so
+long as the fluid is stable (the vertical density gradient is
+negative) the vertical density flux is negative (downward) and hence
+reduces the gravitational PE.
+The vertical density flux associated with the vertical skew-flux always has the same sign as
+the vertical density gradient;
+thus, so long as the fluid is stable (the vertical density gradient is negative)
+the vertical density flux is negative (downward) and hence reduces the gravitational PE.
 For the change in gravitational PE driven by the $k$-flux is
 …
 \frac{-\alpha_i^k \delta_{k+ k_p}[T^i]+ \beta_i^k\delta_{k+ k_p}[S^i]} {{e_{3w}}_{\,i}^{\,k+k_p}},
 \end{align}
+using the definition of the triad slope $\rtriad{R}$,
+\autoref{eq:R} to express $-\alpha _i^k\delta_{i+ i_p}[T^k]+
+\beta_i^k\delta_{i+ i_p}[S^k]$ in terms of  $-\alpha_i^k \delta_{k+
+  k_p}[T^i]+ \beta_i^k\delta_{k+ k_p}[S^i]$.
+using the definition of the triad slope $\rtriad{R}$, \autoref{eq:R} to
+express $-\alpha _i^k\delta_{i+ i_p}[T^k]+\beta_i^k\delta_{i+ i_p}[S^k]$ in terms of
+$-\alpha_i^k \delta_{k+ k_p}[T^i]+ \beta_i^k\delta_{k+ k_p}[S^i]$.
 Where the coordinates slope, the $i$-flux gives a PE change
 …
 \frac{-\alpha_i^k \delta_{k+ k_p}[T^i]+ \beta_i^k\delta_{k+ k_p}[S^i]} {{e_{3w}}_{\,i}^{\,k+k_p}},
 \end{multline}
 (using \autoref{eq:skewfluxu}) and so the total PE change
 \autoref{eq:vert_densityPE} + \autoref{eq:lat_densityPE} associated with the triad fluxes is
+(using \autoref{eq:skewfluxu}) and so the total PE change \autoref{eq:vert_densityPE} +
+\autoref{eq:lat_densityPE} associated with the triad fluxes is
 \begin{multline}
   \label{eq:tot_densityPE}
 …
 \subsection{Treatment of the triads at the boundaries}\label{sec:skew_bdry}
+Triad slopes \rtriadt{R} used for the calculation of the eddy-induced skew-fluxes
+are masked at the boundaries in exactly the same way as are the triad
+slopes \rtriad{R} used for the iso-neutral diffusive fluxes, as
+described in \autoref{sec:iso_bdry} and
+\autoref{fig:bdry_triads}. Thus surface layer triads
+$\triadt{i}{1}{R}{1/2}{-1/2}$ and $\triadt{i+1}{1}{R}{-1/2}{-1/2}$ are
+masked, and both near bottom triad slopes $\triadt{i}{k}{R}{1/2}{1/2}$
+and $\triadt{i+1}{k}{R}{-1/2}{1/2}$ are masked when either of the
+$i,k+1$ or $i+1,k+1$ tracer points is masked, i.e.\ the $i,k+1$
+$u$-point is masked. The namelist parameter \np{ln\_botmix\_triad} has
+no effect on the eddy-induced skew-fluxes.
+Triad slopes \rtriadt{R} used for the calculation of the eddy-induced skew-fluxes are masked at the boundaries
+in exactly the same way as are the triad slopes \rtriad{R} used for the iso-neutral diffusive fluxes,
+as described in \autoref{sec:iso_bdry} and \autoref{fig:bdry_triads}.
+Thus surface layer triads $\triadt{i}{1}{R}{1/2}{-1/2}$ and $\triadt{i+1}{1}{R}{-1/2}{-1/2}$ are masked,
+and both near bottom triad slopes $\triadt{i}{k}{R}{1/2}{1/2}$ and $\triadt{i+1}{k}{R}{-1/2}{1/2}$ are masked when
+either of the $i,k+1$ or $i+1,k+1$ tracer points is masked, i.e.\ the $i,k+1$ $u$-point is masked.
+The namelist parameter \np{ln\_botmix\_triad} has no effect on the eddy-induced skew-fluxes.
 \subsection{Limiting of the slopes within the interior}\label{sec:limitskew}
+Presently, the iso-neutral slopes $\tilde{r}_i$ relative
+to geopotentials are limited to be less than $1/100$, exactly as in
+calculating the iso-neutral diffusion, \S \autoref{sec:limit}. Each
+individual triad \rtriadt{R} is so limited.
+Presently, the iso-neutral slopes $\tilde{r}_i$ relative to geopotentials are limited to be less than $1/100$,
+exactly as in calculating the iso-neutral diffusion, \S \autoref{sec:limit}.
+Each individual triad \rtriadt{R} is so limited.
 \subsection{Tapering within the surface mixed layer}\label{sec:taperskew}
+The slopes $\tilde{r}_i$ relative to
+geopotentials (and thus the individual triads \rtriadt{R}) are always tapered linearly from their value immediately below the mixed layer to zero at the
+surface \autoref{eq:rmtilde}, as described in \autoref{sec:lintaper}. This is
+option (c) of \autoref{fig:eiv_slp}. This linear tapering for the
+slopes used to calculate the eddy-induced fluxes is
+unaffected by the value of \np{ln\_triad\_iso}.
+The justification for this linear slope tapering is that, for $A_e$
+that is constant or varies only in the horizontal (the most commonly
+used options in \NEMO: see \autoref{sec:LDF_coef}), it is
+equivalent to a horizontal eiv (eddy-induced velocity) that is uniform
+within the mixed layer \autoref{eq:eiv_v}. This ensures that the
+eiv velocities do not restratify the mixed layer \citep{Treguier1997,
+  Danabasoglu_al_2008}. Equivantly, in terms
+of the skew-flux formulation we use here, the
+linear slope tapering within the mixed-layer gives a linearly varying
+vertical flux, and so a tracer convergence uniform in depth (the
+horizontal flux convergence is relatively insignificant within the mixed-layer).
+The slopes $\tilde{r}_i$ relative to geopotentials (and thus the individual triads \rtriadt{R})
+are always tapered linearly from their value immediately below the mixed layer to zero at the surface
+\autoref{eq:rmtilde}, as described in \autoref{sec:lintaper}.
+This is option (c) of \autoref{fig:eiv_slp}.
+This linear tapering for the slopes used to calculate the eddy-induced fluxes is unaffected by
+the value of \np{ln\_triad\_iso}.
+The justification for this linear slope tapering is that, for $A_e$ that is constant or varies only in
+the horizontal (the most commonly used options in \NEMO: see \autoref{sec:LDF_coef}),
+it is equivalent to a horizontal eiv (eddy-induced velocity) that is uniform within the mixed layer
+\autoref{eq:eiv_v}.
+This ensures that the eiv velocities do not restratify the mixed layer \citep{Treguier1997,Danabasoglu_al_2008}.
+Equivantly, in terms of the skew-flux formulation we use here,
+the linear slope tapering within the mixed-layer gives a linearly varying vertical flux,
+and so a tracer convergence uniform in depth
+(the horizontal flux convergence is relatively insignificant within the mixed-layer).
 \subsection{Streamfunction diagnostics}\label{sec:sfdiag}
+Where the namelist parameter \np{ln\_traldf\_gdia}\forcode{ = .true.}, diagnosed
+mean eddy-induced velocities are output. Each time step,
+streamfunctions are calculated in the $i$-$k$ and $j$-$k$ planes at
+$uw$ (integer +1/2 $i$, integer $j$, integer +1/2 $k$) and $vw$
+(integer $i$, integer +1/2 $j$, integer +1/2 $k$) points (see Table
+\autoref{tab:cell}) respectively. We follow \citep{Griffies_Bk04} and
+calculate the streamfunction at a given $uw$-point from the
+surrounding four triads according to:
+Where the namelist parameter \np{ln\_traldf\_gdia}\forcode{ = .true.},
+diagnosed mean eddy-induced velocities are output.
+Each time step, streamfunctions are calculated in the $i$-$k$ and $j$-$k$ planes at
+$uw$ (integer +1/2 $i$, integer $j$, integer +1/2 $k$) and $vw$ (integer $i$, integer +1/2 $j$, integer +1/2 $k$)
+points (see Table \autoref{tab:cell}) respectively.
+We follow \citep{Griffies_Bk04} and calculate the streamfunction at a given $uw$-point from
+the surrounding four triads according to:
 \begin{equation}
   \label{eq:sfdiagi}
 …
 \end{equation}
 The streamfunction $\psi_1$ is calculated similarly at $vw$ points.
+The eddy-induced velocities are then calculated from the
+straightforward discretisation of \autoref{eq:eiv_v}:
+The eddy-induced velocities are then calculated from the straightforward discretisation of \autoref{eq:eiv_v}:
 \begin{equation}\label{eq:eiv_v_discrete}
 \begin{split}

NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_ASM.tex

-                      r10146
+                      r10368
 $\ $\newline    % force a new line
 The ASM code adds the functionality to apply increments to the model variables:
 temperature, salinity, sea surface height, velocity and sea ice concentration.
 These are read into the model from a NetCDF file which may be produced by separate data
+assimilation code.  The code can also output model background fields which are used
+as an input to data assimilation code. This is all controlled by the namelist
+\textit{\ngn{nam\_asminc} }.  There is a brief description of all the namelist options
 provided.  To build the ASM code \key{asminc} must be set.
+The ASM code adds the functionality to apply increments to the model variables: temperature, salinity,
+sea surface height, velocity and sea ice concentration.
+These are read into the model from a NetCDF file which may be produced by separate data assimilation code.
+The code can also output model background fields which are used as an input to data assimilation code.
+This is all controlled by the namelist \textit{\ngn{nam\_asminc} }.
+There is a brief description of all the namelist options provided.
+To build the ASM code \key{asminc} must be set.
 %===============================================================
 …
 \label{sec:ASM_DI}
 Direct initialization (DI) refers to the instantaneous correction
 of the model background state using the analysis increment.
+Direct initialization (DI) refers to the instantaneous correction of the model background state using
+the analysis increment.
 DI is used when \np{ln\_asmdin} is set to true.
 …
 Rather than updating the model state directly with the analysis increment,
 it may be preferable to introduce the increment gradually into the ocean
 model in order to minimize spurious adjustment processes. This technique
 is referred to as Incremental Analysis Updates (IAU) \citep{Bloom_al_MWR96}.
+it may be preferable to introduce the increment gradually into the ocean model in order to
+minimize spurious adjustment processes.
+This technique is referred to as Incremental Analysis Updates (IAU) \citep{Bloom_al_MWR96}.
 IAU is a common technique used with 3D assimilation methods such as 3D-Var or OI.
 IAU is used when \np{ln\_asmiau} is set to true.
+With IAU, the model state trajectory ${\bf x}$ in the assimilation window
+($t_{0} \leq t_{i} \leq t_{N}$)
+is corrected by adding the analysis increments for temperature, salinity, horizontal velocity and SSH
+as additional tendency terms to the prognostic equations:
+With IAU, the model state trajectory ${\bf x}$ in the assimilation window ($t_{0} \leq t_{i} \leq t_{N}$)
+is corrected by adding the analysis increments for temperature, salinity, horizontal velocity and SSH as
+additional tendency terms to the prognostic equations:
 \begin{eqnarray}     \label{eq:wa_traj_iau}
 {\bf x}^{a}(t_{i}) = M(t_{i}, t_{0})[{\bf x}^{b}(t_{0})]
 \; + \; F_{i} \delta \tilde{\bf x}^{a}
 \end{eqnarray}
+where $F_{i}$ is a weighting function for applying the increments $\delta
+\tilde{\bf x}^{a}$ defined such that $\sum_{i=1}^{N} F_{i}=1$.
+${\bf x}^b$ denotes the model initial state and ${\bf x}^a$ is the model state
+after the increments are applied.
+where $F_{i}$ is a weighting function for applying the increments $\delta\tilde{\bf x}^{a}$ defined such that
+$\sum_{i=1}^{N} F_{i}=1$.
+${\bf x}^b$ denotes the model initial state and ${\bf x}^a$ is the model state after the increments are applied.
 To control the adjustment time of the model to the increment,
+the increment can be applied over an arbitrary sub-window,
+$t_{m} \leq t_{i} \leq t_{n}$, of the main assimilation window,
+where $t_{0} \leq t_{m} \leq t_{i}$ and $t_{i} \leq t_{n} \leq t_{N}$,
+the increment can be applied over an arbitrary sub-window, $t_{m} \leq t_{i} \leq t_{n}$,
+of the main assimilation window, where $t_{0} \leq t_{m} \leq t_{i}$ and $t_{i} \leq t_{n} \leq t_{N}$.
 Typically the increments are spread evenly over the full window.
 In addition, two different weighting functions have been implemented.
 …
 \end{eqnarray}
 where $M = m-n$.
+The second function employs peaked hat-like weights in order to give maximum
+weight in the centre of the sub-window, with the weighting reduced
+linearly to a small value at the window end-points:
+The second function employs peaked hat-like weights in order to give maximum weight in the centre of the sub-window,
+with the weighting reduced linearly to a small value at the window end-points:
 \begin{eqnarray}     \label{eq:F2_i}
 F^{(2)}_{i}
 …
 \end{eqnarray}
 where $\alpha^{-1} = \sum_{i=1}^{M/2} 2i$ and $M$ is assumed to be even.
+The weights described by \autoref{eq:F2_i} provide a
+smoother transition of the analysis trajectory from one assimilation cycle
+to the next than that described by \autoref{eq:F1_i}.
+The weights described by \autoref{eq:F2_i} provide a smoother transition of the analysis trajectory from
+one assimilation cycle to the next than that described by \autoref{eq:F1_i}.
 %==========================================================================
 …
 \label{sec:ASM_div_dmp}
+The velocity increments may be initialized by the iterative application of
+a divergence damping operator. In iteration step $n$ new estimates of
+velocity increments $u^{n}_I$ and $v^{n}_I$ are updated by:
+The velocity increments may be initialized by the iterative application of a divergence damping operator.
+In iteration step $n$ new estimates of velocity increments $u^{n}_I$ and $v^{n}_I$ are updated by:
 \begin{equation} \label{eq:asm_dmp}
 \left\{ \begin{aligned}
 …
                        +\delta _j \left[ {e_{1v}\,e_{3v}\,v^{n-1}_I} \right]} \right).
 \end{equation}
+By the application of \autoref{eq:asm_dmp} and \autoref{eq:asm_dmp} the divergence is filtered
+in each iteration, and the vorticity is left unchanged. In the presence of coastal boundaries
+with zero velocity increments perpendicular to the coast the divergence is strongly damped.
+This type of the initialisation reduces the vertical velocity magnitude  and alleviates the
+problem of the excessive unphysical vertical mixing in the first steps of the model
+integration \citep{Talagrand_JAS72, Dobricic_al_OS07}. Diffusion coefficients are defined as
+$A_D = \alpha e_{1t} e_{2t}$, where $\alpha = 0.2$. The divergence damping is activated by
+assigning to \np{nn\_divdmp} in the \textit{nam\_asminc} namelist a value greater than zero.
+By choosing this value to be of the order of 100 the increments in the vertical velocity will
+be significantly reduced.
+By the application of \autoref{eq:asm_dmp} and \autoref{eq:asm_dmp} the divergence is filtered in each iteration,
+and the vorticity is left unchanged.
+In the presence of coastal boundaries with zero velocity increments perpendicular to the coast
+the divergence is strongly damped.
+This type of the initialisation reduces the vertical velocity magnitude and
+alleviates the problem of the excessive unphysical vertical mixing in the first steps of the model integration
+\citep{Talagrand_JAS72, Dobricic_al_OS07}.
+Diffusion coefficients are defined as $A_D = \alpha e_{1t} e_{2t}$, where $\alpha = 0.2$.
+The divergence damping is activated by assigning to \np{nn\_divdmp} in the \textit{nam\_asminc} namelist
+a value greater than zero.
+By choosing this value to be of the order of 100 the increments in
+the vertical velocity will be significantly reduced.
 …
 \label{sec:ASM_details}
 Here we show an example \ngn{namasm} namelist and the header of an example assimilation
 increments file on the ORCA2 grid.
+Here we show an example \ngn{namasm} namelist and the header of an example assimilation increments file on
+the ORCA2 grid.
 %------------------------------------------namasm-----------------------------------------------------

NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_CONFIG.tex

-                      r10146
+                      r10368
 The purpose of this part of the manual is to introduce the \NEMO reference configurations.
 These configurations are offered as means to explore various numerical and physical options,
 thus allowing the user to verify that the code is performing in a manner consistent with that
+we are running. This form of verification is critical as one adopts the code for his or her particular
+research purposes. The reference configurations also provide a sense for some of the options available
 in the code, though by no means are all options exercised in the reference configurations.
+The purpose of this part of the manual is to introduce the \NEMO reference configurations.
+These configurations are offered as means to explore various numerical and physical options,
+thus allowing the user to verify that the code is performing in a manner consistent with that we are running.
+This form of verification is critical as one adopts the code for his or her particular research purposes.
+The reference configurations also provide a sense for some of the options available in the code,
+though by no means are all options exercised in the reference configurations.
 %------------------------------------------namcfg----------------------------------------------------
 …
 $\ $\newline
 The 1D model option simulates a stand alone water column within the 3D \NEMO system.
 It can be applied to the ocean alone or to the ocean-ice system and can include passive tracers
+or a biogeochemical model. It is set up by defining the position of the 1D water column in the grid
+The 1D model option simulates a stand alone water column within the 3D \NEMO system.
+It can be applied to the ocean alone or to the ocean-ice system and can include passive tracers or a biogeochemical model.
+It is set up by defining the position of the 1D water column in the grid
 (see \textit{CONFIG/SHARED/namelist\_ref} ).
 The 1D model is a very useful tool
 \textit{(a)} to learn about the physics and numerical treatment of vertical mixing processes ;
 \textit{(b)} to investigate suitable parameterisations of unresolved turbulence (surface wave
+breaking, Langmuir circulation, ...) ;
 \textit{(c)} to compare the behaviour of different vertical mixing schemes  ;
 \textit{(d)} to perform sensitivity studies on the vertical diffusion at a particular point of an ocean domain ;
+The 1D model is a very useful tool
+\textit{(a)} to learn about the physics and numerical treatment of vertical mixing processes;
+\textit{(b)} to investigate suitable parameterisations of unresolved turbulence
+(surface wave breaking, Langmuir circulation, ...);
+\textit{(c)} to compare the behaviour of different vertical mixing schemes;
+\textit{(d)} to perform sensitivity studies on the vertical diffusion at a particular point of an ocean domain;
 \textit{(d)} to produce extra diagnostics, without the large memory requirement of the full 3D model.
 The methodology is based on the use of the zoom functionality over the smallest possible
+domain : a 3x3 domain centered on the grid point of interest,
+with some extra routines. There is no need to define a new mesh, bathymetry,
+initial state or forcing, since the 1D model will use those of the configuration it is a zoom of.
 The chosen grid point is set in \textit{\ngn{namcfg}} namelist by setting the \np{jpizoom} and \np{jpjzoom}
 parameters to the indices of the location of the chosen grid point.
 The 1D model has some specifies. First, all the horizontal derivatives are assumed to be zero, and
 second, the two components of the velocity are moved on a $T$-point.
+The methodology is based on the use of the zoom functionality over the smallest possible domain:
+a 3x3 domain centered on the grid point of interest, with some extra routines.
+There is no need to define a new mesh, bathymetry, initial state or forcing,
+since the 1D model will use those of the configuration it is a zoom of.
+The chosen grid point is set in \textit{\ngn{namcfg}} namelist by
+setting the \np{jpizoom} and \np{jpjzoom} parameters to the indices of the location of the chosen grid point.
+The 1D model has some specifies. First, all the horizontal derivatives are assumed to be zero,
+and second, the two components of the velocity are moved on a $T$-point.
 Therefore, defining \key{c1d} changes five main things in the code behaviour:
 \begin{description}
+\item[(1)] the lateral boundary condition routine (\rou{lbc\_lnk}) set the value of the central column
+of the 3x3 domain is imposed over the whole domain ;
+\item[(3)] a call to \rou{lbc\_lnk} is systematically done when reading input data ($i.e.$ in \mdl{iom}) ;
+\item[(3)] a simplified \rou{stp} routine is used (\rou{stp\_c1d}, see \mdl{step\_c1d} module) in which
+both lateral tendancy terms and lateral physics are not called ;
+\item[(4)] the vertical velocity is zero (so far, no attempt at introducing a Ekman pumping velocity
+has been made) ;
+\item[(5)] a simplified treatment of the Coriolis term is performed as $U$- and $V$-points are the same
+(see \mdl{dyncor\_c1d}).
+\item[(1)]
+  the lateral boundary condition routine (\rou{lbc\_lnk}) set the value of the central column of
+  the 3x3 domain is imposed over the whole domain;
+\item[(3)]
+  a call to \rou{lbc\_lnk} is systematically done when reading input data ($i.e.$ in \mdl{iom});
+\item[(3)]
+  a simplified \rou{stp} routine is used (\rou{stp\_c1d}, see \mdl{step\_c1d} module) in which
+  both lateral tendancy terms and lateral physics are not called;
+\item[(4)]
+  the vertical velocity is zero
+  (so far, no attempt at introducing a Ekman pumping velocity has been made);
+\item[(5)]
+  a simplified treatment of the Coriolis term is performed as $U$- and $V$-points are the same
+  (see \mdl{dyncor\_c1d}).
 \end{description}
 All the relevant \textit{\_c1d} modules can be found in the NEMOGCM/NEMO/OPA\_SRC/C1D directory of
+All the relevant \textit{\_c1d} modules can be found in the NEMOGCM/NEMO/OPA\_SRC/C1D directory of
 the \NEMO distribution.
 …
 \label{sec:CFG_orca}
 The ORCA family is a series of global ocean configurations that are run together with
 the LIM sea-ice model (ORCA-LIM) and possibly with PISCES biogeochemical model
 (ORCA-LIM-PISCES), using various resolutions.
 An appropriate namelist is available in \path{CONFIG/ORCA2_LIM3_PISCES/EXP00/namelist_cfg}
+for ORCA2.
 The domain of ORCA2 configuration is defined in \ifile{ORCA\_R2\_zps\_domcfg} file, this file is available in tar file in the wiki of NEMO : \\
+The ORCA family is a series of global ocean configurations that are run together with
+the LIM sea-ice model (ORCA-LIM) and possibly with PISCES biogeochemical model (ORCA-LIM-PISCES),
+using various resolutions.
+An appropriate namelist is available in \path{CONFIG/ORCA2_LIM3_PISCES/EXP00/namelist_cfg} for ORCA2.
+The domain of ORCA2 configuration is defined in \ifile{ORCA\_R2\_zps\_domcfg} file,
+this file is available in tar file in the wiki of NEMO: \\
 https://forge.ipsl.jussieu.fr/nemo/wiki/Users/ReferenceConfigurations/ORCA2\_LIM3\_PISCES \\
 In this namelist\_cfg the name of domain input file is set in \ngn{namcfg} block of namelist.
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!t]   \begin{center}
+\includegraphics[width=0.98\textwidth]{Fig_ORCA_NH_mesh}
+\caption{  \protect\label{fig:MISC_ORCA_msh}
+ORCA mesh conception. The departure from an isotropic Mercator grid start poleward of 20\degN.
+The two "north pole" are the foci of a series of embedded ellipses (blue curves)
+which are determined analytically and form the i-lines of the ORCA mesh (pseudo latitudes).
+Then, following \citet{Madec_Imbard_CD96}, the normal to the series of ellipses (red curves) is computed
+which provide the j-lines of the mesh (pseudo longitudes).  }
+\end{center}   \end{figure}
+\begin{figure}[!t]
+  \begin{center}
+    \includegraphics[width=0.98\textwidth]{Fig_ORCA_NH_mesh}
+    \caption{  \protect\label{fig:MISC_ORCA_msh}
+      ORCA mesh conception.
+      The departure from an isotropic Mercator grid start poleward of 20\degN.
+      The two "north pole" are the foci of a series of embedded ellipses (blue curves) which
+      are determined analytically and form the i-lines of the ORCA mesh (pseudo latitudes).
+      Then, following \citet{Madec_Imbard_CD96}, the normal to the series of ellipses (red curves) is computed which
+      provides the j-lines of the mesh (pseudo longitudes).  }
+  \end{center}
+\end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 …
 \label{subsec:CFG_orca_grid}
 The ORCA grid is a tripolar is based on the semi-analytical method of \citet{Madec_Imbard_CD96}.
 It allows to construct a global orthogonal curvilinear ocean mesh which has no singularity point inside
+The ORCA grid is a tripolar is based on the semi-analytical method of \citet{Madec_Imbard_CD96}.
+It allows to construct a global orthogonal curvilinear ocean mesh which has no singularity point inside
 the computational domain since two north mesh poles are introduced and placed on lands.
+The method involves defining an analytical set of mesh parallels in the stereographic polar plan,
+computing the associated set of mesh meridians, and projecting the resulting mesh onto the sphere.
+The set of mesh parallels used is a series of embedded ellipses which foci are the two mesh north
+poles (\autoref{fig:MISC_ORCA_msh}). The resulting mesh presents no loss of continuity in
+either the mesh lines or the scale factors, or even the scale factor derivatives over the whole
+ocean domain, as the mesh is not a composite mesh.
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!tbp]  \begin{center}
+\includegraphics[width=1.0\textwidth]{Fig_ORCA_NH_msh05_e1_e2}
+\includegraphics[width=0.80\textwidth]{Fig_ORCA_aniso}
+\caption {  \protect\label{fig:MISC_ORCA_e1e2}
+\textit{Top}: Horizontal scale factors ($e_1$, $e_2$) and
+\textit{Bottom}: ratio of anisotropy ($e_1 / e_2$)
+for ORCA 0.5\deg ~mesh. South of 20\degN a Mercator grid is used ($e_1 = e_2$)
+so that the anisotropy ratio is 1. Poleward of 20\degN, the two "north pole"
+introduce a weak anisotropy over the ocean areas ($< 1.2$) except in vicinity of Victoria Island
+(Canadian Arctic Archipelago). }
+The method involves defining an analytical set of mesh parallels in the stereographic polar plan,
+computing the associated set of mesh meridians, and projecting the resulting mesh onto the sphere.
+The set of mesh parallels used is a series of embedded ellipses which foci are the two mesh north poles
+(\autoref{fig:MISC_ORCA_msh}).
+The resulting mesh presents no loss of continuity in either the mesh lines or the scale factors,
+or even the scale factor derivatives over the whole ocean domain, as the mesh is not a composite mesh.
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!tbp]
+  \begin{center}
+    \includegraphics[width=1.0\textwidth]{Fig_ORCA_NH_msh05_e1_e2}
+    \includegraphics[width=0.80\textwidth]{Fig_ORCA_aniso}
+    \caption {  \protect\label{fig:MISC_ORCA_e1e2}
+      \textit{Top}: Horizontal scale factors ($e_1$, $e_2$) and
+      \textit{Bottom}: ratio of anisotropy ($e_1 / e_2$)
+      for ORCA 0.5\deg ~mesh.
+      South of 20\degN a Mercator grid is used ($e_1 = e_2$) so that the anisotropy ratio is 1.
+      Poleward of 20\degN, the two "north pole" introduce a weak anisotropy over the ocean areas ($< 1.2$) except in
+      vicinity of Victoria Island (Canadian Arctic Archipelago). }
 \end{center}   \end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+The method is applied to Mercator grid ($i.e.$ same zonal and meridional grid spacing) poleward
+of 20\degN, so that the Equator is a mesh line, which provides a better numerical solution
+for equatorial dynamics. The choice of the series of embedded ellipses (position of the foci and
+variation of the ellipses) is a compromise between maintaining  the ratio of mesh anisotropy
+($e_1 / e_2$) close to one in the ocean (especially in area of strong eddy activities such as
+the Gulf Stream) and keeping the smallest scale factor in the northern hemisphere larger
+than the smallest one in the southern hemisphere.
+The resulting mesh is shown in \autoref{fig:MISC_ORCA_msh} and \autoref{fig:MISC_ORCA_e1e2}
+for a half a degree grid (ORCA\_R05).
+The smallest ocean scale factor is found in along  Antarctica, while the ratio of anisotropy remains close to one except near the Victoria Island
+in the Canadian Archipelago.
+The method is applied to Mercator grid ($i.e.$ same zonal and meridional grid spacing) poleward of 20\degN,
+so that the Equator is a mesh line, which provides a better numerical solution for equatorial dynamics.
+The choice of the series of embedded ellipses (position of the foci and variation of the ellipses)
+is a compromise between maintaining the ratio of mesh anisotropy ($e_1 / e_2$) close to one in the ocean
+(especially in area of strong eddy activities such as the Gulf Stream) and keeping the smallest scale factor in
+the northern hemisphere larger than the smallest one in the southern hemisphere.
+The resulting mesh is shown in \autoref{fig:MISC_ORCA_msh} and \autoref{fig:MISC_ORCA_e1e2} for
+a half a degree grid (ORCA\_R05).
+The smallest ocean scale factor is found in along Antarctica,
+while the ratio of anisotropy remains close to one except near the Victoria Island in the Canadian Archipelago.
 % -------------------------------------------------------------------------------------------------------------
 …
+The NEMO system is provided with five built-in ORCA configurations which differ in the
+horizontal resolution. The value of the resolution is given by the resolution at the Equator
+expressed in degrees. Each of configuration is set through the \textit{domain\_cfg} domain configuration file,
+which sets the grid size and configuration name parameters. The NEMO System Team provides only ORCA2 domain input file "\ifile{ORCA\_R2\_zps\_domcfg}" file  (Tab. \autoref{tab:ORCA}).
+The NEMO system is provided with five built-in ORCA configurations which differ in the horizontal resolution.
+The value of the resolution is given by the resolution at the Equator expressed in degrees.
+Each of configuration is set through the \textit{domain\_cfg} domain configuration file,
+which sets the grid size and configuration name parameters.
+The NEMO System Team provides only ORCA2 domain input file "\ifile{ORCA\_R2\_zps\_domcfg}" file
+(Tab. \autoref{tab:ORCA}).
 …
 \hline   \hline
 \end{tabular}
 \caption{ \protect\label{tab:ORCA}
 Domain size of ORCA family configurations.
 The flag for configurations of ORCA family need to be set in \textit{domain\_cfg} file. }
+\caption{ \protect\label{tab:ORCA}
+  Domain size of ORCA family configurations.
+  The flag for configurations of ORCA family need to be set in \textit{domain\_cfg} file. }
 \end{center}
 \end{table}
 …
+The ORCA\_R2 configuration has the following specificity : starting from a 2\deg~ORCA mesh,
+local mesh refinements were applied to the Mediterranean, Red, Black and Caspian Seas,
+so that the resolution is 1\deg \time 1\deg there. A local transformation were also applied
+with in the Tropics in order to refine the meridional resolution up to 0.5\deg at the Equator.
+The ORCA\_R1 configuration has only a local tropical transformation  to refine the meridional
+resolution up to 1/3\deg~at the Equator. Note that the tropical mesh refinements in ORCA\_R2
+and R1 strongly increases the mesh anisotropy there.
+The ORCA\_R2 configuration has the following specificity: starting from a 2\deg~ORCA mesh,
+local mesh refinements were applied to the Mediterranean, Red, Black and Caspian Seas,
+so that the resolution is 1\deg \time 1\deg there.
+A local transformation were also applied with in the Tropics in order to refine the meridional resolution up to
+.5\deg at the Equator.
+The ORCA\_R1 configuration has only a local tropical transformation to refine the meridional resolution up to
+/3\deg~at the Equator.
+Note that the tropical mesh refinements in ORCA\_R2 and R1 strongly increases the mesh anisotropy there.
 The ORCA\_R05 and higher global configurations do not incorporate any regional refinements.
 For ORCA\_R1 and R025, setting the configuration key to 75 allows to use 75 vertical levels,
+otherwise 46 are used. In the other ORCA configurations, 31 levels are used
+For ORCA\_R1 and R025, setting the configuration key to 75 allows to use 75 vertical levels, otherwise 46 are used.
+In the other ORCA configurations, 31 levels are used
 (see \autoref{tab:orca_zgr} \sfcomment{HERE I need to put new table for ORCA2 values} and \autoref{fig:zgr}).
+Only the ORCA\_R2 is provided with all its input files in the \NEMO distribution.
+It is very similar to that used as part of the climate model developed at IPSL for the 4th IPCC
+assessment of climate change (Marti et al., 2009). It is also the basis for the \NEMO contribution
+to the Coordinate Ocean-ice Reference Experiments (COREs) documented in \citet{Griffies_al_OM09}.
+This version of ORCA\_R2 has 31 levels in the vertical, with the highest resolution (10m)
+in the upper 150m (see \autoref{tab:orca_zgr} and \autoref{fig:zgr}).
+Only the ORCA\_R2 is provided with all its input files in the \NEMO distribution.
+It is very similar to that used as part of the climate model developed at IPSL for the 4th IPCC assessment of
+climate change (Marti et al., 2009).
+It is also the basis for the \NEMO contribution to the Coordinate Ocean-ice Reference Experiments (COREs)
+documented in \citet{Griffies_al_OM09}.
+This version of ORCA\_R2 has 31 levels in the vertical, with the highest resolution (10m) in the upper 150m
+(see \autoref{tab:orca_zgr} and \autoref{fig:zgr}).
 The bottom topography and the coastlines are derived from the global atlas of Smith and Sandwell (1997).
 The default forcing uses the boundary forcing from \citet{Large_Yeager_Rep04} (see \autoref{subsec:SBC_blk_core}),
 which was developed for the purpose of running global coupled ocean-ice simulations
+without an interactive atmosphere. This \citet{Large_Yeager_Rep04} dataset is available
+through the \href{http://nomads.gfdl.noaa.gov/nomads/forms/mom4/CORE.html}{GFDL web site}.
+The "normal year" of \citet{Large_Yeager_Rep04} has been chosen of the \NEMO distribution
 since release v3.3.
 ORCA\_R2 pre-defined configuration can also be run with an AGRIF zoom over the Agulhas
+current area ( \key{agrif}  defined) and, by setting the appropriate variables, see \path{CONFIG/SHARED/namelist_ref}
+a regional Arctic or peri-Antarctic configuration is extracted from an ORCA\_R2 or R05 configurations
 using sponge layers at open boundaries.
+which was developed for the purpose of running global coupled ocean-ice simulations without
+an interactive atmosphere.
+This \citet{Large_Yeager_Rep04} dataset is available through
+the \href{http://nomads.gfdl.noaa.gov/nomads/forms/mom4/CORE.html}{GFDL web site}.
+The "normal year" of \citet{Large_Yeager_Rep04} has been chosen of the \NEMO distribution since release v3.3.
+ORCA\_R2 pre-defined configuration can also be run with an AGRIF zoom over the Agulhas current area
+(\key{agrif} defined) and, by setting the appropriate variables, see \path{CONFIG/SHARED/namelist_ref}.
+A regional Arctic or peri-Antarctic configuration is extracted from an ORCA\_R2 or R05 configurations using
+sponge layers at open boundaries.
 % -------------------------------------------------------------------------------------------------------------
 …
 \label{sec:CFG_gyre}
 The GYRE configuration \citep{Levy_al_OM10} has been built to simulate
+the seasonal cycle of a double-gyre box model. It consists in an idealized domain
+similar to that used in the studies of \citet{Drijfhout_JPO94} and \citet{Hazeleger_Drijfhout_JPO98,
+Hazeleger_Drijfhout_JPO99, Hazeleger_Drijfhout_JGR00, Hazeleger_Drijfhout_JPO00},
 over which an analytical seasonal forcing is applied. This allows to investigate the
 spontaneous generation of a large number of interacting, transient mesoscale eddies
+The GYRE configuration \citep{Levy_al_OM10} has been built to
+simulate the seasonal cycle of a double-gyre box model.
+It consists in an idealized domain similar to that used in the studies of \citet{Drijfhout_JPO94} and
+\citet{Hazeleger_Drijfhout_JPO98, Hazeleger_Drijfhout_JPO99, Hazeleger_Drijfhout_JGR00, Hazeleger_Drijfhout_JPO00},
+over which an analytical seasonal forcing is applied.
+This allows to investigate the spontaneous generation of a large number of interacting, transient mesoscale eddies
 and their contribution to the large scale circulation.
+The domain geometry is a closed rectangular basin on the $\beta$-plane centred
+at $\sim$ 30\degN and rotated by 45\deg, 3180~km long, 2120~km wide
+and 4~km deep (\autoref{fig:MISC_strait_hand}).
+The domain is bounded by vertical walls and by a flat bottom. The configuration is
+meant to represent an idealized North Atlantic or North Pacific basin.
+The circulation is forced by analytical profiles of wind and buoyancy fluxes.
+The applied forcings vary seasonally in a sinusoidal manner between winter
+and summer extrema \citep{Levy_al_OM10}.
+The wind stress is zonal and its curl changes sign at 22\degN and 36\degN.
+It forces a subpolar gyre in the north, a subtropical gyre in the wider part of the domain
+and a small recirculation gyre in the southern corner.
+The net heat flux takes the form of a restoring toward a zonal apparent air
+temperature profile. A portion of the net heat flux which comes from the solar radiation
+is allowed to penetrate within the water column.
+The fresh water flux is also prescribed and varies zonally.
+It is determined such as, at each time step, the basin-integrated flux is zero.
+The basin is initialised at rest with vertical profiles of temperature and salinity
+uniformly applied to the whole domain.
+The GYRE configuration is set like an analytical configuration. Through \np{ln\_read\_cfg}\forcode{ = .false.} in \textit{namcfg} namelist defined in the reference configuration \path{CONFIG/GYRE/EXP00/namelist_cfg} anaylitical definition of grid in GYRE is done in usrdef\_hrg, usrdef\_zgr routines. Its horizontal resolution
+(and thus the size of the domain) is determined by setting \np{nn\_GYRE} in  \ngn{namusr\_def}: \\
+The domain geometry is a closed rectangular basin on the $\beta$-plane centred at $\sim$ 30\degN and
+rotated by 45\deg, 3180~km long, 2120~km wide and 4~km deep (\autoref{fig:MISC_strait_hand}).
+The domain is bounded by vertical walls and by a flat bottom.
+The configuration is meant to represent an idealized North Atlantic or North Pacific basin.
+The circulation is forced by analytical profiles of wind and buoyancy fluxes.
+The applied forcings vary seasonally in a sinusoidal manner between winter and summer extrema \citep{Levy_al_OM10}.
+The wind stress is zonal and its curl changes sign at 22\degN and 36\degN.
+It forces a subpolar gyre in the north, a subtropical gyre in the wider part of the domain and
+a small recirculation gyre in the southern corner.
+The net heat flux takes the form of a restoring toward a zonal apparent air temperature profile.
+A portion of the net heat flux which comes from the solar radiation is allowed to penetrate within the water column.
+The fresh water flux is also prescribed and varies zonally.
+It is determined such as, at each time step, the basin-integrated flux is zero.
+The basin is initialised at rest with vertical profiles of temperature and salinity uniformly applied to
+the whole domain.
+The GYRE configuration is set like an analytical configuration.
+Through \np{ln\_read\_cfg}\forcode{ = .false.} in \textit{namcfg} namelist defined in
+the reference configuration \path{CONFIG/GYRE/EXP00/namelist_cfg}
+analytical definition of grid in GYRE is done in usrdef\_hrg, usrdef\_zgr routines.
+Its horizontal resolution (and thus the size of the domain) is determined by
+setting \np{nn\_GYRE} in \ngn{namusr\_def}: \\
 \np{jpiglo} $= 30 \times$ \np{nn\_GYRE} + 2   \\
 \np{jpjglo} $= 20 \times$ \np{nn\_GYRE} + 2   \\
 Obviously, the namelist parameters have to be adjusted to the chosen resolution, see the Configurations
 pages on the NEMO web site (Using NEMO\/Configurations) .
+Obviously, the namelist parameters have to be adjusted to the chosen resolution,
+see the Configurations pages on the NEMO web site (Using NEMO\/Configurations).
 In the vertical, GYRE uses the default 30 ocean levels (\jp{jpk}\forcode{ = 31}) (\autoref{fig:zgr}).
+The GYRE configuration is also used in benchmark test as it is very simple to increase
+its resolution and as it does not requires any input file. For example, keeping a same model size
+on each processor while increasing the number of processor used is very easy, even though the
+physical integrity of the solution can be compromised. Benchmark is activate via \np{ln\_bench}\forcode{ = .true.} in \ngn{namusr\_def} in namelist \path{CONFIG/GYRE/EXP00/namelist_cfg}.
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!t]   \begin{center}
+\includegraphics[width=1.0\textwidth]{Fig_GYRE}
+\caption{  \protect\label{fig:GYRE}
+Snapshot of relative vorticity at the surface of the model domain
+in GYRE R9, R27 and R54. From \citet{Levy_al_OM10}.}
+\end{center}   \end{figure}
+The GYRE configuration is also used in benchmark test as it is very simple to increase its resolution and
+as it does not requires any input file.
+For example, keeping a same model size on each processor while increasing the number of processor used is very easy,
+even though the physical integrity of the solution can be compromised.
+Benchmark is activate via \np{ln\_bench}\forcode{ = .true.} in \ngn{namusr\_def} in
+namelist \path{CONFIG/GYRE/EXP00/namelist_cfg}.
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!t]
+  \begin{center}
+    \includegraphics[width=1.0\textwidth]{Fig_GYRE}
+    \caption{  \protect\label{fig:GYRE}
+      Snapshot of relative vorticity at the surface of the model domain in GYRE R9, R27 and R54.
+      From \citet{Levy_al_OM10}.}
+  \end{center}
+\end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 …
 \label{sec:MISC_config_AMM}
+The AMM, Atlantic Margins Model, is a regional model covering the
+Northwest European Shelf domain on a regular lat-lon grid at
+approximately 12km horizontal resolution. The appropriate
+\textit{\&namcfg} namelist  is available in \textit{CONFIG/AMM12/EXP00/namelist\_cfg}.
+The AMM, Atlantic Margins Model, is a regional model covering the Northwest European Shelf domain on
+a regular lat-lon grid at approximately 12km horizontal resolution.
+The appropriate \textit{\&namcfg} namelist  is available in \textit{CONFIG/AMM12/EXP00/namelist\_cfg}.
 It is used to build the correct dimensions of the AMM domain.
+This configuration tests several features of NEMO functionality specific
+to the shelf seas.
+In particular, the AMM uses $S$-coordinates in the vertical rather than
+$z$-coordinates and is forced with tidal lateral boundary conditions
+using a flather boundary condition from the BDY module.
+The AMM configuration  uses the GLS (\key{zdfgls}) turbulence scheme, the
+VVL non-linear free surface(\key{vvl}) and time-splitting
+(\key{dynspg\_ts}).
+In addition to the tidal boundary condition the model may also take
+open boundary conditions from a North Atlantic model. Boundaries may be
+completely omitted by setting \np{ln\_bdy} to false.
+Sample surface fluxes, river forcing and a sample initial restart file
+are included to test a realistic model run. The Baltic boundary is
+included within the river input file and is specified as a river source.
+Unlike ordinary river points the Baltic inputs also include salinity and
+temperature data.
+This configuration tests several features of NEMO functionality specific to the shelf seas.
+In particular, the AMM uses $S$-coordinates in the vertical rather than $z$-coordinates and
+is forced with tidal lateral boundary conditions using a flather boundary condition from the BDY module.
+The AMM configuration uses the GLS (\key{zdfgls}) turbulence scheme,
+the VVL non-linear free surface(\key{vvl}) and time-splitting (\key{dynspg\_ts}).
+In addition to the tidal boundary condition the model may also take open boundary conditions from
+a North Atlantic model.
+Boundaries may be completely omitted by setting \np{ln\_bdy} to false.
+Sample surface fluxes, river forcing and a sample initial restart file are included to test a realistic model run.
+The Baltic boundary is included within the river input file and is specified as a river source.
+Unlike ordinary river points the Baltic inputs also include salinity and temperature data.
 \end{document}

NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_DIA.tex

-                      r10146
+                      r10368
 \label{sec:DIA_io_old}
+The model outputs are of three types: the restart file, the output listing, and
+the diagnostic output file(s).
+The restart file is used internally by the code when the user wants to start the model with
+The model outputs are of three types: the restart file, the output listing, and the diagnostic output file(s).
+The restart file is used internally by the code when the user wants to start the model with
 initial conditions defined by a previous simulation.
 It contains all the information that is necessary in order for there to be no changes in
+the model results (even at the computer precision) between a run performed with several restarts and
+It contains all the information that is necessary in order for there to be no changes in the model results
+(even at the computer precision) between a run performed with several restarts and
 the same run performed in one step.
+It should be noted that this requires that the restart file contains two consecutive time steps for
+all the prognostic variables, and that it is saved in the same binary format as the one used by
+the computer that is to read it (in particular, 32 bits binary IEEE format must not be used for this file).
+The output listing and file(s) are predefined but should be checked and eventually adapted to
+the user's needs.
+It should be noted that this requires that the restart file contains two consecutive time steps for
+all the prognostic variables, and that it is saved in the same binary format as the one used by the computer that
+is to read it (in particular, 32 bits binary IEEE format must not be used for this file).
+The output listing and file(s) are predefined but should be checked and eventually adapted to the user's needs.
 The output listing is stored in the $ocean.output$ file.
 The information is printed from within the code on the logical unit $numout$.
 …
 By default, diagnostic output files are written in NetCDF format.
 Since version 3.2, when defining \key{iomput}, an I/O server has been added which
 provides more flexibility in the choice of the fields to be written as well as
+how the writing work is distributed over the processors in massively parallel computing.
 A complete description of the use of this I/O server is presented in the next section.
 By default, \key{iomput} is not defined, NEMO produces NetCDF with the old IOIPSL library which
 has been kept for compatibility and its easy installation.
 However, the IOIPSL library is quite inefficient on parallel machines and, since version 3.2,
 many diagnostic options have been added presuming the use of \key{iomput}.
 The usefulness of the default IOIPSL-based option is expected to reduce with each new release.
 If \key{iomput} is not defined, output files and content are defined in the \mdl{diawri} module and
 contain mean (or instantaneous if \key{diainstant} is defined) values over a regular period of
+Since version 3.2, when defining \key{iomput}, an I/O server has been added which
+provides more flexibility in the choice of the fields to be written as well as how
+the writing work is distributed over the processors in massively parallel computing.
+A complete description of the use of this I/O server is presented in the next section.
+By default, \key{iomput} is not defined,
+NEMO produces NetCDF with the old IOIPSL library which has been kept for compatibility and its easy installation.
+However, the IOIPSL library is quite inefficient on parallel machines and, since version 3.2,
+many diagnostic options have been added presuming the use of \key{iomput}.
+The usefulness of the default IOIPSL-based option is expected to reduce with each new release.
+If \key{iomput} is not defined, output files and content are defined in the \mdl{diawri} module and
+contain mean (or instantaneous if \key{diainstant} is defined) values over a regular period of
 nn\_write time-steps (namelist parameter).
 …
 \label{sec:DIA_iom}
 Since version 3.2, iomput is the NEMO output interface of choice.
 It has been designed to be simple to use, flexible and efficient.
+Since version 3.2, iomput is the NEMO output interface of choice.
+It has been designed to be simple to use, flexible and efficient.
 The two main purposes of iomput are:
 \begin{enumerate}
+   \item The complete and flexible control of the output files through external XML files adapted by
+         the user from standard templates.
+   \item To achieve high performance and scalable output through the optional distribution of
+         all diagnostic output related tasks to dedicated processes.
+\item
+  The complete and flexible control of the output files through external XML files adapted by
+  the user from standard templates.
+\item
+  To achieve high performance and scalable output through the optional distribution of
+  all diagnostic output related tasks to dedicated processes.
 \end{enumerate}
 …
 \begin{itemize}
+   \item The choice of output frequencies that can be different for each file
+         (including real months and years).
+   \item The choice of file contents; includes complete flexibility over which data are written in
+         which files (the same data can be written in different files).
+   \item The possibility to split output files at a chosen frequency.
+   \item The possibility to extract a vertical or an horizontal subdomain.
+   \item The choice of the temporal operation to perform,
+         e.g.: average, accumulate, instantaneous, min, max and once.
+   \item Control over metadata via a large XML "database" of possible output fields.
+\item
+  The choice of output frequencies that can be different for each file (including real months and years).
+\item
+  The choice of file contents; includes complete flexibility over which data are written in which files
+  (the same data can be written in different files).
+\item
+  The possibility to split output files at a chosen frequency.
+\item
+  The possibility to extract a vertical or an horizontal subdomain.
+\item
+  The choice of the temporal operation to perform, $e.g.$: average, accumulate, instantaneous, min, max and once.
+\item
+  Control over metadata via a large XML "database" of possible output fields.
 \end{itemize}
 In addition, iomput allows the user to add in the code the output of any new variable (scalar, 2D or
 D) in a very easy way.
+In addition, iomput allows the user to add in the code the output of any new variable (scalar, 2D or 3D)
+in a very easy way.
 All details of iomput functionalities are listed in the following subsections.
+Examples of the XML files that control the outputs can be found in:
+\path{NEMOGCM/CONFIG/ORCA2_LIM/EXP00/iodef.xml}, \path{NEMOGCM/CONFIG/SHARED/field_def.xml} and
+\path{NEMOGCM/CONFIG/SHARED/domain_def.xml}. \\
+Examples of the XML files that control the outputs can be found in: \path{NEMOGCM/CONFIG/ORCA2_LIM/EXP00/iodef.xml},
+\path{NEMOGCM/CONFIG/SHARED/field_def.xml} and \path{NEMOGCM/CONFIG/SHARED/domain_def.xml}. \\
 The second functionality targets output performance when running in parallel (\key{mpp\_mpi}).
 Iomput provides the possibility to specify N dedicated I/O processes (in addition to
 the NEMO processes) to collect and write the outputs.
 With an appropriate choice of N by the user, the bottleneck associated with the writing of
+Iomput provides the possibility to specify N dedicated I/O processes (in addition to the NEMO processes)
+to collect and write the outputs.
+With an appropriate choice of N by the user, the bottleneck associated with the writing of
 the output files can be greatly reduced.
 In version 3.6, the iom\_put interface depends on an external code called
 \href{https://forge.ipsl.jussieu.fr/ioserver/browser/XIOS/branchs/xios-1.0}{XIOS-1.0}
+In version 3.6, the iom\_put interface depends on
+an external code called \href{https://forge.ipsl.jussieu.fr/ioserver/browser/XIOS/branchs/xios-1.0}{XIOS-1.0}
 (use of revision 618 or higher is required).
 This new IO server can take advantage of the parallel I/O functionality of NetCDF4 to
+This new IO server can take advantage of the parallel I/O functionality of NetCDF4 to
 create a single output file and therefore to bypass the rebuilding phase.
+Note that writing in parallel into the same NetCDF files requires that
+your NetCDF4 library is linked to an HDF5 library that has been correctly compiled ($i.e.$ with
+the configure option $--$enable-parallel).
+Note that writing in parallel into the same NetCDF files requires that your NetCDF4 library is linked to
+an HDF5 library that has been correctly compiled ($i.e.$ with the configure option $--$enable-parallel).
 Note that the files created by iomput through XIOS are incompatible with NetCDF3.
+All post-processsing and visualization tools must therefore be compatible with
+NetCDF4 and not only NetCDF3.
+Even if not using the parallel I/O functionality of NetCDF4, using N dedicated I/O servers,
+where N is typically much less than the number of NEMO processors,
+will reduce the number of output files created.
+This can greatly reduce the post-processing burden usually associated with using
+large numbers of NEMO processors.
+Note that for smaller configurations, the rebuilding phase can be avoided, even without
+a parallel-enabled NetCDF4 library, simply by employing only one dedicated I/O server.
+All post-processsing and visualization tools must therefore be compatible with NetCDF4 and not only NetCDF3.
+Even if not using the parallel I/O functionality of NetCDF4, using N dedicated I/O servers,
+where N is typically much less than the number of NEMO processors, will reduce the number of output files created.
+This can greatly reduce the post-processing burden usually associated with using large numbers of NEMO processors.
+Note that for smaller configurations, the rebuilding phase can be avoided,
+even without a parallel-enabled NetCDF4 library, simply by employing only one dedicated I/O server.
 \subsection{XIOS: XML Inputs-Outputs Server}
 …
 \subsubsection{Attached or detached mode?}
 Iomput is based on \href{http://forge.ipsl.jussieu.fr/ioserver/wiki}{XIOS},
+Iomput is based on \href{http://forge.ipsl.jussieu.fr/ioserver/wiki}{XIOS},
 the io\_server developed by Yann Meurdesoif from IPSL.
 The behaviour of the I/O subsystem is controlled by settings in the external XML files listed above.
 …
 \xmlline|<variable id="using_server" type="bool"></variable>|
 The {\tt using\_server} setting determines whether or not the server will be used in
+\textit{attached mode} (as a library) [{\tt> false <}] or in \textit{detached mode} (as
 an external executable on N additional, dedicated cpus) [{\tt > true <}].
+The {\tt using\_server} setting determines whether or not the server will be used in \textit{attached mode}
+(as a library) [{\tt> false <}] or in \textit{detached mode}
+(as an external executable on N additional, dedicated cpus) [{\tt > true <}].
 The \textit{attached mode} is simpler to use but much less efficient for massively parallel applications.
 The type of each file can be either ''multiple\_file'' or ''one\_file''.
 In \textit{attached mode} and if the type of file is ''multiple\_file'',
+In \textit{attached mode} and if the type of file is ''multiple\_file'',
 then each NEMO process will also act as an IO server and produce its own set of output files.
 Superficially, this emulates the standard behaviour in previous versions.
 However, the subdomain written out by each process does not correspond to
+However, the subdomain written out by each process does not correspond to
 the \forcode{jpi x jpj x jpk} domain actually computed by the process (although it may if \forcode{jpni=1}).
 Instead each process will have collected and written out a number of complete longitudinal strips.
 If the ''one\_file'' option is chosen then all processes will collect their longitudinal strips and
+If the ''one\_file'' option is chosen then all processes will collect their longitudinal strips and
 write (in parallel) to a single output file.
 In \textit{detached mode} and if the type of file is ''multiple\_file'', then
+each stand-alone XIOS process will collect data for a range of complete longitudinal strips and
+In \textit{detached mode} and if the type of file is ''multiple\_file'',
+then each stand-alone XIOS process will collect data for a range of complete longitudinal strips and
 write to its own set of output files.
 If the ''one\_file'' option is chosen then all XIOS processes will collect their longitudinal strips and
+If the ''one\_file'' option is chosen then all XIOS processes will collect their longitudinal strips and
 write (in parallel) to a single output file.
 Note running in detached mode requires launching a Multiple Process Multiple Data (MPMD) parallel job.
+The following subsection provides a typical example but the syntax will vary in
+different MPP environments.
+The following subsection provides a typical example but the syntax will vary in different MPP environments.
 \subsubsection{Number of cpu used by XIOS in detached mode}
 …
 The number of cores dedicated to XIOS should be from \texttildelow1/10 to \texttildelow1/50 of the number of
 cores dedicated to NEMO.
 Some manufacturers suggest using O($\sqrt{N}$) dedicated IO processors for N processors but
+Some manufacturers suggest using O($\sqrt{N}$) dedicated IO processors for N processors but
 this is a general recommendation and not specific to NEMO.
 It is difficult to provide precise recommendations because the optimal choice will depend on
+It is difficult to provide precise recommendations because the optimal choice will depend on
 the particular hardware properties of the target system
 (parallel filesystem performance, available memory, memory bandwidth etc.) and
 the volume and frequency of data to be created.
+(parallel filesystem performance, available memory, memory bandwidth etc.)
+and the volume and frequency of data to be created.
 Here is an example of 2 cpus for the io\_server and 62 cpu for nemo using mpirun:
 \cmd|mpirun -np 62 ./nemo.exe : -np 2 ./xios_server.exe|
 …
 \subsubsection{Control of XIOS: the context in iodef.xml}
+As well as the {\tt using\_server} flag, other controls on the use of XIOS are set in
+the XIOS context in iodef.xml.
+As well as the {\tt using\_server} flag, other controls on the use of XIOS are set in the XIOS context in iodef.xml.
 See the XML basics section below for more details on XML syntax and rules.
 …
 \subsubsection{Installation}
+As mentioned, XIOS is supported separately and must be downloaded and compiled before
+it can be used with NEMO.
+See the installation guide on the \href{http://forge.ipsl.jussieu.fr/ioserver/wiki}{XIOS} wiki for
+help and guidance.
+As mentioned, XIOS is supported separately and must be downloaded and compiled before it can be used with NEMO.
+See the installation guide on the \href{http://forge.ipsl.jussieu.fr/ioserver/wiki}{XIOS} wiki for help and guidance.
 NEMO will need to link to the compiled XIOS library.
+The
+\href{https://forge.ipsl.jussieu.fr/nemo/wiki/Users/ModelInterfacing/InputsOutputs#Inputs-OutputsusingXIOS}
+{XIOS with NEMO}
+guide provides an example illustration of how this can be achieved.
+The \href{https://forge.ipsl.jussieu.fr/nemo/wiki/Users/ModelInterfacing/InputsOutputs#Inputs-OutputsusingXIOS}
+{XIOS with NEMO} guide provides an example illustration of how this can be achieved.
 \subsubsection{Add your own outputs}
 It is very easy to add your own outputs with iomput.
 Many standard fields and diagnostics are already prepared ($i.e.$, steps 1 to 3 below have been done) and
+Many standard fields and diagnostics are already prepared ($i.e.$, steps 1 to 3 below have been done) and
 simply need to be activated by including the required output in a file definition in iodef.xml (step 4).
 To add new output variables, all 4 of the following steps must be taken.
 \begin{enumerate}
+   \item[1.] in NEMO code, add a \forcode{CALL iom\_put( 'identifier', array )} where you want to
+               output a 2D or 3D array.
+   \item[2.] If necessary, add \forcode{USE iom ! I/O manager library} to the list of used modules in
+               the upper part of your module.
+   \item[3.] in the field\_def.xml file, add the definition of your variable using
+               the same identifier you used in the f90 code (see subsequent sections for
+               a details of the XML syntax and rules).
+For example:
+\item[1.]
+  in NEMO code, add a \forcode{CALL iom\_put( 'identifier', array )} where you want to output a 2D or 3D array.
+\item[2.]
+  If necessary, add \forcode{USE iom ! I/O manager library} to the list of used modules in
+  the upper part of your module.
+\item[3.]
+  in the field\_def.xml file, add the definition of your variable using the same identifier you used in the f90 code
+  (see subsequent sections for a details of the XML syntax and rules).
+  For example:
 \begin{xmllines}
 …
 \end{xmllines}
 Note your definition must be added to the field\_group whose reference grid is consistent with
 the size of the array passed to iomput.
 The grid\_ref attribute refers to definitions set in iodef.xml which, in turn, reference grids and
+axes either defined in the code (iom\_set\_domain\_attr and iom\_set\_axis\_attr in \mdl{iom}) or
 defined in the domain\_def.xml file.
+Note your definition must be added to the field\_group whose reference grid is consistent with the size of
+the array passed to iomput.
+The grid\_ref attribute refers to definitions set in iodef.xml which, in turn,
+reference grids and axes either defined in the code
+(iom\_set\_domain\_attr and iom\_set\_axis\_attr in \mdl{iom}) or defined in the domain\_def.xml file.
 $e.g.$:
 …
 \end{xmllines}
 Note, if your array is computed within the surface module each \np{nn\_fsbc} time\_step,
+Note, if your array is computed within the surface module each \np{nn\_fsbc} time\_step,
 add the field definition within the field\_group defined with the id "SBC":
+\xmlcode{<field_group id="SBC" ...>} which has been defined with the correct frequency of
+operations (iom\_set\_field\_attr in \mdl{iom})
+   \item[4.] add your field in one of the output files defined in iodef.xml
+(again see subsequent sections for syntax and rules)
+\xmlcode{<field_group id="SBC" ...>} which has been defined with the correct frequency of operations
+(iom\_set\_field\_attr in \mdl{iom})
+\item[4.]
+  add your field in one of the output files defined in iodef.xml
+  (again see subsequent sections for syntax and rules)
 \begin{xmllines}
 …
 XML tags begin with the less-than character ("$<$") and end with the greater-than character ("$>$").
+You use tags to mark the start and end of elements, which are the logical units of information in
+an XML document.
+In addition to marking the beginning of an element, XML start tags also provide a place to
+specify attributes.
+An attribute specifies a single property for an element, using a name/value pair, for example:
+You use tags to mark the start and end of elements, which are the logical units of information in an XML document.
+In addition to marking the beginning of an element, XML start tags also provide a place to specify attributes.
+An attribute specifies a single property for an element, using a name/value pair, for example:
 \xmlcode{<a b="x" c="y" d="z"> ... </a>}.
 See \href{http://www.xmlnews.org/docs/xml-basics.html}{here} for more details.
 …
 \subsubsection{Structure of the XML file used in NEMO}
 The XML file used in XIOS is structured by 7 families of tags:
+The XML file used in XIOS is structured by 7 families of tags:
 context, axis, domain, grid, field, file and variable.
 Each tag family has hierarchy of three flavors (except for context):
 …
 Each element may have several attributes.
+Some attributes are mandatory, other are optional but have a default value and
+other are completely optional.
+Some attributes are mandatory, other are optional but have a default value and other are completely optional.
 Id is a special attribute used to identify an element or a group of elements.
 It must be unique for a kind of element.
 It is optional, but no reference to the corresponding element can be done if it is not defined.
 The XML file is split into context tags that are used to isolate IO definition from different codes or
 different parts of a code.
+The XML file is split into context tags that are used to isolate IO definition from
+different codes or different parts of a code.
 No interference is possible between 2 different contexts.
 Each context has its own calendar and an associated timestep.
 …
 \noindent In NEMO, by default, the field and domain definition is done in 2 separate files:
 \path{NEMOGCM/CONFIG/SHARED/field_def.xml} and \path{NEMOGCM/CONFIG/SHARED/domain_def.xml}
 that are included in the main iodef.xml file through the following commands:
+\path{NEMOGCM/CONFIG/SHARED/field_def.xml} and \path{NEMOGCM/CONFIG/SHARED/domain_def.xml} that
+are included in the main iodef.xml file through the following commands:
 \begin{xmllines}
 <field_definition src="./field_def.xml" />
 …
 XML extensively uses the concept of inheritance.
+XML has a tree based structure with a parent-child oriented relation:
+all children inherit attributes from parent, but an attribute defined in a child replace
+the inherited attribute value.
+Note that the special attribute ''id'' is never inherited. \\ \\
+XML has a tree based structure with a parent-child oriented relation: all children inherit attributes from parent,
+but an attribute defined in a child replace the inherited attribute value.
+Note that the special attribute ''id'' is never inherited.
+\\
+\\
 example 1: Direct inheritance.
 …
 \end{xmllines}
 The field ''sst'' which is part (or a child) of the field\_definition will inherit the value ''average''
 of the attribute ''operation'' from its parent.
+The field ''sst'' which is part (or a child) of the field\_definition will inherit the value ''average'' of
+the attribute ''operation'' from its parent.
 Note that a child can overwrite the attribute definition inherited from its parents.
+In the example above, the field ''sss'' will for example output instantaneous values instead of
+average values. \\ \\
+In the example above, the field ''sss'' will for example output instantaneous values instead of average values.
+\\
+\\
 example 2: Inheritance by reference.
 …
 Groups can be used for 2 purposes.
 Firstly, the group can be used to define common attributes to be shared by the elements of
+Firstly, the group can be used to define common attributes to be shared by the elements of
 the group through inheritance.
 In the following example, we define a group of field that will share a common grid ''grid\_T\_2D''.
+Note that for the field ''toce'', we overwrite the grid definition inherited from the group by
+''grid\_T\_3D''.
+Note that for the field ''toce'', we overwrite the grid definition inherited from the group by ''grid\_T\_3D''.
 \begin{xmllines}
 …
 \subsection{Detailed functionalities}
 The file \path{NEMOGCM/CONFIG/ORCA2_LIM/iodef_demo.xml} provides several examples of the use of
+The file \path{NEMOGCM/CONFIG/ORCA2_LIM/iodef_demo.xml} provides several examples of the use of
 the new functionalities offered by the XML interface of XIOS.
 \subsubsection{Define horizontal subdomains}
 Horizontal subdomains are defined through the attributs zoom\_ibegin, zoom\_jbegin, zoom\_ni, zoom\_nj of
+Horizontal subdomains are defined through the attributs zoom\_ibegin, zoom\_jbegin, zoom\_ni, zoom\_nj of
 the tag family domain.
 It must therefore be done in the domain part of the XML file.
 …
 \end{xmllines}
+The use of this subdomain is done through the redefinition of the attribute domain\_ref of
+the tag family field.
+The use of this subdomain is done through the redefinition of the attribute domain\_ref of the tag family field.
 For example:
 …
 Moorings are seen as an extrem case corresponding to a 1 by 1 subdomain.
 The Equatorial section, the TAO, RAMA and PIRATA moorings are alredy registered in the code and
+The Equatorial section, the TAO, RAMA and PIRATA moorings are already registered in the code and
 can therefore be outputted without taking care of their (i,j) position in the grid.
 These predefined domains can be activated by the use of specific domain\_ref:
 ''EqT'', ''EqU'' or ''EqW'' for the equatorial sections and the mooring position for
 TAO, RAMA and PIRATA followed by ''T'' (for example: ''8s137eT'', ''1.5s80.5eT'' ...)
+''EqT'', ''EqU'' or ''EqW'' for the equatorial sections and
+the mooring position for TAO, RAMA and PIRATA followed by ''T'' (for example: ''8s137eT'', ''1.5s80.5eT'' ...)
 \begin{xmllines}
 …
 \subsubsection{Define vertical zooms}
 Vertical zooms are defined through the attributs zoom\_begin and zoom\_end of the tag family axis.
 It must therefore be done in the axis part of the XML file.
+Vertical zooms are defined through the attributs zoom\_begin and zoom\_end of the tag family axis.
+It must therefore be done in the axis part of the XML file.
 For example, in \path{NEMOGCM/CONFIG/ORCA2_LIM/iodef_demo.xml}, we provide the following example:
 …
 \end{xmllines}
+The use of this vertical zoom is done through the redefinition of the attribute axis\_ref of
+the tag family field.
+The use of this vertical zoom is done through the redefinition of the attribute axis\_ref of the tag family field.
 For example:
 …
 \end{xmllines}
 However it is often very convienent to define the file name with the name of the experiment,
 the output file frequency and the date of the beginning and the end of the simulation
+However it is often very convienent to define the file name with the name of the experiment,
+the output file frequency and the date of the beginning and the end of the simulation
 (which are informations stored either in the namelist or in the XML file).
 To do so, we added the following rule: if the id of the tag file is ''fileN'' (where N = 1 to 999 on
+to 3 digits) or one of the predefined sections or moorings (see next subsection),
+the following part of the name and the name\_suffix (that can be inherited) will be automatically
 replaced by:
+To do so, we added the following rule:
+if the id of the tag file is ''fileN'' (where N = 1 to 999 on 1 to 3 digits) or
+one of the predefined sections or moorings (see next subsection),
+the following part of the name and the name\_suffix (that can be inherited) will be automatically replaced by:
 \begin{table} \scriptsize
 …
 \end{forlines}
+\noindent will give the following file name radical:
+\ifile{myfile\_ORCA2\_19891231\_freq1d}
+\noindent will give the following file name radical: \ifile{myfile\_ORCA2\_19891231\_freq1d}
 \subsubsection{Other controls of the XML attributes from NEMO}
 The values of some attributes are defined by subroutine calls within NEMO
+(calls to iom\_set\_domain\_attr, iom\_set\_axis\_attr and iom\_set\_field\_attr in \mdl{iom}).
+Any definition given in the XML file will be overwritten.
+By convention, these attributes are defined to ''auto'' (for string) or ''0000'' (for integer) in
+the XML file (but this is not necessary). \\
+Here is the list of these attributes: \\
+(calls to iom\_set\_domain\_attr, iom\_set\_axis\_attr and iom\_set\_field\_attr in \mdl{iom}).
+Any definition given in the XML file will be overwritten.
+By convention, these attributes are defined to ''auto'' (for string) or ''0000'' (for integer) in the XML file
+(but this is not necessary).
+\\
+Here is the list of these attributes:
+\\
 \begin{table} \scriptsize
 …
 \begin{enumerate}
 \item Simple computation: directly define the computation when refering to the variable in
 the file definition.
+\item
+  Simple computation: directly define the computation when refering to the variable in the file definition.
 \begin{xmllines}
 …
 \end{xmllines}
+\item Simple computation: define a new variable and use it in the file definition.
+\item
+  Simple computation: define a new variable and use it in the file definition.
 in field\_definition:
 …
 \end{xmllines}
 Note that in this case, the following syntaxe \xmlcode{<field field_ref="sst2" />} is not working as
+Note that in this case, the following syntaxe \xmlcode{<field field_ref="sst2" />} is not working as
 sst2 won't be evaluated.
+\item Change of variable precision:
+\item
+  Change of variable precision:
 \begin{xmllines}
 …
 Note that, then the code is crashing, writting real4 variables forces a numerical convection from
+real8 to real4 which will create an internal error in NetCDF and will avoid the creation of
+the output files.
+Forcing double precision outputs with prec="8" (for example in the field\_definition) will
+avoid this problem.
+\item add user defined attributes:
+real8 to real4 which will create an internal error in NetCDF and will avoid the creation of the output files.
+Forcing double precision outputs with prec="8" (for example in the field\_definition) will avoid this problem.
+\item
+  add user defined attributes:
 \begin{xmllines}
 …
 \end{xmllines}
+\item use of the ``@'' function: example 1, weighted temporal average
+\item
+  use of the ``@'' function: example 1, weighted temporal average
  - define a new variable in field\_definition
 …
 The freq\_op="5d" attribute is used to define the operation frequency of the ``@'' function: here 5 day.
 The temporal operation done by the ``@'' is the one defined in the field definition:
+The temporal operation done by the ``@'' is the one defined in the field definition:
 here we use the default, average.
 So, in the above case, @toce\_e3t will do the 5-day mean of toce*e3t.
 Operation="instant" refers to the temporal operation to be performed on the field''@toce\_e3t / @e3t'':
 here the temporal average is alreday done by the ``@'' function so we just use instant to do the ratio of
+Operation="instant" refers to the temporal operation to be performed on the field''@toce\_e3t / @e3t'':
+here the temporal average is alreday done by the ``@'' function so we just use instant to do the ratio of
 the 2 mean values.
 field\_ref="toce" means that attributes not explicitely defined, are inherited from toce field.
 Note that in this case, freq\_op must be equal to the file output\_freq.
+\item use of the ``@'' function: example 2, monthly SSH standard deviation
+\item
+  use of the ``@'' function: example 2, monthly SSH standard deviation
  - define a new variable in field\_definition
 …
 The freq\_op="1m" attribute is used to define the operation frequency of the ``@'' function: here 1 month.
 The temporal operation done by the ``@'' is the one defined in the field definition:
+The temporal operation done by the ``@'' is the one defined in the field definition:
 here we use the default, average.
 So, in the above case, @ssh2 will do the monthly mean of ssh*ssh.
+Operation="instant" refers to the temporal operation to be performed on
+the field ''sqrt( @ssh2 - @ssh * @ssh )'':
+Operation="instant" refers to the temporal operation to be performed on the field ''sqrt( @ssh2 - @ssh * @ssh )'':
 here the temporal average is alreday done by the ``@'' function so we just use instant.
 field\_ref="ssh" means that attributes not explicitely defined, are inherited from ssh field.
 Note that in this case, freq\_op must be equal to the file output\_freq.
+\item use of the ``@'' function: example 3, monthly average of SST diurnal cycle
+\item
+  use of the ``@'' function: example 3, monthly average of SST diurnal cycle
  - define 2 new variables in field\_definition
 …
 The freq\_op="1d" attribute is used to define the operation frequency of the ``@'' function: here 1 day.
 The temporal operation done by the ``@'' is the one defined in the field definition: here maximum for
 sstmax and minimum for sstmin.
+The temporal operation done by the ``@'' is the one defined in the field definition:
+here maximum for sstmax and minimum for sstmin.
 So, in the above case, @sstmax will do the daily max and @sstmin the daily min.
 Operation="average" refers to the temporal operation to be performed on the field ``@sstmax - @sstmin'':
+Operation="average" refers to the temporal operation to be performed on the field ``@sstmax - @sstmin'':
 here monthly mean (of daily max - daily min of the sst).
 field\_ref="sst" means that attributes not explicitely defined, are inherited from sst field.
 …
 Output from the XIOS-1.0 IO server is compliant with
+\href{http://cfconventions.org/Data/cf-conventions/cf-conventions-1.5/build/cf-conventions.html}{version 1.5}
+of the CF metadata standard.
+Therefore while a user may wish to add their own metadata to the output files (as demonstrated in
+example 4 of section \autoref{subsec:IOM_xmlref}) the metadata should, for the most part, comply with
+the CF-1.5 standard.
+Some metadata that may significantly increase the file size (horizontal cell areas and
+vertices) are controlled by the namelist parameter \np{ln\_cfmeta} in the \ngn{namrun} namelist.
+\href{http://cfconventions.org/Data/cf-conventions/cf-conventions-1.5/build/cf-conventions.html}{version 1.5} of
+the CF metadata standard.
+Therefore while a user may wish to add their own metadata to the output files (as demonstrated in example 4 of
+section \autoref{subsec:IOM_xmlref}) the metadata should, for the most part, comply with the CF-1.5 standard.
+Some metadata that may significantly increase the file size (horizontal cell areas and vertices) are controlled by
+the namelist parameter \np{ln\_cfmeta} in the \ngn{namrun} namelist.
 This must be set to true if these metadata are to be included in the output files.
 …
 Since version 3.3, support for NetCDF4 chunking and (loss-less) compression has been included.
 These options build on the standard NetCDF output and allow the user control over the size of
 the chunks via namelist settings.
+These options build on the standard NetCDF output and allow the user control over the size of the chunks via
+namelist settings.
 Chunking and compression can lead to significant reductions in file sizes for a small runtime overhead.
+For a fuller discussion on chunking and other performance issues the reader is referred to
+the NetCDF4 documentation found
+\href{http://www.unidata.ucar.edu/software/netcdf/docs/netcdf.html#Chunking}{here}.
+The new features are only available when the code has been linked with a NetCDF4 library
+(version 4.1 onwards, recommended) which has been built with HDF5 support
+(version 1.8.4 onwards, recommended).
+Datasets created with chunking and compression are not backwards compatible with
+NetCDF3 "classic" format but most analysis codes can be relinked simply with the new libraries and
+will then read both NetCDF3 and NetCDF4 files.
+NEMO executables linked with NetCDF4 libraries can be made to produce NetCDF3 files by
+For a fuller discussion on chunking and other performance issues the reader is referred to
+the NetCDF4 documentation found \href{http://www.unidata.ucar.edu/software/netcdf/docs/netcdf.html#Chunking}{here}.
+The new features are only available when the code has been linked with a NetCDF4 library
+(version 4.1 onwards, recommended) which has been built with HDF5 support (version 1.8.4 onwards, recommended).
+Datasets created with chunking and compression are not backwards compatible with NetCDF3 "classic" format but
+most analysis codes can be relinked simply with the new libraries and will then read both NetCDF3 and NetCDF4 files.
+NEMO executables linked with NetCDF4 libraries can be made to produce NetCDF3 files by
 setting the \np{ln\_nc4zip} logical to false in the \textit{namnc4} namelist:
 …
 If \key{netcdf4} has not been defined, these namelist parameters are not read.
+In this case, \np{ln\_nc4zip} is set false and dummy routines for
+a few NetCDF4-specific functions are defined.
+These functions will not be used but need to be included so that compilation is possible with
+NetCDF3 libraries.
+When using NetCDF4 libraries, \key{netcdf4} should be defined even if the intention is to
+In this case, \np{ln\_nc4zip} is set false and dummy routines for a few NetCDF4-specific functions are defined.
+These functions will not be used but need to be included so that compilation is possible with NetCDF3 libraries.
+When using NetCDF4 libraries, \key{netcdf4} should be defined even if the intention is to
 create only NetCDF3-compatible files.
+This is necessary to avoid duplication between the dummy routines and
+the actual routines present in the library.
+This is necessary to avoid duplication between the dummy routines and the actual routines present in the library.
 Most compilers will fail at compile time when faced with such duplication.
 Thus when linking with NetCDF4 libraries the user must define \key{netcdf4} and
+Thus when linking with NetCDF4 libraries the user must define \key{netcdf4} and
 control the type of NetCDF file produced via the namelist parameter.
 Chunking and compression is applied only to 4D fields and there is no advantage in
+chunking across more than one time dimension since previously written chunks would have to
 be read back and decompressed before being added to.
+Chunking and compression is applied only to 4D fields and
+there is no advantage in chunking across more than one time dimension since
+previously written chunks would have to be read back and decompressed before being added to.
 Therefore, user control over chunk sizes is provided only for the three space dimensions.
 The user sets an approximate number of chunks along each spatial axis.
 The actual size of the chunks will depend on global domain size for mono-processors or,
 more likely, the local processor domain size for distributed processing.
 The derived values are subject to practical minimum values (to avoid wastefully small chunk sizes) and
+The actual size of the chunks will depend on global domain size for mono-processors or, more likely,
+the local processor domain size for distributed processing.
+The derived values are subject to practical minimum values (to avoid wastefully small chunk sizes) and
 cannot be greater than the domain size in any dimension.
 The algorithm used is:
 …
 \end{forlines}
 \noindent for a standard ORCA2\_LIM configuration gives chunksizes of {\small\tt 46x38x1} respectively in
+\noindent for a standard ORCA2\_LIM configuration gives chunksizes of {\small\tt 46x38x1} respectively in
 the mono-processor case (i.e. global domain of {\small\tt 182x149x31}).
 An illustration of the potential space savings that NetCDF4 chunking and compression provides is given in
 table \autoref{tab:NC4} which compares the results of two short runs of
 the ORCA2\_LIM reference configuration with a 4x2 mpi partitioning.
 Note the variation in the compression ratio achieved which reflects chiefly the dry to
 wet volume ratio of each processing region.
+table \autoref{tab:NC4} which compares the results of two short runs of the ORCA2\_LIM reference configuration with
+a 4x2 mpi partitioning.
+Note the variation in the compression ratio achieved which reflects chiefly the dry to wet volume ratio of
+each processing region.
 %------------------------------------------TABLE----------------------------------------------------
 …
 %----------------------------------------------------------------------------------------------------
 When \key{iomput} is activated with \key{netcdf4} chunking and compression parameters for
+fields produced via \np{iom\_put} calls are set via an equivalent and identically named namelist to
 \textit{namnc4} in \np{xmlio\_server.def}.
 Typically this namelist serves the mean files whilst the \ngn{ namnc4} in
 the main namelist file continues to serve the restart files.
 This duplication is unfortunate but appropriate since, if using io\_servers, the domain sizes of
 the individual files produced by the io\_server processes may be different to those produced by
+When \key{iomput} is activated with \key{netcdf4} chunking and compression parameters for fields produced via
+\np{iom\_put} calls are set via an equivalent and identically named namelist to \textit{namnc4} in
+\np{xmlio\_server.def}.
+Typically this namelist serves the mean files whilst the \ngn{ namnc4} in the main namelist file continues to
+serve the restart files.
+This duplication is unfortunate but appropriate since, if using io\_servers, the domain sizes of
+the individual files produced by the io\_server processes may be different to those produced by
 the invidual processing regions and different chunking choices may be desired.
 …
 %-------------------------------------------------------------------------------------------------------------
 Each trend of the dynamics and/or temperature and salinity time evolution equations can be send to
 \mdl{trddyn} and/or \mdl{trdtra} modules (see TRD directory) just after their computation ($i.e.$ at
 the end of each $dyn\cdots.F90$ and/or $tra\cdots.F90$ routines).
+Each trend of the dynamics and/or temperature and salinity time evolution equations can be send to
+\mdl{trddyn} and/or \mdl{trdtra} modules (see TRD directory) just after their computation
+($i.e.$ at the end of each $dyn\cdots.F90$ and/or $tra\cdots.F90$ routines).
 This capability is controlled by options offered in \ngn{namtrd} namelist.
 Note that the output are done with xIOS, and therefore the \key{IOM} is required.
 …
 \begin{description}
+   \item[\np{ln\_glo\_trd}]: at each \np{nn\_trd} time-step a check of the basin averaged properties of
+                              the momentum and tracer equations is performed.
+                              This also includes a check of $T^2$, $S^2$, $\tfrac{1}{2} (u^2+v2)$, and
+                              potential energy time evolution equations properties;
+   \item[\np{ln\_dyn\_trd}]: each 3D trend of the evolution of the two momentum components is output;
+   \item[\np{ln\_dyn\_mxl}]: each 3D trend of the evolution of the two momentum components averaged over
+                              the mixed layer is output;
+   \item[\np{ln\_vor\_trd}]: a vertical summation of the moment tendencies is performed, then
+                              the curl is computed to obtain the barotropic vorticity tendencies which
+                              are output;
+   \item[\np{ln\_KE\_trd}] : each 3D trend of the Kinetic Energy equation is output ;
+   \item[\np{ln\_tra\_trd}]: each 3D trend of the evolution of temperature and salinity is output ;
+   \item[\np{ln\_tra\_mxl}]: each 2D trend of the evolution of temperature and salinity averaged over
+                              the mixed layer is output;
+\item[\np{ln\_glo\_trd}]:
+  at each \np{nn\_trd} time-step a check of the basin averaged properties of
+  the momentum and tracer equations is performed.
+  This also includes a check of $T^2$, $S^2$, $\tfrac{1}{2} (u^2+v2)$,
+  and potential energy time evolution equations properties;
+\item[\np{ln\_dyn\_trd}]:
+  each 3D trend of the evolution of the two momentum components is output;
+\item[\np{ln\_dyn\_mxl}]:
+  each 3D trend of the evolution of the two momentum components averaged over the mixed layer is output;
+\item[\np{ln\_vor\_trd}]:
+  a vertical summation of the moment tendencies is performed,
+  then the curl is computed to obtain the barotropic vorticity tendencies which are output;
+\item[\np{ln\_KE\_trd}] :
+  each 3D trend of the Kinetic Energy equation is output;
+\item[\np{ln\_tra\_trd}]:
+  each 3D trend of the evolution of temperature and salinity is output;
+\item[\np{ln\_tra\_mxl}]:
+  each 2D trend of the evolution of temperature and salinity averaged over the mixed layer is output;
 \end{description}
 …
 \textbf{Note that} in the current version (v3.6), many changes has been introduced but not fully tested.
 In particular, options associated with \np{ln\_dyn\_mxl}, \np{ln\_vor\_trd}, and \np{ln\_tra\_mxl} are
 not working, and none of the options have been tested with variable volume ($i.e.$ \key{vvl} defined).
+In particular, options associated with \np{ln\_dyn\_mxl}, \np{ln\_vor\_trd}, and \np{ln\_tra\_mxl} are not working,
+and none of the options have been tested with variable volume ($i.e.$ \key{vvl} defined).
 % -------------------------------------------------------------------------------------------------------------
 …
 %--------------------------------------------------------------------------------------------------------------
+The on-line computation of floats advected either by the three dimensional velocity field or
+constraint to remain at a given depth ($w = 0$ in the computation) have been introduced in
+the system during the CLIPPER project.
+The on-line computation of floats advected either by the three dimensional velocity field or constraint to
+remain at a given depth ($w = 0$ in the computation) have been introduced in the system during the CLIPPER project.
 Options are defined by \ngn{namflo} namelis variables.
 The algorithm used is based either on the work of \cite{Blanke_Raynaud_JPO97} (default option), or
 on a $4^th$ Runge-Hutta algorithm (\np{ln\_flork4}\forcode{ = .true.}).
 Note that the \cite{Blanke_Raynaud_JPO97} algorithm have the advantage of providing trajectories which
+The algorithm used is based either on the work of \cite{Blanke_Raynaud_JPO97} (default option),
+or on a $4^th$ Runge-Hutta algorithm (\np{ln\_flork4}\forcode{ = .true.}).
+Note that the \cite{Blanke_Raynaud_JPO97} algorithm have the advantage of providing trajectories which
 are consistent with the numeric of the code, so that the trajectories never intercept the bathymetry.
 \subsubsection{Input data: initial coordinates}
 Initial coordinates can be given with Ariane Tools convention (IJK coordinates,
 (\np{ln\_ariane}\forcode{ = .true.}) ) or with longitude and latitude.
+Initial coordinates can be given with Ariane Tools convention
+(IJK coordinates, (\np{ln\_ariane}\forcode{ = .true.}) ) or with longitude and latitude.
 In case of Ariane convention, input filename is \np{init\_float\_ariane}.
 …
 \np{jpnfl} is the total number of floats during the run.
 When initial positions are read in a restart file (\np{ln\_rstflo}\forcode{ = .true.} ),
+When initial positions are read in a restart file (\np{ln\_rstflo}\forcode{ = .true.} ),
 \np{jpnflnewflo} can be added in the initialization file.
 \subsubsection{Output data}
 \np{nn\_writefl} is the frequency of writing in float output file and \np{nn\_stockfl} is
 the frequency of creation of the float restart file.
+\np{nn\_writefl} is the frequency of writing in float output file and \np{nn\_stockfl} is the frequency of
+creation of the float restart file.
 Output data can be written in ascii files (\np{ln\_flo\_ascii}\forcode{ = .true.}).
 …
 There are 2 possibilities:
  - if (\key{iomput}) is used, outputs are selected in  iodef.xml.
    Here it is an example of specification to put in files description section:
+- if (\key{iomput}) is used, outputs are selected in  iodef.xml.
+Here it is an example of specification to put in files description section:
 \begin{xmllines}
 …
  -  if (\key{iomput}) is not used, a file called \ifile{trajec\_float} will be created by IOIPSL library.
+See also \href{http://stockage.univ-brest.fr/~grima/Ariane/}{here} the web site describing
 the off-line use of this marvellous diagnostic tool.
+ See also \href{http://stockage.univ-brest.fr/~grima/Ariane/}{here} the web site describing the off-line use of
+ this marvellous diagnostic tool.
 % -------------------------------------------------------------------------------------------------------------
 …
 This on-line Harmonic analysis is actived with \key{diaharm}.
 Some parameters are available in namelist \ngn{namdia\_harm} :
+Some parameters are available in namelist \ngn{namdia\_harm}:
  - \np{nit000\_han} is the first time step used for harmonic analysis
 …
  - \np{tname}       is an array with names of tidal constituents to analyse
 \np{nit000\_han} and \np{nitend\_han} must be between \np{nit000} and \np{nitend} of the simulation.
 The restart capability is not implemented.
 The Harmonic analysis solve the following equation:
+ \np{nit000\_han} and \np{nitend\_han} must be between \np{nit000} and \np{nitend} of the simulation.
+ The restart capability is not implemented.
+ The Harmonic analysis solve the following equation:
 \[h_{i} - A_{0} + \sum^{nb\_ana}_{j=1}[A_{j}cos(\nu_{j}t_{j}-\phi_{j})] = e_{i}\]
+With $A_{j}$, $\nu_{j}$, $\phi_{j}$, the amplitude, frequency and phase for each wave and
+$e_{i}$ the error.
+With $A_{j}$, $\nu_{j}$, $\phi_{j}$, the amplitude, frequency and phase for each wave and $e_{i}$ the error.
 $h_{i}$ is the sea level for the time $t_{i}$ and $A_{0}$ is the mean sea level. \\
 We can rewrite this equation:
 …
 %-------------------------------------------------------------------------------------------------------------
 A module is available to compute the transport of volume, heat and salt through sections.
+A module is available to compute the transport of volume, heat and salt through sections.
 This diagnostic is actived with \key{diadct}.
 Each section is defined by the coordinates of its 2 extremities.
+The pathways between them are contructed using tools which can be found in
+\texttt{NEMOGCM/TOOLS/SECTIONS\_DIADCT}
+and are written in a binary file
+\texttt{section\_ijglobal.diadct\_ORCA2\_LIM}
+which is later read in by NEMO to compute on-line transports.
+The pathways between them are contructed using tools which can be found in \texttt{NEMOGCM/TOOLS/SECTIONS\_DIADCT}
+and are written in a binary file \texttt{section\_ijglobal.diadct\_ORCA2\_LIM} which is later read in by
+NEMO to compute on-line transports.
 The on-line transports module creates three output ascii files:
 …
 - \texttt{salt\_transport}   for   salt transports (unit: $10^{9}Kg s^{-1}$) \\
 Namelist variables in \ngn{namdct} control how frequently the flows are summed and the time scales over
 which they are averaged, as well as the level of output for debugging:
+Namelist variables in \ngn{namdct} control how frequently the flows are summed and the time scales over which
+they are averaged, as well as the level of output for debugging:
 \np{nn\_dct}   : frequency of instantaneous transports computing
 \np{nn\_dctwri}: frequency of writing ( mean of instantaneous transports )
 …
 \subsubsection{Creating a binary file containing the pathway of each section}
 In \texttt{NEMOGCM/TOOLS/SECTIONS\_DIADCT/run},
 the file \textit{ {list\_sections.ascii\_global}} contains a list of all the sections that are to
 be computed (this list of sections is based on MERSEA project metrics).
+In \texttt{NEMOGCM/TOOLS/SECTIONS\_DIADCT/run},
+the file \textit{ {list\_sections.ascii\_global}} contains a list of all the sections that are to be computed
+(this list of sections is based on MERSEA project metrics).
 Another file is available for the GYRE configuration (\texttt{ {list\_sections.ascii\_GYRE}}).
 …
  - \texttt{ice}       to compute surface and volume ice transports, \texttt{noice}     if no. \\
+\noindent The results of the computing of transports, and the directions of positive and
 negative flow do not depend on the order of the 2 extremities in this file. \\
 \noindent If nclass $\neq$ 0,the next lines contain the class type and the nclass bounds: \\
+ \noindent The results of the computing of transports, and the directions of positive and
+ negative flow do not depend on the order of the 2 extremities in this file. \\
+\noindent If nclass $\neq$ 0, the next lines contain the class type and the nclass bounds: \\
 {\scriptsize \texttt{
 long1 lat1 long2 lat2 nclass (ok/no)strpond (no)ice section\_name \\
 …
  - \texttt{zsigp} for potential density classes \\
+The script \texttt{job.ksh} computes the pathway for each section and
 creates a binary file \texttt{section\_ijglobal.diadct\_ORCA2\_LIM} which is read by NEMO. \\
+It is possible to use this tools for new configuations: \texttt{job.ksh} has to be updated with
 the coordinates file name and path. \\
+Examples of two sections, the ACC\_Drake\_Passage with no classes, and the ATL\_Cuba\_Florida with
 temperature clases (5 class bounds), are shown: \\
+ The script \texttt{job.ksh} computes the pathway for each section and creates a binary file
+ \texttt{section\_ijglobal.diadct\_ORCA2\_LIM} which is read by NEMO. \\
+ It is possible to use this tools for new configuations: \texttt{job.ksh} has to be updated with
+ the coordinates file name and path. \\
+ Examples of two sections, the ACC\_Drake\_Passage with no classes,
+ and the ATL\_Cuba\_Florida with 4 temperature clases (5 class bounds), are shown: \\
 \noindent {\scriptsize \texttt{
 -68.    -54.5   -60.    -64.7  00 okstrpond noice ACC\_Drake\_Passage \\
 …
 transport\_total}}                                     \\
 For sections with classes, the first \texttt{nclass-1} lines correspond to the transport for
 each class and the last line corresponds to the total transport summed over all classes.
 For sections with no classes, class number \texttt{1} corresponds to \texttt{total class} and
+For sections with classes, the first \texttt{nclass-1} lines correspond to the transport for each class and
+the last line corresponds to the total transport summed over all classes.
+For sections with no classes, class number \texttt{1} corresponds to \texttt{total class} and
 this class is called \texttt{N}, meaning \texttt{none}.
 …
 - \texttt{transport\_direction2} is the negative part of the transport ($\leq$ 0). \\
 \noindent The \texttt{section slope coefficient} gives information about the significance of
 transports signs and direction: \\
+\noindent The \texttt{section slope coefficient} gives information about the significance of transports signs and
+direction: \\
 \begin{table} \scriptsize
 …
 Changes in steric sea level are caused when changes in the density of the water column imply an
 expansion or contraction of the column.
 It is essentially produced through surface heating/cooling and to a lesser extent through
 non-linear effects of the equation of state (cabbeling, thermobaricity...).
+Changes in steric sea level are caused when changes in the density of the water column imply an expansion or
+contraction of the column.
+It is essentially produced through surface heating/cooling and to a lesser extent through non-linear effects of
+the equation of state (cabbeling, thermobaricity...).
 Non-Boussinesq models contain all ocean effects within the ocean acting on the sea level.
 In particular, they include the steric effect.
 In contrast, Boussinesq models, such as \NEMO, conserve volume, rather than mass,
+In contrast, Boussinesq models, such as \NEMO, conserve volume, rather than mass,
 and so do not properly represent expansion or contraction.
 The steric effect is therefore not explicitely represented.
+This approximation does not represent a serious error with respect to the flow field calculated by
+the model \citep{Greatbatch_JGR94}, but extra attention is required when investigating sea level,
+as steric changes are an important contribution to local changes in sea level on seasonal and
+climatic time scales.
+This is especially true for investigation into sea level rise due to global warming.
+Fortunately, the steric contribution to the sea level consists of a spatially uniform component that
+This approximation does not represent a serious error with respect to the flow field calculated by the model
+\citep{Greatbatch_JGR94}, but extra attention is required when investigating sea level,
+as steric changes are an important contribution to local changes in sea level on seasonal and climatic time scales.
+This is especially true for investigation into sea level rise due to global warming.
+Fortunately, the steric contribution to the sea level consists of a spatially uniform component that
 can be diagnosed by considering the mass budget of the world ocean \citep{Greatbatch_JGR94}.
 In order to better understand how global mean sea level evolves and thus how the steric sea level can
 be diagnosed, we compare, in the following, the non-Boussinesq and Boussinesq cases.
+In order to better understand how global mean sea level evolves and thus how the steric sea level can be diagnosed,
+we compare, in the following, the non-Boussinesq and Boussinesq cases.
 Let denote
 …
 \]
 Temporal changes in total mass is obtained from the density conservation equation :
+Temporal changes in total mass is obtained from the density conservation equation:
 \[ \frac{1}{e_3} \partial_t ( e_3\,\rho) + \nabla( \rho \, \textbf{U} )
 …
  \label{eq:Co_nBq} \]
 where $\rho$ is the \textit{in situ} density, and \textit{emp} the surface mass
+exchanges with the other media of the Earth system (atmosphere, sea-ice, land).
+where $\rho$ is the \textit{in situ} density, and \textit{emp} the surface mass exchanges with the other media of
+the Earth system (atmosphere, sea-ice, land).
 Its global averaged leads to the total mass change
 …
 where $\overline{\textit{emp}} = \int_S \textit{emp}\,ds$ is the net mass flux through the ocean surface.
 Bringing \autoref{eq:Mass_nBq} and the time derivative of \autoref{eq:MV_nBq} together leads to
+Bringing \autoref{eq:Mass_nBq} and the time derivative of \autoref{eq:MV_nBq} together leads to
 the evolution equation of the mean sea level
 …
  \label{eq:ssh_nBq} \]
+The first term in equation \autoref{eq:ssh_nBq} alters sea level by adding or
+subtracting mass from the ocean.
+The second term arises from temporal changes in the global mean density; $i.e.$ from steric effects.
+In a Boussinesq fluid, $\rho$ is replaced by $\rho_o$ in all the equation except when
+$\rho$ appears multiplied by the gravity ($i.e.$ in the hydrostatic balance of the primitive Equations).
+In particular, the mass conservation equation, \autoref{eq:Co_nBq}, degenerates into
+the incompressibility equation:
+The first term in equation \autoref{eq:ssh_nBq} alters sea level by adding or subtracting mass from the ocean.
+The second term arises from temporal changes in the global mean density; $i.e.$ from steric effects.
+In a Boussinesq fluid, $\rho$ is replaced by $\rho_o$ in all the equation except when $\rho$ appears multiplied by
+the gravity ($i.e.$ in the hydrostatic balance of the primitive Equations).
+In particular, the mass conservation equation, \autoref{eq:Co_nBq}, degenerates into the incompressibility equation:
 \[ \frac{1}{e_3} \partial_t ( e_3 ) + \nabla( \textbf{U} )
 …
  \label{eq:V_Bq} \]
+Only the volume is conserved, not mass, or, more precisely, the mass which is conserved is
+the Boussinesq mass, $\mathcal{M}_o = \rho_o \mathcal{V}$.
+The total volume (or equivalently the global mean sea level) is altered only by net volume fluxes across
+the ocean surface, not by changes in mean mass of the ocean: the steric effect is missing in
+a Boussinesq fluid.
+Nevertheless, following \citep{Greatbatch_JGR94}, the steric effect on the volume can be diagnosed by
+Only the volume is conserved, not mass, or, more precisely, the mass which is conserved is the Boussinesq mass,
+$\mathcal{M}_o = \rho_o \mathcal{V}$.
+The total volume (or equivalently the global mean sea level) is altered only by net volume fluxes across
+the ocean surface, not by changes in mean mass of the ocean: the steric effect is missing in a Boussinesq fluid.
+Nevertheless, following \citep{Greatbatch_JGR94}, the steric effect on the volume can be diagnosed by
 considering the mass budget of the ocean.
 The apparent changes in $\mathcal{M}$, mass of the ocean, which are not induced by
+surface mass flux must be compensated by a spatially uniform change in the mean sea level due to
 expansion/contraction of the ocean \citep{Greatbatch_JGR94}.
 In others words, the Boussinesq mass, $\mathcal{M}_o$, can be related to $\mathcal{M}$,
 the  total mass of the ocean seen by the Boussinesq model, via the steric contribution to the sea level,
+The apparent changes in $\mathcal{M}$, mass of the ocean, which are not induced by surface mass flux
+must be compensated by a spatially uniform change in the mean sea level due to expansion/contraction of the ocean
+\citep{Greatbatch_JGR94}.
+In others words, the Boussinesq mass, $\mathcal{M}_o$, can be related to $\mathcal{M}$,
+the total mass of the ocean seen by the Boussinesq model, via the steric contribution to the sea level,
 $\eta_s$, a spatially uniform variable, as follows:
 …
  \label{eq:M_Bq} \]
 Any change in $\mathcal{M}$ which cannot be explained by the net mass flux through
 the ocean surface is converted into a mean change in sea level.
 Introducing the total density anomaly, $\mathcal{D}= \int_D d_a \,dv$, where
+$d_a = (\rho -\rho_o ) / \rho_o$ is the density anomaly used in \NEMO (cf. \autoref{subsec:TRA_eos}) in
 \autoref{eq:M_Bq} leads to a very simple form for the steric height:
+Any change in $\mathcal{M}$ which cannot be explained by the net mass flux through the ocean surface
+is converted into a mean change in sea level.
+Introducing the total density anomaly, $\mathcal{D}= \int_D d_a \,dv$,
+where $d_a = (\rho -\rho_o ) / \rho_o$ is the density anomaly used in \NEMO (cf. \autoref{subsec:TRA_eos})
+in \autoref{eq:M_Bq} leads to a very simple form for the steric height:
 \[ \eta_s = - \frac{1}{\mathcal{A}} \mathcal{D}
 …
 We do not recommend that.
 Indeed, in this case $\rho_o$ depends on the initial state of the ocean.
 Since $\rho_o$ has a direct effect on the dynamics of the ocean (it appears in
+the pressure gradient term of the momentum equation) it is definitively not a good idea when
 inter-comparing experiments.
+Since $\rho_o$ has a direct effect on the dynamics of the ocean
+(it appears in the pressure gradient term of the momentum equation)
+it is definitively not a good idea when inter-comparing experiments.
 We better recommend to fixe once for all $\rho_o$ to $1035\;Kg\,m^{-3}$.
 This value is a sensible choice for the reference density used in a Boussinesq ocean climate model since,
 with the exception of only a small percentage of the ocean, density in the World Ocean varies by
 no more than 2$\%$ from this value (\cite{Gill1982}, page 47).
 Second, we have assumed here that the total ocean surface, $\mathcal{A}$, does not change when
+the sea level is changing as it is the case in all global ocean GCMs
+This value is a sensible choice for the reference density used in a Boussinesq ocean climate model since,
+with the exception of only a small percentage of the ocean, density in the World Ocean varies by no more than
+$\%$ from this value (\cite{Gill1982}, page 47).
+Second, we have assumed here that the total ocean surface, $\mathcal{A}$,
+does not change when the sea level is changing as it is the case in all global ocean GCMs
 (wetting and drying of grid point is not allowed).
+Third, the discretisation of \autoref{eq:steric_Bq} depends on the type of
+free surface which is considered.
+Third, the discretisation of \autoref{eq:steric_Bq} depends on the type of free surface which is considered.
 In the non linear free surface case, $i.e.$ \key{vvl} defined, it is given by
 …
  \label{eq:discrete_steric_Bq_nfs} \]
+whereas in the linear free surface, the volume above the \textit{z=0} surface must be explicitly
+taken into account to better approximate the total ocean mass and thus the steric sea level:
+whereas in the linear free surface,
+the volume above the \textit{z=0} surface must be explicitly taken into account to
+better approximate the total ocean mass and thus the steric sea level:
 \[ \eta_s = - \frac{ \sum_{i,\,j,\,k} d_a\; e_{1t}e_{2t}e_{3t} + \sum_{i,\,j} d_a\; e_{1t}e_{2t} \eta }
 …
 In the real ocean, sea ice (and snow above it)  depresses the liquid seawater through its mass loading.
 This depression is a result of the mass of sea ice/snow system acting on the liquid ocean.
 There is, however, no dynamical effect associated with these depressions in the liquid ocean sea level,
+There is, however, no dynamical effect associated with these depressions in the liquid ocean sea level,
 so that there are no associated ocean currents.
+Hence, the dynamically relevant sea level is the effective sea level, $i.e.$ the sea level as if
+sea ice (and snow) were converted to liquid seawater \citep{Campin_al_OM08}.
+However, in the current version of \NEMO the sea-ice is levitating above the ocean without
+mass exchanges between ice and ocean.
+Therefore the model effective sea level is always given by $\eta + \eta_s$,
+whether or not there is sea ice present.
+Hence, the dynamically relevant sea level is the effective sea level,
+$i.e.$ the sea level as if sea ice (and snow) were converted to liquid seawater \citep{Campin_al_OM08}.
+However, in the current version of \NEMO the sea-ice is levitating above the ocean without mass exchanges between
+ice and ocean.
+Therefore the model effective sea level is always given by $\eta + \eta_s$, whether or not there is sea ice present.
 In AR5 outputs, the thermosteric sea level is demanded.
 …
 where $S_o$ and $p_o$ are the initial salinity and pressure, respectively.
 Both steric and thermosteric sea level are computed in \mdl{diaar5} which needs
 the \key{diaar5} defined to be called.
+Both steric and thermosteric sea level are computed in \mdl{diaar5} which needs the \key{diaar5} defined to
+be called.
 % -------------------------------------------------------------------------------------------------------------
 …
 \subsection{Depth of various quantities (\protect\mdl{diahth})}
 Among the available diagnostics the following ones are obtained when defining the \key{diahth} CPP key:
+Among the available diagnostics the following ones are obtained when defining the \key{diahth} CPP key:
 - the mixed layer depth (based on a density criterion \citep{de_Boyer_Montegut_al_JGR04}) (\mdl{diahth})
 …
 %-----------------------------------------------------------------------------------------
 The poleward heat and salt transports, their advective and diffusive component, and
+the meriodional stream function can be computed on-line in \mdl{diaptr} \np{ln\_diaptr} to true
+The poleward heat and salt transports, their advective and diffusive component,
+and the meriodional stream function can be computed on-line in \mdl{diaptr} \np{ln\_diaptr} to true
 (see the \textit{\ngn{namptr} } namelist below).
+When \np{ln\_subbas}\forcode{ = .true.}, transports and stream function are computed for the Atlantic,
+Indian, Pacific and Indo-Pacific Oceans (defined north of 30\deg S) as well as for the World Ocean.
+The sub-basin decomposition requires an input file (\ifile{subbasins}) which contains
+three 2D mask arrays, the Indo-Pacific mask been deduced from the sum of the Indian and
+Pacific mask (\autoref{fig:mask_subasins}).
+When \np{ln\_subbas}\forcode{ = .true.}, transports and stream function are computed for the Atlantic, Indian,
+Pacific and Indo-Pacific Oceans (defined north of 30\deg S) as well as for the World Ocean.
+The sub-basin decomposition requires an input file (\ifile{subbasins}) which contains three 2D mask arrays,
+the Indo-Pacific mask been deduced from the sum of the Indian and Pacific mask (\autoref{fig:mask_subasins}).
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 \begin{figure}[!t] \begin{center}
    \includegraphics[width=1.0\textwidth]{Fig_mask_subasins}
    \caption{   \protect\label{fig:mask_subasins}
+      Decomposition of the World Ocean (here ORCA2) into sub-basin used in to compute the heat and
       salt transports as well as the meridional stream-function: Atlantic basin (red),
+      Pacific basin (green), Indian basin (bleue), Indo-Pacific basin (bleue+green).
+      Note that semi-enclosed seas (Red, Med and Baltic seas) as well as Hudson Bay are removed from
       the sub-basins. Note also that the Arctic Ocean has been split into Atlantic and
       Pacific basins along the North fold line.}
+\begin{figure}[!t]
+  \begin{center}
+    \includegraphics[width=1.0\textwidth]{Fig_mask_subasins}
+    \caption{  \protect\label{fig:mask_subasins}
+      Decomposition of the World Ocean (here ORCA2) into sub-basin used in to
+      compute the heat and salt transports as well as the meridional stream-function:
+      Atlantic basin (red), Pacific basin (green), Indian basin (bleue), Indo-Pacific basin (bleue+green).
+      Note that semi-enclosed seas (Red, Med and Baltic seas) as well as Hudson Bay are removed from the sub-basins.
+      Note also that the Arctic Ocean has been split into Atlantic and Pacific basins along the North fold line.}
 \end{center} \end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 …
 The 25 hour mean is available for daily runs by summing up the 25 hourly instantananeous hourly values from
 midnight at the start of the day to midight at the day end.
 This diagnostic is actived with the logical  $ln\_dia25h$
+This diagnostic is actived with the logical $ln\_dia25h$.
 % -----------------------------------------------------------
 …
 %----------------------------------------------------------------------------------------------------------
+A module is available to output the surface (top), mid water and bed diagnostics of a set of
+standard variables.
+This can be a useful diagnostic when hourly or sub-hourly output is required in
+high resolution tidal outputs.
+A module is available to output the surface (top), mid water and bed diagnostics of a set of standard variables.
+This can be a useful diagnostic when hourly or sub-hourly output is required in high resolution tidal outputs.
 The tidal signal is retained but the overall data usage is cut to just three vertical levels.
 Also the bottom level is calculated for each cell.
 This diagnostic is actived with the logical  $ln\_diatmb$
+This diagnostic is actived with the logical $ln\_diatmb$.
 % -----------------------------------------------------------
 …
 in the zonal, meridional and vertical directions respectively.
 The vertical component is included although it is not strictly valid as the vertical velocity is
 calculated from the continuity equation rather than as a prognostic variable.
+The vertical component is included although it is not strictly valid as the vertical velocity is calculated from
+the continuity equation rather than as a prognostic variable.
 Physically this represents the rate at which information is propogated across a grid cell.
+Values greater than 1 indicate that information is propagated across more than one grid cell in
+a single time step.
+The variables can be activated by setting the \np{nn\_diacfl} namelist parameter to 1 in
+the \ngn{namctl} namelist.
+Values greater than 1 indicate that information is propagated across more than one grid cell in a single time step.
+The variables can be activated by setting the \np{nn\_diacfl} namelist parameter to 1 in the \ngn{namctl} namelist.
 The diagnostics will be written out to an ascii file named cfl\_diagnostics.ascii.
 In this file the maximum value of $C_u$, $C_v$, and $C_w$ are printed at each timestep along with
 the coordinates of where the maximum value occurs.
 At the end of the model run the maximum value of $C_u$, $C_v$, and $C_w$ for the whole model run is
 printed along with the coordinates of each.
+In this file the maximum value of $C_u$, $C_v$, and $C_w$ are printed at each timestep along with the coordinates of
+where the maximum value occurs.
+At the end of the model run the maximum value of $C_u$, $C_v$, and $C_w$ for the whole model run is printed along
+with the coordinates of each.
 The maximum values from the run are also copied to the ocean.output file.

NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_DIU.tex

-                      r10146
+                      r10368
 The skin temperature can be split into three parts:
 \begin{itemize}
+\item A foundation SST which is free from diurnal warming.
+\item A warm layer, typically ~3\,m thick, where heating from solar radiation can
+cause a warm stably stratified layer during the daytime
+\item A cool skin, a thin layer, approximately ~1\,mm thick, where long wave cooling
+is dominant and cools the immediate ocean surface.
+\item
+  A foundation SST which is free from diurnal warming.
+\item
+  A warm layer, typically ~3\,m thick,
+  where heating from solar radiation can cause a warm stably stratified layer during the daytime
+\item
+  A cool skin, a thin layer, approximately ~1\, mm thick,
+  where long wave cooling is dominant and cools the immediate ocean surface.
 \end{itemize}
+Models are provided for both the warm layer, \mdfl{diurnal_bulk}, and the cool skin,
+\mdl{cool_skin}.  Foundation SST is not considered as it can be obtained
+either from the main NEMO model ($i.e.$ from the temperature of the top few model levels)
+or from some other source.
+It must be noted that both the cool skin and warm layer models produce estimates of
+the change in temperature ($\Delta T_{\rm{cs}}$ and $\Delta T_{\rm{wl}}$)
+and both must be added to a foundation SST to obtain the true skin temperature.
+Models are provided for both the warm layer, \mdfl{diurnal_bulk}, and the cool skin, \mdl{cool_skin}.
+Foundation SST is not considered as it can be obtained either from the main NEMO model
+($i.e.$ from the temperature of the top few model levels) or from some other source.
+It must be noted that both the cool skin and warm layer models produce estimates of the change in temperature
+($\Delta T_{\rm{cs}}$ and $\Delta T_{\rm{wl}}$) and
+both must be added to a foundation SST to obtain the true skin temperature.
 Both the cool skin and warm layer models are controlled through the namelist \ngn{namdiu}:
 …
 This namelist contains only two variables:
 \begin{description}
+\item[\np{ln\_diurnal}] A logical switch for turning on/off both the cool skin and warm layer.
+\item[\np{ln\_diurnal\_only}] A logical switch which if \forcode{.true.} will run the diurnal model
+without the other dynamical parts of NEMO.
+\np{ln\_diurnal\_only} must be \forcode{.false.} if \np{ln\_diurnal} is \forcode{.false.}.
+\item[\np{ln\_diurnal}]
+  A logical switch for turning on/off both the cool skin and warm layer.
+\item[\np{ln\_diurnal\_only}]
+  A logical switch which if \forcode{.true.} will run the diurnal model without the other dynamical parts of NEMO.
+  \np{ln\_diurnal\_only} must be \forcode{.false.} if \np{ln\_diurnal} is \forcode{.false.}.
 \end{description}
+Output for the diurnal model is through the variables `sst\_wl' (warm\_layer) and
+`sst\_cs' (cool skin).  These are 2-D variables which will be included in the model
+output if they are specified in the iodef.xml file.
+Output for the diurnal model is through the variables `sst\_wl' (warm\_layer) and `sst\_cs' (cool skin).
+These are 2-D variables which will be included in the model output if they are specified in the iodef.xml file.
+Initialisation is through the restart file.  Specifically the code will expect
+the presence of the 2-D variable ``Dsst'' to initialise the warm layer.
+The cool skin model, which is determined purely by the instantaneous fluxes,
+has no initialisation variable.
+Initialisation is through the restart file.
+Specifically the code will expect the presence of the 2-D variable ``Dsst'' to initialise the warm layer.
+The cool skin model, which is determined purely by the instantaneous fluxes, has no initialisation variable.
 %===============================================================
 …
 %===============================================================
 The warm layer is calculated using the model of \citet{Takaya_al_JGR10} (TAKAYA10 model
 hereafter).  This is a simple flux based model that is defined by the equations
+The warm layer is calculated using the model of \citet{Takaya_al_JGR10} (TAKAYA10 model hereafter).
+This is a simple flux based model that is defined by the equations
 \begin{eqnarray}
 \frac{\partial{\Delta T_{\rm{wl}}}}{\partial{t}}&=&\frac{Q(\nu+1)}{D_T\rho_w c_p
 …
 L&=&\frac{\rho_w c_p u^{*^3}_{w}}{\kappa g \alpha_w Q }\mbox{,}\label{eq:ecmwf2}
 \end{eqnarray}
+where $\Delta T_{\rm{wl}}$ is the temperature difference between the top of the warm
+layer and the depth $D_T=3$\,m at which there is assumed to be no diurnal signal. In
+equation (\autoref{eq:ecmwf1}) $\alpha_w=2\times10^{-4}$ is the thermal expansion
+coefficient of water, $\kappa=0.4$ is von K\'{a}rm\'{a}n's constant, $c_p$ is the heat
+capacity at constant pressure of sea water, $\rho_w$ is the
+water density, and $L$ is the Monin-Obukhov length. The tunable
+variable $\nu$ is a shape parameter that defines the expected
+subskin temperature profile via $T(z)=T(0)-\left(\frac{z}{D_T}\right)^\nu\Delta
+T_{\rm{wl}}$,
+where $T$ is the absolute temperature and $z\le D_T$ is the depth
+below the top of the warm layer.
+The influence of wind on TAKAYA10 comes through the magnitude of the friction velocity
+of the water
+$u^*_{w}$, which can be related to the 10\,m wind speed $u_{10}$ through the relationship
+$u^*_{w} = u_{10}\sqrt{\frac{C_d\rho_a}{\rho_w}}$, where $C_d$ is
+the drag coefficient, and $\rho_a$ is the density of air.  The symbol $Q$ in equation
+(\autoref{eq:ecmwf1}) is the instantaneous total thermal energy
+flux into
+where $\Delta T_{\rm{wl}}$ is the temperature difference between the top of the warm layer and the depth $D_T=3$\,m at which there is assumed to be no diurnal signal.
+In equation (\autoref{eq:ecmwf1}) $\alpha_w=2\times10^{-4}$ is the thermal expansion coefficient of water,
+$\kappa=0.4$ is von K\'{a}rm\'{a}n's constant, $c_p$ is the heat capacity at constant pressure of sea water,
+$\rho_w$ is the water density, and $L$ is the Monin-Obukhov length.
+The tunable variable $\nu$ is a shape parameter that defines the expected subskin temperature profile via
+$T(z)=T(0)-\left(\frac{z}{D_T}\right)^\nu\DeltaT_{\rm{wl}}$,
+where $T$ is the absolute temperature and $z\le D_T$ is the depth below the top of the warm layer.
+The influence of wind on TAKAYA10 comes through the magnitude of the friction velocity of the water $u^*_{w}$,
+which can be related to the 10\,m wind speed $u_{10}$ through
+the relationship $u^*_{w} = u_{10}\sqrt{\frac{C_d\rho_a}{\rho_w}}$, where $C_d$ is the drag coefficient,
+and $\rho_a$ is the density of air.
+The symbol $Q$ in equation (\autoref{eq:ecmwf1}) is the instantaneous total thermal energy flux into
 the diurnal layer, $i.e.$
 \begin{equation}
 Q = Q_{\rm{sol}} + Q_{\rm{lw}} + Q_{\rm{h}}\mbox{,} \label{eq:e_flux_eqn}
 \end{equation}
+where $Q_{\rm{h}}$ is the sensible and latent heat flux, $Q_{\rm{lw}}$ is the long
+wave flux, and $Q_{\rm{sol}}$ is the solar flux absorbed
+within the diurnal warm layer. For $Q_{\rm{sol}}$ the 9 term
+representation of \citet{Gentemann_al_JGR09} is used.  In equation \autoref{eq:ecmwf1}
+the function $f(L_a)=\max(1,L_a^{\frac{2}{3}})$, where $L_a=0.3$\footnote{This
+is a global average value, more accurately $L_a$ could be computed as
+$L_a=(u^*_{w}/u_s)^{\frac{1}{2}}$, where $u_s$ is the stokes drift, but this is not
+currently done} is the turbulent Langmuir number and is a
+parametrization of the effect of waves.
+where $Q_{\rm{h}}$ is the sensible and latent heat flux, $Q_{\rm{lw}}$ is the long wave flux,
+and $Q_{\rm{sol}}$ is the solar flux absorbed within the diurnal warm layer.
+For $Q_{\rm{sol}}$ the 9 term representation of \citet{Gentemann_al_JGR09} is used.
+In equation \autoref{eq:ecmwf1} the function $f(L_a)=\max(1,L_a^{\frac{2}{3}})$,
+where $L_a=0.3$\footnote{
+  This is a global average value, more accurately $L_a$ could be computed as $L_a=(u^*_{w}/u_s)^{\frac{1}{2}}$,
+  where $u_s$ is the stokes drift, but this is not currently done
+} is the turbulent Langmuir number and is a parametrization of the effect of waves.
 The function $\Phi\!\left(\frac{D_T}{L}\right)$ is the similarity function that
+parametrizes the stability of the water column and
+is given by:
+parametrizes the stability of the water column and is given by:
 \begin{equation}
 \Phi(\zeta) = \left\{ \begin{array}{cc} 1 + \frac{5\zeta +
 …
                                     \end{array} \right. \label{eq:stab_func_eqn}
 \end{equation}
 where $\zeta=\frac{D_T}{L}$.  It is clear that the first derivative of
+(\autoref{eq:stab_func_eqn}), and thus of (\autoref{eq:ecmwf1}),
 is discontinuous at $\zeta=0$ ($i.e.$ $Q\rightarrow0$ in equation (\autoref{eq:ecmwf2})).
+where $\zeta=\frac{D_T}{L}$.  It is clear that the first derivative of (\autoref{eq:stab_func_eqn}),
+and thus of (\autoref{eq:ecmwf1}), is discontinuous at $\zeta=0$ ($i.e.$ $Q\rightarrow0$ in
+equation (\autoref{eq:ecmwf2})).
 The two terms on the right hand side of (\autoref{eq:ecmwf1}) represent different processes.
+The first term is simply the diabatic heating or cooling of the
+diurnal warm
+layer due to thermal energy
+fluxes into and out of the layer.  The second term
+parametrizes turbulent fluxes of heat out of the diurnal warm layer due to wind
+induced mixing. In practice the second term acts as a relaxation
+on the temperature.
+The first term is simply the diabatic heating or cooling of the diurnal warm layer due to
+thermal energy fluxes into and out of the layer.
+The second term parametrizes turbulent fluxes of heat out of the diurnal warm layer due to wind induced mixing.
+In practice the second term acts as a relaxation on the temperature.
 %===============================================================
 …
 %===============================================================
+The cool skin is modelled using the framework of \citet{Saunders_JAS82} who used a
+formulation of the near surface temperature difference based upon the heat flux and
+the friction velocity $u^*_{w}$.  As the cool skin
+is so thin (~1\,mm) we ignore the solar flux component to the heat flux and the
+Saunders equation for the cool skin temperature difference $\Delta T_{\rm{cs}}$ becomes
+The cool skin is modelled using the framework of \citet{Saunders_JAS82} who used a formulation of the near surface temperature difference based upon the heat flux and the friction velocity $u^*_{w}$.
+As the cool skin is so thin (~1\,mm) we ignore the solar flux component to the heat flux and the Saunders equation for the cool skin temperature difference $\Delta T_{\rm{cs}}$ becomes
 \begin{equation}
 \label{eq:sunders_eqn}
 …
 \end{equation}
 where $Q_{\rm{ns}}$ is the, usually negative, non-solar heat flux into the ocean and
 $k_t$ is the thermal conductivity of sea water. $\delta$ is the thickness of the
 skin layer and is given by
+$k_t$ is the thermal conductivity of sea water.
+$\delta$ is the thickness of the skin layer and is given by
 \begin{equation}
 \label{eq:sunders_thick_eqn}
 \delta=\frac{\lambda \mu}{u^*_{w}} \mbox{,}
 \end{equation}
 where $\mu$ is the kinematic viscosity of sea water and $\lambda$ is a constant of
 proportionality which \citet{Saunders_JAS82} suggested varied between 5 and 10.
+where $\mu$ is the kinematic viscosity of sea water and $\lambda$ is a constant of proportionality which
+\citet{Saunders_JAS82} suggested varied between 5 and 10.
 The value of $\lambda$ used in equation (\autoref{eq:sunders_thick_eqn}) is that of
+\citet{Artale_al_JGR02},
+which is shown in \citet{Tu_Tsuang_GRL05} to outperform a number of other
 parametrisations at both low and high wind speeds. Specifically,
+The value of $\lambda$ used in equation (\autoref{eq:sunders_thick_eqn}) is that of \citet{Artale_al_JGR02},
+which is shown in \citet{Tu_Tsuang_GRL05} to outperform a number of other parametrisations at
+both low and high wind speeds.
+Specifically,
 \begin{equation}
 \label{eq:artale_lambda_eqn}

NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_DOM.tex

-                      r10146
+                      r10368
 $\ $\newline    % force a new line
+Having defined the continuous equations in \autoref{chap:PE} and chosen a time
+discretization \autoref{chap:STP}, we need to choose a discretization on a grid,
+and numerical algorithms. In the present chapter, we provide a general description
+of the staggered grid used in \NEMO, and other information relevant to the main
+directory routines as well as the DOM (DOMain) directory.
+Having defined the continuous equations in \autoref{chap:PE} and chosen a time discretization \autoref{chap:STP},
+we need to choose a discretization on a grid, and numerical algorithms.
+In the present chapter, we provide a general description of the staggered grid used in \NEMO,
+and other information relevant to the main directory routines as well as the DOM (DOMain) directory.
 $\ $\newline    % force a new line
 …
 \begin{figure}[!tb]    \begin{center}
 \includegraphics[width=0.90\textwidth]{Fig_cell}
 \caption{ \protect\label{fig:cell}
+Arrangement of variables. $t$ indicates scalar points where temperature,
+salinity, density, pressure and horizontal divergence are defined. ($u$,$v$,$w$)
+indicates vector points, and $f$ indicates vorticity points where both relative and
 planetary vorticities are defined}
+\caption{ \protect\label{fig:cell}
+  Arrangement of variables.
+  $t$ indicates scalar points where temperature, salinity, density, pressure and horizontal divergence are defined.
+  ($u$,$v$,$w$) indicates vector points,
+  and $f$ indicates vorticity points where both relative and planetary vorticities are defined}
 \end{center}   \end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 The numerical techniques used to solve the Primitive Equations in this model are
+based on the traditional, centred second-order finite difference approximation.
 Special attention has been given to the homogeneity of the solution in the three
+space directions. The arrangement of variables is the same in all directions.
 It consists of cells centred on scalar points ($t$, $S$, $p$, $\rho$) with vector
+points $(u, v, w)$ defined in the centre of each face of the cells (\autoref{fig:cell}).
 This is the generalisation to three dimensions of the well-known ``C'' grid in
+Arakawa's classification \citep{Mesinger_Arakawa_Bk76}. The relative and
+planetary vorticity, $\zeta$ and $f$, are defined in the centre of each vertical edge
+and the barotropic stream function $\psi$ is defined at horizontal points overlying
+the $\zeta$ and $f$-points.
+The ocean mesh ($i.e.$ the position of all the scalar and vector points) is defined
+by the transformation that gives ($\lambda$ ,$\varphi$ ,$z$) as a function of $(i,j,k)$.
+The grid-points are located at integer or integer and a half value of $(i,j,k)$ as
+indicated on \autoref{tab:cell}. In all the following, subscripts $u$, $v$, $w$,
+$f$, $uw$, $vw$ or $fw$ indicate the position of the grid-point where the scale
+factors are defined. Each scale factor is defined as the local analytical value
+provided by \autoref{eq:scale_factors}. As a result, the mesh on which partial
+derivatives $\frac{\partial}{\partial \lambda}, \frac{\partial}{\partial \varphi}$, and
+$\frac{\partial}{\partial z} $ are evaluated is a uniform mesh with a grid size of unity.
+Discrete partial derivatives are formulated by the traditional, centred second order
+finite difference approximation while the scale factors are chosen equal to their
+local analytical value. An important point here is that the partial derivative of the
+scale factors must be evaluated by centred finite difference approximation, not
+from their analytical expression. This preserves the symmetry of the discrete set
+of equations and therefore satisfies many of the continuous properties (see
+\autoref{apdx:C}). A similar, related remark can be made about the domain
+size: when needed, an area, volume, or the total ocean depth must be evaluated
+as the sum of the relevant scale factors (see \autoref{eq:DOM_bar}) in the next section).
+The numerical techniques used to solve the Primitive Equations in this model are based on the traditional,
+centred second-order finite difference approximation.
+Special attention has been given to the homogeneity of the solution in the three space directions.
+The arrangement of variables is the same in all directions.
+It consists of cells centred on scalar points ($t$, $S$, $p$, $\rho$) with vector points $(u, v, w)$ defined in
+the centre of each face of the cells (\autoref{fig:cell}).
+This is the generalisation to three dimensions of the well-known ``C'' grid in Arakawa's classification
+\citep{Mesinger_Arakawa_Bk76}.
+The relative and planetary vorticity, $\zeta$ and $f$, are defined in the centre of each vertical edge and
+the barotropic stream function $\psi$ is defined at horizontal points overlying the $\zeta$ and $f$-points.
+The ocean mesh ($i.e.$ the position of all the scalar and vector points) is defined by
+the transformation that gives ($\lambda$ ,$\varphi$ ,$z$) as a function of $(i,j,k)$.
+The grid-points are located at integer or integer and a half value of $(i,j,k)$ as indicated on \autoref{tab:cell}.
+In all the following, subscripts $u$, $v$, $w$, $f$, $uw$, $vw$ or $fw$ indicate the position of
+the grid-point where the scale factors are defined.
+Each scale factor is defined as the local analytical value provided by \autoref{eq:scale_factors}.
+As a result,
+the mesh on which partial derivatives $\frac{\partial}{\partial \lambda}, \frac{\partial}{\partial \varphi}$,
+and $\frac{\partial}{\partial z} $ are evaluated is a uniform mesh with a grid size of unity.
+Discrete partial derivatives are formulated by the traditional,
+centred second order finite difference approximation while
+the scale factors are chosen equal to their local analytical value.
+An important point here is that the partial derivative of the scale factors must be evaluated by
+centred finite difference approximation, not from their analytical expression.
+This preserves the symmetry of the discrete set of equations and
+therefore satisfies many of the continuous properties (see \autoref{apdx:C}).
+A similar, related remark can be made about the domain size:
+when needed, an area, volume, or the total ocean depth must be evaluated as the sum of the relevant scale factors
+(see \autoref{eq:DOM_bar}) in the next section).
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 …
 \end{tabular}
 \caption{ \protect\label{tab:cell}
+Location of grid-points as a function of integer or integer and a half value of the column,
+line or level. This indexing is only used for the writing of the semi-discrete equation.
+In the code, the indexing uses integer values only and has a reverse direction
 in the vertical (see \autoref{subsec:DOM_Num_Index})}
+  Location of grid-points as a function of integer or integer and a half value of the column, line or level.
+  This indexing is only used for the writing of the semi-discrete equation.
+  In the code, the indexing uses integer values only and has a reverse direction in the vertical
+  (see \autoref{subsec:DOM_Num_Index})}
 \end{center}
 \end{table}
 …
 \label{subsec:DOM_operators}
 Given the values of a variable $q$ at adjacent points, the differencing and
 averaging operators at the midpoint between them are:
+Given the values of a variable $q$ at adjacent points,
+the differencing and averaging operators at the midpoint between them are:
 \begin{subequations} \label{eq:di_mi}
 \begin{align}
 …
 \end{subequations}
 Similar operators are defined with respect to $i+1/2$, $j$, $j+1/2$, $k$, and
+$k+1/2$. Following \autoref{eq:PE_grad} and \autoref{eq:PE_lap}, the gradient of a
+variable $q$ defined at a $t$-point has its three components defined at $u$-, $v$-
+and $w$-points while its Laplacien is defined at $t$-point. These operators have
 the following discrete forms in the curvilinear $s$-coordinate system:
+Similar operators are defined with respect to $i+1/2$, $j$, $j+1/2$, $k$, and $k+1/2$.
+Following \autoref{eq:PE_grad} and \autoref{eq:PE_lap}, the gradient of a variable $q$ defined at
+a $t$-point has its three components defined at $u$-, $v$- and $w$-points while
+its Laplacien is defined at $t$-point.
+These operators have the following discrete forms in the curvilinear $s$-coordinate system:
 \begin{equation} \label{eq:DOM_grad}
 \nabla q\equiv    \frac{1}{e_{1u} } \delta _{i+1/2 } [q] \;\,\mathbf{i}
 …
 \end{multline}
 Following \autoref{eq:PE_curl} and \autoref{eq:PE_div}, a vector ${\rm {\bf A}}=\left( a_1,a_2,a_3\right)$
 defined at vector points $(u,v,w)$ has its three curl components defined at $vw$-, $uw$,
 and $f$-points, and its divergence defined at $t$-points:
+Following \autoref{eq:PE_curl} and \autoref{eq:PE_div}, a vector ${\rm {\bf A}}=\left( a_1,a_2,a_3\right)$
+defined at vector points $(u,v,w)$ has its three curl components defined at $vw$-, $uw$, and $f$-points,
+and its divergence defined at $t$-points:
 \begin{eqnarray}  \label{eq:DOM_curl}
  \nabla \times {\rm{\bf A}}\equiv &
 …
 \end{eqnarray}
 The vertical average over the whole water column denoted by an overbar becomes
 for a quantity $q$ which is a masked field (i.e. equal to zero inside solid area):
+The vertical average over the whole water column denoted by an overbar becomes for a quantity $q$ which
+is a masked field (i.e. equal to zero inside solid area):
 \begin{equation} \label{eq:DOM_bar}
 \bar q   =         \frac{1}{H}    \int_{k^b}^{k^o} {q\;e_{3q} \,dk}
       \equiv \frac{1}{H_q }\sum\limits_k {q\;e_{3q} }
 \end{equation}
 where $H_q$  is the ocean depth, which is the masked sum of the vertical scale
+factors at $q$ points, $k^b$ and $k^o$ are the bottom and surface $k$-indices,
 and the symbol $k^o$ refers to a summation over all grid points of the same type
 in the direction indicated by the subscript (here $k$).
+where $H_q$  is the ocean depth, which is the masked sum of the vertical scale factors at $q$ points,
+$k^b$ and $k^o$ are the bottom and surface $k$-indices,
+and the symbol $k^o$ refers to a summation over all grid points of the same type in the direction indicated by
+the subscript (here $k$).
 In continuous form, the following properties are satisfied:
 …
 \end{equation}
 It is straightforward to demonstrate that these properties are verified locally in
+discrete form as soon as the scalar $q$ is taken at $t$-points and the vector
 \textbf{A} has its components defined at vector points $(u,v,w)$.
 Let $a$ and $b$ be two fields defined on the mesh, with value zero inside
+continental area. Using integration by parts it can be shown that the differencing
+operators ($\delta_i$, $\delta_j$ and $\delta_k$) are skew-symmetric linear operators,
 and further that the averaging operators $\overline{\,\cdot\,}^{\,i}$,
 $\overline{\,\cdot\,}^{\,k}$ and $\overline{\,\cdot\,}^{\,k}$) are symmetric linear
 operators, $i.e.$
+It is straightforward to demonstrate that these properties are verified locally in discrete form as soon as
+the scalar $q$ is taken at $t$-points and
+the vector \textbf{A} has its components defined at vector points $(u,v,w)$.
+Let $a$ and $b$ be two fields defined on the mesh, with value zero inside continental area.
+Using integration by parts it can be shown that
+the differencing operators ($\delta_i$, $\delta_j$ and $\delta_k$) are skew-symmetric linear operators,
+and further that the averaging operators $\overline{\,\cdot\,}^{\,i}$, $\overline{\,\cdot\,}^{\,k}$ and
+$\overline{\,\cdot\,}^{\,k}$) are symmetric linear operators,
+$i.e.$
 \begin{align}
 \label{eq:DOM_di_adj}
 …
 \end{align}
+In other words, the adjoint of the differencing and averaging operators are
+$\delta_i^*=\delta_{i+1/2}$ and
+In other words, the adjoint of the differencing and averaging operators are $\delta_i^*=\delta_{i+1/2}$ and
 ${(\overline{\,\cdot \,}^{\,i})}^*= \overline{\,\cdot\,}^{\,i+1/2}$, respectively.
 These two properties will be used extensively in the \autoref{apdx:C} to
+These two properties will be used extensively in the \autoref{apdx:C} to
 demonstrate integral conservative properties of the discrete formulation chosen.
 …
 \begin{figure}[!tb]  \begin{center}
 \includegraphics[width=0.90\textwidth]{Fig_index_hor}
 \caption{   \protect\label{fig:index_hor}
+Horizontal integer indexing used in the \textsc{Fortran} code. The dashed area indicates
 the cell in which variables contained in arrays have the same $i$- and $j$-indices}
+\caption{   \protect\label{fig:index_hor}
+  Horizontal integer indexing used in the \textsc{Fortran} code.
+  The dashed area indicates the cell in which variables contained in arrays have the same $i$- and $j$-indices}
 \end{center}   \end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+The array representation used in the \textsc{Fortran} code requires an integer
+indexing while the analytical definition of the mesh (see \autoref{subsec:DOM_cell}) is
+associated with the use of integer values for $t$-points and both integer and
+integer and a half values for all the other points. Therefore a specific integer
+indexing must be defined for points other than $t$-points ($i.e.$ velocity and
+vorticity grid-points). Furthermore, the direction of the vertical indexing has
+been changed so that the surface level is at $k=1$.
+The array representation used in the \textsc{Fortran} code requires an integer indexing while
+the analytical definition of the mesh (see \autoref{subsec:DOM_cell}) is associated with the use of
+integer values for $t$-points and both integer and integer and a half values for all the other points.
+Therefore a specific integer indexing must be defined for points other than $t$-points
+($i.e.$ velocity and vorticity grid-points).
+Furthermore, the direction of the vertical indexing has been changed so that the surface level is at $k=1$.
 % -----------------------------------
 …
 \label{subsec:DOM_Num_Index_hor}
+The indexing in the horizontal plane has been chosen as shown in \autoref{fig:index_hor}.
+For an increasing $i$ index ($j$ index), the $t$-point and the eastward $u$-point
+(northward $v$-point) have the same index (see the dashed area in \autoref{fig:index_hor}).
+The indexing in the horizontal plane has been chosen as shown in \autoref{fig:index_hor}.
+For an increasing $i$ index ($j$ index),
+the $t$-point and the eastward $u$-point (northward $v$-point) have the same index
+(see the dashed area in \autoref{fig:index_hor}).
 A $t$-point and its nearest northeast $f$-point have the same $i$-and $j$-indices.
 …
 \label{subsec:DOM_Num_Index_vertical}
+In the vertical, the chosen indexing requires special attention since the
+$k$-axis is re-orientated downward in the \textsc{Fortran} code compared
+to the indexing used in the semi-discrete equations and given in \autoref{subsec:DOM_cell}.
+The sea surface corresponds to the $w$-level $k=1$ which is the same index
+as $t$-level just below (\autoref{fig:index_vert}). The last $w$-level ($k=jpk$)
+either corresponds to the ocean floor or is inside the bathymetry while the last
+$t$-level is always inside the bathymetry (\autoref{fig:index_vert}). Note that
+for an increasing $k$ index, a $w$-point and the $t$-point just below have the
+same $k$ index, in opposition to what is done in the horizontal plane where
+it is the $t$-point and the nearest velocity points in the direction of the horizontal
+axis that have the same $i$ or $j$ index (compare the dashed area in
+\autoref{fig:index_hor} and \autoref{fig:index_vert}). Since the scale factors are
+chosen to be strictly positive, a \emph{minus sign} appears in the \textsc{Fortran}
+code \emph{before all the vertical derivatives} of the discrete equations given in
+this documentation.
+In the vertical, the chosen indexing requires special attention since
+the $k$-axis is re-orientated downward in the \textsc{Fortran} code compared to
+the indexing used in the semi-discrete equations and given in \autoref{subsec:DOM_cell}.
+The sea surface corresponds to the $w$-level $k=1$ which is the same index as $t$-level just below
+(\autoref{fig:index_vert}).
+The last $w$-level ($k=jpk$) either corresponds to the ocean floor or is inside the bathymetry while
+the last $t$-level is always inside the bathymetry (\autoref{fig:index_vert}).
+Note that for an increasing $k$ index, a $w$-point and the $t$-point just below have the same $k$ index,
+in opposition to what is done in the horizontal plane where
+it is the $t$-point and the nearest velocity points in the direction of the horizontal axis that
+have the same $i$ or $j$ index
+(compare the dashed area in \autoref{fig:index_hor} and \autoref{fig:index_vert}).
+Since the scale factors are chosen to be strictly positive, a \emph{minus sign} appears in the \textsc{Fortran}
+code \emph{before all the vertical derivatives} of the discrete equations given in this documentation.
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 \begin{figure}[!pt]    \begin{center}
 \includegraphics[width=.90\textwidth]{Fig_index_vert}
 \caption{ \protect\label{fig:index_vert}
+Vertical integer indexing used in the \textsc{Fortran } code. Note that
+the $k$-axis is orientated downward. The dashed area indicates the cell in
 which variables contained in arrays have the same $k$-index.}
+\caption{ \protect\label{fig:index_vert}
+  Vertical integer indexing used in the \textsc{Fortran } code.
+  Note that the $k$-axis is orientated downward.
+  The dashed area indicates the cell in which variables contained in arrays have the same $k$-index.}
 \end{center}   \end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 …
 \label{subsec:DOM_size}
 The total size of the computational domain is set by the parameters \np{jpiglo},
 \np{jpjglo} and \np{jpkglo} in the $i$, $j$ and $k$ directions respectively.
+The total size of the computational domain is set by the parameters \np{jpiglo},
+\np{jpjglo} and \np{jpkglo} in the $i$, $j$ and $k$ directions respectively.
 %%%
 %%%
 %%%
+Parameters $jpi$ and $jpj$ refer to the size of each processor subdomain when the code is
+run in parallel using domain decomposition (\key{mpp\_mpi} defined, see \autoref{sec:LBC_mpp}).
+Parameters $jpi$ and $jpj$ refer to the size of each processor subdomain when
+the code is run in parallel using domain decomposition (\key{mpp\_mpi} defined,
+see \autoref{sec:LBC_mpp}).
 …
 \section{Needed fields}
 \label{sec:DOM_fields}
+The ocean mesh ($i.e.$ the position of all the scalar and vector points) is defined
+by the transformation that gives $(\lambda,\varphi,z)$ as a function of $(i,j,k)$.
+The grid-points are located at integer or integer and a half values of as indicated
+in \autoref{tab:cell}. The associated scale factors are defined using the
+analytical first derivative of the transformation \autoref{eq:scale_factors}.
+The ocean mesh ($i.e.$ the position of all the scalar and vector points) is defined by the transformation that gives $(\lambda,\varphi,z)$ as a function of $(i,j,k)$.
+The grid-points are located at integer or integer and a half values of as indicated in \autoref{tab:cell}.
+The associated scale factors are defined using the analytical first derivative of the transformation
+\autoref{eq:scale_factors}.
 Necessary fields for configuration definition are: \\
 Geographic position :
 longitude : glamt , glamu , glamv and glamf  (at T, U, V and F point)
 latitude : gphit , gphiu , gphiv and gphif (at T, U, V and F point)\\
+longitude: glamt, glamu, glamv and glamf (at T, U, V and F point)
+latitude: gphit, gphiu, gphiv and gphif (at T, U, V and F point)\\
 Coriolis parameter (if domain not on the sphere):
 …
  e1t, e1u, e1v and e1f (on i direction),
  e2t, e2u, e2v and e2f (on j direction)
  and ie1e2u\_v, e1e2u , e1e2v
+ e2t, e2u, e2v and e2f (on j direction) and
+ ie1e2u\_v, e1e2u , e1e2v
 e1e2u , e1e2v are u and v surfaces (if gridsize reduction in some straits)\\
 ie1e2u\_v is a flag to flag set u and  v surfaces are neither read nor computed.\\
+These fields can be read in an domain input file which name is setted in \np{cn\_domcfg} parameter specified in \ngn{namcfg}.
+These fields can be read in an domain input file which name is setted in
+\np{cn\_domcfg} parameter specified in \ngn{namcfg}.
 \nlst{namcfg}
 or they can be defined in an analytical way in MY\_SRC directory of the configuration.
+For Reference Configurations of NEMO input domain files are supplied by NEMO System Team. For analytical definition of input fields two routines are supplied: \mdl{userdef\_hgr} and \mdl{userdef\_zgr}. They are an example of GYRE configuration parameters, and they are available in NEMO/OPA\_SRC/USR directory, they provide the horizontal and vertical mesh.
+For Reference Configurations of NEMO input domain files are supplied by NEMO System Team.
+For analytical definition of input fields two routines are supplied: \mdl{userdef\_hgr} and \mdl{userdef\_zgr}.
+They are an example of GYRE configuration parameters, and they are available in NEMO/OPA\_SRC/USR directory,
+they provide the horizontal and vertical mesh.
 % -------------------------------------------------------------------------------------------------------------
 %        Needed fields
 …
 \label{subsec:DOM_hgr_coord_e}
+The ocean mesh ($i.e.$ the position of all the scalar and vector points) is defined
+by the transformation that gives $(\lambda,\varphi,z)$ as a function of $(i,j,k)$.
+The grid-points are located at integer or integer and a half values of as indicated
+in \autoref{tab:cell}. The associated scale factors are defined using the
+analytical first derivative of the transformation \autoref{eq:scale_factors}. These
+definitions are done in two modules, \mdl{domhgr} and \mdl{domzgr}, which
+provide the horizontal and vertical meshes, respectively. This section deals with
+the horizontal mesh parameters.
+In a horizontal plane, the location of all the model grid points is defined from the
+analytical expressions of the longitude $\lambda$ and  latitude $\varphi$ as a
+function of  $(i,j)$. The horizontal scale factors are calculated using
+\autoref{eq:scale_factors}. For example, when the longitude and latitude are
+function of a single value ($i$ and $j$, respectively) (geographical configuration
+of the mesh), the horizontal mesh definition reduces to define the wanted
+$\lambda(i)$, $\varphi(j)$, and their derivatives $\lambda'(i)$ $\varphi'(j)$ in the
+\mdl{domhgr} module. The model computes the grid-point positions and scale
+factors in the horizontal plane as follows:
+The ocean mesh ($i.e.$ the position of all the scalar and vector points) is defined by
+the transformation that gives $(\lambda,\varphi,z)$ as a function of $(i,j,k)$.
+The grid-points are located at integer or integer and a half values of as indicated in \autoref{tab:cell}.
+The associated scale factors are defined using the analytical first derivative of the transformation
+\autoref{eq:scale_factors}.
+These definitions are done in two modules, \mdl{domhgr} and \mdl{domzgr},
+which provide the horizontal and vertical meshes, respectively.
+This section deals with the horizontal mesh parameters.
+In a horizontal plane, the location of all the model grid points is defined from
+the analytical expressions of the longitude $\lambda$ and latitude $\varphi$ as a function of $(i,j)$.
+The horizontal scale factors are calculated using \autoref{eq:scale_factors}.
+For example, when the longitude and latitude are function of a single value
+($i$ and $j$, respectively) (geographical configuration of the mesh),
+the horizontal mesh definition reduces to define the wanted $\lambda(i)$, $\varphi(j)$,
+and their derivatives $\lambda'(i)$ $\varphi'(j)$ in the \mdl{domhgr} module.
+The model computes the grid-point positions and scale factors in the horizontal plane as follows:
 \begin{flalign*}
 \lambda_t &\equiv \text{glamt}= \lambda(i)     & \varphi_t &\equiv \text{gphit} = \varphi(j)\\
 …
 e_{2f} &\equiv \text{e2t} = r_a |\varphi'(j+1/2)|
 \end{flalign*}
 where the last letter of each computational name indicates the grid point
+considered and $r_a$ is the earth radius (defined in \mdl{phycst} along with
+all universal constants). Note that the horizontal position of and scale factors
 at $w$-points are exactly equal to those of $t$-points, thus no specific arrays
+are defined at $w$-points.
+Note that the definition of the scale factors ($i.e.$ as the analytical first derivative
+of the transformation that gives $(\lambda,\varphi,z)$ as a function of $(i,j,k)$) is
+specific to the \NEMO model \citep{Marti_al_JGR92}. As an example, $e_{1t}$ is defined
+locally at a $t$-point, whereas many other models on a C grid choose to define
+such a scale factor as the distance between the $U$-points on each side of the
+$t$-point. Relying on an analytical transformation has two advantages: firstly, there
+is no ambiguity in the scale factors appearing in the discrete equations, since they
+are first introduced in the continuous equations; secondly, analytical transformations
+encourage good practice by the definition of smoothly varying grids (rather than
+allowing the user to set arbitrary jumps in thickness between adjacent layers)
+\citep{Treguier1996}. An example of the effect of such a choice is shown in
 \autoref{fig:zgr_e3}.
+where the last letter of each computational name indicates the grid point considered and
+$r_a$ is the earth radius (defined in \mdl{phycst} along with all universal constants).
+Note that the horizontal position of and scale factors at $w$-points are exactly equal to those of $t$-points,
+thus no specific arrays are defined at $w$-points.
+Note that the definition of the scale factors
+($i.e.$ as the analytical first derivative of the transformation that
+gives $(\lambda,\varphi,z)$ as a function of $(i,j,k)$)
+is specific to the \NEMO model \citep{Marti_al_JGR92}.
+As an example, $e_{1t}$ is defined locally at a $t$-point,
+whereas many other models on a C grid choose to define such a scale factor as
+the distance between the $U$-points on each side of the $t$-point.
+Relying on an analytical transformation has two advantages:
+firstly, there is no ambiguity in the scale factors appearing in the discrete equations,
+since they are first introduced in the continuous equations;
+secondly, analytical transformations encourage good practice by the definition of smoothly varying grids
+(rather than allowing the user to set arbitrary jumps in thickness between adjacent layers) \citep{Treguier1996}.
+An example of the effect of such a choice is shown in \autoref{fig:zgr_e3}.
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 \begin{figure}[!t]     \begin{center}
 \includegraphics[width=0.90\textwidth]{Fig_zgr_e3}
+\caption{ \protect\label{fig:zgr_e3}
+Comparison of (a) traditional definitions of grid-point position and grid-size in the vertical,
+and (b) analytically derived grid-point position and scale factors.
+For both grids here,  the same $w$-point depth has been chosen but in (a) the
+$t$-points are set half way between $w$-points while in (b) they are defined from
+an analytical function: $z(k)=5\,(k-1/2)^3 - 45\,(k-1/2)^2 + 140\,(k-1/2) - 150$.
+Note the resulting difference between the value of the grid-size $\Delta_k$ and
+those of the scale factor $e_k$. }
+\caption{ \protect\label{fig:zgr_e3}
+  Comparison of (a) traditional definitions of grid-point position and grid-size in the vertical,
+  and (b) analytically derived grid-point position and scale factors.
+  For both grids here,
+  the same $w$-point depth has been chosen but in (a) the $t$-points are set half way between $w$-points while
+  in (b) they are defined from an analytical function: $z(k)=5\,(k-1/2)^3 - 45\,(k-1/2)^2 + 140\,(k-1/2) - 150$.
+  Note the resulting difference between the value of the grid-size $\Delta_k$ and those of the scale factor $e_k$. }
 \end{center}   \end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 …
 \label{subsec:DOM_hgr_files}
 All the arrays relating to a particular ocean model configuration (grid-point
+position, scale factors, masks) can be saved in files if \np{nn\_msh} $\not= 0$
+(namelist variable in \ngn{namdom}). This can be particularly useful for plots and off-line
+diagnostics. In some cases, the user may choose to make a local modification
+of a scale factor in the code. This is the case in global configurations when
+restricting the width of a specific strait (usually a one-grid-point strait that
+happens to be too wide due to insufficient model resolution). An example
+is Gibraltar Strait in the ORCA2 configuration. When such modifications are done,
+All the arrays relating to a particular ocean model configuration (grid-point position, scale factors, masks)
+can be saved in files if \np{nn\_msh} $\not= 0$ (namelist variable in \ngn{namdom}).
+This can be particularly useful for plots and off-line diagnostics.
+In some cases, the user may choose to make a local modification of a scale factor in the code.
+This is the case in global configurations when restricting the width of a specific strait
+(usually a one-grid-point strait that happens to be too wide due to insufficient model resolution).
+An example is Gibraltar Strait in the ORCA2 configuration.
+When such modifications are done,
 the output grid written when \np{nn\_msh} $\not= 0$ is no more equal to the input grid.
 …
 Variables are defined through the \ngn{namzgr} and \ngn{namdom} namelists.
 In the vertical, the model mesh is determined by four things:
+(1) the bathymetry given in meters ;
+(2) the number of levels of the model (\jp{jpk}) ;
+(3) the analytical transformation $z(i,j,k)$ and the vertical scale factors
+(derivatives of the transformation) ;
+and (4) the masking system, $i.e.$ the number of wet model levels at each
+(1) the bathymetry given in meters;
+(2) the number of levels of the model (\jp{jpk});
+(3) the analytical transformation $z(i,j,k)$ and the vertical scale factors (derivatives of the transformation); and
+(4) the masking system, $i.e.$ the number of wet model levels at each
 $(i,j)$ column of points.
 …
 \begin{figure}[!tb]    \begin{center}
 \includegraphics[width=1.0\textwidth]{Fig_z_zps_s_sps}
+\caption{  \protect\label{fig:z_zps_s_sps}
+The ocean bottom as seen by the model:
+(a) $z$-coordinate with full step,
+(b) $z$-coordinate with partial step,
+(c) $s$-coordinate: terrain following representation,
+(d) hybrid $s-z$ coordinate,
+(e) hybrid $s-z$ coordinate with partial step, and
+(f) same as (e) but in the non-linear free surface (\protect\np{ln\_linssh}\forcode{ = .false.}).
+Note that the non-linear free surface can be used with any of the
+coordinates (a) to (e).}
+\caption{  \protect\label{fig:z_zps_s_sps}
+  The ocean bottom as seen by the model:
+  (a) $z$-coordinate with full step,
+  (b) $z$-coordinate with partial step,
+  (c) $s$-coordinate: terrain following representation,
+  (d) hybrid $s-z$ coordinate,
+  (e) hybrid $s-z$ coordinate with partial step, and
+  (f) same as (e) but in the non-linear free surface (\protect\np{ln\_linssh}\forcode{ = .false.}).
+  Note that the non-linear free surface can be used with any of the 5 coordinates (a) to (e).}
 \end{center}   \end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 The choice of a vertical coordinate, even if it is made through \ngn{namzgr} namelist parameters,
+must be done once of all at the beginning of an experiment. It is not intended as an
+option which can be enabled or disabled in the middle of an experiment. Three main
+choices are offered (\autoref{fig:z_zps_s_sps}a to c): $z$-coordinate with full step
+bathymetry (\np{ln\_zco}\forcode{ = .true.}), $z$-coordinate with partial step bathymetry
+(\np{ln\_zps}\forcode{ = .true.}), or generalized, $s$-coordinate (\np{ln\_sco}\forcode{ = .true.}).
+Hybridation of the three main coordinates are available: $s-z$ or $s-zps$ coordinate
+(\autoref{fig:z_zps_s_sps}d and \autoref{fig:z_zps_s_sps}e). By default a non-linear free surface is used:
+the coordinate follow the time-variation of the free surface so that the transformation is time dependent:
+$z(i,j,k,t)$ (\autoref{fig:z_zps_s_sps}f). When a linear free surface is assumed (\np{ln\_linssh}\forcode{ = .true.}),
+the vertical coordinate are fixed in time, but the seawater can move up and down across the z=0 surface
+must be done once of all at the beginning of an experiment.
+It is not intended as an option which can be enabled or disabled in the middle of an experiment.
+Three main choices are offered (\autoref{fig:z_zps_s_sps}a to c):
+$z$-coordinate with full step bathymetry (\np{ln\_zco}\forcode{ = .true.}),
+$z$-coordinate with partial step bathymetry (\np{ln\_zps}\forcode{ = .true.}),
+or generalized, $s$-coordinate (\np{ln\_sco}\forcode{ = .true.}).
+Hybridation of the three main coordinates are available:
+$s-z$ or $s-zps$ coordinate (\autoref{fig:z_zps_s_sps} and \autoref{fig:z_zps_s_sps}e).
+By default a non-linear free surface is used: the coordinate follow the time-variation of the free surface so that
+the transformation is time dependent: $z(i,j,k,t)$ (\autoref{fig:z_zps_s_sps}f).
+When a linear free surface is assumed (\np{ln\_linssh}\forcode{ = .true.}),
+the vertical coordinate are fixed in time, but the seawater can move up and down across the z=0 surface
 (in other words, the top of the ocean in not a rigid-lid).
 The last choice in terms of vertical coordinate concerns the presence (or not) in the model domain
+of ocean cavities beneath ice shelves. Setting \np{ln\_isfcav} to true allows to manage ocean cavities,
+otherwise they are filled in. This option is currently only available in $z$- or $zps$-coordinate,
+and partial step are also applied at the ocean/ice shelf interface.
+Contrary to the horizontal grid, the vertical grid is computed in the code and no
+provision is made for reading it from a file. The only input file is the bathymetry
+(in meters) (\ifile{bathy\_meter}).
+\footnote{N.B. in full step $z$-coordinate, a \ifile{bathy\_level} file can replace the
 \ifile{bathy\_meter} file, so that the computation of the number of wet ocean point
+in each water column is by-passed}.
 If \np{ln\_isfcav}\forcode{ = .true.}, an extra file input file describing the ice shelf draft
+(in meters) (\ifile{isf\_draft\_meter}) is needed.
+After reading the bathymetry, the algorithm for vertical grid definition differs
 between the different options:
+The last choice in terms of vertical coordinate concerns the presence (or not) in
+the model domain of ocean cavities beneath ice shelves.
+Setting \np{ln\_isfcav} to true allows to manage ocean cavities, otherwise they are filled in.
+This option is currently only available in $z$- or $zps$-coordinate,
+and partial step are also applied at the ocean/ice shelf interface.
+Contrary to the horizontal grid, the vertical grid is computed in the code and
+no provision is made for reading it from a file.
+The only input file is the bathymetry (in meters) (\ifile{bathy\_meter})
+\footnote{
+  N.B. in full step $z$-coordinate, a \ifile{bathy\_level} file can replace the \ifile{bathy\_meter} file,
+  so that the computation of the number of wet ocean point in each water column is by-passed}.
+If \np{ln\_isfcav}\forcode{ = .true.},
+an extra file input file describing the ice shelf draft (in meters) (\ifile{isf\_draft\_meter}) is needed.
+After reading the bathymetry, the algorithm for vertical grid definition differs between the different options:
 \begin{description}
+\item[\textit{zco}] set a reference coordinate transformation $z_0 (k)$, and set $z(i,j,k,t)=z_0 (k)$.
+\item[\textit{zps}] set a reference coordinate transformation $z_0 (k)$, and
+calculate the thickness of the deepest level at each $(i,j)$ point using the
+bathymetry, to obtain the final three-dimensional depth and scale factor arrays.
+\item[\textit{sco}] smooth the bathymetry to fulfil the hydrostatic consistency
+criteria and set the three-dimensional transformation.
+\item[\textit{s-z} and \textit{s-zps}] smooth the bathymetry to fulfil the hydrostatic
+consistency criteria and set the three-dimensional transformation $z(i,j,k)$, and
+possibly introduce masking of extra land points to better fit the original bathymetry file
+\item[\textit{zco}]
+  set a reference coordinate transformation $z_0 (k)$, and set $z(i,j,k,t)=z_0 (k)$.
+\item[\textit{zps}]
+  set a reference coordinate transformation $z_0 (k)$,
+  and calculate the thickness of the deepest level at each $(i,j)$ point using the bathymetry,
+  to obtain the final three-dimensional depth and scale factor arrays.
+\item[\textit{sco}]
+  smooth the bathymetry to fulfil the hydrostatic consistency criteria and
+  set the three-dimensional transformation.
+\item[\textit{s-z} and \textit{s-zps}]
+  smooth the bathymetry to fulfil the hydrostatic consistency criteria and
+  set the three-dimensional transformation $z(i,j,k)$,
+  and possibly introduce masking of extra land points to better fit the original bathymetry file.
 \end{description}
 %%%
 …
 %%%
+Unless a linear free surface is used (\np{ln\_linssh}\forcode{ = .false.}), the arrays describing
+the grid point depths and vertical scale factors are three set of three dimensional arrays $(i,j,k)$
+defined at \textit{before}, \textit{now} and \textit{after} time step. The time at which they are
+defined is indicated by a suffix:$\_b$, $\_n$, or $\_a$, respectively. They are updated at each model time step
+using a fixed reference coordinate system which computer names have a $\_0$ suffix.
+When the linear free surface option is used (\np{ln\_linssh}\forcode{ = .true.}), \textit{before}, \textit{now}
+and \textit{after} arrays are simply set one for all to their reference counterpart.
+Unless a linear free surface is used (\np{ln\_linssh}\forcode{ = .false.}),
+the arrays describing the grid point depths and vertical scale factors are three set of
+three dimensional arrays $(i,j,k)$ defined at \textit{before}, \textit{now} and \textit{after} time step.
+The time at which they are defined is indicated by a suffix:$\_b$, $\_n$, or $\_a$, respectively.
+They are updated at each model time step using a fixed reference coordinate system which
+computer names have a $\_0$ suffix.
+When the linear free surface option is used (\np{ln\_linssh}\forcode{ = .true.}),
+\textit{before}, \textit{now} and \textit{after} arrays are simply set one for all to their reference counterpart.
 …
 \label{subsec:DOM_bathy}
 Three options are possible for defining the bathymetry, according to the
 namelist variable \np{nn\_bathy} (found in \ngn{namdom} namelist):
+Three options are possible for defining the bathymetry, according to the namelist variable \np{nn\_bathy}
+(found in \ngn{namdom} namelist):
 \begin{description}
+\item[\np{nn\_bathy}\forcode{ = 0}]: a flat-bottom domain is defined. The total depth $z_w (jpk)$
+is given by the coordinate transformation. The domain can either be a closed
+basin or a periodic channel depending on the parameter \np{jperio}.
+\item[\np{nn\_bathy}\forcode{ = -1}]: a domain with a bump of topography one third of the
+domain width at the central latitude. This is meant for the "EEL-R5" configuration,
+a periodic or open boundary channel with a seamount.
+\item[\np{nn\_bathy}\forcode{ = 1}]: read a bathymetry and ice shelf draft (if needed).
+ The \ifile{bathy\_meter} file (Netcdf format) provides the ocean depth (positive, in meters)
+ at each grid point of the model grid. The bathymetry is usually built by interpolating a standard bathymetry product
+($e.g.$ ETOPO2) onto the horizontal ocean mesh. Defining the bathymetry also
+defines the coastline: where the bathymetry is zero, no model levels are defined
+(all levels are masked).
+The \ifile{isfdraft\_meter} file (Netcdf format) provides the ice shelf draft (positive, in meters)
+ at each grid point of the model grid. This file is only needed if \np{ln\_isfcav}\forcode{ = .true.}.
+Defining the ice shelf draft will also define the ice shelf edge and the grounding line position.
+\item[\np{nn\_bathy}\forcode{ = 0}]:
+  a flat-bottom domain is defined.
+  The total depth $z_w (jpk)$ is given by the coordinate transformation.
+  The domain can either be a closed basin or a periodic channel depending on the parameter \np{jperio}.
+\item[\np{nn\_bathy}\forcode{ = -1}]:
+  a domain with a bump of topography one third of the domain width at the central latitude.
+  This is meant for the "EEL-R5" configuration, a periodic or open boundary channel with a seamount.
+\item[\np{nn\_bathy}\forcode{ = 1}]:
+  read a bathymetry and ice shelf draft (if needed).
+  The \ifile{bathy\_meter} file (Netcdf format) provides the ocean depth (positive, in meters) at
+  each grid point of the model grid.
+  The bathymetry is usually built by interpolating a standard bathymetry product ($e.g.$ ETOPO2) onto
+  the horizontal ocean mesh.
+  Defining the bathymetry also defines the coastline: where the bathymetry is zero,
+  no model levels are defined (all levels are masked).
+  The \ifile{isfdraft\_meter} file (Netcdf format) provides the ice shelf draft (positive, in meters) at
+  each grid point of the model grid.
+  This file is only needed if \np{ln\_isfcav}\forcode{ = .true.}.
+  Defining the ice shelf draft will also define the ice shelf edge and the grounding line position.
 \end{description}
 When a global ocean is coupled to an atmospheric model it is better to represent
+all large water bodies (e.g, great lakes, Caspian sea...) even if the model
+resolution does not allow their communication with the rest of the ocean.
 This is unnecessary when the ocean is forced by fixed atmospheric conditions,
 so these seas can be removed from the ocean domain. The user has the option
+to set the bathymetry in closed seas to zero (see \autoref{sec:MISC_closea}), but the
 code has to be adapted to the user's configuration.
+When a global ocean is coupled to an atmospheric model it is better to represent all large water bodies
+(e.g, great lakes, Caspian sea...)
+even if the model resolution does not allow their communication with the rest of the ocean.
+This is unnecessary when the ocean is forced by fixed atmospheric conditions,
+so these seas can be removed from the ocean domain.
+The user has the option to set the bathymetry in closed seas to zero (see \autoref{sec:MISC_closea}),
+but the code has to be adapted to the user's configuration.
 % -------------------------------------------------------------------------------------------------------------
 …
 \begin{figure}[!tb]    \begin{center}
 \includegraphics[width=0.90\textwidth]{Fig_zgr}
 \caption{ \protect\label{fig:zgr}
+Default vertical mesh for ORCA2: 30 ocean levels (L30). Vertical level functions for
+(a) T-point depth and (b) the associated scale factor as computed
 from \autoref{eq:DOM_zgr_ana_1} using \autoref{eq:DOM_zgr_coef} in $z$-coordinate.}
+\caption{ \protect\label{fig:zgr}
+  Default vertical mesh for ORCA2: 30 ocean levels (L30).
+  Vertical level functions for (a) T-point depth and (b) the associated scale factor as computed from
+  \autoref{eq:DOM_zgr_ana_1} using \autoref{eq:DOM_zgr_coef} in $z$-coordinate.}
 \end{center}   \end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+The reference coordinate transformation $z_0 (k)$ defines the arrays $gdept_0$
+and $gdepw_0$ for $t$- and $w$-points, respectively. As indicated on
+\autoref{fig:index_vert} \jp{jpk} is the number of $w$-levels. $gdepw_0(1)$ is the
+ocean surface. There are at most \jp{jpk}-1 $t$-points inside the ocean, the
+additional $t$-point at $jk=jpk$ is below the sea floor and is not used.
+The vertical location of $w$- and $t$-levels is defined from the analytic expression
+of the depth $z_0(k)$ whose analytical derivative with respect to $k$ provides the
+vertical scale factors. The user must provide the analytical expression of both
+$z_0$ and its first derivative with respect to $k$. This is done in routine \mdl{domzgr}
+through statement functions, using parameters provided in the \ngn{namcfg} namelist.
+It is possible to define a simple regular vertical grid by giving zero stretching (\np{ppacr=0}).
+In that case, the parameters \jp{jpk} (number of $w$-levels) and \np{pphmax}
+(total ocean depth in meters) fully define the grid.
+For climate-related studies it is often desirable to concentrate the vertical resolution
+near the ocean surface. The following function is proposed as a standard for a
+$z$-coordinate (with either full or partial steps):
+The reference coordinate transformation $z_0 (k)$ defines the arrays $gdept_0$ and $gdepw_0$ for
+$t$- and $w$-points, respectively.
+As indicated on \autoref{fig:index_vert} \jp{jpk} is the number of $w$-levels. $gdepw_0(1)$ is the ocean surface.
+There are at most \jp{jpk}-1 $t$-points inside the ocean,
+the additional $t$-point at $jk=jpk$ is below the sea floor and is not used.
+The vertical location of $w$- and $t$-levels is defined from the analytic expression of the depth $z_0(k)$ whose
+analytical derivative with respect to $k$ provides the vertical scale factors.
+The user must provide the analytical expression of both $z_0$ and its first derivative with respect to $k$.
+This is done in routine \mdl{domzgr} through statement functions,
+using parameters provided in the \ngn{namcfg} namelist.
+It is possible to define a simple regular vertical grid by giving zero stretching (\np{ppacr=0}).
+In that case,
+the parameters \jp{jpk} (number of $w$-levels) and \np{pphmax} (total ocean depth in meters) fully define the grid.
+For climate-related studies it is often desirable to concentrate the vertical resolution near the ocean surface.
+The following function is proposed as a standard for a $z$-coordinate (with either full or partial steps):
 \begin{equation} \label{eq:DOM_zgr_ana_1}
 \begin{split}
 …
 \end{split}
 \end{equation}
+where $k=1$ to \jp{jpk} for $w$-levels and $k=1$ to $k=1$ for $T-$levels. Such an
+expression allows us to define a nearly uniform vertical location of levels at the
+ocean top and bottom with a smooth hyperbolic tangent transition in between
+(\autoref{fig:zgr}).
+If the ice shelf cavities are opened (\np{ln\_isfcav}\forcode{ = .true.}), the definition of $z_0$ is the same.
+where $k=1$ to \jp{jpk} for $w$-levels and $k=1$ to $k=1$ for $T-$levels.
+Such an expression allows us to define a nearly uniform vertical location of levels at the ocean top and bottom with
+a smooth hyperbolic tangent transition in between (\autoref{fig:zgr}).
+If the ice shelf cavities are opened (\np{ln\_isfcav}\forcode{ = .true.}), the definition of $z_0$ is the same.
 However, definition of $e_3^0$ at $t$- and $w$-points is respectively changed to:
 \begin{equation} \label{eq:DOM_zgr_ana_2}
 …
 The most used vertical grid for ORCA2 has $10~m$ ($500~m)$ resolution in the
+surface (bottom) layers and a depth which varies from 0 at the sea surface to a
 minimum of $-5000~m$. This leads to the following conditions:
+The most used vertical grid for ORCA2 has $10~m$ ($500~m)$ resolution in the surface (bottom) layers and
+a depth which varies from 0 at the sea surface to a minimum of $-5000~m$.
+This leads to the following conditions:
 \begin{equation} \label{eq:DOM_zgr_coef}
 \begin{split}
 …
 \end{equation}
+With the choice of the stretching $h_{cr} =3$ and the number of levels
+\jp{jpk}=$31$, the four coefficients $h_{sur}$, $h_{0}$, $h_{1}$, and $h_{th}$ in
+\autoref{eq:DOM_zgr_ana_2} have been determined such that \autoref{eq:DOM_zgr_coef} is
+satisfied, through an optimisation procedure using a bisection method. For the first
+standard ORCA2 vertical grid this led to the following values: $h_{sur} =4762.96$,
+$h_0 =255.58, h_1 =245.5813$, and $h_{th} =21.43336$. The resulting depths and
+scale factors as a function of the model levels are shown in \autoref{fig:zgr} and
+given in \autoref{tab:orca_zgr}. Those values correspond to the parameters
+\np{ppsur}, \np{ppa0}, \np{ppa1}, \np{ppkth} in \ngn{namcfg} namelist.
+Rather than entering parameters $h_{sur}$, $h_{0}$, and $h_{1}$ directly, it is
+possible to recalculate them. In that case the user sets
+\np{ppsur}\forcode{ = }\np{ppa0}\forcode{ = }\np{ppa1}\forcode{ = 999999}., in \ngn{namcfg} namelist,
+and specifies instead the four following parameters:
+With the choice of the stretching $h_{cr} =3$ and the number of levels \jp{jpk}=$31$,
+the four coefficients $h_{sur}$, $h_{0}$, $h_{1}$, and $h_{th}$ in
+\autoref{eq:DOM_zgr_ana_2} have been determined such that
+\autoref{eq:DOM_zgr_coef} is satisfied, through an optimisation procedure using a bisection method.
+For the first standard ORCA2 vertical grid this led to the following values:
+$h_{sur} =4762.96$, $h_0 =255.58, h_1 =245.5813$, and $h_{th} =21.43336$.
+The resulting depths and scale factors as a function of the model levels are shown in
+\autoref{fig:zgr} and given in \autoref{tab:orca_zgr}.
+Those values correspond to the parameters \np{ppsur}, \np{ppa0}, \np{ppa1}, \np{ppkth} in \ngn{namcfg} namelist.
+Rather than entering parameters $h_{sur}$, $h_{0}$, and $h_{1}$ directly, it is possible to recalculate them.
+In that case the user sets \np{ppsur}\forcode{ = }\np{ppa0}\forcode{ = }\np{ppa1}\forcode{ = 999999}.,
+in \ngn{namcfg} namelist, and specifies instead the four following parameters:
 \begin{itemize}
+\item    \np{ppacr}=$h_{cr} $: stretching factor (nondimensional). The larger
+\np{ppacr}, the smaller the stretching. Values from $3$ to $10$ are usual.
+\item    \np{ppkth}=$h_{th} $: is approximately the model level at which maximum
+stretching occurs (nondimensional, usually of order 1/2 or 2/3 of \jp{jpk})
+\item    \np{ppdzmin}: minimum thickness for the top layer (in meters)
+\item    \np{pphmax}: total depth of the ocean (meters).
+\item
+  \np{ppacr}=$h_{cr} $: stretching factor (nondimensional).
+  The larger \np{ppacr}, the smaller the stretching.
+  Values from $3$ to $10$ are usual.
+\item
+  \np{ppkth}=$h_{th} $: is approximately the model level at which maximum stretching occurs
+  (nondimensional, usually of order 1/2 or 2/3 of \jp{jpk})
+\item
+  \np{ppdzmin}: minimum thickness for the top layer (in meters).
+\item
+  \np{pphmax}: total depth of the ocean (meters).
 \end{itemize}
+As an example, for the $45$ layers used in the DRAKKAR configuration those
+parameters are: \jp{jpk}\forcode{ = 46}, \np{ppacr}\forcode{ = 9}, \np{ppkth}\forcode{ = 23.563}, \np{ppdzmin}\forcode{ = 6}m, \np{pphmax}\forcode{ = 5750}m.
+As an example, for the $45$ layers used in the DRAKKAR configuration those parameters are:
+\jp{jpk}\forcode{ = 46}, \np{ppacr}\forcode{ = 9}, \np{ppkth}\forcode{ = 23.563},
+\np{ppdzmin}\forcode{ = 6}m, \np{pphmax}\forcode{ = 5750}m.
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 …
 &  \textbf{5250.23}& 5000.00 &   \textbf{500.56} & 500.33 \\ \hline
 \end{tabular} \end{center}
 \caption{ \protect\label{tab:orca_zgr}
+Default vertical mesh in $z$-coordinate for 30 layers ORCA2 configuration as computed
 from \autoref{eq:DOM_zgr_ana_2} using the coefficients given in \autoref{eq:DOM_zgr_coef}}
+\caption{ \protect\label{tab:orca_zgr}
+  Default vertical mesh in $z$-coordinate for 30 layers ORCA2 configuration as computed from
+  \autoref{eq:DOM_zgr_ana_2} using the coefficients given in \autoref{eq:DOM_zgr_coef}}
 \end{table}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 …
 %--------------------------------------------------------------------------------------------------------------
+In $z$-coordinate partial step, the depths of the model levels are defined by the
+reference analytical function $z_0 (k)$ as described in the previous
+section, \emph{except} in the bottom layer. The thickness of the bottom layer is
+allowed to vary as a function of geographical location $(\lambda,\varphi)$ to allow a
+better representation of the bathymetry, especially in the case of small
+slopes (where the bathymetry varies by less than one level thickness from
+one grid point to the next). The reference layer thicknesses $e_{3t}^0$ have been
+defined in the absence of bathymetry. With partial steps, layers from 1 to
+\jp{jpk}-2 can have a thickness smaller than $e_{3t}(jk)$. The model deepest layer (\jp{jpk}-1)
+is allowed to have either a smaller or larger thickness than $e_{3t}(jpk)$: the
+maximum thickness allowed is $2*e_{3t}(jpk-1)$. This has to be kept in mind when
+specifying values in \ngn{namdom} namelist, as the maximum depth \np{pphmax}
+in partial steps: for example, with
+\np{pphmax}$=5750~m$ for the DRAKKAR 45 layer grid, the maximum ocean depth
+allowed is actually $6000~m$ (the default thickness $e_{3t}(jpk-1)$ being $250~m$).
+Two variables in the namdom namelist are used to define the partial step
+vertical grid. The mimimum water thickness (in meters) allowed for a cell
+partially filled with bathymetry at level jk is the minimum of \np{rn\_e3zps\_min}
+(thickness in meters, usually $20~m$) or $e_{3t}(jk)*$\np{rn\_e3zps\_rat} (a fraction,
+usually 10\%, of the default thickness $e_{3t}(jk)$).
+In $z$-coordinate partial step,
+the depths of the model levels are defined by the reference analytical function $z_0 (k)$ as described in
+the previous section, \emph{except} in the bottom layer.
+The thickness of the bottom layer is allowed to vary as a function of geographical location $(\lambda,\varphi)$ to
+allow a better representation of the bathymetry, especially in the case of small slopes
+(where the bathymetry varies by less than one level thickness from one grid point to the next).
+The reference layer thicknesses $e_{3t}^0$ have been defined in the absence of bathymetry.
+With partial steps, layers from 1 to \jp{jpk}-2 can have a thickness smaller than $e_{3t}(jk)$.
+The model deepest layer (\jp{jpk}-1) is allowed to have either a smaller or larger thickness than $e_{3t}(jpk)$:
+the maximum thickness allowed is $2*e_{3t}(jpk-1)$.
+This has to be kept in mind when specifying values in \ngn{namdom} namelist,
+as the maximum depth \np{pphmax} in partial steps:
+for example, with \np{pphmax}$=5750~m$ for the DRAKKAR 45 layer grid,
+the maximum ocean depth allowed is actually $6000~m$ (the default thickness $e_{3t}(jpk-1)$ being $250~m$).
+Two variables in the namdom namelist are used to define the partial step vertical grid.
+The mimimum water thickness (in meters) allowed for a cell partially filled with bathymetry at level jk is
+the minimum of \np{rn\_e3zps\_min} (thickness in meters, usually $20~m$) or $e_{3t}(jk)*$\np{rn\_e3zps\_rat}
+(a fraction, usually 10\%, of the default thickness $e_{3t}(jk)$).
 \gmcomment{ \colorbox{yellow}{Add a figure here of pstep especially at last ocean level }  }
 …
 %--------------------------------------------------------------------------------------------------------------
 Options are defined in \ngn{namzgr\_sco}.
+In $s$-coordinate (\np{ln\_sco}\forcode{ = .true.}), the depth and thickness of the model
+levels are defined from the product of a depth field and either a stretching
+function or its derivative, respectively:
+In $s$-coordinate (\np{ln\_sco}\forcode{ = .true.}), the depth and thickness of the model levels are defined from
+the product of a depth field and either a stretching function or its derivative, respectively:
 \begin{equation} \label{eq:DOM_sco_ana}
 …
 \end{equation}
 where $h$ is the depth of the last $w$-level ($z_0(k)$) defined at the $t$-point
+location in the horizontal and $z_0(k)$ is a function which varies from $0$ at the sea
+surface to $1$ at the ocean bottom. The depth field $h$ is not necessary the ocean
+depth, since a mixed step-like and bottom-following representation of the
 topography can be used (\autoref{fig:z_zps_s_sps}d-e) or an envelop bathymetry can be defined (\autoref{fig:z_zps_s_sps}f).
 The namelist parameter \np{rn\_rmax} determines the slope at which the terrain-following coordinate intersects
 the sea bed and becomes a pseudo z-coordinate.
 The coordinate can also be hybridised by specifying \np{rn\_sbot\_min} and \np{rn\_sbot\_max}
 as the minimum and maximum depths at which the terrain-following vertical coordinate is calculated.
 Options for stretching the coordinate are provided as examples, but care must be taken to ensure
 that the vertical stretch used is appropriate for the application.
 The original default NEMO s-coordinate stretching is available if neither of the other options
 are specified as true (\np{ln\_s\_SH94}\forcode{ = .false.} and \np{ln\_s\_SF12}\forcode{ = .false.}).
+where $h$ is the depth of the last $w$-level ($z_0(k)$) defined at the $t$-point location in the horizontal and
+$z_0(k)$ is a function which varies from $0$ at the sea surface to $1$ at the ocean bottom.
+The depth field $h$ is not necessary the ocean depth,
+since a mixed step-like and bottom-following representation of the topography can be used
+(\autoref{fig:z_zps_s_sps}d-e) or an envelop bathymetry can be defined (\autoref{fig:z_zps_s_sps}f).
+The namelist parameter \np{rn\_rmax} determines the slope at which
+the terrain-following coordinate intersects the sea bed and becomes a pseudo z-coordinate.
+The coordinate can also be hybridised by specifying \np{rn\_sbot\_min} and \np{rn\_sbot\_max} as
+the minimum and maximum depths at which the terrain-following vertical coordinate is calculated.
+Options for stretching the coordinate are provided as examples,
+but care must be taken to ensure that the vertical stretch used is appropriate for the application.
+The original default NEMO s-coordinate stretching is available if neither of the other options are specified as true
+(\np{ln\_s\_SH94}\forcode{ = .false.} and \np{ln\_s\_SF12}\forcode{ = .false.}).
 This uses a depth independent $\tanh$ function for the stretching \citep{Madec_al_JPO96}:
 …
 \end{equation}
 where $s_{min}$ is the depth at which the $s$-coordinate stretching starts and
 allows a $z$-coordinate to placed on top of the stretched coordinate,
+where $s_{min}$ is the depth at which the $s$-coordinate stretching starts and
+allows a $z$-coordinate to placed on top of the stretched coordinate,
 and $z$ is the depth (negative down from the asea surface).
 …
 \end{equation}
+A stretching function, modified from the commonly used \citet{Song_Haidvogel_JCP94}
+stretching (\np{ln\_s\_SH94}\forcode{ = .true.}), is also available and is more commonly used for shelf seas modelling:
+A stretching function,
+modified from the commonly used \citet{Song_Haidvogel_JCP94} stretching (\np{ln\_s\_SH94}\forcode{ = .true.}),
+is also available and is more commonly used for shelf seas modelling:
 \begin{equation}
 …
 \begin{figure}[!ht]    \begin{center}
 \includegraphics[width=1.0\textwidth]{Fig_sco_function}
 \caption{  \protect\label{fig:sco_function}
+Examples of the stretching function applied to a seamount; from left to right:
 surface, surface and bottom, and bottom intensified resolutions}
+\caption{  \protect\label{fig:sco_function}
+  Examples of the stretching function applied to a seamount;
+  from left to right: surface, surface and bottom, and bottom intensified resolutions}
 \end{center}   \end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+where $H_c$ is the critical depth (\np{rn\_hc}) at which the coordinate transitions from
+pure $\sigma$ to the stretched coordinate,  and $\theta$ (\np{rn\_theta}) and $b$ (\np{rn\_bb})
+are the surface and bottom control parameters such that $0\leqslant \theta \leqslant 20$, and
+$0\leqslant b\leqslant 1$. $b$ has been designed to allow surface and/or bottom
+increase of the vertical resolution (\autoref{fig:sco_function}).
+Another example has been provided at version 3.5 (\np{ln\_s\_SF12}) that allows
+a fixed surface resolution in an analytical terrain-following stretching \citet{Siddorn_Furner_OM12}.
+where $H_c$ is the critical depth (\np{rn\_hc}) at which
+the coordinate transitions from pure $\sigma$ to the stretched coordinate,
+and $\theta$ (\np{rn\_theta}) and $b$ (\np{rn\_bb}) are the surface and bottom control parameters such that
+$0\leqslant \theta \leqslant 20$, and $0\leqslant b\leqslant 1$.
+$b$ has been designed to allow surface and/or bottom increase of the vertical resolution
+(\autoref{fig:sco_function}).
+Another example has been provided at version 3.5 (\np{ln\_s\_SF12}) that allows a fixed surface resolution in
+an analytical terrain-following stretching \citet{Siddorn_Furner_OM12}.
 In this case the a stretching function $\gamma$ is defined such that:
 …
 \end{equation}
+This gives an analytical stretching of $\sigma$ that is solvable in $A$ and $B$ as a function of
+the user prescribed stretching parameter $\alpha$ (\np{rn\_alpha}) that stretches towards
+the surface ($\alpha > 1.0$) or the bottom ($\alpha < 1.0$) and user prescribed surface (\np{rn\_zs})
+and bottom depths. The bottom cell depth in this example is given as a function of water depth:
+This gives an analytical stretching of $\sigma$ that is solvable in $A$ and $B$ as a function of
+the user prescribed stretching parameter $\alpha$ (\np{rn\_alpha}) that stretches towards
+the surface ($\alpha > 1.0$) or the bottom ($\alpha < 1.0$) and
+user prescribed surface (\np{rn\_zs}) and bottom depths.
+The bottom cell depth in this example is given as a function of water depth:
 \begin{equation} \label{eq:DOM_zb}
 …
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 \begin{figure}[!ht]
+   \includegraphics[width=1.0\textwidth]{FIG_DOM_compare_coordinates_surface}
+        \caption{A comparison of the \citet{Song_Haidvogel_JCP94} $S$-coordinate (solid lines), a 50 level $Z$-coordinate (contoured surfaces) and the \citet{Siddorn_Furner_OM12} $S$-coordinate (dashed lines) in the surface 100m for a idealised bathymetry that goes from 50m to 5500m depth. For clarity every third coordinate surface is shown.}
+   \includegraphics[width=1.0\textwidth]{Fig_DOM_compare_coordinates_surface}
+   \caption{
+     A comparison of the \citet{Song_Haidvogel_JCP94} $S$-coordinate (solid lines),
+     a 50 level $Z$-coordinate (contoured surfaces) and
+     the \citet{Siddorn_Furner_OM12} $S$-coordinate (dashed lines) in
+     the surface 100m for a idealised bathymetry that goes from 50m to 5500m depth.
+     For clarity every third coordinate surface is shown.}
     \label{fig:fig_compare_coordinates_surface}
 \end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+This gives a smooth analytical stretching in computational space that is constrained to given specified surface and bottom grid cell thicknesses in real space. This is not to be confused with the hybrid schemes that superimpose geopotential coordinates on terrain following coordinates thus creating a non-analytical vertical coordinate that therefore may suffer from large gradients in the vertical resolutions. This stretching is less straightforward to implement than the \citet{Song_Haidvogel_JCP94} stretching, but has the advantage of resolving diurnal processes in deep water and has generally flatter slopes.
+As with the \citet{Song_Haidvogel_JCP94} stretching the stretch is only applied at depths greater than the critical depth $h_c$. In this example two options are available in depths shallower than $h_c$, with pure sigma being applied if the \np{ln\_sigcrit} is true and pure z-coordinates if it is false (the z-coordinate being equal to the depths of the stretched coordinate at $h_c$.
+Minimising the horizontal slope of the vertical coordinate is important in terrain-following systems as large slopes lead to hydrostatic consistency. A hydrostatic consistency parameter diagnostic following \citet{Haney1991} has been implemented, and is output as part of the model mesh file at the start of the run.
+This gives a smooth analytical stretching in computational space that is constrained to
+given specified surface and bottom grid cell thicknesses in real space.
+This is not to be confused with the hybrid schemes that
+superimpose geopotential coordinates on terrain following coordinates thus
+creating a non-analytical vertical coordinate that
+therefore may suffer from large gradients in the vertical resolutions.
+This stretching is less straightforward to implement than the \citet{Song_Haidvogel_JCP94} stretching,
+but has the advantage of resolving diurnal processes in deep water and has generally flatter slopes.
+As with the \citet{Song_Haidvogel_JCP94} stretching the stretch is only applied at depths greater than
+the critical depth $h_c$.
+In this example two options are available in depths shallower than $h_c$,
+with pure sigma being applied if the \np{ln\_sigcrit} is true and pure z-coordinates if it is false
+(the z-coordinate being equal to the depths of the stretched coordinate at $h_c$).
+Minimising the horizontal slope of the vertical coordinate is important in terrain-following systems as
+large slopes lead to hydrostatic consistency.
+A hydrostatic consistency parameter diagnostic following \citet{Haney1991} has been implemented,
+and is output as part of the model mesh file at the start of the run.
 % -------------------------------------------------------------------------------------------------------------
 …
 \label{subsec:DOM_msk}
+Whatever the vertical coordinate used, the model offers the possibility of
+representing the bottom topography with steps that follow the face of the
+model cells (step like topography) \citep{Madec_al_JPO96}. The distribution of
+the steps in the horizontal is defined in a 2D integer array, mbathy, which
+gives the number of ocean levels ($i.e.$ those that are not masked) at each
+$t$-point. mbathy is computed from the meter bathymetry using the definiton of
+gdept as the number of $t$-points which gdept $\leq$ bathy.
+Modifications of the model bathymetry are performed in the \textit{bat\_ctl}
+routine (see \mdl{domzgr} module) after mbathy is computed. Isolated grid points
+that do not communicate with another ocean point at the same level are eliminated.
+As for the representation of bathymetry, a 2D integer array, misfdep, is created.
+misfdep defines the level of the first wet $t$-point. All the cells between $k=1$ and $misfdep(i,j)-1$ are masked.
+Whatever the vertical coordinate used,
+the model offers the possibility of representing the bottom topography with steps that
+follow the face of the model cells (step like topography) \citep{Madec_al_JPO96}.
+The distribution of the steps in the horizontal is defined in a 2D integer array, mbathy,
+which gives the number of ocean levels ($i.e.$ those that are not masked) at each $t$-point.
+mbathy is computed from the meter bathymetry using the definiton of gdept as
+the number of $t$-points which gdept $\leq$ bathy.
+Modifications of the model bathymetry are performed in the \textit{bat\_ctl} routine (see \mdl{domzgr} module) after
+mbathy is computed.
+Isolated grid points that do not communicate with another ocean point at the same level are eliminated.
+As for the representation of bathymetry, a 2D integer array, misfdep, is created.
+misfdep defines the level of the first wet $t$-point.
+All the cells between $k=1$ and $misfdep(i,j)-1$ are masked.
 By default, misfdep(:,:)=1 and no cells are masked.
 In case of ice shelf cavities, modifications of the model bathymetry and ice shelf draft into
+the cavities are performed in the \textit{zgr\_isf} routine. The compatibility between ice shelf draft and bathymetry is checked.
+the cavities are performed in the \textit{zgr\_isf} routine.
+The compatibility between ice shelf draft and bathymetry is checked.
 All the locations where the isf cavity is thinnest than \np{rn\_isfhmin} meters are grounded ($i.e.$ masked).
+If only one cell on the water column is opened at $t$-, $u$- or $v$-points, the bathymetry or the ice shelf draft is dug to fit this constrain.
+If only one cell on the water column is opened at $t$-, $u$- or $v$-points,
+the bathymetry or the ice shelf draft is dug to fit this constrain.
 If the incompatibility is too strong (need to dig more than 1 cell), the cell is masked.\\
 …
 \end{align*}
 Note that, without ice shelves cavities, masks at $t-$ and $w-$points are identical with
+the numerical indexing used (\autoref{subsec:DOM_Num_Index}). Nevertheless, $wmask$ are required
 with oceean cavities to deal with the top boundary (ice shelf/ocean interface)
+Note that, without ice shelves cavities,
+masks at $t-$ and $w-$points are identical with the numerical indexing used (\autoref{subsec:DOM_Num_Index}).
+Nevertheless, $wmask$ are required with ocean cavities to deal with the top boundary (ice shelf/ocean interface)
 exactly in the same way as for the bottom boundary.
 The specification of closed lateral boundaries requires that at least the first and last
+rows and columns of the \textit{mbathy} array are set to zero. In the particular
+case of an east-west cyclical boundary condition, \textit{mbathy} has its last
 column equal to the second one and its first column equal to the last but one
+The specification of closed lateral boundaries requires that at least
+the first and last rows and columns of the \textit{mbathy} array are set to zero.
+In the particular case of an east-west cyclical boundary condition,
+\textit{mbathy} has its last column equal to the second one and its first column equal to the last but one
 (and so too the mask arrays) (see \autoref{fig:LBC_jperio}).
 …
 Options are defined in \ngn{namtsd}.
+By default, the ocean start from rest (the velocity field is set to zero) and the initialization of
+temperature and salinity fields is controlled through the \np{ln\_tsd\_ini} namelist parameter.
+By default, the ocean start from rest (the velocity field is set to zero) and the initialization of temperature and salinity fields is controlled through the \np{ln\_tsd\_ini} namelist parameter.
 \begin{description}
+\item[\np{ln\_tsd\_init}\forcode{ = .true.}] use a T and S input files that can be given on the model grid itself or
+on their native input data grid. In the latter case, the data will be interpolated on-the-fly both in the
+horizontal and the vertical to the model grid (see \autoref{subsec:SBC_iof}). The information relative to the
+input files are given in the \np{sn\_tem} and \np{sn\_sal} structures.
+The computation is done in the \mdl{dtatsd} module.
+\item[\np{ln\_tsd\_init}\forcode{ = .false.}] use constant salinity value of 35.5 psu and an analytical profile of temperature
+(typical of the tropical ocean), see \rou{istate\_t\_s} subroutine called from \mdl{istate} module.
+\item[\np{ln\_tsd\_init}\forcode{ = .true.}]
+  use a T and S input files that can be given on the model grid itself or on their native input data grid.
+  In the latter case,
+  the data will be interpolated on-the-fly both in the horizontal and the vertical to the model grid
+  (see \autoref{subsec:SBC_iof}).
+  The information relative to the input files are given in the \np{sn\_tem} and \np{sn\_sal} structures.
+  The computation is done in the \mdl{dtatsd} module.
+\item[\np{ln\_tsd\_init}\forcode{ = .false.}]
+  use constant salinity value of 35.5 psu and an analytical profile of temperature (typical of the tropical ocean),
+  see \rou{istate\_t\_s} subroutine called from \mdl{istate} module.
 \end{description}
 \end{document}

NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_DYN.tex

-                      r10146
+                      r10368
 $\ $\newline      %force an empty line
 Using the representation described in \autoref{chap:DOM}, several semi-discrete
 space forms of the dynamical equations are available depending on the vertical
+coordinate used and on the conservation properties of the vorticity term. In all
+the equations presented here, the masking has been omitted for simplicity.
 One must be aware that all the quantities are masked fields and that each time an
 average or difference operator is used, the resulting field is multiplied by a mask.
+Using the representation described in \autoref{chap:DOM},
+several semi-discrete space forms of the dynamical equations are available depending on
+the vertical coordinate used and on the conservation properties of the vorticity term.
+In all the equations presented here, the masking has been omitted for simplicity.
+One must be aware that all the quantities are masked fields and
+that each time an average or difference operator is used, the resulting field is multiplied by a mask.
 The prognostic ocean dynamics equation can be summarized as follows:
 …
          + \text{HPG} + \text{SPG} + \text{LDF} + \text{ZDF}
 \end{equation*}
 NXT stands for next, referring to the time-stepping. The first group of terms on
+the rhs of this equation corresponds to the Coriolis and advection
+terms that are decomposed into either a vorticity part (VOR), a kinetic energy part (KEG)
 and a vertical advection part (ZAD) in the vector invariant formulation, or a Coriolis
+and advection part (COR+ADV) in the flux formulation. The terms following these
+are the pressure gradient contributions (HPG, Hydrostatic Pressure Gradient,
+and SPG, Surface Pressure Gradient); and contributions from lateral diffusion
+(LDF) and vertical diffusion (ZDF), which are added to the rhs in the \mdl{dynldf}
+and \mdl{dynzdf} modules. The vertical diffusion term includes the surface and
+bottom stresses. The external forcings and parameterisations require complex
+inputs (surface wind stress calculation using bulk formulae, estimation of mixing
+coefficients) that are carried out in modules SBC, LDF and ZDF and are described
+in \autoref{chap:SBC}, \autoref{chap:LDF} and \autoref{chap:ZDF}, respectively.
+In the present chapter we also describe the diagnostic equations used to compute
+the horizontal divergence, curl of the velocities (\emph{divcur} module) and
 the vertical velocity (\emph{wzvmod} module).
+NXT stands for next, referring to the time-stepping.
+The first group of terms on the rhs of this equation corresponds to the Coriolis and advection terms that
+are decomposed into either a vorticity part (VOR), a kinetic energy part (KEG) and
+a vertical advection part (ZAD) in the vector invariant formulation,
+or a Coriolis and advection part (COR+ADV) in the flux formulation.
+The terms following these are the pressure gradient contributions
+(HPG, Hydrostatic Pressure Gradient, and SPG, Surface Pressure Gradient);
+and contributions from lateral diffusion (LDF) and vertical diffusion (ZDF),
+which are added to the rhs in the \mdl{dynldf} and \mdl{dynzdf} modules.
+The vertical diffusion term includes the surface and bottom stresses.
+The external forcings and parameterisations require complex inputs
+(surface wind stress calculation using bulk formulae, estimation of mixing coefficients)
+that are carried out in modules SBC, LDF and ZDF and are described in
+\autoref{chap:SBC}, \autoref{chap:LDF} and \autoref{chap:ZDF}, respectively.
+In the present chapter we also describe the diagnostic equations used to compute the horizontal divergence,
+curl of the velocities (\emph{divcur} module) and the vertical velocity (\emph{wzvmod} module).
 The different options available to the user are managed by namelist variables.
 For term \textit{ttt} in the momentum equations, the logical namelist variables are \textit{ln\_dynttt\_xxx},
+where \textit{xxx} is a 3 or 4 letter acronym corresponding to each optional scheme.
+If a CPP key is used for this term its name is \key{ttt}. The corresponding
+code can be found in the \textit{dynttt\_xxx} module in the DYN directory, and it is
+usually computed in the \textit{dyn\_ttt\_xxx} subroutine.
+The user has the option of extracting and outputting each tendency term from the
+D momentum equations (\key{trddyn} defined), as described in
+\autoref{chap:MISC}.  Furthermore, the tendency terms associated with the 2D
+barotropic vorticity balance (when \key{trdvor} is defined) can be derived from the
+D terms.
+where \textit{xxx} is a 3 or 4 letter acronym corresponding to each optional scheme.
+If a CPP key is used for this term its name is \key{ttt}.
+The corresponding code can be found in the \textit{dynttt\_xxx} module in the DYN directory,
+and it is usually computed in the \textit{dyn\_ttt\_xxx} subroutine.
+The user has the option of extracting and outputting each tendency term from the 3D momentum equations
+(\key{trddyn} defined), as described in \autoref{chap:MISC}.
+Furthermore, the tendency terms associated with the 2D barotropic vorticity balance (when \key{trdvor} is defined)
+can be derived from the 3D terms.
 %%%
 \gmcomment{STEVEN: not quite sure I've got the sense of the last sentence. does
 …
 \end{equation}
+The horizontal divergence is defined at a $T$-point. It is given by:
+The horizontal divergence is defined at a $T$-point.
+It is given by:
 \begin{equation} \label{eq:divcur_div}
 \chi =\frac{1}{e_{1t}\,e_{2t}\,e_{3t} }
 …
 \end{equation}
+Note that although the vorticity has the same discrete expression in $z$-
+and $s$-coordinates, its physical meaning is not identical. $\zeta$ is a pseudo
+vorticity along $s$-surfaces (only pseudo because $(u,v)$ are still defined along
+geopotential surfaces, but are not necessarily defined at the same depth).
+The vorticity and divergence at the \textit{before} step are used in the computation
+of the horizontal diffusion of momentum. Note that because they have been
+calculated prior to the Asselin filtering of the \textit{before} velocities, the
+\textit{before} vorticity and divergence arrays must be included in the restart file
+to ensure perfect restartability. The vorticity and divergence at the \textit{now}
+time step are used for the computation of the nonlinear advection and of the
+vertical velocity respectively.
+Note that although the vorticity has the same discrete expression in $z$- and $s$-coordinates,
+its physical meaning is not identical.
+$\zeta$ is a pseudo vorticity along $s$-surfaces
+(only pseudo because $(u,v)$ are still defined along geopotential surfaces,
+but are not necessarily defined at the same depth).
+The vorticity and divergence at the \textit{before} step are used in the computation of
+the horizontal diffusion of momentum.
+Note that because they have been calculated prior to the Asselin filtering of the \textit{before} velocities,
+the \textit{before} vorticity and divergence arrays must be included in the restart file to
+ensure perfect restartability.
+The vorticity and divergence at the \textit{now} time step are used for the computation of
+the nonlinear advection and of the vertical velocity respectively.
 %--------------------------------------------------------------------------------------------------------------
 …
 \label{subsec:DYN_sshwzv}
 The sea surface height is given by :
+The sea surface height is given by:
 \begin{equation} \label{eq:dynspg_ssh}
 \begin{aligned}
 …
 \end{equation}
 where \textit{emp} is the surface freshwater budget (evaporation minus precipitation),
+expressed in Kg/m$^2$/s (which is equal to mm/s), and $\rho _w$=1,035~Kg/m$^3$
+is the reference density of sea water (Boussinesq approximation). If river runoff is
+expressed as a surface freshwater flux (see \autoref{chap:SBC}) then \textit{emp} can be
+written as the evaporation minus precipitation, minus the river runoff.
+The sea-surface height is evaluated using exactly the same time stepping scheme
+as the tracer equation \autoref{eq:tra_nxt}:
+a leapfrog scheme in combination with an Asselin time filter, $i.e.$ the velocity appearing
+in \autoref{eq:dynspg_ssh} is centred in time (\textit{now} velocity).
+This is of paramount importance. Replacing $T$ by the number $1$ in the tracer equation and summing
+over the water column must lead to the sea surface height equation otherwise tracer content
+will not be conserved \citep{Griffies_al_MWR01, Leclair_Madec_OM09}.
+The vertical velocity is computed by an upward integration of the horizontal
+divergence starting at the bottom, taking into account the change of the thickness of the levels :
+expressed in Kg/m$^2$/s (which is equal to mm/s),
+and $\rho _w$=1,035~Kg/m$^3$ is the reference density of sea water (Boussinesq approximation).
+If river runoff is expressed as a surface freshwater flux (see \autoref{chap:SBC}) then
+\textit{emp} can be written as the evaporation minus precipitation, minus the river runoff.
+The sea-surface height is evaluated using exactly the same time stepping scheme as
+the tracer equation \autoref{eq:tra_nxt}:
+a leapfrog scheme in combination with an Asselin time filter,
+$i.e.$ the velocity appearing in \autoref{eq:dynspg_ssh} is centred in time (\textit{now} velocity).
+This is of paramount importance.
+Replacing $T$ by the number $1$ in the tracer equation and summing over the water column must lead to
+the sea surface height equation otherwise tracer content will not be conserved
+\citep{Griffies_al_MWR01, Leclair_Madec_OM09}.
+The vertical velocity is computed by an upward integration of the horizontal divergence starting at the bottom,
+taking into account the change of the thickness of the levels:
 \begin{equation} \label{eq:wzv}
 \left\{   \begin{aligned}
 …
 In the case of a non-linear free surface (\key{vvl}), the top vertical velocity is $-\textit{emp}/\rho_w$,
 as changes in the divergence of the barotropic transport are absorbed into the change
 of the level thicknesses, re-orientated downward.
+as changes in the divergence of the barotropic transport are absorbed into the change of the level thicknesses,
+re-orientated downward.
 \gmcomment{not sure of this...  to be modified with the change in emp setting}
 In the case of a linear free surface, the time derivative in \autoref{eq:wzv} disappears.
 The upper boundary condition applies at a fixed level $z=0$. The top vertical velocity
+is thus equal to the divergence of the barotropic transport ($i.e.$ the first term in the
 right-hand-side of \autoref{eq:dynspg_ssh}).
 Note also that whereas the vertical velocity has the same discrete
+expression in $z$- and $s$-coordinates, its physical meaning is not the same:
 in the second case, $w$ is the velocity normal to the $s$-surfaces.
 Note also that the $k$-axis is re-orientated downwards in the \textsc{fortran} code compared
 to the indexing used in the semi-discrete equations such as \autoref{eq:wzv}
+The upper boundary condition applies at a fixed level $z=0$.
+The top vertical velocity is thus equal to the divergence of the barotropic transport
+($i.e.$ the first term in the right-hand-side of \autoref{eq:dynspg_ssh}).
+Note also that whereas the vertical velocity has the same discrete expression in $z$- and $s$-coordinates,
+its physical meaning is not the same:
+in the second case, $w$ is the velocity normal to the $s$-surfaces.
+Note also that the $k$-axis is re-orientated downwards in the \textsc{fortran} code compared to
+the indexing used in the semi-discrete equations such as \autoref{eq:wzv}
 (see \autoref{subsec:DOM_Num_Index_vertical}).
 …
 %-------------------------------------------------------------------------------------------------------------
+The vector invariant form of the momentum equations is the one most
+often used in applications of the \NEMO ocean model. The flux form option
+(see next section) has been present since version $2$. Options are defined
+through the \ngn{namdyn\_adv} namelist variables
+Coriolis and momentum advection terms are evaluated using a leapfrog
+scheme, $i.e.$ the velocity appearing in these expressions is centred in
+time (\textit{now} velocity).
+At the lateral boundaries either free slip, no slip or partial slip boundary
+conditions are applied following \autoref{chap:LBC}.
+The vector invariant form of the momentum equations is the one most often used in
+applications of the \NEMO ocean model.
+The flux form option (see next section) has been present since version $2$.
+Options are defined through the \ngn{namdyn\_adv} namelist variables Coriolis and
+momentum advection terms are evaluated using a leapfrog scheme,
+$i.e.$ the velocity appearing in these expressions is centred in time (\textit{now} velocity).
+At the lateral boundaries either free slip, no slip or partial slip boundary conditions are applied following
+\autoref{chap:LBC}.
 % -------------------------------------------------------------------------------------------------------------
 …
 Options are defined through the \ngn{namdyn\_vor} namelist variables.
 Four discretisations of the vorticity term (\np{ln\_dynvor\_xxx}\forcode{ = .true.}) are available:
 conserving potential enstrophy of horizontally non-divergent flow (ENS scheme) ;
 conserving horizontal kinetic energy (ENE scheme) ; conserving potential enstrophy for
+the relative vorticity term and horizontal kinetic energy for the planetary vorticity
+term (MIX scheme) ; or conserving both the potential enstrophy of horizontally non-divergent
+flow and horizontal kinetic energy (EEN scheme) (see \autoref{subsec:C_vorEEN}). In the
+case of ENS, ENE or MIX schemes the land sea mask may be slightly modified to ensure the
+consistency of vorticity term with analytical equations (\np{ln\_dynvor\_con}\forcode{ = .true.}).
+The vorticity terms are all computed in dedicated routines that can be found in
 the \mdl{dynvor} module.
+Four discretisations of the vorticity term (\np{ln\_dynvor\_xxx}\forcode{ = .true.}) are available:
+conserving potential enstrophy of horizontally non-divergent flow (ENS scheme);
+conserving horizontal kinetic energy (ENE scheme);
+conserving potential enstrophy for the relative vorticity term and
+horizontal kinetic energy for the planetary vorticity term (MIX scheme);
+or conserving both the potential enstrophy of horizontally non-divergent flow and horizontal kinetic energy
+(EEN scheme) (see \autoref{subsec:C_vorEEN}).
+In the case of ENS, ENE or MIX schemes the land sea mask may be slightly modified to ensure the consistency of
+vorticity term with analytical equations (\np{ln\_dynvor\_con}\forcode{ = .true.}).
+The vorticity terms are all computed in dedicated routines that can be found in the \mdl{dynvor} module.
 %-------------------------------------------------------------
 …
 \label{subsec:DYN_vor_ens}
+In the enstrophy conserving case (ENS scheme), the discrete formulation of the
+vorticity term provides a global conservation of the enstrophy
+($ [ (\zeta +f ) / e_{3f} ]^2 $ in $s$-coordinates) for a horizontally non-divergent
+flow ($i.e.$ $\chi$=$0$), but does not conserve the total kinetic energy. It is given by:
+In the enstrophy conserving case (ENS scheme),
+the discrete formulation of the vorticity term provides a global conservation of the enstrophy
+($ [ (\zeta +f ) / e_{3f} ]^2 $ in $s$-coordinates) for a horizontally non-divergent flow ($i.e.$ $\chi$=$0$),
+but does not conserve the total kinetic energy.
+It is given by:
 \begin{equation} \label{eq:dynvor_ens}
 \left\{
 …
 \label{subsec:DYN_vor_ene}
 The kinetic energy conserving scheme (ENE scheme) conserves the global
 kinetic energy but not the global enstrophy. It is given by:
+The kinetic energy conserving scheme (ENE scheme) conserves the global kinetic energy but not the global enstrophy.
+It is given by:
 \begin{equation} \label{eq:dynvor_ene}
 \left\{   \begin{aligned}
 …
 \label{subsec:DYN_vor_mix}
+For the mixed energy/enstrophy conserving scheme (MIX scheme), a mixture of the
+two previous schemes is used. It consists of the ENS scheme (\autoref{eq:dynvor_ens})
+for the relative vorticity term, and of the ENE scheme (\autoref{eq:dynvor_ene}) applied
+to the planetary vorticity term.
+For the mixed energy/enstrophy conserving scheme (MIX scheme), a mixture of the two previous schemes is used.
+It consists of the ENS scheme (\autoref{eq:dynvor_ens}) for the relative vorticity term,
+and of the ENE scheme (\autoref{eq:dynvor_ene}) applied to the planetary vorticity term.
 \begin{equation} \label{eq:dynvor_mix}
 \left\{ {     \begin{aligned}
 …
 \label{subsec:DYN_vor_een}
 In both the ENS and ENE schemes, it is apparent that the combination of $i$ and $j$
+averages of the velocity allows for the presence of grid point oscillation structures
 that will be invisible to the operator. These structures are \textit{computational modes}
+that will be at least partly damped by the momentum diffusion operator ($i.e.$ the
+subgrid-scale advection), but not by the resolved advection term. The ENS and ENE schemes
 therefore do not contribute to dump any grid point noise in the horizontal velocity field.
 Such noise would result in more noise in the vertical velocity field, an undesirable feature.
 This is a well-known characteristic of $C$-grid discretization where $u$ and $v$ are located
+at different grid points, a price worth paying to avoid a double averaging in the pressure
+gradient term as in the $B$-grid.
+In both the ENS and ENE schemes,
+it is apparent that the combination of $i$ and $j$ averages of the velocity allows for
+the presence of grid point oscillation structures that will be invisible to the operator.
+These structures are \textit{computational modes} that will be at least partly damped by
+the momentum diffusion operator ($i.e.$ the subgrid-scale advection), but not by the resolved advection term.
+The ENS and ENE schemes therefore do not contribute to dump any grid point noise in the horizontal velocity field.
+Such noise would result in more noise in the vertical velocity field, an undesirable feature.
+This is a well-known characteristic of $C$-grid discretization where
+$u$ and $v$ are located at different grid points,
+a price worth paying to avoid a double averaging in the pressure gradient term as in the $B$-grid.
 \gmcomment{ To circumvent this, Adcroft (ADD REF HERE)
 Nevertheless, this technique strongly distort the phase and group velocity of Rossby waves....}
 A very nice solution to the problem of double averaging was proposed by \citet{Arakawa_Hsu_MWR90}.
 The idea is to get rid of the double averaging by considering triad combinations of vorticity.
+A very nice solution to the problem of double averaging was proposed by \citet{Arakawa_Hsu_MWR90}.
+The idea is to get rid of the double averaging by considering triad combinations of vorticity.
 It is noteworthy that this solution is conceptually quite similar to the one proposed by
 \citep{Griffies_al_JPO98} for the discretization of the iso-neutral diffusion operator (see \autoref{apdx:C}).
 …
 q  = \frac{\zeta +f} {e_{3f} }
 \end{equation}
 where the relative vorticity is defined by (\autoref{eq:divcur_cur}), the Coriolis parameter
 is given by $f=2 \,\Omega \;\sin \varphi _f $ and the layer thickness at $f$-points is:
+where the relative vorticity is defined by (\autoref{eq:divcur_cur}),
+the Coriolis parameter is given by $f=2 \,\Omega \;\sin \varphi _f $ and the layer thickness at $f$-points is:
 \begin{equation} \label{eq:een_e3f}
 e_{3f} = \overline{\overline {e_{3t} }} ^{\,i+1/2,j+1/2}
 …
 \begin{figure}[!ht]    \begin{center}
 \includegraphics[width=0.70\textwidth]{Fig_DYN_een_triad}
 \caption{ \protect\label{fig:DYN_een_triad}
+Triads used in the energy and enstrophy conserving scheme (een) for
 $u$-component (upper panel) and $v$-component (lower panel).}
+\caption{ \protect\label{fig:DYN_een_triad}
+  Triads used in the energy and enstrophy conserving scheme (een) for
+  $u$-component (upper panel) and $v$-component (lower panel).}
 \end{center}   \end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 A key point in \autoref{eq:een_e3f} is how the averaging in the \textbf{i}- and \textbf{j}- directions is made.
 It uses the sum of masked t-point vertical scale factor divided either
+by the sum of the four t-point masks (\np{nn\_een\_e3f}\forcode{ = 1}),
+or  just by $4$ (\np{nn\_een\_e3f}\forcode{ = .true.}).
+The latter case preserves the continuity of $e_{3f}$ when one or more of the neighbouring $e_{3t}$
+tends to zero and extends by continuity the value of $e_{3f}$ into the land areas.
+This case introduces a sub-grid-scale topography at f-points (with a systematic reduction of $e_{3f}$
+when a model level intercept the bathymetry) that tends to reinforce the topostrophy of the flow
+It uses the sum of masked t-point vertical scale factor divided either by the sum of the four t-point masks
+(\np{nn\_een\_e3f}\forcode{ = 1}), or just by $4$ (\np{nn\_een\_e3f}\forcode{ = .true.}).
+The latter case preserves the continuity of $e_{3f}$ when one or more of the neighbouring $e_{3t}$ tends to zero and
+extends by continuity the value of $e_{3f}$ into the land areas.
+This case introduces a sub-grid-scale topography at f-points
+(with a systematic reduction of $e_{3f}$ when a model level intercept the bathymetry)
+that tends to reinforce the topostrophy of the flow
 ($i.e.$ the tendency of the flow to follow the isobaths) \citep{Penduff_al_OS07}.
 Next, the vorticity triads, $ {^i_j}\mathbb{Q}^{i_p}_{j_p}$ can be defined at a $T$-point as
 the following triad combinations of the neighbouring potential vorticities defined at f-points
+Next, the vorticity triads, $ {^i_j}\mathbb{Q}^{i_p}_{j_p}$ can be defined at a $T$-point as
+the following triad combinations of the neighbouring potential vorticities defined at f-points
 (\autoref{fig:DYN_een_triad}):
 \begin{equation} \label{eq:Q_triads}
 …
 \end{equation}
 This EEN scheme in fact combines the conservation properties of the ENS and ENE schemes.
 It conserves both total energy and potential enstrophy in the limit of horizontally
 nondivergent flow ($i.e.$ $\chi$=$0$) (see \autoref{subsec:C_vorEEN}).
 Applied to a realistic ocean configuration, it has been shown that it leads to a significant
+reduction of the noise in the vertical velocity field \citep{Le_Sommer_al_OM09}.
+This EEN scheme in fact combines the conservation properties of the ENS and ENE schemes.
+It conserves both total energy and potential enstrophy in the limit of horizontally nondivergent flow
+($i.e.$ $\chi$=$0$) (see \autoref{subsec:C_vorEEN}).
+Applied to a realistic ocean configuration, it has been shown that it leads to a significant reduction of
+the noise in the vertical velocity field \citep{Le_Sommer_al_OM09}.
 Furthermore, used in combination with a partial steps representation of bottom topography,
 it improves the interaction between current and topography, leading to a larger
 topostrophy of the flow  \citep{Barnier_al_OD06, Penduff_al_OS07}.
+it improves the interaction between current and topography,
+leading to a larger topostrophy of the flow \citep{Barnier_al_OD06, Penduff_al_OS07}.
 %--------------------------------------------------------------------------------------------------------------
 …
 \label{subsec:DYN_keg}
+As demonstrated in \autoref{apdx:C}, there is a single discrete formulation
+of the kinetic energy gradient term that, together with the formulation chosen for
+the vertical advection (see below), conserves the total kinetic energy:
+As demonstrated in \autoref{apdx:C},
+there is a single discrete formulation of the kinetic energy gradient term that,
+together with the formulation chosen for the vertical advection (see below),
+conserves the total kinetic energy:
 \begin{equation} \label{eq:dynkeg}
 \left\{ \begin{aligned}
 …
 \label{subsec:DYN_zad}
+The discrete formulation of the vertical advection, together with the formulation
+chosen for the gradient of kinetic energy (KE) term, conserves the total kinetic
+energy. Indeed, the change of KE due to the vertical advection is exactly
+balanced by the change of KE due to the gradient of KE (see \autoref{apdx:C}).
+The discrete formulation of the vertical advection, t
+ogether with the formulation chosen for the gradient of kinetic energy (KE) term,
+conserves the total kinetic energy.
+Indeed, the change of KE due to the vertical advection is exactly balanced by
+the change of KE due to the gradient of KE (see \autoref{apdx:C}).
 \begin{equation} \label{eq:dynzad}
 \left\{     \begin{aligned}
 …
 \end{aligned}         \right.
 \end{equation}
 When \np{ln\_dynzad\_zts}\forcode{ = .true.}, a split-explicit time stepping with 5 sub-timesteps is used
 on the vertical advection term.
+When \np{ln\_dynzad\_zts}\forcode{ = .true.},
+a split-explicit time stepping with 5 sub-timesteps is used on the vertical advection term.
 This option can be useful when the value of the timestep is limited by vertical advection \citep{Lemarie_OM2015}.
 Note that in this case, a similar split-explicit time stepping should be used on
+vertical advection of tracer to ensure a better stability,
+Note that in this case,
+a similar split-explicit time stepping should be used on vertical advection of tracer to ensure a better stability,
 an option which is only available with a TVD scheme (see \np{ln\_traadv\_tvd\_zts} in \autoref{subsec:TRA_adv_tvd}).
 …
 Options are defined through the \ngn{namdyn\_adv} namelist variables.
 In the flux form (as in the vector invariant form), the Coriolis and momentum
+advection terms are evaluated using a leapfrog scheme, $i.e.$ the velocity
+appearing in their expressions is centred in time (\textit{now} velocity). At the
+lateral boundaries either free slip, no slip or partial slip boundary conditions
 are applied following \autoref{chap:LBC}.
+In the flux form (as in the vector invariant form),
+the Coriolis and momentum advection terms are evaluated using a leapfrog scheme,
+$i.e.$ the velocity appearing in their expressions is centred in time (\textit{now} velocity).
+At the lateral boundaries either free slip,
+no slip or partial slip boundary conditions are applied following \autoref{chap:LBC}.
 …
 \label{subsec:DYN_cor_flux}
 In flux form, the vorticity term reduces to a Coriolis term in which the Coriolis
+parameter has been modified to account for the "metric" term. This altered
 Coriolis parameter is thus discretised at $f$-points. It is given by:
+In flux form, the vorticity term reduces to a Coriolis term in which the Coriolis parameter has been modified to account for the "metric" term.
+This altered Coriolis parameter is thus discretised at $f$-points.
+It is given by:
 \begin{multline} \label{eq:dyncor_metric}
 f+\frac{1}{e_1 e_2 }\left( {v\frac{\partial e_2 }{\partial i}  -  u\frac{\partial e_1 }{\partial j}} \right)  \\
 …
 \end{multline}
+Any of the (\autoref{eq:dynvor_ens}), (\autoref{eq:dynvor_ene}) and (\autoref{eq:dynvor_een})
+schemes can be used to compute the product of the Coriolis parameter and the
+vorticity. However, the energy-conserving scheme (\autoref{eq:dynvor_een}) has
+exclusively been used to date. This term is evaluated using a leapfrog scheme,
+$i.e.$ the velocity is centred in time (\textit{now} velocity).
+Any of the (\autoref{eq:dynvor_ens}), (\autoref{eq:dynvor_ene}) and (\autoref{eq:dynvor_een}) schemes can be used to
+compute the product of the Coriolis parameter and the vorticity.
+However, the energy-conserving scheme (\autoref{eq:dynvor_een}) has exclusively been used to date.
+This term is evaluated using a leapfrog scheme, $i.e.$ the velocity is centred in time (\textit{now} velocity).
 %--------------------------------------------------------------------------------------------------------------
 …
 \label{subsec:DYN_adv_flux}
 The discrete expression of the advection term is given by :
+The discrete expression of the advection term is given by:
 \begin{equation} \label{eq:dynadv}
 \left\{
 …
 \end{equation}
 Two advection schemes are available: a $2^{nd}$ order centered finite
+difference scheme, CEN2, or a $3^{rd}$ order upstream biased scheme, UBS.
+The latter is described in \citet{Shchepetkin_McWilliams_OM05}. The schemes are
+selected using the namelist logicals \np{ln\_dynadv\_cen2} and \np{ln\_dynadv\_ubs}.
 In flux form, the schemes differ by the choice of a space and time interpolation to
+define the value of $u$ and $v$ at the centre of each face of $u$- and $v$-cells,
 $i.e.$ at the $T$-, $f$-, and $uw$-points for $u$ and at the $f$-, $T$- and
 $vw$-points for $v$.
+Two advection schemes are available:
+a $2^{nd}$ order centered finite difference scheme, CEN2,
+or a $3^{rd}$ order upstream biased scheme, UBS.
+The latter is described in \citet{Shchepetkin_McWilliams_OM05}.
+The schemes are selected using the namelist logicals \np{ln\_dynadv\_cen2} and \np{ln\_dynadv\_ubs}.
+In flux form, the schemes differ by the choice of a space and time interpolation to define the value of
+$u$ and $v$ at the centre of each face of $u$- and $v$-cells, $i.e.$ at the $T$-, $f$-,
+and $uw$-points for $u$ and at the $f$-, $T$- and $vw$-points for $v$.
 %-------------------------------------------------------------
 …
 \label{subsec:DYN_adv_cen2}
+In the centered $2^{nd}$ order formulation, the velocity is evaluated as the
+mean of the two neighbouring points :
+In the centered $2^{nd}$ order formulation, the velocity is evaluated as the mean of the two neighbouring points:
 \begin{equation} \label{eq:dynadv_cen2}
 \left\{     \begin{aligned}
 …
 \end{equation}
 The scheme is non diffusive (i.e. conserves the kinetic energy) but dispersive
+($i.e.$ it may create false extrema). It is therefore notoriously noisy and must be
+used in conjunction with an explicit diffusion operator to produce a sensible solution.
 The associated time-stepping is performed using a leapfrog scheme in conjunction
 with an Asselin time-filter, so $u$ and $v$ are the \emph{now} velocities.
+The scheme is non diffusive (i.e. conserves the kinetic energy) but dispersive ($i.e.$ it may create false extrema).
+It is therefore notoriously noisy and must be used in conjunction with an explicit diffusion operator to
+produce a sensible solution.
+The associated time-stepping is performed using a leapfrog scheme in conjunction with an Asselin time-filter,
+so $u$ and $v$ are the \emph{now} velocities.
 %-------------------------------------------------------------
 …
 \label{subsec:DYN_adv_ubs}
 The UBS advection scheme is an upstream biased third order scheme based on
 an upstream-biased parabolic interpolation. For example, the evaluation of
 $u_T^{ubs} $ is done as follows:
+The UBS advection scheme is an upstream biased third order scheme based on
+an upstream-biased parabolic interpolation.
+For example, the evaluation of $u_T^{ubs} $ is done as follows:
 \begin{equation} \label{eq:dynadv_ubs}
 u_T^{ubs} =\overline u ^i-\;\frac{1}{6}   \begin{cases}
 …
 \end{cases}
 \end{equation}
+where $u"_{i+1/2} =\delta _{i+1/2} \left[ {\delta _i \left[ u \right]} \right]$. This results
+in a dissipatively dominant ($i.e.$ hyper-diffusive) truncation error \citep{Shchepetkin_McWilliams_OM05}.
+The overall performance of the advection scheme is similar to that reported in
+\citet{Farrow1995}. It is a relatively good compromise between accuracy and
+smoothness. It is not a \emph{positive} scheme, meaning that false extrema are
+permitted. But the amplitudes of the false extrema are significantly reduced over
+those in the centred second order method. As the scheme already includes
+a diffusion component, it can be used without explicit lateral diffusion on momentum
+($i.e.$ \np{ln\_dynldf\_lap}\forcode{ = }\np{ln\_dynldf\_bilap}\forcode{ = .false.}), and it is recommended to do so.
+The UBS scheme is not used in all directions. In the vertical, the centred $2^{nd}$
+order evaluation of the advection is preferred, $i.e.$ $u_{uw}^{ubs}$ and
+$u_{vw}^{ubs}$ in \autoref{eq:dynadv_cen2} are used. UBS is diffusive and is
+associated with vertical mixing of momentum. \gmcomment{ gm  pursue the
+where $u"_{i+1/2} =\delta _{i+1/2} \left[ {\delta _i \left[ u \right]} \right]$.
+This results in a dissipatively dominant ($i.e.$ hyper-diffusive) truncation error
+\citep{Shchepetkin_McWilliams_OM05}.
+The overall performance of the advection scheme is similar to that reported in \citet{Farrow1995}.
+It is a relatively good compromise between accuracy and smoothness.
+It is not a \emph{positive} scheme, meaning that false extrema are permitted.
+But the amplitudes of the false extrema are significantly reduced over those in the centred second order method.
+As the scheme already includes a diffusion component, it can be used without explicit lateral diffusion on momentum
+($i.e.$ \np{ln\_dynldf\_lap}\forcode{ = }\np{ln\_dynldf\_bilap}\forcode{ = .false.}),
+and it is recommended to do so.
+The UBS scheme is not used in all directions.
+In the vertical, the centred $2^{nd}$ order evaluation of the advection is preferred, $i.e.$ $u_{uw}^{ubs}$ and
+$u_{vw}^{ubs}$ in \autoref{eq:dynadv_cen2} are used.
+UBS is diffusive and is associated with vertical mixing of momentum. \gmcomment{ gm  pursue the
 sentence:Since vertical mixing of momentum is a source term of the TKE equation...  }
+For stability reasons, the first term in (\autoref{eq:dynadv_ubs}), which corresponds
+to a second order centred scheme, is evaluated using the \textit{now} velocity
+(centred in time), while the second term, which is the diffusion part of the scheme,
+is evaluated using the \textit{before} velocity (forward in time). This is discussed
+by \citet{Webb_al_JAOT98} in the context of the Quick advection scheme.
+Note that the UBS and QUICK (Quadratic Upstream Interpolation for Convective Kinematics)
+schemes only differ by one coefficient. Replacing $1/6$ by $1/8$ in
+(\autoref{eq:dynadv_ubs}) leads to the QUICK advection scheme \citep{Webb_al_JAOT98}.
+This option is not available through a namelist parameter, since the $1/6$ coefficient
+is hard coded. Nevertheless it is quite easy to make the substitution in the
+\mdl{dynadv\_ubs} module and obtain a QUICK scheme.
+Note also that in the current version of \mdl{dynadv\_ubs}, there is also the
+possibility of using a $4^{th}$ order evaluation of the advective velocity as in
+ROMS. This is an error and should be suppressed soon.
+For stability reasons, the first term in (\autoref{eq:dynadv_ubs}),
+which corresponds to a second order centred scheme, is evaluated using the \textit{now} velocity (centred in time),
+while the second term, which is the diffusion part of the scheme,
+is evaluated using the \textit{before} velocity (forward in time).
+This is discussed by \citet{Webb_al_JAOT98} in the context of the Quick advection scheme.
+Note that the UBS and QUICK (Quadratic Upstream Interpolation for Convective Kinematics) schemes only differ by
+one coefficient.
+Replacing $1/6$ by $1/8$ in (\autoref{eq:dynadv_ubs}) leads to the QUICK advection scheme \citep{Webb_al_JAOT98}.
+This option is not available through a namelist parameter, since the $1/6$ coefficient is hard coded.
+Nevertheless it is quite easy to make the substitution in the \mdl{dynadv\_ubs} module and obtain a QUICK scheme.
+Note also that in the current version of \mdl{dynadv\_ubs},
+there is also the possibility of using a $4^{th}$ order evaluation of the advective velocity as in ROMS.
+This is an error and should be suppressed soon.
 %%%
 \gmcomment{action :  this have to be done}
 …
 Options are defined through the \ngn{namdyn\_hpg} namelist variables.
 The key distinction between the different algorithms used for the hydrostatic
+pressure gradient is the vertical coordinate used, since HPG is a \emph{horizontal}
+pressure gradient, $i.e.$ computed along geopotential surfaces. As a result, any
+tilt of the surface of the computational levels will require a specific treatment to
+The key distinction between the different algorithms used for
+the hydrostatic pressure gradient is the vertical coordinate used,
+since HPG is a \emph{horizontal} pressure gradient, $i.e.$ computed along geopotential surfaces.
+As a result, any tilt of the surface of the computational levels will require a specific treatment to
 compute the hydrostatic pressure gradient.
 The hydrostatic pressure gradient term is evaluated either using a leapfrog scheme,
 $i.e.$ the density appearing in its expression is centred in time (\emph{now} $\rho$), or
+a semi-implcit scheme. At the lateral boundaries either free slip, no slip or partial slip
 boundary conditions are applied.
+The hydrostatic pressure gradient term is evaluated either using a leapfrog scheme,
+$i.e.$ the density appearing in its expression is centred in time (\emph{now} $\rho$),
+or a semi-implcit scheme.
+At the lateral boundaries either free slip, no slip or partial slip boundary conditions are applied.
 %--------------------------------------------------------------------------------------------------------------
 …
 \label{subsec:DYN_hpg_zco}
+The hydrostatic pressure can be obtained by integrating the hydrostatic equation
+vertically from the surface. However, the pressure is large at great depth while its
+horizontal gradient is several orders of magnitude smaller. This may lead to large
+truncation errors in the pressure gradient terms. Thus, the two horizontal components
+of the hydrostatic pressure gradient are computed directly as follows:
+The hydrostatic pressure can be obtained by integrating the hydrostatic equation vertically from the surface.
+However, the pressure is large at great depth while its horizontal gradient is several orders of magnitude smaller.
+This may lead to large truncation errors in the pressure gradient terms.
+Thus, the two horizontal components of the hydrostatic pressure gradient are computed directly as follows:
 for $k=km$ (surface layer, $jk=1$ in the code)
 …
 \end{equation}
 Note that the $1/2$ factor in (\autoref{eq:dynhpg_zco_surf}) is adequate because of
 the definition of $e_{3w}$ as the vertical derivative of the scale factor at the surface
+level ($z=0$). Note also that in case of variable volume level (\key{vvl} defined), the
+surface pressure gradient is included in \autoref{eq:dynhpg_zco_surf} and \autoref{eq:dynhpg_zco}
 through the space and time variations of the vertical scale factor $e_{3w}$.
+Note that the $1/2$ factor in (\autoref{eq:dynhpg_zco_surf}) is adequate because of the definition of $e_{3w}$ as
+the vertical derivative of the scale factor at the surface level ($z=0$).
+Note also that in case of variable volume level (\key{vvl} defined),
+the surface pressure gradient is included in \autoref{eq:dynhpg_zco_surf} and
+\autoref{eq:dynhpg_zco} through the space and time variations of the vertical scale factor $e_{3w}$.
 %--------------------------------------------------------------------------------------------------------------
 …
 \label{subsec:DYN_hpg_zps}
+With partial bottom cells, tracers in horizontally adjacent cells generally live at
+different depths. Before taking horizontal gradients between these tracer points,
+a linear interpolation is used to approximate the deeper tracer as if it actually lived
+at the depth of the shallower tracer point.
+Apart from this modification, the horizontal hydrostatic pressure gradient evaluated
+in the $z$-coordinate with partial step is exactly as in the pure $z$-coordinate case.
+As explained in detail in section \autoref{sec:TRA_zpshde}, the nonlinearity of pressure
+effects in the equation of state is such that it is better to interpolate temperature and
+salinity vertically before computing the density. Horizontal gradients of temperature
+and salinity are needed for the TRA modules, which is the reason why the horizontal
+gradients of density at the deepest model level are computed in module \mdl{zpsdhe}
+located in the TRA directory and described in \autoref{sec:TRA_zpshde}.
+With partial bottom cells, tracers in horizontally adjacent cells generally live at different depths.
+Before taking horizontal gradients between these tracer points,
+a linear interpolation is used to approximate the deeper tracer as if
+it actually lived at the depth of the shallower tracer point.
+Apart from this modification,
+the horizontal hydrostatic pressure gradient evaluated in the $z$-coordinate with partial step is exactly as in
+the pure $z$-coordinate case.
+As explained in detail in section \autoref{sec:TRA_zpshde},
+the nonlinearity of pressure effects in the equation of state is such that
+it is better to interpolate temperature and salinity vertically before computing the density.
+Horizontal gradients of temperature and salinity are needed for the TRA modules,
+which is the reason why the horizontal gradients of density at the deepest model level are computed in
+module \mdl{zpsdhe} located in the TRA directory and described in \autoref{sec:TRA_zpshde}.
 %--------------------------------------------------------------------------------------------------------------
 …
 \label{subsec:DYN_hpg_sco}
 Pressure gradient formulations in an $s$-coordinate have been the subject of a vast
 number of papers ($e.g.$, \citet{Song1998, Shchepetkin_McWilliams_OM05}).
 A number of different pressure gradient options are coded but the ROMS-like, density Jacobian with
 cubic polynomial method is currently disabled whilst known bugs are under investigation.
+Pressure gradient formulations in an $s$-coordinate have been the subject of a vast number of papers
+($e.g.$, \citet{Song1998, Shchepetkin_McWilliams_OM05}).
+A number of different pressure gradient options are coded but the ROMS-like,
+density Jacobian with cubic polynomial method is currently disabled whilst known bugs are under investigation.
 $\bullet$ Traditional coding (see for example \citet{Madec_al_JPO96}: (\np{ln\_dynhpg\_sco}\forcode{ = .true.})
 …
 \end{equation}
 Where the first term is the pressure gradient along coordinates, computed as in
+\autoref{eq:dynhpg_zco_surf} - \autoref{eq:dynhpg_zco}, and $z_T$ is the depth of
 the $T$-point evaluated from the sum of the vertical scale factors at the $w$-point
+Where the first term is the pressure gradient along coordinates,
+computed as in \autoref{eq:dynhpg_zco_surf} - \autoref{eq:dynhpg_zco},
+and $z_T$ is the depth of the $T$-point evaluated from the sum of the vertical scale factors at the $w$-point
 ($e_{3w}$).
 …
 (\np{ln\_dynhpg\_djc}\forcode{ = .true.}) (currently disabled; under development)
+Note that expression \autoref{eq:dynhpg_sco} is commonly used when the variable volume formulation is
+activated (\key{vvl}) because in that case, even with a flat bottom, the coordinate surfaces are not
+horizontal but follow the free surface \citep{Levier2007}. The pressure jacobian scheme
+(\np{ln\_dynhpg\_prj}\forcode{ = .true.}) is available as an improved option to \np{ln\_dynhpg\_sco}\forcode{ = .true.} when
+\key{vvl} is active.  The pressure Jacobian scheme uses a constrained cubic spline to reconstruct
+the density profile across the water column. This method maintains the monotonicity between the
+density nodes  The pressure can be calculated by analytical integration of the density profile and a
+pressure Jacobian method is used to solve the horizontal pressure gradient. This method can provide
+a more accurate calculation of the horizontal pressure gradient than the standard scheme.
+Note that expression \autoref{eq:dynhpg_sco} is commonly used when the variable volume formulation is activated
+(\key{vvl}) because in that case, even with a flat bottom,
+the coordinate surfaces are not horizontal but follow the free surface \citep{Levier2007}.
+The pressure jacobian scheme (\np{ln\_dynhpg\_prj}\forcode{ = .true.}) is available as
+an improved option to \np{ln\_dynhpg\_sco}\forcode{ = .true.} when \key{vvl} is active.
+The pressure Jacobian scheme uses a constrained cubic spline to
+reconstruct the density profile across the water column.
+This method maintains the monotonicity between the density nodes.
+The pressure can be calculated by analytical integration of the density profile and
+a pressure Jacobian method is used to solve the horizontal pressure gradient.
+This method can provide a more accurate calculation of the horizontal pressure gradient than the standard scheme.
 \subsection{Ice shelf cavity}
 \label{subsec:DYN_hpg_isf}
 Beneath an ice shelf, the total pressure gradient is the sum of the pressure gradient due to the ice shelf load and
+ the pressure gradient due to the ocean load. If cavity opened (\np{ln\_isfcav}\forcode{ = .true.}) these 2 terms can be
+ calculated by setting \np{ln\_dynhpg\_isf}\forcode{ = .true.}. No other scheme are working with the ice shelf.\\
+the pressure gradient due to the ocean load.
+If cavity opened (\np{ln\_isfcav}\forcode{ = .true.}) these 2 terms can be calculated by
+setting \np{ln\_dynhpg\_isf}\forcode{ = .true.}.
+No other scheme are working with the ice shelf.\\
 $\bullet$ The main hypothesis to compute the ice shelf load is that the ice shelf is in an isostatic equilibrium.
+ The top pressure is computed integrating from surface to the base of the ice shelf a reference density profile
+(prescribed as density of a water at 34.4 PSU and -1.9\degC) and corresponds to the water replaced by the ice shelf.
+This top pressure is constant over time. A detailed description of this method is described in \citet{Losch2008}.\\
+$\bullet$ The ocean load is computed using the expression \autoref{eq:dynhpg_sco} described in \autoref{subsec:DYN_hpg_sco}.
+The top pressure is computed integrating from surface to the base of the ice shelf a reference density profile
+(prescribed as density of a water at 34.4 PSU and -1.9\degC) and
+corresponds to the water replaced by the ice shelf.
+This top pressure is constant over time.
+A detailed description of this method is described in \citet{Losch2008}.\\
+$\bullet$ The ocean load is computed using the expression \autoref{eq:dynhpg_sco} described in
+\autoref{subsec:DYN_hpg_sco}.
 %--------------------------------------------------------------------------------------------------------------
 …
 \label{subsec:DYN_hpg_imp}
+The default time differencing scheme used for the horizontal pressure gradient is
+a leapfrog scheme and therefore the density used in all discrete expressions given
+above is the  \textit{now} density, computed from the \textit{now} temperature and
+salinity. In some specific cases (usually high resolution simulations over an ocean
+domain which includes weakly stratified regions) the physical phenomenon that
+controls the time-step is internal gravity waves (IGWs). A semi-implicit scheme for
+doubling the stability limit associated with IGWs can be used \citep{Brown_Campana_MWR78,
+Maltrud1998}. It involves the evaluation of the hydrostatic pressure gradient as an
+average over the three time levels $t-\rdt$, $t$, and $t+\rdt$ ($i.e.$
+\textit{before},  \textit{now} and  \textit{after} time-steps), rather than at the central
+time level $t$ only, as in the standard leapfrog scheme.
+The default time differencing scheme used for the horizontal pressure gradient is a leapfrog scheme and
+therefore the density used in all discrete expressions given above is the  \textit{now} density,
+computed from the \textit{now} temperature and salinity.
+In some specific cases
+(usually high resolution simulations over an ocean domain which includes weakly stratified regions)
+the physical phenomenon that controls the time-step is internal gravity waves (IGWs).
+A semi-implicit scheme for doubling the stability limit associated with IGWs can be used
+\citep{Brown_Campana_MWR78, Maltrud1998}.
+It involves the evaluation of the hydrostatic pressure gradient as
+an average over the three time levels $t-\rdt$, $t$, and $t+\rdt$
+($i.e.$  \textit{before},  \textit{now} and  \textit{after} time-steps),
+rather than at the central time level $t$ only, as in the standard leapfrog scheme.
 $\bullet$ leapfrog scheme (\np{ln\_dynhpg\_imp}\forcode{ = .true.}):
 …
 \end{equation}
+The semi-implicit time scheme \autoref{eq:dynhpg_imp} is made possible without
+significant additional computation since the density can be updated to time level
+$t+\rdt$ before computing the horizontal hydrostatic pressure gradient. It can
+be easily shown that the stability limit associated with the hydrostatic pressure
+gradient doubles using \autoref{eq:dynhpg_imp} compared to that using the
+standard leapfrog scheme \autoref{eq:dynhpg_lf}. Note that \autoref{eq:dynhpg_imp}
+is equivalent to applying a time filter to the pressure gradient to eliminate high
+frequency IGWs. Obviously, when using \autoref{eq:dynhpg_imp}, the doubling of
+the time-step is achievable only if no other factors control the time-step, such as
+the stability limits associated with advection or diffusion.
+In practice, the semi-implicit scheme is used when \np{ln\_dynhpg\_imp}\forcode{ = .true.}.
+In this case, we choose to apply the time filter to temperature and salinity used in
+the equation of state, instead of applying it to the hydrostatic pressure or to the
+density, so that no additional storage array has to be defined. The density used to
+compute the hydrostatic pressure gradient (whatever the formulation) is evaluated
+as follows:
+The semi-implicit time scheme \autoref{eq:dynhpg_imp} is made possible without
+significant additional computation since the density can be updated to time level $t+\rdt$ before
+computing the horizontal hydrostatic pressure gradient.
+It can be easily shown that the stability limit associated with the hydrostatic pressure gradient doubles using
+\autoref{eq:dynhpg_imp} compared to that using the standard leapfrog scheme \autoref{eq:dynhpg_lf}.
+Note that \autoref{eq:dynhpg_imp} is equivalent to applying a time filter to the pressure gradient to
+eliminate high frequency IGWs.
+Obviously, when using \autoref{eq:dynhpg_imp},
+the doubling of the time-step is achievable only if no other factors control the time-step,
+such as the stability limits associated with advection or diffusion.
+In practice, the semi-implicit scheme is used when \np{ln\_dynhpg\_imp}\forcode{ = .true.}.
+In this case, we choose to apply the time filter to temperature and salinity used in the equation of state,
+instead of applying it to the hydrostatic pressure or to the density,
+so that no additional storage array has to be defined.
+The density used to compute the hydrostatic pressure gradient (whatever the formulation) is evaluated as follows:
 \begin{equation} \label{eq:rho_flt}
    \rho^t = \rho( \widetilde{T},\widetilde {S},z_t)
 …
 \end{equation}
 Note that in the semi-implicit case, it is necessary to save the filtered density, an
+extra three-dimensional field, in the restart file to restart the model with exact
 reproducibility. This option is controlled by  \np{nn\_dynhpg\_rst}, a namelist parameter.
+Note that in the semi-implicit case, it is necessary to save the filtered density,
+an extra three-dimensional field, in the restart file to restart the model with exact reproducibility.
+This option is controlled by  \np{nn\_dynhpg\_rst}, a namelist parameter.
 % ================================================================
 …
 Options are defined through the \ngn{namdyn\_spg} namelist variables.
+The surface pressure gradient term is related to the representation of the free surface (\autoref{sec:PE_hor_pg}).
+The main distinction is between the fixed volume case (linear free surface) and the variable volume case
+(nonlinear free surface, \key{vvl} is defined). In the linear free surface case (\autoref{subsec:PE_free_surface})
+the vertical scale factors $e_{3}$ are fixed in time, while they are time-dependent in the nonlinear case
+(\autoref{subsec:PE_free_surface}).
+The surface pressure gradient term is related to the representation of the free surface (\autoref{sec:PE_hor_pg}).
+The main distinction is between the fixed volume case (linear free surface) and
+the variable volume case (nonlinear free surface, \key{vvl} is defined).
+In the linear free surface case (\autoref{subsec:PE_free_surface})
+the vertical scale factors $e_{3}$ are fixed in time,
+while they are time-dependent in the nonlinear case (\autoref{subsec:PE_free_surface}).
 With both linear and nonlinear free surface, external gravity waves are allowed in the equations,
 which imposes a very small time step when an explicit time stepping is used.
+which imposes a very small time step when an explicit time stepping is used.
 Two methods are proposed to allow a longer time step for the three-dimensional equations:
 the filtered free surface, which is a modification of the continuous equations (see \autoref{eq:PE_flt}),
 and the split-explicit free surface described below.
+and the split-explicit free surface described below.
 The extra term introduced in the filtered method is calculated implicitly,
 so that the update of the next velocities is done in module \mdl{dynspg\_flt} and not in \mdl{dynnxt}.
 The form of the surface pressure gradient term depends on how the user wants to handle
+the fast external gravity waves that are a solution of the analytical equation (\autoref{sec:PE_hor_pg}).
+The form of the surface pressure gradient term depends on how the user wants to
+handle the fast external gravity waves that are a solution of the analytical equation (\autoref{sec:PE_hor_pg}).
 Three formulations are available, all controlled by a CPP key (ln\_dynspg\_xxx):
 an explicit formulation which requires a small time step ;
 a filtered free surface formulation which allows a larger time step by adding a filtering
 term into the momentum equation ;
+an explicit formulation which requires a small time step;
+a filtered free surface formulation which allows a larger time step by
+adding a filtering term into the momentum equation;
 and a split-explicit free surface formulation, described below, which also allows a larger time step.
+The extra term introduced in the filtered method is calculated
+implicitly, so that a solver is used to compute it. As a consequence the update of the $next$
+velocities is done in module \mdl{dynspg\_flt} and not in \mdl{dynnxt}.
+The extra term introduced in the filtered method is calculated implicitly, so that a solver is used to compute it.
+As a consequence the update of the $next$ velocities is done in module \mdl{dynspg\_flt} and not in \mdl{dynnxt}.
 …
 \label{subsec:DYN_spg_exp}
+In the explicit free surface formulation (\key{dynspg\_exp} defined), the model time step
+is chosen to be small enough to resolve the external gravity waves (typically a few tens of seconds).
+In the explicit free surface formulation (\key{dynspg\_exp} defined),
+the model time step is chosen to be small enough to resolve the external gravity waves
+(typically a few tens of seconds).
 The surface pressure gradient, evaluated using a leap-frog scheme ($i.e.$ centered in time),
 is thus simply given by :
 …
 \end{equation}
+Note that in the non-linear free surface case ($i.e.$ \key{vvl} defined), the surface pressure
+gradient is already included in the momentum tendency  through the level thickness variation
+allowed in the computation of the hydrostatic pressure gradient. Thus, nothing is done in the \mdl{dynspg\_exp} module.
+Note that in the non-linear free surface case ($i.e.$ \key{vvl} defined),
+the surface pressure gradient is already included in the momentum tendency through
+the level thickness variation allowed in the computation of the hydrostatic pressure gradient.
+Thus, nothing is done in the \mdl{dynspg\_exp} module.
 %--------------------------------------------------------------------------------------------------------------
 …
 The split-explicit free surface formulation used in \NEMO (\key{dynspg\_ts} defined),
+also called the time-splitting formulation, follows the one
+proposed by \citet{Shchepetkin_McWilliams_OM05}. The general idea is to solve the free surface
+equation and the associated barotropic velocity equations with a smaller time
+step than $\rdt$, the time step used for the three dimensional prognostic
+variables (\autoref{fig:DYN_dynspg_ts}).
+The size of the small time step, $\rdt_e$ (the external mode or barotropic time step)
+ is provided through the \np{nn\_baro} namelist parameter as:
+$\rdt_e = \rdt / nn\_baro$. This parameter can be optionally defined automatically (\np{ln\_bt\_nn\_auto}\forcode{ = .true.})
+considering that the stability of the barotropic system is essentially controled by external waves propagation.
+also called the time-splitting formulation, follows the one proposed by \citet{Shchepetkin_McWilliams_OM05}.
+The general idea is to solve the free surface equation and the associated barotropic velocity equations with
+a smaller time step than $\rdt$, the time step used for the three dimensional prognostic variables
+(\autoref{fig:DYN_dynspg_ts}).
+The size of the small time step, $\rdt_e$ (the external mode or barotropic time step) is provided through
+the \np{nn\_baro} namelist parameter as: $\rdt_e = \rdt / nn\_baro$.
+This parameter can be optionally defined automatically (\np{ln\_bt\_nn\_auto}\forcode{ = .true.}) considering that
+the stability of the barotropic system is essentially controled by external waves propagation.
 Maximum Courant number is in that case time independent, and easily computed online from the input bathymetry.
 Therefore, $\rdt_e$ is adjusted so that the Maximum allowed Courant number is smaller than \np{rn\_bt\_cmax}.
 …
   \end{equation}
 \end{subequations}
+where $\rm {\overline{\bf G}}$ is a forcing term held constant, containing coupling term between modes, surface atmospheric forcing as well as slowly varying barotropic terms not explicitly computed to gain efficiency. The third term on the right hand side of \autoref{eq:BT_dyn} represents the bottom stress (see section \autoref{sec:ZDF_bfr}), explicitly accounted for at each barotropic iteration. Temporal discretization of the system above follows a three-time step Generalized Forward Backward algorithm detailed in \citet{Shchepetkin_McWilliams_OM05}. AB3-AM4 coefficients used in \NEMO follow the second-order accurate, "multi-purpose" stability compromise as defined in \citet{Shchepetkin_McWilliams_Bk08} (see their figure 12, lower left).
+where $\rm {\overline{\bf G}}$ is a forcing term held constant, containing coupling term between modes,
+surface atmospheric forcing as well as slowly varying barotropic terms not explicitly computed to gain efficiency.
+The third term on the right hand side of \autoref{eq:BT_dyn} represents the bottom stress
+(see section \autoref{sec:ZDF_bfr}), explicitly accounted for at each barotropic iteration.
+Temporal discretization of the system above follows a three-time step Generalized Forward Backward algorithm
+detailed in \citet{Shchepetkin_McWilliams_OM05}.
+AB3-AM4 coefficients used in \NEMO follow the second-order accurate,
+"multi-purpose" stability compromise as defined in \citet{Shchepetkin_McWilliams_Bk08}
+(see their figure 12, lower left).
 %>   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >
 …
 \includegraphics[width=0.7\textwidth]{Fig_DYN_dynspg_ts}
 \caption{  \protect\label{fig:DYN_dynspg_ts}
+Schematic of the split-explicit time stepping scheme for the external
+and internal modes. Time increases to the right. In this particular exemple,
+a boxcar averaging window over $nn\_baro$ barotropic time steps is used ($nn\_bt\_flt=1$) and $nn\_baro=5$.
+Internal mode time steps (which are also the model time steps) are denoted
+by $t-\rdt$, $t$ and $t+\rdt$. Variables with $k$ superscript refer to instantaneous barotropic variables,
+$< >$ and $<< >>$ operator refer to time filtered variables using respectively primary (red vertical bars) and secondary weights (blue vertical bars).
+The former are used to obtain time filtered quantities at $t+\rdt$ while the latter are used to obtain time averaged
+transports to advect tracers.
+a) Forward time integration: \protect\np{ln\_bt\_fw}\forcode{ = .true.},  \protect\np{ln\_bt\_av}\forcode{ = .true.}.
+b) Centred time integration: \protect\np{ln\_bt\_fw}\forcode{ = .false.}, \protect\np{ln\_bt\_av}\forcode{ = .true.}.
+c) Forward time integration with no time filtering (POM-like scheme): \protect\np{ln\_bt\_fw}\forcode{ = .true.}, \protect\np{ln\_bt\_av}\forcode{ = .false.}. }
+  Schematic of the split-explicit time stepping scheme for the external and internal modes.
+  Time increases to the right. In this particular exemple,
+  a boxcar averaging window over $nn\_baro$ barotropic time steps is used ($nn\_bt\_flt=1$) and $nn\_baro=5$.
+  Internal mode time steps (which are also the model time steps) are denoted by $t-\rdt$, $t$ and $t+\rdt$.
+  Variables with $k$ superscript refer to instantaneous barotropic variables,
+  $< >$ and $<< >>$ operator refer to time filtered variables using respectively primary (red vertical bars) and
+  secondary weights (blue vertical bars).
+  The former are used to obtain time filtered quantities at $t+\rdt$ while
+  the latter are used to obtain time averaged transports to advect tracers.
+  a) Forward time integration: \protect\np{ln\_bt\_fw}\forcode{ = .true.},
+  \protect\np{ln\_bt\_av}\forcode{ = .true.}.
+  b) Centred time integration: \protect\np{ln\_bt\_fw}\forcode{ = .false.},
+  \protect\np{ln\_bt\_av}\forcode{ = .true.}.
+  c) Forward time integration with no time filtering (POM-like scheme):
+  \protect\np{ln\_bt\_fw}\forcode{ = .true.}, \protect\np{ln\_bt\_av}\forcode{ = .false.}. }
 \end{center}    \end{figure}
 %>   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >
+In the default case (\np{ln\_bt\_fw}\forcode{ = .true.}), the external mode is integrated
+between \textit{now} and  \textit{after} baroclinic time-steps (\autoref{fig:DYN_dynspg_ts}a). To avoid aliasing of fast barotropic motions into three dimensional equations, time filtering is eventually applied on barotropic
+quantities (\np{ln\_bt\_av}\forcode{ = .true.}). In that case, the integration is extended slightly beyond  \textit{after} time step to provide time filtered quantities.
+These are used for the subsequent initialization of the barotropic mode in the following baroclinic step.
+In the default case (\np{ln\_bt\_fw}\forcode{ = .true.}),
+the external mode is integrated between \textit{now} and \textit{after} baroclinic time-steps
+(\autoref{fig:DYN_dynspg_ts}a).
+To avoid aliasing of fast barotropic motions into three dimensional equations,
+time filtering is eventually applied on barotropic quantities (\np{ln\_bt\_av}\forcode{ = .true.}).
+In that case, the integration is extended slightly beyond \textit{after} time step to
+provide time filtered quantities.
+These are used for the subsequent initialization of the barotropic mode in the following baroclinic step.
 Since external mode equations written at baroclinic time steps finally follow a forward time stepping scheme,
+asselin filtering is not applied to barotropic quantities. \\
+Alternatively, one can choose to integrate barotropic equations starting
+from \textit{before} time step (\np{ln\_bt\_fw}\forcode{ = .false.}). Although more computationaly expensive ( \np{nn\_baro} additional iterations are indeed necessary), the baroclinic to barotropic forcing term given at \textit{now} time step
+become centred in the middle of the integration window. It can easily be shown that this property
+removes part of splitting errors between modes, which increases the overall numerical robustness.
+asselin filtering is not applied to barotropic quantities.\\
+Alternatively, one can choose to integrate barotropic equations starting from \textit{before} time step
+(\np{ln\_bt\_fw}\forcode{ = .false.}).
+Although more computationaly expensive ( \np{nn\_baro} additional iterations are indeed necessary),
+the baroclinic to barotropic forcing term given at \textit{now} time step become centred in
+the middle of the integration window.
+It can easily be shown that this property removes part of splitting errors between modes,
+which increases the overall numerical robustness.
 %references to Patrick Marsaleix' work here. Also work done by SHOM group.
 %%%
+As far as tracer conservation is concerned, barotropic velocities used to advect tracers must also be updated
+at \textit{now} time step. This implies to change the traditional order of computations in \NEMO: most of momentum
+trends (including the barotropic mode calculation) updated first, tracers' after. This \textit{de facto} makes semi-implicit hydrostatic
+pressure gradient (see section \autoref{subsec:DYN_hpg_imp}) and time splitting not compatible.
+Advective barotropic velocities are obtained by using a secondary set of filtering weights, uniquely defined from the filter
+coefficients used for the time averaging (\citet{Shchepetkin_McWilliams_OM05}). Consistency between the time averaged continuity equation and the time stepping of tracers is here the key to obtain exact conservation.
+As far as tracer conservation is concerned,
+barotropic velocities used to advect tracers must also be updated at \textit{now} time step.
+This implies to change the traditional order of computations in \NEMO:
+most of momentum trends (including the barotropic mode calculation) updated first, tracers' after.
+This \textit{de facto} makes semi-implicit hydrostatic pressure gradient
+(see section \autoref{subsec:DYN_hpg_imp})
+and time splitting not compatible.
+Advective barotropic velocities are obtained by using a secondary set of filtering weights,
+uniquely defined from the filter coefficients used for the time averaging (\citet{Shchepetkin_McWilliams_OM05}).
+Consistency between the time averaged continuity equation and the time stepping of tracers is here the key to
+obtain exact conservation.
 %%%
+One can eventually choose to feedback instantaneous values by not using any time filter (\np{ln\_bt\_av}\forcode{ = .false.}).
+In that case, external mode equations are continuous in time, ie they are not re-initialized when starting a new
+sub-stepping sequence. This is the method used so far in the POM model, the stability being maintained by refreshing at (almost)
+each barotropic time step advection and horizontal diffusion terms. Since the latter terms have not been added in \NEMO for
+computational efficiency, removing time filtering is not recommended except for debugging purposes.
+This may be used for instance to appreciate the damping effect of the standard formulation on external gravity waves in idealized or weakly non-linear cases. Although the damping is lower than for the filtered free surface, it is still significant as shown by \citet{Levier2007} in the case of an analytical barotropic Kelvin wave.
+One can eventually choose to feedback instantaneous values by not using any time filter
+(\np{ln\_bt\_av}\forcode{ = .false.}).
+In that case, external mode equations are continuous in time,
+$i.e.$ they are not re-initialized when starting a new sub-stepping sequence.
+This is the method used so far in the POM model, the stability being maintained by
+refreshing at (almost) each barotropic time step advection and horizontal diffusion terms.
+Since the latter terms have not been added in \NEMO for computational efficiency,
+removing time filtering is not recommended except for debugging purposes.
+This may be used for instance to appreciate the damping effect of the standard formulation on
+external gravity waves in idealized or weakly non-linear cases.
+Although the damping is lower than for the filtered free surface,
+it is still significant as shown by \citet{Levier2007} in the case of an analytical barotropic Kelvin wave.
 %>>>>>===============
 …
 \textbf{title: Time stepping the barotropic system }
+Assume knowledge of the full velocity and tracer fields at baroclinic time $\tau$. Hence,
+we can update the surface height and vertically integrated velocity with a leap-frog
+scheme using the small barotropic time step $\rdt$. We have
+Assume knowledge of the full velocity and tracer fields at baroclinic time $\tau$.
+Hence, we can update the surface height and vertically integrated velocity with a leap-frog scheme using
+the small barotropic time step $\rdt$.
+We have
 \begin{equation} \label{eq:DYN_spg_ts_eta}
 …
+\
+In these equations, araised (b) denotes values of surface height and vertically integrated velocity updated with the barotropic time steps. The $\tau$ time label on $\eta^{(b)}$
+and $U^{(b)}$ denotes the baroclinic time at which the vertically integrated forcing $\textbf{M}(\tau)$ (note that this forcing includes the surface freshwater forcing), the tracer fields, the freshwater flux $\text{EMP}_w(\tau)$, and total depth of the ocean $H(\tau)$ are held for the duration of the barotropic time stepping over a single cycle. This is also the time
+that sets the barotropic time steps via
+In these equations, araised (b) denotes values of surface height and vertically integrated velocity updated with
+the barotropic time steps.
+The $\tau$ time label on $\eta^{(b)}$ and $U^{(b)}$ denotes the baroclinic time at which
+the vertically integrated forcing $\textbf{M}(\tau)$
+(note that this forcing includes the surface freshwater forcing),
+the tracer fields, the freshwater flux $\text{EMP}_w(\tau)$,
+and total depth of the ocean $H(\tau)$ are held for the duration of the barotropic time stepping over
+a single cycle.
+This is also the time that sets the barotropic time steps via
 \begin{equation} \label{eq:DYN_spg_ts_t}
 t_n=\tau+n\rdt
 \end{equation}
+with $n$ an integer. The density scaled surface pressure is evaluated via
+with $n$ an integer.
+The density scaled surface pressure is evaluated via
 \begin{equation} \label{eq:DYN_spg_ts_ps}
 p_s^{(b)}(\tau,t_{n}) = \begin{cases}
 …
  \overline{\eta^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N \eta^{(b)}(\tau-\rdt,t_{n})
 \end{equation}
+the time averaged surface height taken from the previous barotropic cycle. Likewise,
+the time averaged surface height taken from the previous barotropic cycle.
+Likewise,
 \begin{equation} \label{eq:DYN_spg_ts_u}
 \textbf{U}^{(b)}(\tau,t_{n=0}) = \overline{\textbf{U}^{(b)}(\tau)}   \\
 …
    = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau-\rdt,t_{n})
 \end{equation}
+the time averaged vertically integrated transport. Notably, there is no Robert-Asselin time filter used in the barotropic portion of the integration.
+Upon reaching $t_{n=N} = \tau + 2\rdt \tau$ , the vertically integrated velocity is time averaged to produce the updated vertically integrated velocity at baroclinic time $\tau + \rdt \tau$
+the time averaged vertically integrated transport.
+Notably, there is no Robert-Asselin time filter used in the barotropic portion of the integration.
+Upon reaching $t_{n=N} = \tau + 2\rdt \tau$ ,
+the vertically integrated velocity is time averaged to produce the updated vertically integrated velocity at
+baroclinic time $\tau + \rdt \tau$
 \begin{equation} \label{eq:DYN_spg_ts_u}
 \textbf{U}(\tau+\rdt) = \overline{\textbf{U}^{(b)}(\tau+\rdt)}
    = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau,t_{n})
 \end{equation}
+The surface height on the new baroclinic time step is then determined via a baroclinic leap-frog using the following form
+The surface height on the new baroclinic time step is then determined via a baroclinic leap-frog using
+the following form
 \begin{equation} \label{eq:DYN_spg_ts_ssh}
 …
 \end{equation}
+ The use of this "big-leap-frog" scheme for the surface height ensures compatibility between the mass/volume budgets and the tracer budgets. More discussion of this point is provided in Chapter 10 (see in particular Section 10.2).
+The use of this "big-leap-frog" scheme for the surface height ensures compatibility between
+the mass/volume budgets and the tracer budgets.
+More discussion of this point is provided in Chapter 10 (see in particular Section 10.2).
 In general, some form of time filter is needed to maintain integrity of the surface
+height field due to the leap-frog splitting mode in equation \autoref{eq:DYN_spg_ts_ssh}. We
+have tried various forms of such filtering, with the following method discussed in
+\cite{Griffies_al_MWR01} chosen due to its stability and reasonably good maintenance of
+tracer conservation properties (see ??)
+In general, some form of time filter is needed to maintain integrity of the surface height field due to
+the leap-frog splitting mode in equation \autoref{eq:DYN_spg_ts_ssh}.
+We have tried various forms of such filtering,
+with the following method discussed in \cite{Griffies_al_MWR01} chosen due to
+its stability and reasonably good maintenance of tracer conservation properties (see ??).
 \begin{equation} \label{eq:DYN_spg_ts_sshf}
 …
 \end{equation}
+which is useful since it isolates all the time filtering aspects into the term multiplied
+by $\alpha$. This isolation allows for an easy check that tracer conservation is exact when
+eliminating tracer and surface height time filtering (see ?? for more complete discussion). However, in the general case with a non-zero $\alpha$, the filter \autoref{eq:DYN_spg_ts_sshf} was found to be more conservative, and so is recommended.
+which is useful since it isolates all the time filtering aspects into the term multiplied by $\alpha$.
+This isolation allows for an easy check that tracer conservation is exact when
+eliminating tracer and surface height time filtering (see ?? for more complete discussion).
+However, in the general case with a non-zero $\alpha$,
+the filter \autoref{eq:DYN_spg_ts_sshf} was found to be more conservative, and so is recommended.
 }            %%end gm comment (copy of griffies book)
 …
 \end{equation}
 where $T_c$, is a parameter with dimensions of time which characterizes the force,
 $\widetilde{\rho} = \rho / \rho_o$ is the dimensionless density, and $\rm {\bf M}$
 represents the collected contributions of the Coriolis, hydrostatic pressure gradient,
+$\widetilde{\rho} = \rho / \rho_o$ is the dimensionless density,
+and $\rm {\bf M}$ represents the collected contributions of the Coriolis, hydrostatic pressure gradient,
 non-linear and viscous terms in \autoref{eq:PE_dyn}.
 }   %end gmcomment
+Note that in the linear free surface formulation (\key{vvl} not defined), the ocean depth
+is time-independent and so is the matrix to be inverted. It is computed once and for all and applies to all ocean time steps.
+Note that in the linear free surface formulation (\key{vvl} not defined),
+the ocean depth is time-independent and so is the matrix to be inverted.
+It is computed once and for all and applies to all ocean time steps.
 % ================================================================
 …
 Options are defined through the \ngn{namdyn\_ldf} namelist variables.
+The options available for lateral diffusion are to use either laplacian
+(rotated or not) or biharmonic operators. The coefficients may be constant
+or spatially variable; the description of the coefficients is found in the chapter
+on lateral physics (\autoref{chap:LDF}). The lateral diffusion of momentum is
+evaluated using a forward scheme, $i.e.$ the velocity appearing in its expression
+is the \textit{before} velocity in time, except for the pure vertical component
+that appears when a tensor of rotation is used. This latter term is solved
+implicitly together with the vertical diffusion term (see \autoref{chap:STP})
+At the lateral boundaries either free slip, no slip or partial slip boundary
+conditions are applied according to the user's choice (see \autoref{chap:LBC}).
+The options available for lateral diffusion are to use either laplacian (rotated or not) or biharmonic operators.
+The coefficients may be constant or spatially variable;
+the description of the coefficients is found in the chapter on lateral physics (\autoref{chap:LDF}).
+The lateral diffusion of momentum is evaluated using a forward scheme,
+$i.e.$ the velocity appearing in its expression is the \textit{before} velocity in time,
+except for the pure vertical component that appears when a tensor of rotation is used.
+This latter term is solved implicitly together with the vertical diffusion term (see \autoref{chap:STP}).
+At the lateral boundaries either free slip,
+no slip or partial slip boundary conditions are applied according to the user's choice (see \autoref{chap:LBC}).
 \gmcomment{
+Hyperviscous operators are frequently used in the simulation of turbulent flows to control
+the dissipation of unresolved small scale features.
+Their primary role is to provide strong dissipation at the smallest scale supported by the grid
+while minimizing the impact on the larger scale features.
+Hyperviscous operators are thus designed to be more scale selective than the traditional,
+physically motivated Laplace operator.
+In finite difference methods, the biharmonic operator is frequently the method of choice to achieve
+this scale selective dissipation since its damping time ($i.e.$ its spin down time)
+scale like $\lambda^{-4}$ for disturbances of wavelength $\lambda$
+(so that short waves damped more rapidelly than long ones),
 whereas the Laplace operator damping time scales only like $\lambda^{-2}$.
+  Hyperviscous operators are frequently used in the simulation of turbulent flows to
+  control the dissipation of unresolved small scale features.
+  Their primary role is to provide strong dissipation at the smallest scale supported by
+  the grid while minimizing the impact on the larger scale features.
+  Hyperviscous operators are thus designed to be more scale selective than the traditional,
+  physically motivated Laplace operator.
+  In finite difference methods,
+  the biharmonic operator is frequently the method of choice to achieve this scale selective dissipation since
+  its damping time ($i.e.$ its spin down time) scale like $\lambda^{-4}$ for disturbances of wavelength $\lambda$
+  (so that short waves damped more rapidelly than long ones),
+  whereas the Laplace operator damping time scales only like $\lambda^{-2}$.
+}
 …
 \end{equation}
 As explained in \autoref{subsec:PE_ldf}, this formulation (as the gradient of a divergence
+and curl of the vorticity) preserves symmetry and ensures a complete
 separation between the vorticity and divergence parts of the momentum diffusion.
+As explained in \autoref{subsec:PE_ldf},
+this formulation (as the gradient of a divergence and curl of the vorticity) preserves symmetry and
+ensures a complete separation between the vorticity and divergence parts of the momentum diffusion.
 %--------------------------------------------------------------------------------------------------------------
 …
 \label{subsec:DYN_ldf_iso}
 A rotation of the lateral momentum diffusion operator is needed in several cases:
 for iso-neutral diffusion in the $z$-coordinate (\np{ln\_dynldf\_iso}\forcode{ = .true.}) and for
+either iso-neutral (\np{ln\_dynldf\_iso}\forcode{ = .true.}) or geopotential
+(\np{ln\_dynldf\_hor}\forcode{ = .true.}) diffusion in the $s$-coordinate. In the partial step
+case, coordinates are horizontal except at the deepest level and no
+rotation is performed when \np{ln\_dynldf\_hor}\forcode{ = .true.}. The diffusion operator
+is defined simply as the divergence of down gradient momentum fluxes on each
+momentum component. It must be emphasized that this formulation ignores
+constraints on the stress tensor such as symmetry. The resulting discrete
 representation is:
+A rotation of the lateral momentum diffusion operator is needed in several cases:
+for iso-neutral diffusion in the $z$-coordinate (\np{ln\_dynldf\_iso}\forcode{ = .true.}) and
+for either iso-neutral (\np{ln\_dynldf\_iso}\forcode{ = .true.}) or
+geopotential (\np{ln\_dynldf\_hor}\forcode{ = .true.}) diffusion in the $s$-coordinate.
+In the partial step case, coordinates are horizontal except at the deepest level and
+no rotation is performed when \np{ln\_dynldf\_hor}\forcode{ = .true.}.
+The diffusion operator is defined simply as the divergence of down gradient momentum fluxes on
+each momentum component.
+It must be emphasized that this formulation ignores constraints on the stress tensor such as symmetry.
+The resulting discrete representation is:
 \begin{equation} \label{eq:dyn_ldf_iso}
 \begin{split}
 …
  \end{split}
 \end{equation}
+where $r_1$ and $r_2$ are the slopes between the surface along which the
+diffusion operator acts and the surface of computation ($z$- or $s$-surfaces).
+The way these slopes are evaluated is given in the lateral physics chapter
+(\autoref{chap:LDF}).
+where $r_1$ and $r_2$ are the slopes between the surface along which the diffusion operator acts and
+the surface of computation ($z$- or $s$-surfaces).
+The way these slopes are evaluated is given in the lateral physics chapter (\autoref{chap:LDF}).
 %--------------------------------------------------------------------------------------------------------------
 …
 \label{subsec:DYN_ldf_bilap}
+The lateral fourth order operator formulation on momentum is obtained by
+applying \autoref{eq:dynldf_lap} twice. It requires an additional assumption on
+boundary conditions: the first derivative term normal to the coast depends on
+the free or no-slip lateral boundary conditions chosen, while the third
+derivative terms normal to the coast are set to zero (see \autoref{chap:LBC}).
+The lateral fourth order operator formulation on momentum is obtained by applying \autoref{eq:dynldf_lap} twice.
+It requires an additional assumption on boundary conditions:
+the first derivative term normal to the coast depends on the free or no-slip lateral boundary conditions chosen,
+while the third derivative terms normal to the coast are set to zero (see \autoref{chap:LBC}).
 %%%
 \gmcomment{add a remark on the the change in the position of the coefficient}
 …
 Options are defined through the \ngn{namzdf} namelist variables.
+The large vertical diffusion coefficient found in the surface mixed layer together
+with high vertical resolution implies that in the case of explicit time stepping there
+would be too restrictive a constraint on the time step. Two time stepping schemes
+can be used for the vertical diffusion term : $(a)$ a forward time differencing
+scheme (\np{ln\_zdfexp}\forcode{ = .true.}) using a time splitting technique
+(\np{nn\_zdfexp} $>$ 1) or $(b)$ a backward (or implicit) time differencing scheme
+(\np{ln\_zdfexp}\forcode{ = .false.}) (see \autoref{chap:STP}). Note that namelist variables
+\np{ln\_zdfexp} and \np{nn\_zdfexp} apply to both tracers and dynamics.
+The formulation of the vertical subgrid scale physics is the same whatever
+the vertical coordinate is. The vertical diffusion operators given by
+\autoref{eq:PE_zdf} take the following semi-discrete space form:
+The large vertical diffusion coefficient found in the surface mixed layer together with high vertical resolution implies that in the case of explicit time stepping there would be too restrictive a constraint on the time step.
+Two time stepping schemes can be used for the vertical diffusion term:
+$(a)$ a forward time differencing scheme
+(\np{ln\_zdfexp}\forcode{ = .true.}) using a time splitting technique (\np{nn\_zdfexp} $>$ 1) or
+$(b)$ a backward (or implicit) time differencing scheme (\np{ln\_zdfexp}\forcode{ = .false.})
+(see \autoref{chap:STP}).
+Note that namelist variables \np{ln\_zdfexp} and \np{nn\_zdfexp} apply to both tracers and dynamics.
+The formulation of the vertical subgrid scale physics is the same whatever the vertical coordinate is.
+The vertical diffusion operators given by \autoref{eq:PE_zdf} take the following semi-discrete space form:
 \begin{equation} \label{eq:dynzdf}
 \left\{   \begin{aligned}
 …
 \end{aligned}   \right.
 \end{equation}
+where $A_{uw}^{vm} $ and $A_{vw}^{vm} $ are the vertical eddy viscosity and
+diffusivity coefficients. The way these coefficients are evaluated
+depends on the vertical physics used (see \autoref{chap:ZDF}).
+The surface boundary condition on momentum is the stress exerted by
+the wind. At the surface, the momentum fluxes are prescribed as the boundary
+condition on the vertical turbulent momentum fluxes,
+where $A_{uw}^{vm} $ and $A_{vw}^{vm} $ are the vertical eddy viscosity and diffusivity coefficients.
+The way these coefficients are evaluated depends on the vertical physics used (see \autoref{chap:ZDF}).
+The surface boundary condition on momentum is the stress exerted by the wind.
+At the surface, the momentum fluxes are prescribed as the boundary condition on
+the vertical turbulent momentum fluxes,
 \begin{equation} \label{eq:dynzdf_sbc}
 \left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{z=1}
     = \frac{1}{\rho _o} \binom{\tau _u}{\tau _v }
 \end{equation}
 where $\left( \tau _u ,\tau _v \right)$ are the two components of the wind stress
+vector in the (\textbf{i},\textbf{j}) coordinate system. The high mixing coefficients
 in the surface mixed layer ensure that the surface wind stress is distributed in
 the vertical over the mixed layer depth. If the vertical mixing coefficient
+is small (when no mixed layer scheme is used) the surface stress enters only
 the top model level, as a body force. The surface wind stress is calculated
+in the surface module routines (SBC, see \autoref{chap:SBC})
 The turbulent flux of momentum at the bottom of the ocean is specified through
 a bottom friction parameterisation (see \autoref{sec:ZDF_bfr})
+where $\left( \tau _u ,\tau _v \right)$ are the two components of the wind stress vector in
+the (\textbf{i},\textbf{j}) coordinate system.
+The high mixing coefficients in the surface mixed layer ensure that the surface wind stress is distributed in
+the vertical over the mixed layer depth.
+If the vertical mixing coefficient is small (when no mixed layer scheme is used)
+the surface stress enters only the top model level, as a body force.
+The surface wind stress is calculated in the surface module routines (SBC, see \autoref{chap:SBC}).
+The turbulent flux of momentum at the bottom of the ocean is specified through a bottom friction parameterisation
+(see \autoref{sec:ZDF_bfr})
 % ================================================================
 …
 \label{sec:DYN_forcing}
 Besides the surface and bottom stresses (see the above section) which are
+introduced as boundary conditions on the vertical mixing, three other forcings
 may enter the dynamical equations by affecting the surface pressure gradient.
 (1) When \np{ln\_apr\_dyn}\forcode{ = .true.} (see \autoref{sec:SBC_apr}), the atmospheric pressure is taken
 into account when computing the surface pressure gradient.
 (2) When \np{ln\_tide\_pot}\forcode{ = .true.} and \np{ln\_tide}\forcode{ = .true.} (see \autoref{sec:SBC_tide}),
+Besides the surface and bottom stresses (see the above section)
+which are introduced as boundary conditions on the vertical mixing,
+three other forcings may enter the dynamical equations by affecting the surface pressure gradient.
+(1) When \np{ln\_apr\_dyn}\forcode{ = .true.} (see \autoref{sec:SBC_apr}),
+the atmospheric pressure is taken into account when computing the surface pressure gradient.
+(2) When \np{ln\_tide\_pot}\forcode{ = .true.} and \np{ln\_tide}\forcode{ = .true.} (see \autoref{sec:SBC_tide}),
 the tidal potential is taken into account when computing the surface pressure gradient.
+(3) When \np{nn\_ice\_embd}\forcode{ = 2} and LIM or CICE is used ($i.e.$ when the sea-ice is embedded in the ocean),
+(3) When \np{nn\_ice\_embd}\forcode{ = 2} and LIM or CICE is used
+($i.e.$ when the sea-ice is embedded in the ocean),
 the snow-ice mass is taken into account when computing the surface pressure gradient.
 …
 Options are defined through the \ngn{namdom} namelist variables.
+The general framework for dynamics time stepping is a leap-frog scheme,
+$i.e.$ a three level centred time scheme associated with an Asselin time filter
+(cf. \autoref{chap:STP}). The scheme is applied to the velocity, except when using
+the flux form of momentum advection (cf. \autoref{sec:DYN_adv_cor_flux}) in the variable
+volume case (\key{vvl} defined), where it has to be applied to the thickness
+weighted velocity (see \autoref{sec:A_momentum})
+$\bullet$ vector invariant form or linear free surface (\np{ln\_dynhpg\_vec}\forcode{ = .true.} ; \key{vvl} not defined):
+The general framework for dynamics time stepping is a leap-frog scheme,
+$i.e.$ a three level centred time scheme associated with an Asselin time filter (cf. \autoref{chap:STP}).
+The scheme is applied to the velocity, except when
+using the flux form of momentum advection (cf. \autoref{sec:DYN_adv_cor_flux})
+in the variable volume case (\key{vvl} defined),
+where it has to be applied to the thickness weighted velocity (see \autoref{sec:A_momentum})
+$\bullet$ vector invariant form or linear free surface
+(\np{ln\_dynhpg\_vec}\forcode{ = .true.} ; \key{vvl} not defined):
 \begin{equation} \label{eq:dynnxt_vec}
 \left\{   \begin{aligned}
 …
 \end{equation}
+$\bullet$ flux form and nonlinear free surface (\np{ln\_dynhpg\_vec}\forcode{ = .false.} ; \key{vvl} defined):
+$\bullet$ flux form and nonlinear free surface
+(\np{ln\_dynhpg\_vec}\forcode{ = .false.} ; \key{vvl} defined):
 \begin{equation} \label{eq:dynnxt_flux}
 \left\{   \begin{aligned}
 …
 \end{aligned}   \right.
 \end{equation}
+where RHS is the right hand side of the momentum equation, the subscript $f$
+denotes filtered values and $\gamma$ is the Asselin coefficient. $\gamma$ is
+initialized as \np{nn\_atfp} (namelist parameter). Its default value is \np{nn\_atfp}\forcode{ = 10.e-3}.
+In both cases, the modified Asselin filter is not applied since perfect conservation
+is not an issue for the momentum equations.
+Note that with the filtered free surface, the update of the \textit{after} velocities
+is done in the \mdl{dynsp\_flt} module, and only array swapping
+and Asselin filtering is done in \mdl{dynnxt}.
+where RHS is the right hand side of the momentum equation,
+the subscript $f$ denotes filtered values and $\gamma$ is the Asselin coefficient.
+$\gamma$ is initialized as \np{nn\_atfp} (namelist parameter).
+Its default value is \np{nn\_atfp}\forcode{ = 10.e-3}.
+In both cases, the modified Asselin filter is not applied since perfect conservation is not an issue for
+the momentum equations.
+Note that with the filtered free surface,
+the update of the \textit{after} velocities is done in the \mdl{dynsp\_flt} module,
+and only array swapping and Asselin filtering is done in \mdl{dynnxt}.
 % ================================================================

NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_LBC.tex

-                      r10146
+                      r10368
 Options are defined through the \ngn{namlbc} namelist variables.
+The discrete representation of a domain with complex boundaries (coastlines and
+bottom topography) leads to arrays that include large portions where a computation
+is not required as the model variables remain at zero. Nevertheless, vectorial
+supercomputers are far more efficient when computing over a whole array, and the
+readability of a code is greatly improved when boundary conditions are applied in
+an automatic way rather than by a specific computation before or after each
+computational loop. An efficient way to work over the whole domain while specifying
+the boundary conditions, is to use multiplication by mask arrays in the computation.
+A mask array is a matrix whose elements are $1$ in the ocean domain and $0$
+elsewhere. A simple multiplication of a variable by its own mask ensures that it will
+remain zero over land areas. Since most of the boundary conditions consist of a
+zero flux across the solid boundaries, they can be simply applied by multiplying
+variables by the correct mask arrays, $i.e.$ the mask array of the grid point where
+the flux is evaluated. For example, the heat flux in the \textbf{i}-direction is evaluated
+at $u$-points. Evaluating this quantity as,
+The discrete representation of a domain with complex boundaries (coastlines and bottom topography) leads to
+arrays that include large portions where a computation is not required as the model variables remain at zero.
+Nevertheless, vectorial supercomputers are far more efficient when computing over a whole array,
+and the readability of a code is greatly improved when boundary conditions are applied in
+an automatic way rather than by a specific computation before or after each computational loop.
+An efficient way to work over the whole domain while specifying the boundary conditions,
+is to use multiplication by mask arrays in the computation.
+A mask array is a matrix whose elements are $1$ in the ocean domain and $0$ elsewhere.
+A simple multiplication of a variable by its own mask ensures that it will remain zero over land areas.
+Since most of the boundary conditions consist of a zero flux across the solid boundaries,
+they can be simply applied by multiplying variables by the correct mask arrays,
+$i.e.$ the mask array of the grid point where the flux is evaluated.
+For example, the heat flux in the \textbf{i}-direction is evaluated at $u$-points.
+Evaluating this quantity as,
 \begin{equation} \label{eq:lbc_aaaa}
 …
 }{e_{1u} } \; \delta _{i+1 / 2} \left[ T \right]\;\;mask_u
 \end{equation}
+(where mask$_{u}$ is the mask array at a $u$-point) ensures that the heat flux is
+zero inside land and at the boundaries, since mask$_{u}$ is zero at solid boundaries
+which in this case are defined at $u$-points (normal velocity $u$ remains zero at
+the coast) (\autoref{fig:LBC_uv}).
+(where mask$_{u}$ is the mask array at a $u$-point) ensures that the heat flux is zero inside land and
+at the boundaries, since mask$_{u}$ is zero at solid boundaries which in this case are defined at $u$-points
+(normal velocity $u$ remains zero at the coast) (\autoref{fig:LBC_uv}).
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 …
 \includegraphics[width=0.90\textwidth]{Fig_LBC_uv}
 \caption{  \protect\label{fig:LBC_uv}
+Lateral boundary (thick line) at T-level. The velocity normal to the boundary is set to zero.}
+  Lateral boundary (thick line) at T-level.
+  The velocity normal to the boundary is set to zero.}
 \end{center}   \end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+For momentum the situation is a bit more complex as two boundary conditions
+must be provided along the coast (one each for the normal and tangential velocities).
+The boundary of the ocean in the C-grid is defined by the velocity-faces.
+For example, at a given $T$-level, the lateral boundary (a coastline or an intersection
+with the bottom topography) is made of segments joining $f$-points, and normal
+velocity points are located between two $f-$points (\autoref{fig:LBC_uv}).
+The boundary condition on the normal velocity (no flux through solid boundaries)
+can thus be easily implemented using the mask system. The boundary condition
+on the tangential velocity requires a more specific treatment. This boundary
+condition influences the relative vorticity and momentum diffusive trends, and is
+required in order to compute the vorticity at the coast. Four different types of
+lateral boundary condition are available, controlled by the value of the \np{rn\_shlat}
+namelist parameter. (The value of the mask$_{f}$ array along the coastline is set
+equal to this parameter.) These are:
+For momentum the situation is a bit more complex as two boundary conditions must be provided along the coast
+(one each for the normal and tangential velocities).
+The boundary of the ocean in the C-grid is defined by the velocity-faces.
+For example, at a given $T$-level,
+the lateral boundary (a coastline or an intersection with the bottom topography) is made of
+segments joining $f$-points, and normal velocity points are located between two $f-$points (\autoref{fig:LBC_uv}).
+The boundary condition on the normal velocity (no flux through solid boundaries)
+can thus be easily implemented using the mask system.
+The boundary condition on the tangential velocity requires a more specific treatment.
+This boundary condition influences the relative vorticity and momentum diffusive trends,
+and is required in order to compute the vorticity at the coast.
+Four different types of lateral boundary condition are available,
+controlled by the value of the \np{rn\_shlat} namelist parameter
+(The value of the mask$_{f}$ array along the coastline is set equal to this parameter).
+These are:
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 …
 \includegraphics[width=0.90\textwidth]{Fig_LBC_shlat}
 \caption{     \protect\label{fig:LBC_shlat}
+lateral boundary condition (a) free-slip ($rn\_shlat=0$) ; (b) no-slip ($rn\_shlat=2$)
+; (c) "partial" free-slip ($0<rn\_shlat<2$) and (d) "strong" no-slip ($2<rn\_shlat$).
+Implied "ghost" velocity inside land area is display in grey. }
+  lateral boundary condition
+  (a) free-slip ($rn\_shlat=0$);
+  (b) no-slip ($rn\_shlat=2$);
+  (c) "partial" free-slip ($0<rn\_shlat<2$) and
+  (d) "strong" no-slip ($2<rn\_shlat$).
+  Implied "ghost" velocity inside land area is display in grey. }
 \end{center}    \end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 …
 \begin{description}
+\item[free-slip boundary condition (\np{rn\_shlat}\forcode{ = 0}): ]  the tangential velocity at the
+coastline is equal to the offshore velocity, $i.e.$ the normal derivative of the
+tangential velocity is zero at the coast, so the vorticity: mask$_{f}$ array is set
+to zero inside the land and just at the coast (\autoref{fig:LBC_shlat}-a).
+\item[no-slip boundary condition (\np{rn\_shlat}\forcode{ = 2}): ] the tangential velocity vanishes
+at the coastline. Assuming that the tangential velocity decreases linearly from
+the closest ocean velocity grid point to the coastline, the normal derivative is
+evaluated as if the velocities at the closest land velocity gridpoint and the closest
+ocean velocity gridpoint were of the same magnitude but in the opposite direction
+(\autoref{fig:LBC_shlat}-b). Therefore, the vorticity along the coastlines is given by:
+\item[free-slip boundary condition (\np{rn\_shlat}\forcode{ = 0}):] the tangential velocity at
+  the coastline is equal to the offshore velocity,
+  $i.e.$ the normal derivative of the tangential velocity is zero at the coast,
+  so the vorticity: mask$_{f}$ array is set to zero inside the land and just at the coast
+  (\autoref{fig:LBC_shlat}-a).
+\item[no-slip boundary condition (\np{rn\_shlat}\forcode{ = 2}):] the tangential velocity vanishes at the coastline.
+  Assuming that the tangential velocity decreases linearly from
+  the closest ocean velocity grid point to the coastline,
+  the normal derivative is evaluated as if the velocities at the closest land velocity gridpoint and
+  the closest ocean velocity gridpoint were of the same magnitude but in the opposite direction
+  (\autoref{fig:LBC_shlat}-b).
+  Therefore, the vorticity along the coastlines is given by:
 \begin{equation*}
 \zeta \equiv 2 \left(\delta_{i+1/2} \left[e_{2v} v \right] - \delta_{j+1/2} \left[e_{1u} u \right] \right) / \left(e_{1f} e_{2f} \right) \ ,
 \end{equation*}
 where $u$ and $v$ are masked fields. Setting the mask$_{f}$ array to $2$ along
+the coastline provides a vorticity field computed with the no-slip boundary condition,
 simply by multiplying it by the mask$_{f}$ :
+where $u$ and $v$ are masked fields.
+Setting the mask$_{f}$ array to $2$ along the coastline provides a vorticity field computed with
+the no-slip boundary condition, simply by multiplying it by the mask$_{f}$ :
 \begin{equation} \label{eq:lbc_bbbb}
 \zeta \equiv \frac{1}{e_{1f} {\kern 1pt}e_{2f} }\left( {\delta _{i+1/2}
 …
 \end{equation}
+\item["partial" free-slip boundary condition (0$<$\np{rn\_shlat}$<$2): ] the tangential
+velocity at the coastline is smaller than the offshore velocity, $i.e.$ there is a lateral
+friction but not strong enough to make the tangential velocity at the coast vanish
+(\autoref{fig:LBC_shlat}-c). This can be selected by providing a value of mask$_{f}$
+strictly inbetween $0$ and $2$.
+\item["strong" no-slip boundary condition (2$<$\np{rn\_shlat}): ] the viscous boundary
+layer is assumed to be smaller than half the grid size (\autoref{fig:LBC_shlat}-d).
+The friction is thus larger than in the no-slip case.
+\item["partial" free-slip boundary condition (0$<$\np{rn\_shlat}$<$2):] the tangential velocity at
+  the coastline is smaller than the offshore velocity, $i.e.$ there is a lateral friction but
+  not strong enough to make the tangential velocity at the coast vanish (\autoref{fig:LBC_shlat}-c).
+  This can be selected by providing a value of mask$_{f}$ strictly inbetween $0$ and $2$.
+\item["strong" no-slip boundary condition (2$<$\np{rn\_shlat}):] the viscous boundary layer is assumed to
+  be smaller than half the grid size (\autoref{fig:LBC_shlat}-d).
+  The friction is thus larger than in the no-slip case.
 \end{description}
+Note that when the bottom topography is entirely represented by the $s$-coor-dinates
+(pure $s$-coordinate), the lateral boundary condition on tangential velocity is of much
+less importance as it is only applied next to the coast where the minimum water depth
+can be quite shallow.
+Note that when the bottom topography is entirely represented by the $s$-coor-dinates (pure $s$-coordinate),
+the lateral boundary condition on tangential velocity is of much less importance as
+it is only applied next to the coast where the minimum water depth can be quite shallow.
 …
 \label{sec:LBC_jperio}
+At the model domain boundaries several choices are offered: closed, cyclic east-west,
+cyclic north-south, a north-fold, and combination closed-north fold
+or bi-cyclic east-west and north-fold. The north-fold boundary condition is associated with the 3-pole ORCA mesh.
+At the model domain boundaries several choices are offered:
+closed, cyclic east-west, cyclic north-south, a north-fold, and combination closed-north fold or
+bi-cyclic east-west and north-fold.
+The north-fold boundary condition is associated with the 3-pole ORCA mesh.
 % -------------------------------------------------------------------------------------------------------------
 …
 \label{subsec:LBC_jperio012}
+The choice of closed or cyclic model domain boundary condition is made
+by setting \np{jperio} to 0, 1, 2 or 7 in namelist \ngn{namcfg}. Each time such a boundary
+condition is needed, it is set by a call to routine \mdl{lbclnk}. The computation of
+momentum and tracer trends proceeds from $i=2$ to $i=jpi-1$ and from $j=2$ to
+$j=jpj-1$, $i.e.$ in the model interior. To choose a lateral model boundary condition
+is to specify the first and last rows and columns of the model variables.
+The choice of closed or cyclic model domain boundary condition is made by
+setting \np{jperio} to 0, 1, 2 or 7 in namelist \ngn{namcfg}.
+Each time such a boundary condition is needed, it is set by a call to routine \mdl{lbclnk}.
+The computation of momentum and tracer trends proceeds from $i=2$ to $i=jpi-1$ and from $j=2$ to $j=jpj-1$,
+$i.e.$ in the model interior.
+To choose a lateral model boundary condition is to specify the first and last rows and columns of
+the model variables.
 \begin{description}
+\item[For closed boundary (\np{jperio}\forcode{ = 0})], solid walls are imposed at all model
+boundaries: first and last rows and columns are set to zero.
+\item[For cyclic east-west boundary (\np{jperio}\forcode{ = 1})], first and last rows are set
+to zero (closed) whilst the first column is set to the value of the last-but-one column
+and the last column to the value of the second one (\autoref{fig:LBC_jperio}-a).
+Whatever flows out of the eastern (western) end of the basin enters the western
+(eastern) end.
+\item[For cyclic north-south boundary (\np{jperio}\forcode{ = 2})], first and last columns are set
+to zero (closed) whilst the first row is set to the value of the last-but-one row
+and the last row to the value of the second one (\autoref{fig:LBC_jperio}-a).
+Whatever flows out of the northern (southern) end of the basin enters the southern
+(northern) end.
+\item[For closed boundary (\np{jperio}\forcode{ = 0})],
+  solid walls are imposed at all model boundaries:
+  first and last rows and columns are set to zero.
+\item[For cyclic east-west boundary (\np{jperio}\forcode{ = 1})],
+  first and last rows are set to zero (closed) whilst the first column is set to
+  the value of the last-but-one column and the last column to the value of the second one
+  (\autoref{fig:LBC_jperio}-a).
+  Whatever flows out of the eastern (western) end of the basin enters the western (eastern) end.
+\item[For cyclic north-south boundary (\np{jperio}\forcode{ = 2})],
+  first and last columns are set to zero (closed) whilst the first row is set to
+  the value of the last-but-one row and the last row to the value of the second one
+  (\autoref{fig:LBC_jperio}-a).
+  Whatever flows out of the northern (southern) end of the basin enters the southern (northern) end.
 \item[Bi-cyclic east-west and north-south boundary (\np{jperio}\forcode{ = 7})] combines cases 1 and 2.
 …
 \includegraphics[width=1.0\textwidth]{Fig_LBC_jperio}
 \caption{    \protect\label{fig:LBC_jperio}
 setting of (a) east-west cyclic  (b) symmetric across the equator boundary conditions.}
+  setting of (a) east-west cyclic  (b) symmetric across the equator boundary conditions.}
 \end{center}   \end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 …
 \label{subsec:LBC_north_fold}
+The north fold boundary condition has been introduced in order to handle the north
+boundary of a three-polar ORCA grid. Such a grid has two poles in the northern hemisphere
+(\autoref{fig:MISC_ORCA_msh}, and thus requires a specific treatment illustrated in \autoref{fig:North_Fold_T}.
+The north fold boundary condition has been introduced in order to handle the north boundary of
+a three-polar ORCA grid.
+Such a grid has two poles in the northern hemisphere (\autoref{fig:MISC_ORCA_msh},
+and thus requires a specific treatment illustrated in \autoref{fig:North_Fold_T}.
 Further information can be found in \mdl{lbcnfd} module which applies the north fold boundary condition.
 …
 \begin{figure}[!t]    \begin{center}
 \includegraphics[width=0.90\textwidth]{Fig_North_Fold_T}
 \caption{    \protect\label{fig:North_Fold_T}
+North fold boundary with a $T$-point pivot and cyclic east-west boundary condition
+($jperio=4$), as used in ORCA 2, 1/4, and 1/12. Pink shaded area corresponds
 to the inner domain mask (see text). }
+\caption{    \protect\label{fig:North_Fold_T}
+  North fold boundary with a $T$-point pivot and cyclic east-west boundary condition ($jperio=4$),
+  as used in ORCA 2, 1/4, and 1/12.
+  Pink shaded area corresponds to the inner domain mask (see text). }
 \end{center}   \end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 …
 \label{sec:LBC_mpp}
+For massively parallel processing (mpp), a domain decomposition method is used.
+The basic idea of the method is to split the large computation domain of a numerical
+experiment into several smaller domains and solve the set of equations by addressing
+independent local problems. Each processor has its own local memory and computes
+the model equation over a subdomain of the whole model domain. The subdomain
+boundary conditions are specified through communications between processors
+which are organized by explicit statements (message passing method).
+A big advantage is that the method does not need many modifications of the initial
+FORTRAN code. From the modeller's point of view, each sub domain running on
+a processor is identical to the "mono-domain" code. In addition, the programmer
+manages the communications between subdomains, and the code is faster when
+the number of processors is increased. The porting of OPA code on an iPSC860
+was achieved during Guyon's PhD [Guyon et al. 1994, 1995] in collaboration with
+CETIIS and ONERA. The implementation in the operational context and the studies
+of performance on a T3D and T3E Cray computers have been made in collaboration
+with IDRIS and CNRS. The present implementation is largely inspired by Guyon's
+work  [Guyon 1995].
+The parallelization strategy is defined by the physical characteristics of the
+ocean model. Second order finite difference schemes lead to local discrete
+operators that depend at the very most on one neighbouring point. The only
+non-local computations concern the vertical physics (implicit diffusion,
+turbulent closure scheme, ...) (delocalization over the whole water column),
+and the solving of the elliptic equation associated with the surface pressure
+gradient computation (delocalization over the whole horizontal domain).
+Therefore, a pencil strategy is used for the data sub-structuration
+: the 3D initial domain is laid out on local processor
+memories following a 2D horizontal topological splitting. Each sub-domain
+computes its own surface and bottom boundary conditions and has a side
+wall overlapping interface which defines the lateral boundary conditions for
+computations in the inner sub-domain. The overlapping area consists of the
+two rows at each edge of the sub-domain. After a computation, a communication
+phase starts: each processor sends to its neighbouring processors the update
+values of the points corresponding to the interior overlapping area to its
+neighbouring sub-domain ($i.e.$ the innermost of the two overlapping rows).
+The communication is done through the Message Passing Interface (MPI).
+The data exchanges between processors are required at the very
+place where lateral domain boundary conditions are set in the mono-domain
+computation : the \rou{lbc\_lnk} routine (found in \mdl{lbclnk} module)
+which manages such conditions is interfaced with routines found in \mdl{lib\_mpp} module
+when running on an MPP computer ($i.e.$ when \key{mpp\_mpi} defined).
+It has to be pointed out that when using the MPP version of the model,
+the east-west cyclic boundary condition is done implicitly,
+For massively parallel processing (mpp), a domain decomposition method is used.
+The basic idea of the method is to split the large computation domain of a numerical experiment into
+several smaller domains and solve the set of equations by addressing independent local problems.
+Each processor has its own local memory and computes the model equation over a subdomain of the whole model domain.
+The subdomain boundary conditions are specified through communications between processors which
+are organized by explicit statements (message passing method).
+A big advantage is that the method does not need many modifications of the initial FORTRAN code.
+From the modeller's point of view, each sub domain running on a processor is identical to the "mono-domain" code.
+In addition, the programmer manages the communications between subdomains,
+and the code is faster when the number of processors is increased.
+The porting of OPA code on an iPSC860 was achieved during Guyon's PhD [Guyon et al. 1994, 1995]
+in collaboration with CETIIS and ONERA.
+The implementation in the operational context and the studies of performance on
+a T3D and T3E Cray computers have been made in collaboration with IDRIS and CNRS.
+The present implementation is largely inspired by Guyon's work [Guyon 1995].
+The parallelization strategy is defined by the physical characteristics of the ocean model.
+Second order finite difference schemes lead to local discrete operators that
+depend at the very most on one neighbouring point.
+The only non-local computations concern the vertical physics
+(implicit diffusion, turbulent closure scheme, ...) (delocalization over the whole water column),
+and the solving of the elliptic equation associated with the surface pressure gradient computation
+(delocalization over the whole horizontal domain).
+Therefore, a pencil strategy is used for the data sub-structuration:
+the 3D initial domain is laid out on local processor memories following a 2D horizontal topological splitting.
+Each sub-domain computes its own surface and bottom boundary conditions and
+has a side wall overlapping interface which defines the lateral boundary conditions for
+computations in the inner sub-domain.
+The overlapping area consists of the two rows at each edge of the sub-domain.
+After a computation, a communication phase starts:
+each processor sends to its neighbouring processors the update values of the points corresponding to
+the interior overlapping area to its neighbouring sub-domain ($i.e.$ the innermost of the two overlapping rows).
+The communication is done through the Message Passing Interface (MPI).
+The data exchanges between processors are required at the very place where
+lateral domain boundary conditions are set in the mono-domain computation:
+the \rou{lbc\_lnk} routine (found in \mdl{lbclnk} module) which manages such conditions is interfaced with
+routines found in \mdl{lib\_mpp} module when running on an MPP computer ($i.e.$ when \key{mpp\_mpi} defined).
+It has to be pointed out that when using the MPP version of the model,
+the east-west cyclic boundary condition is done implicitly,
 whilst the south-symmetric boundary condition option is not available.
 …
 \begin{figure}[!t]    \begin{center}
 \includegraphics[width=0.90\textwidth]{Fig_mpp}
 \caption{   \protect\label{fig:mpp}
 Positioning of a sub-domain when massively parallel processing is used. }
+\caption{   \protect\label{fig:mpp}
+  Positioning of a sub-domain when massively parallel processing is used. }
 \end{center}   \end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 In the standard version of \NEMO, the splitting is regular and arithmetic.
+The i-axis is divided by \jp{jpni} and the j-axis by \jp{jpnj} for a number of processors
+\jp{jpnij} most often equal to $jpni \times jpnj$ (parameters set in
+ \ngn{nammpp} namelist). Each processor is independent and without message passing
+ or synchronous process, programs run alone and access just its own local memory.
+ For this reason, the main model dimensions are now the local dimensions of the subdomain (pencil)
+ that are named \jp{jpi}, \jp{jpj}, \jp{jpk}. These dimensions include the internal
+ domain and the overlapping rows. The number of rows to exchange (known as
+ the halo) is usually set to one (\jp{jpreci}=1, in \mdl{par\_oce}). The whole domain
+ dimensions are named \np{jpiglo}, \np{jpjglo} and \jp{jpk}. The relationship between
+ the whole domain and a sub-domain is:
+The i-axis is divided by \jp{jpni} and
+the j-axis by \jp{jpnj} for a number of processors \jp{jpnij} most often equal to $jpni \times jpnj$
+(parameters set in  \ngn{nammpp} namelist).
+Each processor is independent and without message passing or synchronous process,
+programs run alone and access just its own local memory.
+For this reason, the main model dimensions are now the local dimensions of the subdomain (pencil) that
+are named \jp{jpi}, \jp{jpj}, \jp{jpk}.
+These dimensions include the internal domain and the overlapping rows.
+The number of rows to exchange (known as the halo) is usually set to one (\jp{jpreci}=1, in \mdl{par\_oce}).
+The whole domain dimensions are named \np{jpiglo}, \np{jpjglo} and \jp{jpk}.
+The relationship between the whole domain and a sub-domain is:
 \begin{eqnarray}
       jpi & = & ( jpiglo-2*jpreci + (jpni-1) ) / jpni + 2*jpreci  \nonumber \\
 …
 One also defines variables nldi and nlei which correspond to the internal domain bounds,
 and the variables nimpp and njmpp which are the position of the (1,1) grid-point in the global domain.
 An element of $T_{l}$, a local array (subdomain) corresponds to an element of $T_{g}$,
+An element of $T_{l}$, a local array (subdomain) corresponds to an element of $T_{g}$,
 a global array (whole domain) by the relationship:
 \begin{equation} \label{eq:lbc_nimpp}
 …
 with  $1 \leq i \leq jpi$, $1  \leq j \leq jpj $ , and  $1  \leq k \leq jpk$.
 Processors are numbered from 0 to $jpnij-1$, the number is saved in the variable
+nproc. In the standard version, a processor has no more than four neighbouring
+processors named nono (for north), noea (east), noso (south) and nowe (west)
 and two variables, nbondi and nbondj, indicate the relative position of the processor :
+Processors are numbered from 0 to $jpnij-1$, the number is saved in the variable nproc.
+In the standard version, a processor has no more than
+four neighbouring processors named nono (for north), noea (east), noso (south) and nowe (west) and
+two variables, nbondi and nbondj, indicate the relative position of the processor:
 \begin{itemize}
 \item       nbondi = -1    an east neighbour, no west processor,
 …
 \item       nbondi =  2    no splitting following the i-axis.
 \end{itemize}
+During the simulation, processors exchange data with their neighbours.
+If there is effectively a neighbour, the processor receives variables from this
+processor on its overlapping row, and sends the data issued from internal
+domain corresponding to the overlapping row of the other processor.
+The \NEMO model computes equation terms with the help of mask arrays (0 on land
+points and 1 on sea points). It is easily readable and very efficient in the context of
+a computer with vectorial architecture. However, in the case of a scalar processor,
+computations over the land regions become more expensive in terms of CPU time.
+It is worse when we use a complex configuration with a realistic bathymetry like the
+global ocean where more than 50 \% of points are land points. For this reason, a
+pre-processing tool can be used to choose the mpp domain decomposition with a
+maximum number of only land points processors, which can then be eliminated (\autoref{fig:mppini2})
+(For example, the mpp\_optimiz tools, available from the DRAKKAR web site).
+This optimisation is dependent on the specific bathymetry employed. The user
+then chooses optimal parameters \jp{jpni}, \jp{jpnj} and \jp{jpnij} with
+$jpnij < jpni \times jpnj$, leading to the elimination of $jpni \times jpnj - jpnij$
+land processors. When those parameters are specified in \ngn{nammpp} namelist,
+the algorithm in the \rou{inimpp2} routine sets each processor's parameters (nbound,
+nono, noea,...) so that the land-only processors are not taken into account.
+During the simulation, processors exchange data with their neighbours.
+If there is effectively a neighbour, the processor receives variables from this processor on its overlapping row,
+and sends the data issued from internal domain corresponding to the overlapping row of the other processor.
+The \NEMO model computes equation terms with the help of mask arrays (0 on land points and 1 on sea points).
+It is easily readable and very efficient in the context of a computer with vectorial architecture.
+However, in the case of a scalar processor, computations over the land regions become more expensive in
+terms of CPU time.
+It is worse when we use a complex configuration with a realistic bathymetry like the global ocean where
+more than 50 \% of points are land points.
+For this reason, a pre-processing tool can be used to choose the mpp domain decomposition with a maximum number of
+only land points processors, which can then be eliminated (\autoref{fig:mppini2})
+(For example, the mpp\_optimiz tools, available from the DRAKKAR web site).
+This optimisation is dependent on the specific bathymetry employed.
+The user then chooses optimal parameters \jp{jpni}, \jp{jpnj} and \jp{jpnij} with $jpnij < jpni \times jpnj$,
+leading to the elimination of $jpni \times jpnj - jpnij$ land processors.
+When those parameters are specified in \ngn{nammpp} namelist,
+the algorithm in the \rou{inimpp2} routine sets each processor's parameters (nbound, nono, noea,...) so that
+the land-only processors are not taken into account.
 \gmcomment{Note that the inimpp2 routine is general so that the original inimpp
 routine should be suppressed from the code.}
+When land processors are eliminated, the value corresponding to these locations in
+the model output files is undefined. Note that this is a problem for the meshmask file
+which requires to be defined over the whole domain. Therefore, user should not eliminate
+land processors when creating a meshmask file ($i.e.$ when setting a non-zero value to \np{nn\_msh}).
+When land processors are eliminated,
+the value corresponding to these locations in the model output files is undefined.
+Note that this is a problem for the meshmask file which requires to be defined over the whole domain.
+Therefore, user should not eliminate land processors when creating a meshmask file
+($i.e.$ when setting a non-zero value to \np{nn\_msh}).
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 …
 \includegraphics[width=0.90\textwidth]{Fig_mppini2}
 \caption {    \protect\label{fig:mppini2}
+Example of Atlantic domain defined for the CLIPPER projet. Initial grid is
+composed of 773 x 1236 horizontal points.
+(a) the domain is split onto 9 \time 20 subdomains (jpni=9, jpnj=20).
+subdomains are land areas.
+(b) 52 subdomains are eliminated (white rectangles) and the resulting number
 of processors really used during the computation is jpnij=128.}
+  Example of Atlantic domain defined for the CLIPPER projet.
+  Initial grid is composed of 773 x 1236 horizontal points.
+  (a) the domain is split onto 9 \time 20 subdomains (jpni=9, jpnj=20).
+subdomains are land areas.
+  (b) 52 subdomains are eliminated (white rectangles) and
+  the resulting number of processors really used during the computation is jpnij=128.}
 \end{center}   \end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 …
 \nlst{nambdy}
 %-----------------------------------------------------------------------------------------------
-%-----------------------------------------nambdy_index--------------------------------------------
+%
-%\nlst{nambdy_index}
-%-----------------------------------------------------------------------------------------------
 %-----------------------------------------nambdy_dta--------------------------------------------
 \nlst{nambdy_dta}
 %-----------------------------------------------------------------------------------------------
+%-----------------------------------------nambdy_dta--------------------------------------------
+%
+%\nlst{nambdy_dta2}
+%-----------------------------------------------------------------------------------------------
+Options are defined through the \ngn{nambdy} \ngn{nambdy\_index}
+\ngn{nambdy\_dta} \ngn{nambdy\_dta2} namelist variables.
+The BDY module is the core implementation of open boundary
+conditions for regional configurations. It implements the Flow
+Relaxation Scheme algorithm for temperature, salinity, velocities and
+ice fields, and the Flather radiation condition for the depth-mean
+transports. The specification of the location of the open boundary is
+completely flexible and allows for example the open boundary to follow
+an isobath or other irregular contour.
+The BDY module was modelled on the OBC module (see NEMO 3.4) and shares many
+features and a similar coding structure \citep{Chanut2005}.
+Boundary data files used with earlier versions of NEMO may need
+to be re-ordered to work with this version. See the
+section on the Input Boundary Data Files for details.
+Options are defined through the \ngn{nambdy} \ngn{nambdy\_dta} namelist variables.
+The BDY module is the core implementation of open boundary conditions for regional configurations.
+It implements the Flow Relaxation Scheme algorithm for temperature, salinity, velocities and ice fields, and
+the Flather radiation condition for the depth-mean transports.
+The specification of the location of the open boundary is completely flexible and
+allows for example the open boundary to follow an isobath or other irregular contour.
+The BDY module was modelled on the OBC module (see NEMO 3.4) and shares many features and
+a similar coding structure \citep{Chanut2005}.
+Boundary data files used with earlier versions of NEMO may need to be re-ordered to work with this version.
+See the section on the Input Boundary Data Files for details.
 %----------------------------------------------
 …
 The BDY module is activated by setting \np{ln\_bdy} to true.
+It is possible to define more than one boundary ``set'' and apply
+different boundary conditions to each set. The number of boundary
+sets is defined by \np{nb\_bdy}.  Each boundary set may be defined
+as a set of straight line segments in a namelist
+(\np{ln\_coords\_file}\forcode{ = .false.}) or read in from a file
+(\np{ln\_coords\_file}\forcode{ = .true.}). If the set is defined in a namelist,
+then the namelists nambdy\_index must be included separately, one for
+each set. If the set is defined by a file, then a
+``\ifile{coordinates.bdy}'' file must be provided. The coordinates.bdy file
+is analagous to the usual NEMO ``\ifile{coordinates}'' file. In the example
+above, there are two boundary sets, the first of which is defined via
+a file and the second is defined in a namelist. For more details of
+the definition of the boundary geometry see section
+\autoref{subsec:BDY_geometry}.
+For each boundary set a boundary
+condition has to be chosen for the barotropic solution (``u2d'':
+sea-surface height and barotropic velocities), for the baroclinic
+velocities (``u3d''), and for the active tracers\footnote{The BDY
+  module does not deal with passive tracers at this version}
+(``tra''). For each set of variables there is a choice of algorithm
+and a choice for the data, eg. for the active tracers the algorithm is
+set by \np{nn\_tra} and the choice of data is set by
+\np{nn\_tra\_dta}.
+It is possible to define more than one boundary ``set'' and apply different boundary conditions to each set.
+The number of boundary sets is defined by \np{nb\_bdy}.
+Each boundary set may be defined as a set of straight line segments in a namelist
+(\np{ln\_coords\_file}\forcode{ = .false.}) or read in from a file (\np{ln\_coords\_file}\forcode{ = .true.}).
+If the set is defined in a namelist, then the namelists nambdy\_index must be included separately, one for each set.
+If the set is defined by a file, then a ``\ifile{coordinates.bdy}'' file must be provided.
+The coordinates.bdy file is analagous to the usual NEMO ``\ifile{coordinates}'' file.
+In the example above, there are two boundary sets, the first of which is defined via a file and
+the second is defined in a namelist.
+For more details of the definition of the boundary geometry see section \autoref{subsec:BDY_geometry}.
+For each boundary set a boundary condition has to be chosen for the barotropic solution
+(``u2d'':sea-surface height and barotropic velocities), for the baroclinic velocities (``u3d''), and
+for the active tracers \footnote{The BDY module does not deal with passive tracers at this version} (``tra'').
+For each set of variables there is a choice of algorithm and a choice for the data,
+eg. for the active tracers the algorithm is set by \np{nn\_tra} and the choice of data is set by \np{nn\_tra\_dta}.
 The choice of algorithm is currently as follows:
 …
 \begin{itemize}
 \item[0.] No boundary condition applied. So the solution will ``see''
   the land points around the edge of the edge of the domain.
 \item[1.] Flow Relaxation Scheme (FRS) available for all variables.
 \item[2.] Flather radiation scheme for the barotropic variables. The
   Flather scheme is not compatible with the filtered free surface
+\item[0.] No boundary condition applied.
+  So the solution will ``see'' the land points around the edge of the edge of the domain.
+\item[1.] Flow Relaxation Scheme (FRS) available for all variables.
+\item[2.] Flather radiation scheme for the barotropic variables.
+  The Flather scheme is not compatible with the filtered free surface
   ({\it dynspg\_ts}).
 \end{itemize}
 …
 \mbox{}
+The main choice for the boundary data is
+to use initial conditions as boundary data (\np{nn\_tra\_dta}\forcode{ = 0}) or to
+use external data from a file (\np{nn\_tra\_dta}\forcode{ = 1}). For the
+barotropic solution there is also the option to use tidal
+harmonic forcing either by itself or in addition to other external
+data.
+If external boundary data is required then the nambdy\_dta namelist
+must be defined. One nambdy\_dta namelist is required for each boundary
+set in the order in which the boundary sets are defined in nambdy. In
+the example given, two boundary sets have been defined and so there
+are two nambdy\_dta namelists. The boundary data is read in using the
+fldread module, so the nambdy\_dta namelist is in the format required
+for fldread. For each variable required, the filename, the frequency
+of the files and the frequency of the data in the files is given. Also
+whether or not time-interpolation is required and whether the data is
+climatological (time-cyclic) data. Note that on-the-fly spatial
+interpolation of boundary data is not available at this version.
+In the example namelists given, two boundary sets are defined. The
+first set is defined via a file and applies FRS conditions to
+temperature and salinity and Flather conditions to the barotropic
+variables. External data is provided in daily files (from a
+large-scale model). Tidal harmonic forcing is also used. The second
+set is defined in a namelist. FRS conditions are applied on
+temperature and salinity and climatological data is read from external
+files.
+The main choice for the boundary data is to use initial conditions as boundary data
+(\np{nn\_tra\_dta}\forcode{ = 0}) or to use external data from a file (\np{nn\_tra\_dta}\forcode{ = 1}).
+For the barotropic solution there is also the option to use tidal harmonic forcing either by
+itself or in addition to other external data.
+If external boundary data is required then the nambdy\_dta namelist must be defined.
+One nambdy\_dta namelist is required for each boundary set in the order in which
+the boundary sets are defined in nambdy.
+In the example given, two boundary sets have been defined and so there are two nambdy\_dta namelists.
+The boundary data is read in using the fldread module,
+so the nambdy\_dta namelist is in the format required for fldread.
+For each variable required, the filename, the frequency of the files and
+the frequency of the data in the files is given.
+Also whether or not time-interpolation is required and whether the data is climatological (time-cyclic) data.
+Note that on-the-fly spatial interpolation of boundary data is not available at this version.
+In the example namelists given, two boundary sets are defined.
+The first set is defined via a file and applies FRS conditions to temperature and salinity and
+Flather conditions to the barotropic variables.
+External data is provided in daily files (from a large-scale model).
+Tidal harmonic forcing is also used.
+The second set is defined in a namelist.
+FRS conditions are applied on temperature and salinity and climatological data is read from external files.
 %----------------------------------------------
 …
 The Flow Relaxation Scheme (FRS) \citep{Davies_QJRMS76,Engerdahl_Tel95},
 applies a simple relaxation of the model fields to
+externally-specified values over a zone next to the edge of the model
+domain. Given a model prognostic variable $\Phi$
+applies a simple relaxation of the model fields to externally-specified values over
+a zone next to the edge of the model domain.
+Given a model prognostic variable $\Phi$
 \begin{equation}  \label{eq:bdy_frs1}
 \Phi(d) = \alpha(d)\Phi_{e}(d) + (1-\alpha(d))\Phi_{m}(d)\;\;\;\;\; d=1,N
 \end{equation}
+where $\Phi_{m}$ is the model solution and $\Phi_{e}$ is the specified
+external field, $d$ gives the discrete distance from the model
+boundary  and $\alpha$ is a parameter that varies from $1$ at $d=1$ to
+a small value at $d=N$. It can be shown that this scheme is equivalent
+to adding a relaxation term to the prognostic equation for $\Phi$ of
+the form:
+where $\Phi_{m}$ is the model solution and $\Phi_{e}$ is the specified external field,
+$d$ gives the discrete distance from the model boundary and
+$\alpha$ is a parameter that varies from $1$ at $d=1$ to a small value at $d=N$.
+It can be shown that this scheme is equivalent to adding a relaxation term to
+the prognostic equation for $\Phi$ of the form:
 \begin{equation}  \label{eq:bdy_frs2}
 -\frac{1}{\tau}\left(\Phi - \Phi_{e}\right)
 \end{equation}
+where the relaxation time scale $\tau$ is given by a function of
+$\alpha$ and the model time step $\Delta t$:
+where the relaxation time scale $\tau$ is given by a function of $\alpha$ and the model time step $\Delta t$:
 \begin{equation}  \label{eq:bdy_frs3}
 \tau = \frac{1-\alpha}{\alpha}  \,\rdt
 \end{equation}
+Thus the model solution is completely prescribed by the external
+conditions at the edge of the model domain and is relaxed towards the
+external conditions over the rest of the FRS zone. The application of
+a relaxation zone helps to prevent spurious reflection of outgoing
+signals from the model boundary.
+Thus the model solution is completely prescribed by the external conditions at the edge of the model domain and
+is relaxed towards the external conditions over the rest of the FRS zone.
+The application of a relaxation zone helps to prevent spurious reflection of
+outgoing signals from the model boundary.
 The function $\alpha$ is specified as a $tanh$ function:
 …
 \alpha(d) = 1 - \tanh\left(\frac{d-1}{2}\right),       \quad d=1,N
 \end{equation}
 The width of the FRS zone is specified in the namelist as
+\np{nn\_rimwidth}. This is typically set to a value between 8 and 10.
+The width of the FRS zone is specified in the namelist as \np{nn\_rimwidth}.
+This is typically set to a value between 8 and 10.
 %----------------------------------------------
 …
 \label{subsec:BDY_flather_scheme}
+The \citet{Flather_JPO94} scheme is a radiation condition on the normal, depth-mean
+transport across the open boundary. It takes the form
+The \citet{Flather_JPO94} scheme is a radiation condition on the normal,
+depth-mean transport across the open boundary.
+It takes the form
 \begin{equation}  \label{eq:bdy_fla1}
 U = U_{e} + \frac{c}{h}\left(\eta - \eta_{e}\right),
 \end{equation}
+where $U$ is the depth-mean velocity normal to the boundary and $\eta$
+is the sea surface height, both from the model. The subscript $e$
+indicates the same fields from external sources. The speed of external
+gravity waves is given by $c = \sqrt{gh}$, and $h$ is the depth of the
+water column. The depth-mean normal velocity along the edge of the
+model domain is set equal to the
+external depth-mean normal velocity, plus a correction term that
+allows gravity waves generated internally to exit the model boundary.
+Note that the sea-surface height gradient in \autoref{eq:bdy_fla1}
+is a spatial gradient across the model boundary, so that $\eta_{e}$ is
+defined on the $T$ points with $nbr=1$ and $\eta$ is defined on the
+$T$ points with $nbr=2$. $U$ and $U_{e}$ are defined on the $U$ or
+$V$ points with $nbr=1$, $i.e.$ between the two $T$ grid points.
+where $U$ is the depth-mean velocity normal to the boundary and $\eta$ is the sea surface height,
+both from the model.
+The subscript $e$ indicates the same fields from external sources.
+The speed of external gravity waves is given by $c = \sqrt{gh}$, and $h$ is the depth of the water column.
+The depth-mean normal velocity along the edge of the model domain is set equal to
+the external depth-mean normal velocity,
+plus a correction term that allows gravity waves generated internally to exit the model boundary.
+Note that the sea-surface height gradient in \autoref{eq:bdy_fla1} is a spatial gradient across the model boundary,
+so that $\eta_{e}$ is defined on the $T$ points with $nbr=1$ and $\eta$ is defined on the $T$ points with $nbr=2$.
+$U$ and $U_{e}$ are defined on the $U$ or $V$ points with $nbr=1$, $i.e.$ between the two $T$ grid points.
 %----------------------------------------------
 …
 \label{subsec:BDY_geometry}
+Each open boundary set is defined as a list of points. The information
+is stored in the arrays $nbi$, $nbj$, and $nbr$ in the $idx\_bdy$
+structure.  The $nbi$ and $nbj$ arrays
+define the local $(i,j)$ indices of each point in the boundary zone
+and the $nbr$ array defines the discrete distance from the boundary
+with $nbr=1$ meaning that the point is next to the edge of the
+model domain and $nbr>1$ showing that the point is increasingly
+further away from the edge of the model domain. A set of $nbi$, $nbj$,
+and $nbr$ arrays is defined for each of the $T$, $U$ and $V$
+grids. Figure \autoref{fig:LBC_bdy_geom} shows an example of an irregular
+boundary.
+The boundary geometry for each set may be defined in a namelist
+nambdy\_index or by reading in a ``\ifile{coordinates.bdy}'' file. The
+nambdy\_index namelist defines a series of straight-line segments for
+north, east, south and west boundaries. For the northern boundary,
+\np{nbdysegn} gives the number of segments, \np{jpjnob} gives the $j$
+index for each segment and \np{jpindt} and \np{jpinft} give the start
+and end $i$ indices for each segment with similar for the other
+boundaries. These segments define a list of $T$ grid points along the
+outermost row of the boundary ($nbr\,=\, 1$). The code deduces the $U$ and
+$V$ points and also the points for $nbr\,>\, 1$ if
+$nn\_rimwidth\,>\,1$.
+The boundary geometry may also be defined from a
+``\ifile{coordinates.bdy}'' file. Figure \autoref{fig:LBC_nc_header}
+gives an example of the header information from such a file. The file
+should contain the index arrays for each of the $T$, $U$ and $V$
+grids. The arrays must be in order of increasing $nbr$. Note that the
+$nbi$, $nbj$ values in the file are global values and are converted to
+local values in the code. Typically this file will be used to generate
+external boundary data via interpolation and so will also contain the
+latitudes and longitudes of each point as shown. However, this is not
+necessary to run the model.
+For some choices of irregular boundary the model domain may contain
+areas of ocean which are not part of the computational domain. For
+example if an open boundary is defined along an isobath, say at the
+shelf break, then the areas of ocean outside of this boundary will
+need to be masked out. This can be done by reading a mask file defined
+as \np{cn\_mask\_file} in the nam\_bdy namelist. Only one mask file is
+used even if multiple boundary sets are defined.
+Each open boundary set is defined as a list of points.
+The information is stored in the arrays $nbi$, $nbj$, and $nbr$ in the $idx\_bdy$ structure.
+The $nbi$ and $nbj$ arrays define the local $(i,j)$ indices of each point in the boundary zone and
+the $nbr$ array defines the discrete distance from the boundary with $nbr=1$ meaning that
+the point is next to the edge of the model domain and $nbr>1$ showing that
+the point is increasingly further away from the edge of the model domain.
+A set of $nbi$, $nbj$, and $nbr$ arrays is defined for each of the $T$, $U$ and $V$ grids.
+Figure \autoref{fig:LBC_bdy_geom} shows an example of an irregular boundary.
+The boundary geometry for each set may be defined in a namelist nambdy\_index or
+by reading in a ``\ifile{coordinates.bdy}'' file.
+The nambdy\_index namelist defines a series of straight-line segments for north, east, south and west boundaries.
+For the northern boundary, \np{nbdysegn} gives the number of segments,
+\np{jpjnob} gives the $j$ index for each segment and \np{jpindt} and
+\np{jpinft} give the start and end $i$ indices for each segment with similar for the other boundaries.
+These segments define a list of $T$ grid points along the outermost row of the boundary ($nbr\,=\, 1$).
+The code deduces the $U$ and $V$ points and also the points for $nbr\,>\, 1$ if $nn\_rimwidth\,>\,1$.
+The boundary geometry may also be defined from a ``\ifile{coordinates.bdy}'' file.
+Figure \autoref{fig:LBC_nc_header} gives an example of the header information from such a file.
+The file should contain the index arrays for each of the $T$, $U$ and $V$ grids.
+The arrays must be in order of increasing $nbr$.
+Note that the $nbi$, $nbj$ values in the file are global values and are converted to local values in the code.
+Typically this file will be used to generate external boundary data via interpolation and so
+will also contain the latitudes and longitudes of each point as shown.
+However, this is not necessary to run the model.
+For some choices of irregular boundary the model domain may contain areas of ocean which
+are not part of the computational domain.
+For example if an open boundary is defined along an isobath, say at the shelf break,
+then the areas of ocean outside of this boundary will need to be masked out.
+This can be done by reading a mask file defined as \np{cn\_mask\_file} in the nam\_bdy namelist.
+Only one mask file is used even if multiple boundary sets are defined.
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 …
 \includegraphics[width=1.0\textwidth]{Fig_LBC_bdy_geom}
 \caption {      \protect\label{fig:LBC_bdy_geom}
 Example of geometry of unstructured open boundary}
+  Example of geometry of unstructured open boundary}
 \end{center}   \end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 …
 \label{subsec:BDY_data}
 The data files contain the data arrays
+in the order in which the points are defined in the $nbi$ and $nbj$
 arrays. The data arrays are dimensioned on: a time dimension;
+The data files contain the data arrays in the order in which the points are defined in the $nbi$ and $nbj$ arrays.
+The data arrays are dimensioned on:
+a time dimension;
 $xb$ which is the index of the boundary data point in the horizontal;
+and $yb$ which is a degenerate dimension of 1 to enable the file to be
+read by the standard NEMO I/O routines. The 3D fields also have a
+depth dimension.
+At Version 3.4 there are new restrictions on the order in which the
+boundary points are defined (and therefore restrictions on the order
+of the data in the file). In particular:
+and $yb$ which is a degenerate dimension of 1 to enable the file to be read by the standard NEMO I/O routines.
+The 3D fields also have a depth dimension.
+At Version 3.4 there are new restrictions on the order in which the boundary points are defined
+(and therefore restrictions on the order of the data in the file).
+In particular:
 \mbox{}
 \begin{enumerate}
 \item The data points must be in order of increasing $nbr$, ie. all
   the $nbr=1$ points, then all the $nbr=2$ points etc.
 \item All the data for a particular boundary set must be in the same
   order. (Prior to 3.4 it was possible to define barotropic data in a
   different order to the data for tracers and baroclinic velocities).
+\item The data points must be in order of increasing $nbr$,
+  ie. all the $nbr=1$ points, then all the $nbr=2$ points etc.
+\item All the data for a particular boundary set must be in the same order.
+  (Prior to 3.4 it was possible to define barotropic data in a different order to
+  the data for tracers and baroclinic velocities).
 \end{enumerate}
 \mbox{}
+These restrictions mean that data files used with previous versions of
+the model may not work with version 3.4. A fortran utility
+{\it bdy\_reorder} exists in the TOOLS directory which will re-order the
+data in old BDY data files.
+These restrictions mean that data files used with previous versions of the model may not work with version 3.4.
+A fortran utility {\it bdy\_reorder} exists in the TOOLS directory which
+will re-order the data in old BDY data files.
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 \begin{figure}[!t]     \begin{center}
 \includegraphics[width=1.0\textwidth]{Fig_LBC_nc_header}
 \caption {     \protect\label{fig:LBC_nc_header}
 Example of the header for a \protect\ifile{coordinates.bdy} file}
+\caption {     \protect\label{fig:LBC_nc_header}
+  Example of the header for a \protect\ifile{coordinates.bdy} file}
 \end{center}   \end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 …
 \label{subsec:BDY_vol_corr}
+There is an option to force the total volume in the regional model to be constant,
+similar to the option in the OBC module. This is controlled  by the \np{nn\_volctl}
+parameter in the namelist. A value of \np{nn\_volctl}\forcode{ = 0} indicates that this option is not used.
+If  \np{nn\_volctl}\forcode{ = 1} then a correction is applied to the normal velocities
+around the boundary at each timestep to ensure that the integrated volume flow
+through the boundary is zero. If \np{nn\_volctl}\forcode{ = 2} then the calculation of
+the volume change on the timestep includes the change due to the freshwater
+flux across the surface and the correction velocity corrects for this as well.
+There is an option to force the total volume in the regional model to be constant,
+similar to the option in the OBC module.
+This is controlled  by the \np{nn\_volctl} parameter in the namelist.
+A value of \np{nn\_volctl}\forcode{ = 0} indicates that this option is not used.
+If \np{nn\_volctl}\forcode{ = 1} then a correction is applied to the normal velocities around the boundary at
+each timestep to ensure that the integrated volume flow through the boundary is zero.
+If \np{nn\_volctl}\forcode{ = 2} then the calculation of the volume change on
+the timestep includes the change due to the freshwater flux across the surface and
+the correction velocity corrects for this as well.
 If more than one boundary set is used then volume correction is

NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_LDF.tex

-                      r10146
+                      r10368
+The lateral physics terms in the momentum and tracer equations have been
+described in \autoref{eq:PE_zdf} and their discrete formulation in \autoref{sec:TRA_ldf}
+and \autoref{sec:DYN_ldf}). In this section we further discuss each lateral physics option.
+Choosing one lateral physics scheme means for the user defining,
+(1) the type of operator used (laplacian or bilaplacian operators, or no lateral mixing term) ;
+(2) the direction along which the lateral diffusive fluxes are evaluated (model level, geopotential or isopycnal surfaces) ; and
+(3) the space and time variations of the eddy coefficients.
+These three aspects of the lateral diffusion are set through namelist parameters
+(see the \textit{\ngn{nam\_traldf}} and \textit{\ngn{nam\_dynldf}} below).
+Note that this chapter describes the standard implementation of iso-neutral
+tracer mixing, and Griffies's implementation, which is used if
+\np{traldf\_grif}\forcode{ = .true.}, is described in Appdx\autoref{apdx:triad}
+The lateral physics terms in the momentum and tracer equations have been described in \autoref{eq:PE_zdf} and
+their discrete formulation in \autoref{sec:TRA_ldf} and \autoref{sec:DYN_ldf}).
+In this section we further discuss each lateral physics option.
+Choosing one lateral physics scheme means for the user defining,
+(1) the type of operator used (laplacian or bilaplacian operators, or no lateral mixing term);
+(2) the direction along which the lateral diffusive fluxes are evaluated
+(model level, geopotential or isopycnal surfaces); and
+(3) the space and time variations of the eddy coefficients.
+These three aspects of the lateral diffusion are set through namelist parameters
+(see the \textit{\ngn{nam\_traldf}} and \textit{\ngn{nam\_dynldf}} below).
+Note that this chapter describes the standard implementation of iso-neutral tracer mixing,
+and Griffies's implementation, which is used if \np{traldf\_grif}\forcode{ = .true.},
+is described in Appdx\autoref{apdx:triad}
 %-----------------------------------nam_traldf - nam_dynldf--------------------------------------------
 …
 Better work can be achieved by using \citet{Griffies_al_JPO98, Griffies_Bk04} iso-neutral scheme. }
 A direction for lateral mixing has to be defined when the desired operator does
+not act along the model levels. This occurs when $(a)$ horizontal mixing is
+required on tracer or momentum (\np{ln\_traldf\_hor} or \np{ln\_dynldf\_hor})
+in $s$- or mixed $s$-$z$- coordinates, and $(b)$ isoneutral mixing is required
+whatever the vertical coordinate is. This direction of mixing is defined by its
+slopes in the \textbf{i}- and \textbf{j}-directions at the face of the cell of the
+quantity to be diffused. For a tracer, this leads to the following four slopes :
 $r_{1u}$, $r_{1w}$, $r_{2v}$, $r_{2w}$ (see \autoref{eq:tra_ldf_iso}), while
+for momentum the slopes are  $r_{1t}$, $r_{1uw}$, $r_{2f}$, $r_{2uw}$ for
 $u$ and  $r_{1f}$, $r_{1vw}$, $r_{2t}$, $r_{2vw}$ for $v$.
+A direction for lateral mixing has to be defined when the desired operator does not act along the model levels.
+This occurs when $(a)$ horizontal mixing is required on tracer or momentum
+(\np{ln\_traldf\_hor} or \np{ln\_dynldf\_hor}) in $s$- or mixed $s$-$z$- coordinates,
+and $(b)$ isoneutral mixing is required whatever the vertical coordinate is.
+This direction of mixing is defined by its slopes in the \textbf{i}- and \textbf{j}-directions at the face of
+the cell of the quantity to be diffused.
+For a tracer, this leads to the following four slopes:
+$r_{1u}$, $r_{1w}$, $r_{2v}$, $r_{2w}$ (see \autoref{eq:tra_ldf_iso}),
+while for momentum the slopes are  $r_{1t}$, $r_{1uw}$, $r_{2f}$, $r_{2uw}$ for $u$ and
+$r_{1f}$, $r_{1vw}$, $r_{2t}$, $r_{2vw}$ for $v$.
 %gm% add here afigure of the slope in i-direction
 …
 \subsection{Slopes for tracer geopotential mixing in the $s$-coordinate}
+In $s$-coordinates, geopotential mixing ($i.e.$ horizontal mixing) $r_1$ and
+$r_2$ are the slopes between the geopotential and computational surfaces.
+Their discrete formulation is found by locally solving \autoref{eq:tra_ldf_iso}
+when the diffusive fluxes in the three directions are set to zero and $T$ is
+assumed to be horizontally uniform, $i.e.$ a linear function of $z_T$, the
+depth of a $T$-point.
+In $s$-coordinates, geopotential mixing ($i.e.$ horizontal mixing) $r_1$ and $r_2$ are the slopes between
+the geopotential and computational surfaces.
+Their discrete formulation is found by locally solving \autoref{eq:tra_ldf_iso} when
+the diffusive fluxes in the three directions are set to zero and $T$ is assumed to be horizontally uniform,
+$i.e.$ a linear function of $z_T$, the depth of a $T$-point.
 %gm { Steven : My version is obviously wrong since I'm left with an arbitrary constant which is the local vertical temperature gradient}
 …
 %gm%  caution I'm not sure the simplification was a good idea!
 These slopes are computed once in \rou{ldfslp\_init} when \np{ln\_sco}\forcode{ = .true.}rue,
 and either \np{ln\_traldf\_hor}\forcode{ = .true.}rue or \np{ln\_dynldf\_hor}\forcode{ = .true.}rue.
+These slopes are computed once in \rou{ldfslp\_init} when \np{ln\_sco}\forcode{ = .true.}rue,
+and either \np{ln\_traldf\_hor}\forcode{ = .true.} or \np{ln\_dynldf\_hor}\forcode{ = .true.}.
 \subsection{Slopes for tracer iso-neutral mixing}
 \label{subsec:LDF_slp_iso}
+In iso-neutral mixing  $r_1$ and $r_2$ are the slopes between the iso-neutral
+and computational surfaces. Their formulation does not depend on the vertical
+coordinate used. Their discrete formulation is found using the fact that the
+diffusive fluxes of locally referenced potential density ($i.e.$ $in situ$ density)
+vanish. So, substituting $T$ by $\rho$ in \autoref{eq:tra_ldf_iso} and setting the
+diffusive fluxes in the three directions to zero leads to the following definition for
+the neutral slopes:
+In iso-neutral mixing  $r_1$ and $r_2$ are the slopes between the iso-neutral and computational surfaces.
+Their formulation does not depend on the vertical coordinate used.
+Their discrete formulation is found using the fact that the diffusive fluxes of
+locally referenced potential density ($i.e.$ $in situ$ density) vanish.
+So, substituting $T$ by $\rho$ in \autoref{eq:tra_ldf_iso} and setting the diffusive fluxes in
+the three directions to zero leads to the following definition for the neutral slopes:
 \begin{equation} \label{eq:ldfslp_iso}
 …
 %In the $z$-coordinate, the derivative of the  \autoref{eq:ldfslp_iso} numerator is evaluated at the same depth \nocite{as what?} ($T$-level, which is the same as the $u$- and $v$-levels), so  the $in situ$ density can be used for its evaluation.
 As the mixing is performed along neutral surfaces, the gradient of $\rho$ in
+\autoref{eq:ldfslp_iso} has to be evaluated at the same local pressure (which,
+in decibars, is approximated by the depth in meters in the model). Therefore
+\autoref{eq:ldfslp_iso} cannot be used as such, but further transformation is
 needed depending on the vertical coordinate used:
+As the mixing is performed along neutral surfaces, the gradient of $\rho$ in \autoref{eq:ldfslp_iso} has to
+be evaluated at the same local pressure
+(which, in decibars, is approximated by the depth in meters in the model).
+Therefore \autoref{eq:ldfslp_iso} cannot be used as such,
+but further transformation is needed depending on the vertical coordinate used:
 \begin{description}
 \item[$z$-coordinate with full step : ] in \autoref{eq:ldfslp_iso} the densities
+appearing in the $i$ and $j$ derivatives  are taken at the same depth, thus
+the $in situ$ density can be used. This is not the case for the vertical
+derivatives: $\delta_{k+1/2}[\rho]$ is replaced by $-\rho N^2/g$, where $N^2$
+is the local Brunt-Vais\"{a}l\"{a} frequency evaluated following
 \citet{McDougall1987} (see \autoref{subsec:TRA_bn2}).
 \item[$z$-coordinate with partial step : ] this case is identical to the full step
+case except that at partial step level, the \emph{horizontal} density gradient
 is evaluated as described in \autoref{sec:TRA_zpshde}.
 \item[$s$- or hybrid $s$-$z$- coordinate : ] in the current release of \NEMO,
+iso-neutral mixing is only employed for $s$-coordinates if the
+Griffies scheme is used (\np{traldf\_grif}\forcode{ = .true.}; see Appdx \autoref{apdx:triad}).
+In other words, iso-neutral mixing will only be accurately represented with a
+linear equation of state (\np{nn\_eos}\forcode{ = 1..2}). In the case of a "true" equation
+of state, the evaluation of $i$ and $j$ derivatives in \autoref{eq:ldfslp_iso}
+will include a pressure dependent part, leading to the wrong evaluation of
 the neutral slopes.
+\item[$z$-coordinate with full step: ]
+  in \autoref{eq:ldfslp_iso} the densities appearing in the $i$ and $j$ derivatives  are taken at the same depth,
+  thus the $in situ$ density can be used.
+  This is not the case for the vertical derivatives: $\delta_{k+1/2}[\rho]$ is replaced by $-\rho N^2/g$,
+  where $N^2$ is the local Brunt-Vais\"{a}l\"{a} frequency evaluated following \citet{McDougall1987}
+  (see \autoref{subsec:TRA_bn2}).
+\item[$z$-coordinate with partial step: ]
+  this case is identical to the full step case except that at partial step level,
+  the \emph{horizontal} density gradient is evaluated as described in \autoref{sec:TRA_zpshde}.
+\item[$s$- or hybrid $s$-$z$- coordinate: ]
+  in the current release of \NEMO, iso-neutral mixing is only employed for $s$-coordinates if
+  the Griffies scheme is used (\np{traldf\_grif}\forcode{ = .true.};
+  see Appdx \autoref{apdx:triad}).
+  In other words, iso-neutral mixing will only be accurately represented with a linear equation of state
+  (\np{nn\_eos}\forcode{ = 1..2}).
+  In the case of a "true" equation of state, the evaluation of $i$ and $j$ derivatives in \autoref{eq:ldfslp_iso}
+  will include a pressure dependent part, leading to the wrong evaluation of the neutral slopes.
 %gm%
+Note: The solution for $s$-coordinate passes trough the use of different
+(and better) expression for the constraint on iso-neutral fluxes. Following
+\citet{Griffies_Bk04}, instead of specifying directly that there is a zero neutral
+diffusive flux of locally referenced potential density, we stay in the $T$-$S$
+plane and consider the balance between the neutral direction diffusive fluxes
+of potential temperature and salinity:
+  Note: The solution for $s$-coordinate passes trough the use of different (and better) expression for
+  the constraint on iso-neutral fluxes.
+  Following \citet{Griffies_Bk04}, instead of specifying directly that there is a zero neutral diffusive flux of
+  locally referenced potential density, we stay in the $T$-$S$ plane and consider the balance between
+  the neutral direction diffusive fluxes of potential temperature and salinity:
 \begin{equation}
 \alpha \ \textbf{F}(T) = \beta \ \textbf{F}(S)
 …
 \end{split}
 \end{equation}
+where $\alpha$ and $\beta$, the thermal expansion and saline contraction
+coefficients introduced in \autoref{subsec:TRA_bn2}, have to be evaluated at the three
+velocity points. In order to save computation time, they should be approximated
+by the mean of their values at $T$-points (for example in the case of $\alpha$:
+$\alpha_u=\overline{\alpha_T}^{i+1/2}$,  $\alpha_v=\overline{\alpha_T}^{j+1/2}$
+and $\alpha_w=\overline{\alpha_T}^{k+1/2}$).
+Note that such a formulation could be also used in the $z$-coordinate and
+$z$-coordinate with partial steps cases.
+where $\alpha$ and $\beta$, the thermal expansion and saline contraction coefficients introduced in
+\autoref{subsec:TRA_bn2}, have to be evaluated at the three velocity points.
+In order to save computation time, they should be approximated by the mean of their values at $T$-points
+(for example in the case of $\alpha$:
+$\alpha_u=\overline{\alpha_T}^{i+1/2}$,  $\alpha_v=\overline{\alpha_T}^{j+1/2}$ and
+$\alpha_w=\overline{\alpha_T}^{k+1/2}$).
+Note that such a formulation could be also used in the $z$-coordinate and $z$-coordinate with partial steps cases.
 \end{description}
+This implementation is a rather old one. It is similar to the one
+proposed by Cox [1987], except for the background horizontal
+diffusion. Indeed, the Cox implementation of isopycnal diffusion in
+GFDL-type models requires a minimum background horizontal diffusion
+for numerical stability reasons.  To overcome this problem, several
+techniques have been proposed in which the numerical schemes of the
+ocean model are modified \citep{Weaver_Eby_JPO97,
+  Griffies_al_JPO98}. Griffies's scheme is now available in \NEMO if
+\np{traldf\_grif\_iso} is set true; see Appdx \autoref{apdx:triad}. Here,
+another strategy is presented \citep{Lazar_PhD97}: a local
+filtering of the iso-neutral slopes (made on 9 grid-points) prevents
+the development of grid point noise generated by the iso-neutral
+diffusion operator (\autoref{fig:LDF_ZDF1}). This allows an
+iso-neutral diffusion scheme without additional background horizontal
+mixing. This technique can be viewed as a diffusion operator that acts
+along large-scale (2~$\Delta$x) \gmcomment{2deltax doesnt seem very
+  large scale} iso-neutral surfaces. The diapycnal diffusion required
+for numerical stability is thus minimized and its net effect on the
+flow is quite small when compared to the effect of an horizontal
+background mixing.
+Nevertheless, this iso-neutral operator does not ensure that variance cannot increase,
+This implementation is a rather old one.
+It is similar to the one proposed by Cox [1987], except for the background horizontal diffusion.
+Indeed, the Cox implementation of isopycnal diffusion in GFDL-type models requires
+a minimum background horizontal diffusion for numerical stability reasons.
+To overcome this problem, several techniques have been proposed in which the numerical schemes of
+the ocean model are modified \citep{Weaver_Eby_JPO97, Griffies_al_JPO98}.
+Griffies's scheme is now available in \NEMO if \np{traldf\_grif\_iso} is set true; see Appdx \autoref{apdx:triad}.
+Here, another strategy is presented \citep{Lazar_PhD97}:
+a local filtering of the iso-neutral slopes (made on 9 grid-points) prevents the development of
+grid point noise generated by the iso-neutral diffusion operator (\autoref{fig:LDF_ZDF1}).
+This allows an iso-neutral diffusion scheme without additional background horizontal mixing.
+This technique can be viewed as a diffusion operator that acts along large-scale
+(2~$\Delta$x) \gmcomment{2deltax doesnt seem very large scale} iso-neutral surfaces.
+The diapycnal diffusion required for numerical stability is thus minimized and its net effect on the flow is quite small when compared to the effect of an horizontal background mixing.
+Nevertheless, this iso-neutral operator does not ensure that variance cannot increase,
 contrary to the \citet{Griffies_al_JPO98} operator which has that property.
 …
 \includegraphics[width=0.70\textwidth]{Fig_LDF_ZDF1}
 \caption {    \protect\label{fig:LDF_ZDF1}
 averaging procedure for isopycnal slope computation.}
+  averaging procedure for isopycnal slope computation.}
 \end{center}    \end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 …
 % surface motivates this flattening of isopycnals near the surface).
 For numerical stability reasons \citep{Cox1987, Griffies_Bk04}, the slopes must also
+be bounded by $1/100$ everywhere. This constraint is applied in a piecewise linear
+fashion, increasing from zero at the surface to $1/100$ at $70$ metres and thereafter
+decreasing to zero at the bottom of the ocean. (the fact that the eddies "feel" the
 surface motivates this flattening of isopycnals near the surface).
+For numerical stability reasons \citep{Cox1987, Griffies_Bk04}, the slopes must also be bounded by
+$1/100$ everywhere.
+This constraint is applied in a piecewise linear fashion, increasing from zero at the surface to
+$1/100$ at $70$ metres and thereafter decreasing to zero at the bottom of the ocean
+(the fact that the eddies "feel" the surface motivates this flattening of isopycnals near the surface).
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!ht]     \begin{center}
+\includegraphics[width=0.70\textwidth]{Fig_eiv_slp}
+\caption {     \protect\label{fig:eiv_slp}
+Vertical profile of the slope used for lateral mixing in the mixed layer :
+\textit{(a)} in the real ocean the slope is the iso-neutral slope in the ocean interior,
+which has to be adjusted at the surface boundary (i.e. it must tend to zero at the
+surface since there is no mixing across the air-sea interface: wall boundary
+condition). Nevertheless, the profile between the surface zero value and the interior
+iso-neutral one is unknown, and especially the value at the base of the mixed layer ;
+\textit{(b)} profile of slope using a linear tapering of the slope near the surface and
+imposing a maximum slope of 1/100 ; \textit{(c)} profile of slope actually used in
+\NEMO: a linear decrease of the slope from zero at the surface to its ocean interior
+value computed just below the mixed layer. Note the huge change in the slope at the
+base of the mixed layer between  \textit{(b)}  and \textit{(c)}.}
+\end{center}   \end{figure}
+\begin{figure}[!ht]
+  \begin{center}
+    \includegraphics[width=0.70\textwidth]{Fig_eiv_slp}
+    \caption {     \protect\label{fig:eiv_slp}
+      Vertical profile of the slope used for lateral mixing in the mixed layer:
+      \textit{(a)} in the real ocean the slope is the iso-neutral slope in the ocean interior,
+      which has to be adjusted at the surface boundary
+      (i.e. it must tend to zero at the surface since there is no mixing across the air-sea interface:
+      wall boundary condition).
+      Nevertheless, the profile between the surface zero value and the interior iso-neutral one is unknown,
+      and especially the value at the base of the mixed layer;
+      \textit{(b)} profile of slope using a linear tapering of the slope near the surface and
+      imposing a maximum slope of 1/100;
+      \textit{(c)} profile of slope actually used in \NEMO: a linear decrease of the slope from
+      zero at the surface to its ocean interior value computed just below the mixed layer.
+      Note the huge change in the slope at the base of the mixed layer between \textit{(b)} and \textit{(c)}.}
+  \end{center}
+\end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 \colorbox{yellow}{add here a discussion about the flattening of the slopes, vs  tapering the coefficient.}
+\colorbox{yellow}{add here a discussion about the flattening of the slopes, vs tapering the coefficient.}
 \subsection{Slopes for momentum iso-neutral mixing}
 The iso-neutral diffusion operator on momentum is the same as the one used on
+tracers but applied to each component of the velocity separately (see
+\autoref{eq:dyn_ldf_iso} in section~\autoref{subsec:DYN_ldf_iso}). The slopes between the
+surface along which the diffusion operator acts and the surface of computation
 ($z$- or $s$-surfaces) are defined at $T$-, $f$-, and \textit{uw}- points for the
+$u$-component, and $T$-, $f$- and \textit{vw}- points for the $v$-component.
 They are computed from the slopes used for tracer diffusion, $i.e.$
 \autoref{eq:ldfslp_geo} and \autoref{eq:ldfslp_iso} :
+The iso-neutral diffusion operator on momentum is the same as the one used on tracers but
+applied to each component of the velocity separately
+(see \autoref{eq:dyn_ldf_iso} in section~\autoref{subsec:DYN_ldf_iso}).
+The slopes between the surface along which the diffusion operator acts and the surface of computation
+($z$- or $s$-surfaces) are defined at $T$-, $f$-, and \textit{uw}- points for the $u$-component, and $T$-, $f$- and
+\textit{vw}- points for the $v$-component.
+They are computed from the slopes used for tracer diffusion,
+$i.e.$ \autoref{eq:ldfslp_geo} and \autoref{eq:ldfslp_iso} :
 \begin{equation} \label{eq:ldfslp_dyn}
 …
 \end{equation}
+The major issue remaining is in the specification of the boundary conditions.
+The same boundary conditions are chosen as those used for lateral
+diffusion along model level surfaces, i.e. using the shear computed along
+the model levels and with no additional friction at the ocean bottom (see
+\autoref{sec:LBC_coast}).
+The major issue remaining is in the specification of the boundary conditions.
+The same boundary conditions are chosen as those used for lateral diffusion along model level surfaces,
+$i.e.$ using the shear computed along the model levels and with no additional friction at the ocean bottom
+(see \autoref{sec:LBC_coast}).
 …
 \label{sec:LDF_coef}
 Introducing a space variation in the lateral eddy mixing coefficients changes
 the model core memory requirement, adding up to four extra three-dimensional
+arrays for the geopotential or isopycnal second order operator applied to
+momentum. Six CPP keys control the space variation of eddy coefficients:
+three for momentum and three for tracer. The three choices allow:
 a space variation in the three space directions (\key{traldf\_c3d},  \key{dynldf\_c3d}),
 in the horizontal plane (\key{traldf\_c2d},  \key{dynldf\_c2d}),
 or in the vertical only (\key{traldf\_c1d},  \key{dynldf\_c1d}).
+Introducing a space variation in the lateral eddy mixing coefficients changes the model core memory requirement,
+adding up to four extra three-dimensional arrays for the geopotential or isopycnal second order operator applied to
+momentum.
+Six CPP keys control the space variation of eddy coefficients: three for momentum and three for tracer.
+The three choices allow:
+a space variation in the three space directions (\key{traldf\_c3d},  \key{dynldf\_c3d}),
+in the horizontal plane (\key{traldf\_c2d},  \key{dynldf\_c2d}),
+or in the vertical only (\key{traldf\_c1d},  \key{dynldf\_c1d}).
 The default option is a constant value over the whole ocean on both momentum and tracers.
+The number of additional arrays that have to be defined and the gridpoint
+position at which they are defined depend on both the space variation chosen
+and the type of operator used. The resulting eddy viscosity and diffusivity
+coefficients can be a function of more than one variable. Changes in the
+computer code when switching from one option to another have been
+minimized by introducing the eddy coefficients as statement functions
+(include file \hf{ldftra\_substitute} and \hf{ldfdyn\_substitute}). The functions
+are replaced by their actual meaning during the preprocessing step (CPP).
+The specification of the space variation of the coefficient is made in
+\mdl{ldftra} and \mdl{ldfdyn}, or more precisely in include files
+\hf{traldf\_cNd} and \hf{dynldf\_cNd}, with N=1, 2 or 3.
+The user can modify these include files as he/she wishes. The way the
+mixing coefficient are set in the reference version can be briefly described
+as follows:
+The number of additional arrays that have to be defined and the gridpoint position at which
+they are defined depend on both the space variation chosen and the type of operator used.
+The resulting eddy viscosity and diffusivity coefficients can be a function of more than one variable.
+Changes in the computer code when switching from one option to another have been minimized by
+introducing the eddy coefficients as statement functions
+(include file \hf{ldftra\_substitute} and \hf{ldfdyn\_substitute}).
+The functions are replaced by their actual meaning during the preprocessing step (CPP).
+The specification of the space variation of the coefficient is made in \mdl{ldftra} and \mdl{ldfdyn},
+or more precisely in include files \hf{traldf\_cNd} and \hf{dynldf\_cNd}, with N=1, 2 or 3.
+The user can modify these include files as he/she wishes.
+The way the mixing coefficient are set in the reference version can be briefly described as follows:
 \subsubsection{Constant mixing coefficients (default option)}
+When none of the \key{dynldf\_...} and \key{traldf\_...} keys are
+defined, a constant value is used over the whole ocean for momentum and
+tracers, which is specified through the \np{rn\_ahm0} and \np{rn\_aht0} namelist
+parameters.
+When none of the \key{dynldf\_...} and \key{traldf\_...} keys are defined,
+a constant value is used over the whole ocean for momentum and tracers,
+which is specified through the \np{rn\_ahm0} and \np{rn\_aht0} namelist parameters.
 \subsubsection{Vertically varying mixing coefficients (\protect\key{traldf\_c1d} and \key{dynldf\_c1d})}
 The 1D option is only available when using the $z$-coordinate with full step.
 Indeed in all the other types of vertical coordinate, the depth is a 3D function
+of (\textbf{i},\textbf{j},\textbf{k}) and therefore, introducing depth-dependent
+mixing coefficients will require 3D arrays. In the 1D option, a hyperbolic variation
+of the lateral mixing coefficient is introduced in which the surface value is
+\np{rn\_aht0} (\np{rn\_ahm0}), the bottom value is 1/4 of the surface value,
 and the transition takes place around z=300~m with a width of 300~m
 ($i.e.$ both the depth and the width of the inflection point are set to 300~m).
+The 1D option is only available when using the $z$-coordinate with full step.
+Indeed in all the other types of vertical coordinate,
+the depth is a 3D function of (\textbf{i},\textbf{j},\textbf{k}) and therefore,
+introducing depth-dependent mixing coefficients will require 3D arrays.
+In the 1D option, a hyperbolic variation of the lateral mixing coefficient is introduced in which
+the surface value is \np{rn\_aht0} (\np{rn\_ahm0}), the bottom value is 1/4 of the surface value,
+and the transition takes place around z=300~m with a width of 300~m
+($i.e.$ both the depth and the width of the inflection point are set to 300~m).
 This profile is hard coded in file \hf{traldf\_c1d}, but can be easily modified by users.
 \subsubsection{Horizontally varying mixing coefficients (\protect\key{traldf\_c2d} and \protect\key{dynldf\_c2d})}
 By default the horizontal variation of the eddy coefficient depends on the local mesh
 size and the type of operator used:
+By default the horizontal variation of the eddy coefficient depends on the local mesh size and
+the type of operator used:
 \begin{equation} \label{eq:title}
   A_l = \left\{
 …
    \end{aligned}    \right.
 \end{equation}
+where $e_{max}$ is the maximum of $e_1$ and $e_2$ taken over the whole masked
+ocean domain, and $A_o^l$ is the \np{rn\_ahm0} (momentum) or \np{rn\_aht0} (tracer)
+namelist parameter. This variation is intended to reflect the lesser need for subgrid
+scale eddy mixing where the grid size is smaller in the domain. It was introduced in
+the context of the DYNAMO modelling project \citep{Willebrand_al_PO01}.
+Note that such a grid scale dependance of mixing coefficients significantly increase
+the range of stability of model configurations presenting large changes in grid pacing
+such as global ocean models. Indeed, in such a case, a constant mixing coefficient
+can lead to a blow up of the model due to large coefficient compare to the smallest
+grid size (see \autoref{sec:STP_forward_imp}), especially when using a bilaplacian operator.
+Other formulations can be introduced by the user for a given configuration.
+For example, in the ORCA2 global ocean model (see Configurations), the laplacian
+viscosity operator uses \np{rn\_ahm0}~= 4.10$^4$ m$^2$/s poleward of 20$^{\circ}$
+north and south and decreases linearly to \np{rn\_aht0}~= 2.10$^3$ m$^2$/s
+at the equator \citep{Madec_al_JPO96, Delecluse_Madec_Bk00}. This modification
+can be found in routine \rou{ldf\_dyn\_c2d\_orca} defined in \mdl{ldfdyn\_c2d}.
+Similar modified horizontal variations can be found with the Antarctic or Arctic
+sub-domain options of ORCA2 and ORCA05 (see \&namcfg namelist).
+where $e_{max}$ is the maximum of $e_1$ and $e_2$ taken over the whole masked ocean domain,
+and $A_o^l$ is the \np{rn\_ahm0} (momentum) or \np{rn\_aht0} (tracer) namelist parameter.
+This variation is intended to reflect the lesser need for subgrid scale eddy mixing where
+the grid size is smaller in the domain.
+It was introduced in the context of the DYNAMO modelling project \citep{Willebrand_al_PO01}.
+Note that such a grid scale dependance of mixing coefficients significantly increase the range of stability of
+model configurations presenting large changes in grid pacing such as global ocean models.
+Indeed, in such a case, a constant mixing coefficient can lead to a blow up of the model due to
+large coefficient compare to the smallest grid size (see \autoref{sec:STP_forward_imp}),
+especially when using a bilaplacian operator.
+Other formulations can be introduced by the user for a given configuration.
+For example, in the ORCA2 global ocean model (see Configurations),
+the laplacian viscosity operator uses \np{rn\_ahm0}~= 4.10$^4$ m$^2$/s poleward of 20$^{\circ}$ north and south and
+decreases linearly to \np{rn\_aht0}~= 2.10$^3$ m$^2$/s at the equator \citep{Madec_al_JPO96, Delecluse_Madec_Bk00}.
+This modification can be found in routine \rou{ldf\_dyn\_c2d\_orca} defined in \mdl{ldfdyn\_c2d}.
+Similar modified horizontal variations can be found with the Antarctic or Arctic sub-domain options of
+ORCA2 and ORCA05 (see \&namcfg namelist).
 \subsubsection{Space varying mixing coefficients (\protect\key{traldf\_c3d} and \key{dynldf\_c3d})}
 The 3D space variation of the mixing coefficient is simply the combination of the
+D and 2D cases, $i.e.$ a hyperbolic tangent variation with depth associated with
 a grid size dependence of the magnitude of the coefficient.
+The 3D space variation of the mixing coefficient is simply the combination of the 1D and 2D cases,
+$i.e.$ a hyperbolic tangent variation with depth associated with a grid size dependence of
+the magnitude of the coefficient.
 \subsubsection{Space and time varying mixing coefficients}
 There is no default specification of space and time varying mixing coefficient.
+The only case available is specific to the ORCA2 and ORCA05 global ocean
+configurations. It provides only a tracer
+mixing coefficient for eddy induced velocity (ORCA2) or both iso-neutral and
+eddy induced velocity (ORCA05) that depends on the local growth rate of
+baroclinic instability. This specification is actually used when an ORCA key
+and both \key{traldf\_eiv} and \key{traldf\_c2d} are defined.
+The only case available is specific to the ORCA2 and ORCA05 global ocean configurations.
+It provides only a tracer mixing coefficient for eddy induced velocity (ORCA2) or both iso-neutral and
+eddy induced velocity (ORCA05) that depends on the local growth rate of baroclinic instability.
+This specification is actually used when an ORCA key and both \key{traldf\_eiv} and \key{traldf\_c2d} are defined.
 $\ $\newline    % force a new ligne
 …
 The following points are relevant when the eddy coefficient varies spatially:
+(1) the momentum diffusion operator acting along model level surfaces is
+written in terms of curl and divergent components of the horizontal current
+(see \autoref{subsec:PE_ldf}). Although the eddy coefficient could be set to different values
+in these two terms, this option is not currently available.
+(2) with an horizontally varying viscosity, the quadratic integral constraints
+on enstrophy and on the square of the horizontal divergence for operators
+acting along model-surfaces are no longer satisfied
+(1) the momentum diffusion operator acting along model level surfaces is written in terms of curl and
+divergent components of the horizontal current (see \autoref{subsec:PE_ldf}).
+Although the eddy coefficient could be set to different values in these two terms,
+this option is not currently available.
+(2) with an horizontally varying viscosity, the quadratic integral constraints on enstrophy and on the square of
+the horizontal divergence for operators acting along model-surfaces are no longer satisfied
 (\autoref{sec:dynldf_properties}).
+(3) for isopycnal diffusion on momentum or tracers, an additional purely
+horizontal background diffusion with uniform coefficient can be added by
+setting a non zero value of \np{rn\_ahmb0} or \np{rn\_ahtb0}, a background horizontal
+eddy viscosity or diffusivity coefficient (namelist parameters whose default
+values are $0$). However, the technique used to compute the isopycnal
+slopes is intended to get rid of such a background diffusion, since it introduces
+spurious diapycnal diffusion (see \autoref{sec:LDF_slp}).
+(4) when an eddy induced advection term is used (\key{traldf\_eiv}), $A^{eiv}$,
+the eddy induced coefficient has to be defined. Its space variations are controlled
+by the same CPP variable as for the eddy diffusivity coefficient ($i.e.$
+\key{traldf\_cNd}).
+(3) for isopycnal diffusion on momentum or tracers, an additional purely horizontal background diffusion with
+uniform coefficient can be added by setting a non zero value of \np{rn\_ahmb0} or \np{rn\_ahtb0},
+a background horizontal eddy viscosity or diffusivity coefficient
+(namelist parameters whose default values are $0$).
+However, the technique used to compute the isopycnal slopes is intended to get rid of such a background diffusion,
+since it introduces spurious diapycnal diffusion (see \autoref{sec:LDF_slp}).
+(4) when an eddy induced advection term is used (\key{traldf\_eiv}),
+$A^{eiv}$, the eddy induced coefficient has to be defined.
+Its space variations are controlled by the same CPP variable as for the eddy diffusivity coefficient
+($i.e.$ \key{traldf\_cNd}).
 (5) the eddy coefficient associated with a biharmonic operator must be set to a \emph{negative} value.
 …
 (6) it is possible to use both the laplacian and biharmonic operators concurrently.
 (7) it is possible to run without explicit lateral diffusion on momentum (\np{ln\_dynldf\_lap}\forcode{ =
+}\np{ln\_dynldf\_bilap}\forcode{ = .false.}). This is recommended when using the UBS advection
+scheme on momentum (\np{ln\_dynadv\_ubs}\forcode{ = .true.}, see \autoref{subsec:DYN_adv_ubs})
 and can be useful for testing purposes.
+(7) it is possible to run without explicit lateral diffusion on momentum
+(\np{ln\_dynldf\_lap}\forcode{ = .?.}\np{ln\_dynldf\_bilap}\forcode{ = .false.}).
+This is recommended when using the UBS advection scheme on momentum (\np{ln\_dynadv\_ubs}\forcode{ = .true.},
+see \autoref{subsec:DYN_adv_ubs}) and can be useful for testing purposes.
 % ================================================================
 …
 %%gm  from Triad appendix  : to be incorporated....
 \gmcomment{
+Values of iso-neutral diffusivity and GM coefficient are set as
+described in \autoref{sec:LDF_coef}. If none of the keys \key{traldf\_cNd},
+N=1,2,3 is set (the default), spatially constant iso-neutral $A_l$ and
+GM diffusivity $A_e$ are directly set by \np{rn\_aeih\_0} and
+\np{rn\_aeiv\_0}. If 2D-varying coefficients are set with
+\key{traldf\_c2d} then $A_l$ is reduced in proportion with horizontal
+scale factor according to \autoref{eq:title} \footnote{Except in global ORCA
+  $0.5^{\circ}$ runs with \key{traldf\_eiv}, where
+  $A_l$ is set like $A_e$ but with a minimum vale of
+  $100\;\mathrm{m}^2\;\mathrm{s}^{-1}$}. In idealised setups with
+\key{traldf\_c2d}, $A_e$ is reduced similarly, but if \key{traldf\_eiv}
+is set in the global configurations with \key{traldf\_c2d}, a horizontally varying $A_e$ is
+instead set from the Held-Larichev parameterisation\footnote{In this
+  case, $A_e$ at low latitudes $|\theta|<20^{\circ}$ is further
+  reduced by a factor $|f/f_{20}|$, where $f_{20}$ is the value of $f$
+  at $20^{\circ}$~N} (\mdl{ldfeiv}) and \np{rn\_aeiv\_0} is ignored
+unless it is zero.
+  Values of iso-neutral diffusivity and GM coefficient are set as described in \autoref{sec:LDF_coef}.
+  If none of the keys \key{traldf\_cNd}, N=1,2,3 is set (the default), spatially constant iso-neutral $A_l$ and
+  GM diffusivity $A_e$ are directly set by \np{rn\_aeih\_0} and \np{rn\_aeiv\_0}.
+  If 2D-varying coefficients are set with \key{traldf\_c2d} then $A_l$ is reduced in proportion with horizontal
+  scale factor according to \autoref{eq:title} \footnote{
+    Except in global ORCA  $0.5^{\circ}$ runs with \key{traldf\_eiv},
+    where $A_l$ is set like $A_e$ but with a minimum vale of $100\;\mathrm{m}^2\;\mathrm{s}^{-1}$
+  }.
+  In idealised setups with \key{traldf\_c2d}, $A_e$ is reduced similarly, but if \key{traldf\_eiv} is set in
+  the global configurations with \key{traldf\_c2d}, a horizontally varying $A_e$ is instead set from
+  the Held-Larichev parameterisation \footnote{
+    In this case, $A_e$ at low latitudes $|\theta|<20^{\circ}$ is further reduced by a factor $|f/f_{20}|$,
+    where $f_{20}$ is the value of $f$ at $20^{\circ}$~N
+  } (\mdl{ldfeiv}) and \np{rn\_aeiv\_0} is ignored unless it is zero.
+}
+When Gent and McWilliams [1990] diffusion is used (\key{traldf\_eiv} defined),
+an eddy induced tracer advection term is added, the formulation of which
+depends on the slopes of iso-neutral surfaces. Contrary to the case of iso-neutral
+mixing, the slopes used here are referenced to the geopotential surfaces, $i.e.$
+\autoref{eq:ldfslp_geo} is used in $z$-coordinates, and the sum \autoref{eq:ldfslp_geo}
++ \autoref{eq:ldfslp_iso} in $s$-coordinates. The eddy induced velocity is given by:
+When Gent and McWilliams [1990] diffusion is used (\key{traldf\_eiv} defined),
+an eddy induced tracer advection term is added,
+the formulation of which depends on the slopes of iso-neutral surfaces.
+Contrary to the case of iso-neutral mixing, the slopes used here are referenced to the geopotential surfaces,
+$i.e.$ \autoref{eq:ldfslp_geo} is used in $z$-coordinates,
+and the sum \autoref{eq:ldfslp_geo} + \autoref{eq:ldfslp_iso} in $s$-coordinates.
+The eddy induced velocity is given by:
 \begin{equation} \label{eq:ldfeiv}
 \begin{split}
 …
 \end{split}
 \end{equation}
 where $A^{eiv}$ is the eddy induced velocity coefficient whose value is set
+through \np{rn\_aeiv}, a \textit{nam\_traldf} namelist parameter.
 The three components of the eddy induced velocity are computed and add
+to the eulerian velocity in \mdl{traadv\_eiv}. This has been preferred to a
+separate computation of the advective trends associated with the eiv velocity,
 since it allows us to take advantage of all the advection schemes offered for
+the tracers (see \autoref{sec:TRA_adv}) and not just the $2^{nd}$ order advection
+scheme as in previous releases of OPA \citep{Madec1998}. This is particularly
+useful for passive tracers where \emph{positivity} of the advection scheme is
 of paramount importance.
 At the surface, lateral and bottom boundaries, the eddy induced velocity,
+where $A^{eiv}$ is the eddy induced velocity coefficient whose value is set through \np{rn\_aeiv},
+a \textit{nam\_traldf} namelist parameter.
+The three components of the eddy induced velocity are computed and
+add to the eulerian velocity in \mdl{traadv\_eiv}.
+This has been preferred to a separate computation of the advective trends associated with the eiv velocity,
+since it allows us to take advantage of all the advection schemes offered for the tracers
+(see \autoref{sec:TRA_adv}) and not just the $2^{nd}$ order advection scheme as in
+previous releases of OPA \citep{Madec1998}.
+This is particularly useful for passive tracers where \emph{positivity} of the advection scheme is of
+paramount importance.
+At the surface, lateral and bottom boundaries, the eddy induced velocity,
 and thus the advective eddy fluxes of heat and salt, are set to zero.

NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_OBS.tex

-                      r10146
+                      r10368
 $\ $\newline    % force a new line
+The observation and model comparison code (OBS) reads in observation files (profile
+temperature and salinity, sea surface temperature, sea level anomaly, sea ice concentration,
+and velocity) and calculates  an interpolated model equivalent value at the observation
+location and nearest model timestep. The resulting data are saved in a ``feedback'' file (or
+files). The code was originally developed for use with the NEMOVAR data assimilation code, but
+can be used for validation or verification of the model or with any other data assimilation system.
+The OBS code is called from \mdl{nemogcm} for model initialisation and to calculate the model
+equivalent values for observations on the 0th timestep. The code is then called again after
+each timestep from \mdl{step}. The code is only activated if the namelist logical \np{ln\_diaobs}
+is set to true.
+For all data types a 2D horizontal interpolator or averager is needed to interpolate/average the model fields to
+the observation location. For {\em in situ} profiles, a 1D vertical interpolator is needed in
+addition to provide model fields at the observation depths. This now works in a generalised vertical
+coordinate system.
+The observation and model comparison code (OBS) reads in observation files
+(profile temperature and salinity, sea surface temperature, sea level anomaly, sea ice concentration, and velocity) and calculates an interpolated model equivalent value at the observation location and nearest model timestep.
+The resulting data are saved in a ``feedback'' file (or files).
+The code was originally developed for use with the NEMOVAR data assimilation code,
+but can be used for validation or verification of the model or with any other data assimilation system.
+The OBS code is called from \mdl{nemogcm} for model initialisation and to calculate the model equivalent values for observations on the 0th timestep.
+The code is then called again after each timestep from \mdl{step}.
+The code is only activated if the namelist logical \np{ln\_diaobs} is set to true.
+For all data types a 2D horizontal interpolator or averager is needed to
+interpolate/average the model fields to the observation location.
+For {\em in situ} profiles, a 1D vertical interpolator is needed in addition to
+provide model fields at the observation depths.
+This now works in a generalised vertical coordinate system.
 Some profile observation types (e.g. tropical moored buoys) are made available as daily averaged quantities.
+The observation operator code can be set-up to calculate the equivalent daily average model temperature fields
+using the \np{nn\_profdavtypes} namelist array. Some SST observations are equivalent to a night-time
+average value and the observation operator code can calculate equivalent night-time average model SST fields by
+setting the namelist value \np{ln\_sstnight} to true. Otherwise the model value from the nearest timestep to the
+observation time is used.
+The code is controlled by the namelist \textit{namobs}. See the following sections for more
+details on setting up the namelist.
+The observation operator code can be set-up to calculate the equivalent daily average model temperature fields using
+the \np{nn\_profdavtypes} namelist array.
+Some SST observations are equivalent to a night-time average value and
+the observation operator code can calculate equivalent night-time average model SST fields by
+setting the namelist value \np{ln\_sstnight} to true.
+Otherwise the model value from the nearest timestep to the observation time is used.
+The code is controlled by the namelist \textit{namobs}.
+See the following sections for more details on setting up the namelist.
 \autoref{sec:OBS_example} introduces a test example of the observation operator code including
+where to obtain data and how to setup the namelist. \autoref{sec:OBS_details} introduces some
+more technical details of the different observation types used and also shows a more complete
+namelist. \autoref{sec:OBS_theory} introduces some of the theoretical aspects of the observation
+operator including interpolation methods and running on multiple processors.
+where to obtain data and how to setup the namelist.
+\autoref{sec:OBS_details} introduces some more technical details of the different observation types used and
+also shows a more complete namelist.
+\autoref{sec:OBS_theory} introduces some of the theoretical aspects of the observation operator including
+interpolation methods and running on multiple processors.
 \autoref{sec:OBS_ooo} describes the offline observation operator code.
+\autoref{sec:OBS_obsutils} introduces some utilities to help working with the files
+produced by the OBS code.
+\autoref{sec:OBS_obsutils} introduces some utilities to help working with the files produced by the OBS code.
 % ================================================================
 …
 This section describes an example of running the observation operator code using
 profile data which can be freely downloaded. It shows how to adapt an
 existing run and build of NEMO to run the observation operator.
+profile data which can be freely downloaded.
+It shows how to adapt an existing run and build of NEMO to run the observation operator.
 \begin{enumerate}
 \item Compile NEMO.
+\item Download some EN4 data from
+\href{http://www.metoffice.gov.uk/hadobs}{www.metoffice.gov.uk/hadobs}. Choose observations which are
+valid for the period of your test run because the observation operator compares
+the model and observations for a matching date and time.
+\item Download some EN4 data from \href{http://www.metoffice.gov.uk/hadobs}{www.metoffice.gov.uk/hadobs}.
+  Choose observations which are valid for the period of your test run because
+  the observation operator compares the model and observations for a matching date and time.
 \item Compile the OBSTOOLS code using:
 …
 \end{cmds}
+\item Include the following in the NEMO namelist to run the observation
+operator on this data:
+\item Include the following in the NEMO namelist to run the observation operator on this data:
 \end{enumerate}
+%------------------------------------------namobs_example-----------------------------------------------------
+%
+%\nlst{namobs_example}
+%-------------------------------------------------------------------------------------------------------------
+Options are defined through the  \ngn{namobs} namelist variables.
+The options \np{ln\_t3d} and \np{ln\_s3d} switch on the temperature and salinity
+profile observation operator code. The filename or array of filenames are
+specified using the \np{cn\_profbfiles} variable. The model grid points for a
+particular  observation latitude and longitude are found using the grid
+searching part of the code. This can be expensive, particularly for large
+numbers of observations, setting \np{ln\_grid\_search\_lookup} allows the use of
+a lookup table which is saved into an ``xypos`` file (or files). This will need
+to be generated the first time if it does not exist in the run directory.
+Options are defined through the \ngn{namobs} namelist variables.
+The options \np{ln\_t3d} and \np{ln\_s3d} switch on the temperature and salinity profile observation operator code.
+The filename or array of filenames are specified using the \np{cn\_profbfiles} variable.
+The model grid points for a particular observation latitude and longitude are found using
+the grid searching part of the code.
+This can be expensive, particularly for large numbers of observations,
+setting \np{ln\_grid\_search\_lookup} allows the use of a lookup table which
+is saved into an ``xypos`` file (or files).
+This will need to be generated the first time if it does not exist in the run directory.
 However, once produced it will significantly speed up future grid searches.
 Setting \np{ln\_grid\_global} means that the code distributes the observations
+evenly between processors. Alternatively each processor will work with
 observations located within the model subdomain (see section~\autoref{subsec:OBS_parallel}).
 A number of utilities are now provided to plot the feedback files, convert and
 recombine the files. These are explained in more detail in section~\autoref{sec:OBS_obsutils}.
 Utilites to convert other input data formats into the feedback format are also
 described in section~\autoref{sec:OBS_obsutils}.
+Setting \np{ln\_grid\_global} means that the code distributes the observations evenly between processors.
+Alternatively each processor will work with observations located within the model subdomain
+(see section~\autoref{subsec:OBS_parallel}).
+A number of utilities are now provided to plot the feedback files, convert and recombine the files.
+These are explained in more detail in section~\autoref{sec:OBS_obsutils}.
+Utilites to convert other input data formats into the feedback format are also described in
+section~\autoref{sec:OBS_obsutils}.
 \section{Technical details (feedback type observation file headers)}
 \label{sec:OBS_details}
+Here we show a more complete example namelist  \ngn{namobs} and also show the NetCDF headers
+of the observation
+files that may be used with the observation operator
+Here we show a more complete example namelist \ngn{namobs} and also show the NetCDF headers of
+the observation files that may be used with the observation operator.
 %------------------------------------------namobs--------------------------------------------------------
 …
 %-------------------------------------------------------------------------------------------------------------
+The observation operator code uses the "feedback" observation file format for
+all data types. All the
+observation files must be in NetCDF format. Some example headers (produced using
+\mbox{\textit{ncdump~-h}}) for profile
+data, sea level anomaly and sea surface temperature are in the following
+subsections.
+The observation operator code uses the "feedback" observation file format for all data types.
+All the observation files must be in NetCDF format.
+Some example headers (produced using \mbox{\textit{ncdump~-h}}) for profile data, sea level anomaly and
+sea surface temperature are in the following subsections.
 \subsection{Profile feedback}
 …
 \end{clines}
+The mean dynamic
+topography (MDT) must be provided in a separate file defined on the model grid
+ called \ifile{slaReferenceLevel}. The MDT is required in
+order to produce the model equivalent sea level anomaly from the model sea
+surface height. Below is an example header for this file (on the ORCA025 grid).
+The mean dynamic topography (MDT) must be provided in a separate file defined on
+the model grid called \ifile{slaReferenceLevel}.
+The MDT is required in order to produce the model equivalent sea level anomaly from the model sea surface height.
+Below is an example header for this file (on the ORCA025 grid).
 \begin{clines}
 …
 \subsection{Horizontal interpolation and averaging methods}
+For most observation types, the horizontal extent of the observation is small compared to the model grid size
+and so the model equivalent of the observation is calculated by interpolating from the four surrounding grid
+points to the observation location. Some satellite observations (e.g. microwave satellite SST data, or satellite SSS data)
+have a footprint which is similar in size or larger than the model grid size (particularly when the grid size is small).
+In those cases the model counterpart should be calculated by averaging the model grid points over the same size as the footprint.
+NEMO therefore has the capability to specify either an interpolation or an averaging (for surface observation types only).
+The main namelist option associated with the interpolation/averaging is \np{nn\_2dint}. This default option can be set to values from 0 to 6.
+For most observation types, the horizontal extent of the observation is small compared to the model grid size and so
+the model equivalent of the observation is calculated by interpolating from
+the four surrounding grid points to the observation location.
+Some satellite observations (e.g. microwave satellite SST data, or satellite SSS data) have a footprint which
+is similar in size or larger than the model grid size (particularly when the grid size is small).
+In those cases the model counterpart should be calculated by averaging the model grid points over
+the same size as the footprint.
+NEMO therefore has the capability to specify either an interpolation or an averaging
+(for surface observation types only).
+The main namelist option associated with the interpolation/averaging is \np{nn\_2dint}.
+This default option can be set to values from 0 to 6.
 Values between 0 to 4 are associated with interpolation while values 5 or 6 are associated with averaging.
 \begin{itemize}
 …
 \item \np{nn\_2dint}\forcode{ = 3}: Bilinear remapping interpolation (general grid)
 \item \np{nn\_2dint}\forcode{ = 4}: Polynomial interpolation
+\item \np{nn\_2dint}\forcode{ = 5}: Radial footprint averaging with diameter specified in the namelist as \np{rn\_???\_avglamscl} in degrees or metres (set using \np{ln\_???\_fp\_indegs})
+\item \np{nn\_2dint}\forcode{ = 6}: Rectangular footprint averaging with E/W and N/S size specified in the namelist as \np{rn\_???\_avglamscl} and \np{rn\_???\_avgphiscl} in degrees or metres (set using \np{ln\_???\_fp\_indegs})
+\item \np{nn\_2dint}\forcode{ = 5}: Radial footprint averaging with diameter specified in the namelist as
+  \np{rn\_???\_avglamscl} in degrees or metres (set using \np{ln\_???\_fp\_indegs})
+\item \np{nn\_2dint}\forcode{ = 6}: Rectangular footprint averaging with E/W and N/S size specified in
+  the namelist as \np{rn\_???\_avglamscl} and \np{rn\_???\_avgphiscl} in degrees or metres
+  (set using \np{ln\_???\_fp\_indegs})
 \end{itemize}
+The ??? in the last two options indicate these options should be specified for each observation type for which the averaging is to be performed (see namelist example above).
+The \np{nn\_2dint} default option can be overridden for surface observation types using namelist values \np{nn\_2dint\_???} where ??? is one of sla,sst,sss,sic.
+The ??? in the last two options indicate these options should be specified for each observation type for
+which the averaging is to be performed (see namelist example above).
+The \np{nn\_2dint} default option can be overridden for surface observation types using
+namelist values \np{nn\_2dint\_???} where ??? is one of sla,sst,sss,sic.
 Below is some more detail on the various options for interpolation and averaging available in NEMO.
 \subsubsection{Horizontal interpolation}
+Consider an observation point ${\rm P}$ with
+with longitude and latitude $({\lambda_{}}_{\rm P}, \phi_{\rm P})$ and the
+four nearest neighbouring model grid points ${\rm A}$, ${\rm B}$, ${\rm C}$
+and ${\rm D}$ with longitude and latitude ($\lambda_{\rm A}$, $\phi_{\rm A}$),
+($\lambda_{\rm B}$, $\phi_{\rm B}$) etc.
+All horizontal interpolation methods implemented in NEMO
+estimate the value of a model variable $x$ at point $P$ as
+a weighted linear combination of the values of the model
+variables at the grid points ${\rm A}$, ${\rm B}$ etc.:
+Consider an observation point ${\rm P}$ with with longitude and latitude $({\lambda_{}}_{\rm P}, \phi_{\rm P})$ and
+the four nearest neighbouring model grid points ${\rm A}$, ${\rm B}$, ${\rm C}$ and ${\rm D}$ with
+longitude and latitude ($\lambda_{\rm A}$, $\phi_{\rm A}$),($\lambda_{\rm B}$, $\phi_{\rm B}$) etc.
+All horizontal interpolation methods implemented in NEMO estimate the value of a model variable $x$ at point $P$ as
+a weighted linear combination of the values of the model variables at the grid points ${\rm A}$, ${\rm B}$ etc.:
 \begin{eqnarray}
 {x_{}}_{\rm P} & \hspace{-2mm} = \hspace{-2mm} &
 …
                    {w_{}}_{\rm D} {x_{}}_{\rm D} \right)
 \end{eqnarray}
+where ${w_{}}_{\rm A}$, ${w_{}}_{\rm B}$ etc. are the respective weights for the
+model field at points ${\rm A}$, ${\rm B}$ etc., and
+$w = {w_{}}_{\rm A} + {w_{}}_{\rm B} + {w_{}}_{\rm C} + {w_{}}_{\rm D}$.
+where ${w_{}}_{\rm A}$, ${w_{}}_{\rm B}$ etc. are the respective weights for the model field at
+points ${\rm A}$, ${\rm B}$ etc., and $w = {w_{}}_{\rm A} + {w_{}}_{\rm B} + {w_{}}_{\rm C} + {w_{}}_{\rm D}$.
 Four different possibilities are available for computing the weights.
 …
 \begin{enumerate}
 \item[1.] {\bf Great-Circle distance-weighted interpolation.} The weights
   are computed as a function of the great-circle distance $s(P, \cdot)$
   between $P$ and the model grid points $A$, $B$ etc. For example,
   the weight given to the field ${x_{}}_{\rm A}$ is specified as the
   product of the distances from ${\rm P}$ to the other points:
+\item[1.] {\bf Great-Circle distance-weighted interpolation.}
+  The weights are computed as a function of the great-circle distance $s(P, \cdot)$ between $P$ and
+  the model grid points $A$, $B$ etc.
+  For example, the weight given to the field ${x_{}}_{\rm A}$ is specified as the product of the distances
+  from ${\rm P}$ to the other points:
   \begin{eqnarray}
   {w_{}}_{\rm A} = s({\rm P}, {\rm B}) \, s({\rm P}, {\rm C}) \, s({\rm P}, {\rm D})
 …
    \end{eqnarray}
    and $M$ corresponds to $B$, $C$ or $D$.
+   A more stable form of the great-circle distance formula for
+   small distances ($x$ near 1) involves the arcsine function
+   ($e.g.$ see p.~101 of \citet{Daley_Barker_Bk01}:
+   A more stable form of the great-circle distance formula for small distances ($x$ near 1)
+   involves the arcsine function ($e.g.$ see p.~101 of \citet{Daley_Barker_Bk01}:
    \begin{eqnarray}
    s\left( {\rm P}, {\rm M} \right)
 …
   \end{eqnarray}
+\item[2.] {\bf Great-Circle distance-weighted interpolation with small angle
+  approximation.} Similar to the previous interpolation but with the
+  distance $s$ computed as
+\item[2.] {\bf Great-Circle distance-weighted interpolation with small angle approximation.}
+  Similar to the previous interpolation but with the distance $s$ computed as
   \begin{eqnarray}
     s\left( {\rm P}, {\rm M} \right)
 …
   where $M$ corresponds to $A$, $B$, $C$ or $D$.
+\item[3.] {\bf Bilinear interpolation for a regular spaced grid.} The
+  interpolation is split into two 1D interpolations in the longitude
+  and latitude directions, respectively.
+\item[4.] {\bf Bilinear remapping interpolation for a general grid.} An
+  iterative scheme that involves first mapping a quadrilateral cell
+  into a cell with coordinates (0,0), (1,0), (0,1) and (1,1). This
+  method is based on the SCRIP interpolation package \citep{Jones_1998}.
+\item[3.] {\bf Bilinear interpolation for a regular spaced grid.}
+  The interpolation is split into two 1D interpolations in the longitude and latitude directions, respectively.
+\item[4.] {\bf Bilinear remapping interpolation for a general grid.}
+  An iterative scheme that involves first mapping a quadrilateral cell into
+  a cell with coordinates (0,0), (1,0), (0,1) and (1,1).
+  This method is based on the SCRIP interpolation package \citep{Jones_1998}.
 \end{enumerate}
 …
 For each surface observation type:
 \begin{itemize}
+\item The standard grid-searching code is used to find the nearest model grid point to the observation location (see next subsection).
+\item The maximum number of grid points is calculated in the local grid domain for which the averaging is likely need to cover.
+\item The lats/longs of the grid points surrounding the nearest model grid box are extracted using existing mpi routines.
+\item The weights for each grid point associated with each observation are calculated, either for radial or rectangular footprints. For grid points completely within the footprint, the weight is one; for grid points completely outside the footprint, the weight is zero. For grid points which are partly within the footprint the ratio between the area of the footprint within the grid box and the total area of the grid box is used as the weight.
+\item The weighted average of the model grid points associated with each observation is calculated, and this is then given as the model counterpart of the observation.
+\item The standard grid-searching code is used to find the nearest model grid point to the observation location
+  (see next subsection).
+\item The maximum number of grid points is calculated in the local grid domain for which
+  the averaging is likely need to cover.
+\item The lats/longs of the grid points surrounding the nearest model grid box are extracted using
+  existing mpi routines.
+\item The weights for each grid point associated with each observation are calculated,
+  either for radial or rectangular footprints.
+  For grid points completely within the footprint, the weight is one;
+  for grid points completely outside the footprint, the weight is zero.
+  For grid points which are partly within the footprint the ratio between the area of the footprint within
+  the grid box and the total area of the grid box is used as the weight.
+\item The weighted average of the model grid points associated with each observation is calculated,
+  and this is then given as the model counterpart of the observation.
 \end{itemize}
+Examples of the weights calculated for an observation with rectangular and radial footprints are shown in Figs.~\autoref{fig:obsavgrec} and~\autoref{fig:obsavgrad}.
+Examples of the weights calculated for an observation with rectangular and radial footprints are shown in
+Figs.~\autoref{fig:obsavgrec} and~\autoref{fig:obsavgrad}.
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 …
 \includegraphics[width=0.90\textwidth]{Fig_OBS_avg_rec}
 \caption{      \protect\label{fig:obsavgrec}
+Weights associated with each model grid box (blue lines and numbers) for an observation at -170.5E, 56.0N with a rectangular footprint of 1\deg x 1\deg.}
+  Weights associated with each model grid box (blue lines and numbers)
+  for an observation at -170.5E, 56.0N with a rectangular footprint of 1\deg x 1\deg.}
 \end{center}      \end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 …
 \includegraphics[width=0.90\textwidth]{Fig_OBS_avg_rad}
 \caption{      \protect\label{fig:obsavgrad}
+Weights associated with each model grid box (blue lines and numbers) for an observation at -170.5E, 56.0N with a radial footprint with diameter 1\deg.}
+  Weights associated with each model grid box (blue lines and numbers)
+  for an observation at -170.5E, 56.0N with a radial footprint with diameter 1\deg.}
 \end{center}      \end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 …
 \subsection{Grid search}
+For many grids used by the NEMO model, such as the ORCA family,
+the horizontal grid coordinates $i$ and $j$ are not simple functions
+of latitude and longitude. Therefore, it is not always straightforward
+to determine the grid points surrounding any given observational position.
+Before the interpolation can be performed, a search
+algorithm is then required to determine the corner points of
+For many grids used by the NEMO model, such as the ORCA family, the horizontal grid coordinates $i$ and $j$ are not simple functions of latitude and longitude.
+Therefore, it is not always straightforward to determine the grid points surrounding any given observational position.
+Before the interpolation can be performed, a search algorithm is then required to determine the corner points of
 the quadrilateral cell in which the observation is located.
+This is the most difficult and time consuming part of the
+D interpolation procedure.
+A robust test for determining if an observation falls
+within a given quadrilateral cell is as follows. Let
+${\rm P}({\lambda_{}}_{\rm P} ,{\phi_{}}_{\rm P} )$ denote the observation point,
+and let ${\rm A}({\lambda_{}}_{\rm A} ,{\phi_{}}_{\rm A} )$,
+${\rm B}({\lambda_{}}_{\rm B} ,{\phi_{}}_{\rm B} )$,
+${\rm C}({\lambda_{}}_{\rm C} ,{\phi_{}}_{\rm C} )$
+and
+${\rm D}({\lambda_{}}_{\rm D} ,{\phi_{}}_{\rm D} )$ denote
+the bottom left, bottom right, top left and top right
+corner points of the cell, respectively.
+To determine if P is inside
+the cell, we verify that the cross-products
+This is the most difficult and time consuming part of the 2D interpolation procedure.
+A robust test for determining if an observation falls within a given quadrilateral cell is as follows.
+Let ${\rm P}({\lambda_{}}_{\rm P} ,{\phi_{}}_{\rm P} )$ denote the observation point,
+and let ${\rm A}({\lambda_{}}_{\rm A} ,{\phi_{}}_{\rm A} )$, ${\rm B}({\lambda_{}}_{\rm B} ,{\phi_{}}_{\rm B} )$,
+${\rm C}({\lambda_{}}_{\rm C} ,{\phi_{}}_{\rm C} )$ and ${\rm D}({\lambda_{}}_{\rm D} ,{\phi_{}}_{\rm D} )$
+denote the bottom left, bottom right, top left and top right corner points of the cell, respectively.
+To determine if P is inside the cell, we verify that the cross-products
 \begin{eqnarray}
 \begin{array}{lllll}
 …
 \label{eq:cross}
 \end{eqnarray}
+point in the opposite direction to the unit normal
+$\widehat{\bf k}$ (i.e., that the coefficients of
+$\widehat{\bf k}$ are negative),
+where ${{\bf r}_{}}_{\rm PA}$, ${{\bf r}_{}}_{\rm PB}$,
+etc. correspond to the vectors between points P and A,
+P and B, etc.. The method used is
+similar to the method used in
+the SCRIP interpolation package \citep{Jones_1998}.
+In order to speed up the grid search, there is the possibility to construct
+a lookup table for a user specified resolution. This lookup
+table contains the lower and upper bounds on the $i$ and $j$ indices
+to be searched for on a regular grid. For each observation position,
+the closest point on the regular grid of this position is computed and
+the $i$ and $j$ ranges of this point searched to determine the precise
+four points surrounding the observation.
+point in the opposite direction to the unit normal $\widehat{\bf k}$
+(i.e., that the coefficients of $\widehat{\bf k}$ are negative),
+where ${{\bf r}_{}}_{\rm PA}$, ${{\bf r}_{}}_{\rm PB}$, etc. correspond to
+the vectors between points P and A, P and B, etc..
+The method used is similar to the method used in the SCRIP interpolation package \citep{Jones_1998}.
+In order to speed up the grid search, there is the possibility to construct a lookup table for a user specified resolution.
+This lookup table contains the lower and upper bounds on the $i$ and $j$ indices to
+be searched for on a regular grid.
+For each observation position, the closest point on the regular grid of this position is computed and
+the $i$ and $j$ ranges of this point searched to determine the precise four points surrounding the observation.
 \subsection{Parallel aspects of horizontal interpolation}
 \label{subsec:OBS_parallel}
+For horizontal interpolation, there is the basic problem that the
+observations are unevenly distributed on the globe. In numerical
+models, it is common to divide the model grid into subgrids (or
+domains) where each subgrid is executed on a single processing element
+with explicit message passing for exchange of information along the
+domain boundaries when running on a massively parallel processor (MPP)
+system. This approach is used by \NEMO.
+For observations there is no natural distribution since the
+observations are not equally distributed on the globe.
+Two options have been made available: 1) geographical distribution;
+For horizontal interpolation, there is the basic problem that
+the observations are unevenly distributed on the globe.
+In numerical models, it is common to divide the model grid into subgrids (or domains) where
+each subgrid is executed on a single processing element with explicit message passing for
+exchange of information along the domain boundaries when running on a massively parallel processor (MPP) system.
+This approach is used by \NEMO.
+For observations there is no natural distribution since the observations are not equally distributed on the globe.
+Two options have been made available:
+) geographical distribution;
 and 2) round-robin.
 …
 \includegraphics[width=10cm,height=12cm,angle=-90.]{Fig_ASM_obsdist_local}
 \caption{      \protect\label{fig:obslocal}
 Example of the distribution of observations with the geographical distribution of observational data.}
+  Example of the distribution of observations with the geographical distribution of observational data.}
 \end{center}      \end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+This is the simplest option in which the observations are distributed according
+to the domain of the grid-point parallelization. \autoref{fig:obslocal}
+shows an example of the distribution of the {\em in situ} data on processors
+with a different colour for each observation
+on a given processor for a 4 $\times$ 2 decomposition with ORCA2.
+This is the simplest option in which the observations are distributed according to
+the domain of the grid-point parallelization.
+\autoref{fig:obslocal} shows an example of the distribution of the {\em in situ} data on processors with
+a different colour for each observation on a given processor for a 4 $\times$ 2 decomposition with ORCA2.
 The grid-point domain decomposition is clearly visible on the plot.
+The advantage of this approach is that all
+information needed for horizontal interpolation is available without
+any MPP communication. Of course, this is under the assumption that
+we are only using a $2 \times 2$ grid-point stencil for the interpolation
+(e.g., bilinear interpolation). For higher order interpolation schemes this
+is no longer valid. A disadvantage with the above scheme is that the number of
+observations on each processor can be very different. If the cost of
+the actual interpolation is expensive relative to the communication of
+data needed for interpolation, this could lead to load imbalance.
+The advantage of this approach is that all information needed for horizontal interpolation is available without
+any MPP communication.
+Of course, this is under the assumption that we are only using a $2 \times 2$ grid-point stencil for
+the interpolation (e.g., bilinear interpolation).
+For higher order interpolation schemes this is no longer valid.
+A disadvantage with the above scheme is that the number of observations on each processor can be very different.
+If the cost of the actual interpolation is expensive relative to the communication of data needed for interpolation,
+this could lead to load imbalance.
 \subsubsection{Round-robin distribution of observations among processors}
 …
 \includegraphics[width=10cm,height=12cm,angle=-90.]{Fig_ASM_obsdist_global}
 \caption{      \protect\label{fig:obsglobal}
 Example of the distribution of observations with the round-robin distribution of observational data.}
+  Example of the distribution of observations with the round-robin distribution of observational data.}
 \end{center}     \end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+An alternative approach is to distribute the observations equally
+among processors and use message passing in order to retrieve
+the stencil for interpolation. The simplest distribution of the observations
+is to distribute them using a round-robin scheme. \autoref{fig:obsglobal}
+shows the distribution of the {\em in situ} data on processors for the
+round-robin distribution of observations with a different colour for
+each observation on a given processor for a 4 $\times$ 2 decomposition
+with ORCA2 for the same input data as in \autoref{fig:obslocal}.
+An alternative approach is to distribute the observations equally among processors and
+use message passing in order to retrieve the stencil for interpolation.
+The simplest distribution of the observations is to distribute them using a round-robin scheme.
+\autoref{fig:obsglobal} shows the distribution of the {\em in situ} data on processors for
+the round-robin distribution of observations with a different colour for each observation on a given processor for
+a 4 $\times$ 2 decomposition with ORCA2 for the same input data as in \autoref{fig:obslocal}.
 The observations are now clearly randomly distributed on the globe.
+In order to be able to perform horizontal interpolation in this case,
+a subroutine has been developed that retrieves any grid points in the
+global space.
+In order to be able to perform horizontal interpolation in this case,
+a subroutine has been developed that retrieves any grid points in the global space.
 \subsection{Vertical interpolation operator}
+Vertical interpolation is achieved using either a cubic spline or
+linear interpolation. For the cubic spline, the top and
+bottom boundary conditions for the second derivative of the
+interpolating polynomial in the spline are set to zero.
+Vertical interpolation is achieved using either a cubic spline or linear interpolation.
+For the cubic spline, the top and bottom boundary conditions for the second derivative of
+the interpolating polynomial in the spline are set to zero.
 At the bottom boundary, this is done using the land-ocean mask.
 …
 \subsection{Concept}
+The obs oper maps model variables to observation space. It is possible to apply this mapping
+without running the model. The software which performs this functionality is known as the
+\textbf{offline obs oper}. The obs oper is divided into three stages. An initialisation phase,
+an interpolation phase and an output phase. The implementation of which is outlined in the
+previous sections. During the interpolation phase the offline obs oper populates the model
+arrays by reading saved model fields from disk.
+There are two ways of exploiting this offline capacity. The first is to mimic the behaviour of
+the online system by supplying model fields at regular intervals between the start and the end
+of the run. This approach results in a single model counterpart per observation. This kind of
+usage produces feedback files the same file format as the online obs oper.
+The second is to take advantage of the offline setting in which multiple model counterparts can
+be calculated per observation. In this case it is possible to consider all forecasts verifying
+at the same time. By forecast, I mean any method which produces an estimate of physical reality
+which is not an observed value. In the case of class 4 files this means forecasts, analyses, persisted
+analyses and climatological values verifying at the same time. Although the class 4 file format
+doesn't account for multiple ensemble members or multiple experiments per observation, it is possible
+to include these components in the same or multiple files.
+The obs oper maps model variables to observation space.
+It is possible to apply this mapping without running the model.
+The software which performs this functionality is known as the \textbf{offline obs oper}.
+The obs oper is divided into three stages.
+An initialisation phase, an interpolation phase and an output phase.
+The implementation of which is outlined in the previous sections.
+During the interpolation phase the offline obs oper populates the model arrays by
+reading saved model fields from disk.
+There are two ways of exploiting this offline capacity.
+The first is to mimic the behaviour of the online system by supplying model fields at
+regular intervals between the start and the end of the run.
+This approach results in a single model counterpart per observation.
+This kind of usage produces feedback files the same file format as the online obs oper.
+The second is to take advantage of the offline setting in which
+multiple model counterparts can be calculated per observation.
+In this case it is possible to consider all forecasts verifying at the same time.
+By forecast, I mean any method which produces an estimate of physical reality which is not an observed value.
+In the case of class 4 files this means forecasts, analyses, persisted analyses and
+climatological values verifying at the same time.
+Although the class 4 file format doesn't account for multiple ensemble members or
+multiple experiments per observation, it is possible to include these components in the same or multiple files.
 %--------------------------------------------------------------------------------------------------------
 …
 \subsubsection{Building}
 In addition to \emph{OPA\_SRC} the offline obs oper requires the inclusion
+of the \emph{OOO\_SRC} directory. \emph{OOO\_SRC} contains a replacement \mdl{nemo} and
+\mdl{nemogcm} which overwrites the resultant \textbf{nemo.exe}. This is the approach taken
 by \emph{SAS\_SRC} and \emph{OFF\_SRC}.
+In addition to \emph{OPA\_SRC} the offline obs oper requires the inclusion of the \emph{OOO\_SRC} directory.
+\emph{OOO\_SRC} contains a replacement \mdl{nemo} and \mdl{nemogcm} which
+overwrites the resultant \textbf{nemo.exe}.
+This is the approach taken by \emph{SAS\_SRC} and \emph{OFF\_SRC}.
 %--------------------------------------------------------------------------------------------------------
 …
 \subsubsection{Quick script}
 A useful Python utility to control the namelist options can be found in \textbf{OBSTOOLS/OOO}. The
+functions which locate model fields and observation files can be manually specified. The package
 can be installed by appropriate use of the included setup.py script.
+A useful Python utility to control the namelist options can be found in \textbf{OBSTOOLS/OOO}.
+The functions which locate model fields and observation files can be manually specified.
+The package can be installed by appropriate use of the included setup.py script.
 Documentation can be auto-generated by Sphinx by running \emph{make html} in the \textbf{doc} directory.
 …
 %--------------------------------------------------------------------------------------------------------
 \subsection{Configuring the offline observation operator}
 The observation files and settings understood by \textbf{namobs} have been outlined in the online
+obs oper section. In addition there are two further namelists wich control the operation of the offline
+obs oper. \textbf{namooo} which controls the input model fields and \textbf{namcl4} which controls the
 production of class 4 files.
+The observation files and settings understood by \textbf{namobs} have been outlined in the online obs oper section.
+In addition there are two further namelists wich control the operation of the offline obs oper.
+\textbf{namooo} which controls the input model fields and \textbf{namcl4} which
+controls the production of class 4 files.
 \subsubsection{Single field}
 …
 In offline mode model arrays are populated at appropriate time steps via input files.
 At present, \textbf{tsn} and \textbf{sshn} are populated by the default read routines.
 These routines will be expanded upon in future versions to allow the specification of any
 model variable. As such, input files must be global versions of the model domain with
+These routines will be expanded upon in future versions to allow the specification of any model variable.
+As such, input files must be global versions of the model domain with
 \textbf{votemper}, \textbf{vosaline} and optionally \textbf{sshn} present.
 For each field read there must be an entry in the \textbf{namooo} namelist specifying the
+name of the file to read and the index along the \emph{time\_counter}. For example, to
 read the second time counter from a single file the namelist would be.
+For each field read there must be an entry in the \textbf{namooo} namelist specifying
+the name of the file to read and the index along the \emph{time\_counter}.
+For example, to read the second time counter from a single file the namelist would be.
 \begin{forlines}
 …
 \subsubsection{Multiple fields per run}
 Model field iteration is controlled via \textbf{nn\_ooo\_freq} which specifies
+the number of model steps at which the next field gets read. For example, if
 hourly fields are to be interpolated in a setup where 288 steps equals 24 hours.
+Model field iteration is controlled via \textbf{nn\_ooo\_freq} which
+specifies the number of model steps at which the next field gets read.
+For example, if 12 hourly fields are to be interpolated in a setup where 288 steps equals 24 hours.
 \begin{forlines}
 …
 \end{forlines}
 The above namelist will result in feedback files whose first 12 hours contain
 the first field of foo.nc and the second 12 hours contain the second field.
+The above namelist will result in feedback files whose first 12 hours contain the first field of foo.nc and
+the second 12 hours contain the second field.
 %\begin{framed}
 …
 %\end{framed}
+It is easy to see how a collection of fields taken fron a number of files
+at different indices can be combined at a particular frequency in time to
+generate a pseudo model evolution. As long as all that is needed is a single
+model counterpart at a regular interval then namooo is all that needs to
+be edited. However, a far more interesting approach can be taken in which
+multiple forecasts, analyses, persisted analyses and climatologies are
+considered against the same set of observations. For this a slightly more
+complicated approach is needed. It is referred to as \emph{Class 4} since
+it is the fourth metric defined by the GODAE intercomparison project.
+It is easy to see how a collection of fields taken fron a number of files at different indices can be combined at
+a particular frequency in time to generate a pseudo model evolution.
+As long as all that is needed is a single model counterpart at a regular interval then
+namooo is all that needs to be edited.
+However, a far more interesting approach can be taken in which multiple forecasts, analyses, persisted analyses and
+climatologies are considered against the same set of observations.
+For this a slightly more complicated approach is needed.
+It is referred to as \emph{Class 4} since it is the fourth metric defined by the GODAE intercomparison project.
 %--------------------------------------------------------------------------------------------------------
 …
 \subsubsection{Multiple model counterparts per observation a.k.a Class 4}
+A generalisation of feedback files to allow multiple model components per observation. For a single
+observation, as well as previous forecasts verifying at the same time there are also analyses, persisted
+analyses and climatologies.
+The above namelist performs two basic functions. It organises the fields
+given in \textbf{namooo} into groups so that observations can be matched
+up multiple times. It also controls the metadata and the output variable
+of the class 4 file when a write routine is called.
+A generalisation of feedback files to allow multiple model components per observation.
+For a single observation, as well as previous forecasts verifying at the same time
+there are also analyses, persisted analyses and climatologies.
+The above namelist performs two basic functions.
+It organises the fields given in \textbf{namooo} into groups so that observations can be matched up multiple times.
+It also controls the metadata and the output variable of the class 4 file when a write routine is called.
 %\begin{framed}
+\textbf{Note: ln\_cl4} must be set to \forcode{.true.} in \textbf{namobs}
+to use class 4 outputs.
+\textbf{Note: ln\_cl4} must be set to \forcode{.true.} in \textbf{namobs} to use class 4 outputs.
 %\end{framed}
 …
 \noindent
 \linebreak
 Much of the namelist is devoted to specifying this convention. The
+following namelist settings control the elements of the output
 file names. Each should be specified as a single string of character data.
+Much of the namelist is devoted to specifying this convention.
+The following namelist settings control the elements of the output file names.
+Each should be specified as a single string of character data.
 \begin{description}
 \item[cl4\_prefix]
 Prefix for class 4 files e.g. class4
+  Prefix for class 4 files e.g. class4
 \item[cl4\_date]
 YYYYMMDD validity date
+  YYYYMMDD validity date
 \item[cl4\_sys]
 The name of the class 4 model system e.g. FOAM
+  The name of the class 4 model system e.g. FOAM
 \item[cl4\_cfg]
 The name of the class 4 model configuration e.g. orca025
+  The name of the class 4 model configuration e.g. orca025
 \item[cl4\_vn]
 The name of the class 4 model version e.g. 12.0
+  The name of the class 4 model version e.g. 12.0
 \end{description}
 \noindent
 The kind is specified by the observation type internally to the obs oper. The processor
 number is specified internally in NEMO.
+The kind is specified by the observation type internally to the obs oper.
+The processor number is specified internally in NEMO.
 \subsubsection{Class 4 file global attributes}
+Global attributes necessary to fulfill the class 4 file definition. These
+are also useful pieces of information when collaborating with external
+partners.
+Global attributes necessary to fulfill the class 4 file definition.
+These are also useful pieces of information when collaborating with external partners.
 \begin{description}
 \item[cl4\_contact]
 Contact email for class 4 files.
+  Contact email for class 4 files.
 \item[cl4\_inst]
 The name of the producers institution.
+  The name of the producers institution.
 \item[cl4\_cfg]
 The name of the class 4 model configuration e.g. orca025
+  The name of the class 4 model configuration e.g. orca025
 \item[cl4\_vn]
 The name of the class 4 model version e.g. 12.0
+  The name of the class 4 model version e.g. 12.0
 \end{description}
 \noindent
+The obs\_type,
+creation date and validity time are specified internally to the obs oper.
+The obs\_type, creation date and validity time are specified internally to the obs oper.
 \subsubsection{Class 4 model counterpart configuration}
+As seen previously it is possible to perform a single sweep of the
+obs oper and specify a collection of model fields equally spaced
+along that sweep. In the class 4 case the single sweep is replaced
+with multiple sweeps and a certain ammount of book keeping is
+needed to ensure each model counterpart makes its way to the
+correct piece of memory in the output files.
+As seen previously it is possible to perform a single sweep of the obs oper and
+specify a collection of model fields equally spaced along that sweep.
+In the class 4 case the single sweep is replaced with multiple sweeps and
+a certain ammount of book keeping is needed to ensure each model counterpart makes its way to
+the correct piece of memory in the output files.
 \noindent
 \linebreak
+In terms of book keeping, the offline obs oper needs to know how many
+full sweeps need to be performed. This is specified via the
+\textbf{cl4\_match\_len} variable and is the total number of model
+counterparts per observation. For example, a 3 forecasts plus 3 persistence
+fields plus an analysis field would be 7 counterparts per observation.
+In terms of book keeping, the offline obs oper needs to know how many full sweeps need to be performed.
+This is specified via the \textbf{cl4\_match\_len} variable and
+is the total number of model counterparts per observation.
+For example, a 3 forecasts plus 3 persistence fields plus an analysis field would be 7 counterparts per observation.
 \begin{forlines}
 …
 \end{forlines}
 Then to correctly allocate a class 4 file the forecast axis must be defined. This
 is controlled via \textbf{cl4\_fcst\_len}, which in out above example would be 3.
+Then to correctly allocate a class 4 file the forecast axis must be defined.
+This is controlled via \textbf{cl4\_fcst\_len}, which in out above example would be 3.
 \begin{forlines}
 …
 \end{forlines}
+Then for each model field it is necessary to designate what class 4 variable and
+index along the forecast dimension the model counterpart should be stored in the
+output file. As well as a value for that lead time in hours, this will be useful
+when interpreting the data afterwards.
+Then for each model field it is necessary to designate what class 4 variable and index along
+the forecast dimension the model counterpart should be stored in the output file.
+As well as a value for that lead time in hours, this will be useful when interpreting the data afterwards.
 \begin{forlines}
 …
 \end{forlines}
 In terms of files and indices of fields inside each file the class 4 approach
+makes use of the \textbf{namooo} namelist. If our fields are in separate files
 with a single field per file our example inputs will be specified.
+In terms of files and indices of fields inside each file the class 4 approach makes use of
+the \textbf{namooo} namelist.
+If our fields are in separate files with a single field per file our example inputs will be specified.
 \begin{forlines}
 …
 \end{forlines}
+When we combine all of the naming conventions, global attributes and i/o instructions
+the class 4 namelist becomes.
+When we combine all of the naming conventions, global attributes and i/o instructions the class 4 namelist becomes.
 \begin{forlines}
 …
 \subsubsection{Climatology interpolation}
 The climatological counterpart is generated at the start of the run by restarting
+the model from climatology through appropriate use of \textbf{namtsd}. To override
+the offline observation operator read routine and to take advantage of the restart
 settings, specify the first entry in \textbf{cl4\_vars} as "climatology". This will then
+pipe the restart from climatology into the output class 4 file. As in every other
 class 4 matchup the input file, input index and output index must be specified.
 These can be replaced with dummy data since they are not used but they must be
 present to cycle through the matchups correctly.
+The climatological counterpart is generated at the start of the run by
+restarting the model from climatology through appropriate use of \textbf{namtsd}.
+To override the offline observation operator read routine and to take advantage of the restart settings,
+specify the first entry in \textbf{cl4\_vars} as "climatology".
+This will then pipe the restart from climatology into the output class 4 file.
+As in every other class 4 matchup the input file, input index and output index must be specified.
+These can be replaced with dummy data since they are not used but
+they must be present to cycle through the matchups correctly.
 \subsection{Advanced usage}
 In certain cases it may be desirable to include both multiple model fields per
+observation window with multiple match ups per observation. This can be achieved
+by specifying \textbf{nn\_ooo\_freq} as well as the class 4 settings. Care must
 be taken in generating the ooo\_files list such that the files are arranged into
 consecutive blocks of single match ups. For example, 2 forecast fields
+of 12 hourly data would result in 4 separate read operations but only 2 write
 operations, 1 per forecast.
+In certain cases it may be desirable to include both multiple model fields per observation window with
+multiple match ups per observation.
+This can be achieved by specifying \textbf{nn\_ooo\_freq} as well as the class 4 settings.
+Care must be taken in generating the ooo\_files list such that the files are arranged into
+consecutive blocks of single match ups.
+For example, 2 forecast fields of 12 hourly data would result in 4 separate read operations but
+only 2 write operations, 1 per forecast.
 \begin{forlines}
 …
 \end{forlines}
+The above notation reveals the internal split between match up iterators and file
+iterators. This technique has not been used before so experimentation is needed
+before results can be trusted.
+The above notation reveals the internal split between match up iterators and file iterators.
+This technique has not been used before so experimentation is needed before results can be trusted.
 …
 \label{sec:OBS_obsutils}
+Some tools for viewing and processing of observation and feedback files are provided in the
+NEMO repository for convenience. These include OBSTOOLS which are a collection of Fortran
+programs which are helpful to deal with feedback files. They do such tasks as observation file
+conversion, printing of file contents, some basic statistical analysis of feedback files. The
+other tool is an IDL program called dataplot which uses a graphical interface to visualise
+observations and feedback files. OBSTOOLS and dataplot are described in more detail below.
+Some tools for viewing and processing of observation and feedback files are provided in
+the NEMO repository for convenience.
+These include OBSTOOLS which are a collection of Fortran programs which are helpful to deal with feedback files.
+They do such tasks as observation file conversion, printing of file contents,
+some basic statistical analysis of feedback files.
+The other tool is an IDL program called dataplot which uses a graphical interface to
+visualise observations and feedback files.
+OBSTOOLS and dataplot are described in more detail below.
 \subsection{Obstools}
 A series of Fortran utilities is provided with NEMO called OBSTOOLS. This are helpful in
 handling observation files and the feedback file output from the NEMO observation operator.
+A series of Fortran utilities is provided with NEMO called OBSTOOLS.
+This are helpful in handling observation files and the feedback file output from the NEMO observation operator.
 The utilities are as follows
 \subsubsection{c4comb}
+The program c4comb combines multiple class 4 files produced by individual processors in an
+MPI run of NEMO offline obs\_oper into a single class 4 file. The program is called in the following way:
+The program c4comb combines multiple class 4 files produced by individual processors in
+an MPI run of NEMO offline obs\_oper into a single class 4 file.
+The program is called in the following way:
 …
 \subsubsection{corio2fb}
 The program corio2fb converts profile observation files from the Coriolis format to the
 standard feedback format. The program is called in the following way:
+The program corio2fb converts profile observation files from the Coriolis format to the standard feedback format.
+The program is called in the following way:
 \footnotesize
 …
 \subsubsection{enact2fb}
 The program enact2fb converts profile observation files from the ENACT format to the standard
 feedback format. The program is called in the following way:
+The program enact2fb converts profile observation files from the ENACT format to the standard feedback format.
+The program is called in the following way:
 \footnotesize
 …
 \subsubsection{fbcomb}
+The program fbcomb combines multiple feedback files produced by individual processors in an
+MPI run of NEMO into a single feedback file. The program is called in the following way:
+The program fbcomb combines multiple feedback files produced by individual processors in
+an MPI run of NEMO into a single feedback file.
+The program is called in the following way:
 \footnotesize
 …
 \subsubsection{fbmatchup}
 The program fbmatchup will match observations from two feedback files. The program is called
 in the following way:
+The program fbmatchup will match observations from two feedback files.
+The program is called in the following way:
 \footnotesize
 …
 The program fbprint will print the contents of a feedback file or files to standard output.
 Selected information can be output using optional arguments. The program is called in the
 following way:
+Selected information can be output using optional arguments.
+The program is called in the following way:
 \footnotesize
 …
 \subsubsection{fbsel}
 The program fbsel will select or subsample observations. The program is called in the
 following way:
+The program fbsel will select or subsample observations.
+The program is called in the following way:
 \footnotesize
 …
 \subsubsection{fbstat}
 The program fbstat will output summary statistics in different global areas into a number of
 files. The program is called in the following way:
+The program fbstat will output summary statistics in different global areas into a number of files.
+The program is called in the following way:
 \footnotesize
 …
 \subsubsection{fbthin}
+The program fbthin will thin the data to 1 degree resolution. The code could easily be
+modified to thin to a different resolution. The program is called in the following way:
+The program fbthin will thin the data to 1 degree resolution.
+The code could easily be modified to thin to a different resolution.
+The program is called in the following way:
 \footnotesize
 …
 \subsubsection{sla2fb}
 The program sla2fb will convert an AVISO SLA format file to feedback format. The program is
 called in the following way:
+The program sla2fb will convert an AVISO SLA format file to feedback format.
+The program is called in the following way:
 \footnotesize
 …
 \subsubsection{vel2fb}
 The program vel2fb will convert TAO/PIRATA/RAMA currents files to feedback format. The program
 is called in the following way:
+The program vel2fb will convert TAO/PIRATA/RAMA currents files to feedback format.
+The program is called in the following way:
 \footnotesize
 …
 \subsection{Dataplot}
+An IDL program called dataplot is included which uses a graphical interface to visualise
+observations and feedback files. It is possible to zoom in, plot individual profiles and
+calculate some basic statistics. To plot some data run IDL and then:
+An IDL program called dataplot is included which uses a graphical interface to
+visualise observations and feedback files.
+It is possible to zoom in, plot individual profiles and calculate some basic statistics.
+To plot some data run IDL and then:
 \footnotesize
 \begin{minted}{idl}
 …
 \end{minted}
 To read multiple files into dataplot, for example multiple feedback files from different
+processors or from different days, the easiest method is to use the spawn command to generate
 a list of files which can then be passed to dataplot.
+To read multiple files into dataplot,
+for example multiple feedback files from different processors or from different days,
+the easiest method is to use the spawn command to generate a list of files which can then be passed to dataplot.
 \footnotesize
 \begin{minted}{idl}
 …
 \autoref{fig:obsdataplotmain} shows the main window which is launched when dataplot starts.
+This is split into three parts. At the top there is a menu bar which contains a variety of
+drop down menus. Areas - zooms into prespecified regions; plot - plots the data as a
+timeseries or a T-S diagram if appropriate; Find - allows data to be searched; Config - sets
+various configuration options.
+The middle part is a plot of the geographical location of the observations. This will plot the
+observation value, the model background value or observation minus background value depending
+on the option selected in the radio button at the bottom of the window. The plotting colour
+range can be changed by clicking on the colour bar. The title of the plot gives some basic
+information about the date range and depth range shown, the extreme values, and the mean and
+rms values. It is possible to zoom in using a drag-box. You may also zoom in or out using the
+mouse wheel.
+The bottom part of the window controls what is visible in the plot above. There are two bars
+which select the level range plotted (for profile data). The other bars below select the date
+range shown. The bottom of the figure allows the option to plot the mean, root mean square,
+standard deviation or mean square values. As mentioned above you can choose to plot the
+observation value, the model background value or observation minus background value. The next
+group of radio buttons selects the map projection. This can either be regular latitude
+longitude grid, or north or south polar stereographic. The next group of radio buttons will
+plot bad observations, switch to salinity and plot density for profile observations. The
+rightmost group of buttons will print the plot window as a postscript, save it as png, or exit
+from dataplot.
+This is split into three parts.
+At the top there is a menu bar which contains a variety of drop down menus.
+Areas - zooms into prespecified regions;
+plot - plots the data as a timeseries or a T-S diagram if appropriate;
+Find - allows data to be searched;
+Config - sets various configuration options.
+The middle part is a plot of the geographical location of the observations.
+This will plot the observation value, the model background value or observation minus background value depending on
+the option selected in the radio button at the bottom of the window.
+The plotting colour range can be changed by clicking on the colour bar.
+The title of the plot gives some basic information about the date range and depth range shown,
+the extreme values, and the mean and rms values.
+It is possible to zoom in using a drag-box.
+You may also zoom in or out using the mouse wheel.
+The bottom part of the window controls what is visible in the plot above.
+There are two bars which select the level range plotted (for profile data).
+The other bars below select the date range shown.
+The bottom of the figure allows the option to plot the mean, root mean square, standard deviation or
+mean square values.
+As mentioned above you can choose to plot the observation value, the model background value or
+observation minus background value.
+The next group of radio buttons selects the map projection.
+This can either be regular latitude longitude grid, or north or south polar stereographic.
+The next group of radio buttons will plot bad observations, switch to salinity and
+plot density for profile observations.
+The rightmost group of buttons will print the plot window as a postscript, save it as png, or exit from dataplot.
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 …
 \includegraphics[width=9cm,angle=-90.]{Fig_OBS_dataplot_main}
 \caption{      \protect\label{fig:obsdataplotmain}
 Main window of dataplot.}
+  Main window of dataplot.}
 \end{center}     \end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 If a profile point is clicked with the mouse button a plot of the observation and background
 values as a function of depth (\autoref{fig:obsdataplotprofile}).
+If a profile point is clicked with the mouse button a plot of the observation and background values as
+a function of depth (\autoref{fig:obsdataplotprofile}).
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 …
 \includegraphics[width=7cm,angle=-90.]{Fig_OBS_dataplot_prof}
 \caption{      \protect\label{fig:obsdataplotprofile}
 Profile plot from dataplot produced by right clicking on a point in the main window.}
+  Profile plot from dataplot produced by right clicking on a point in the main window.}
 \end{center}     \end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>

NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_SBC.tex

-                      r10146
+                      r10368
 The ocean needs six fields as surface boundary condition:
 \begin{itemize}
+   \item the two components of the surface ocean stress $\left( {\tau _u \;,\;\tau _v} \right)$
+   \item the incoming solar and non solar heat fluxes $\left( {Q_{ns} \;,\;Q_{sr} } \right)$
+   \item the surface freshwater budget $\left( {\textit{emp}} \right)$
+   \item the surface salt flux associated with freezing/melting of seawater $\left( {\textit{sfx}} \right)$
+\item
+  the two components of the surface ocean stress $\left( {\tau _u \;,\;\tau _v} \right)$
+\item
+  the incoming solar and non solar heat fluxes $\left( {Q_{ns} \;,\;Q_{sr} } \right)$
+\item
+  the surface freshwater budget $\left( {\textit{emp}} \right)$
+\item
+  the surface salt flux associated with freezing/melting of seawater $\left( {\textit{sfx}} \right)$
 \end{itemize}
 plus an optional field:
 …
 \end{itemize}
+Five different ways to provide the first six fields to the ocean are available which
+are controlled by namelist \ngn{namsbc} variables: an analytical formulation (\np{ln\_ana}\forcode{ = .true.}),
+a flux formulation (\np{ln\_flx}\forcode{ = .true.}), a bulk formulae formulation (CORE
+(\np{ln\_blk\_core}\forcode{ = .true.}), CLIO (\np{ln\_blk\_clio}\forcode{ = .true.}) or MFS
+\footnote { Note that MFS bulk formulae compute fluxes only for the ocean component}
+(\np{ln\_blk\_mfs}\forcode{ = .true.}) bulk formulae) and a coupled or mixed forced/coupled formulation
+(exchanges with a atmospheric model via the OASIS coupler) (\np{ln\_cpl} or \np{ln\_mixcpl}\forcode{ = .true.}).
+When used ($i.e.$ \np{ln\_apr\_dyn}\forcode{ = .true.}), the atmospheric pressure forces both ocean and ice dynamics.
+The frequency at which the forcing fields have to be updated is given by the \np{nn\_fsbc} namelist parameter.
+When the fields are supplied from data files (flux and bulk formulations), the input fields
+need not be supplied on the model grid. Instead a file of coordinates and weights can
+be supplied which maps the data from the supplied grid to the model points
+(so called "Interpolation on the Fly", see \autoref{subsec:SBC_iof}).
+Five different ways to provide the first six fields to the ocean are available which are controlled by
+namelist \ngn{namsbc} variables:
+an analytical formulation (\np{ln\_ana}\forcode{ = .true.}),
+a flux formulation (\np{ln\_flx}\forcode{ = .true.}),
+a bulk formulae formulation (CORE (\np{ln\_blk\_core}\forcode{ = .true.}),
+CLIO (\np{ln\_blk\_clio}\forcode{ = .true.}) or
+MFS \footnote { Note that MFS bulk formulae compute fluxes only for the ocean component}
+(\np{ln\_blk\_mfs}\forcode{ = .true.}) bulk formulae) and
+a coupled or mixed forced/coupled formulation (exchanges with a atmospheric model via the OASIS coupler)
+(\np{ln\_cpl} or \np{ln\_mixcpl}\forcode{ = .true.}).
+When used ($i.e.$ \np{ln\_apr\_dyn}\forcode{ = .true.}),
+the atmospheric pressure forces both ocean and ice dynamics.
+The frequency at which the forcing fields have to be updated is given by the \np{nn\_fsbc} namelist parameter.
+When the fields are supplied from data files (flux and bulk formulations),
+the input fields need not be supplied on the model grid.
+Instead a file of coordinates and weights can be supplied which maps the data from the supplied grid to
+the model points (so called "Interpolation on the Fly", see \autoref{subsec:SBC_iof}).
 If the Interpolation on the Fly option is used, input data belonging to land points (in the native grid),
 can be masked to avoid spurious results in proximity of the coasts  as large sea-land gradients characterize
 most of the atmospheric variables.
 In addition, the resulting fields can be further modified using several namelist options.
+can be masked to avoid spurious results in proximity of the coasts as
+large sea-land gradients characterize most of the atmospheric variables.
+In addition, the resulting fields can be further modified using several namelist options.
 These options control
 \begin{itemize}
+\item the rotation of vector components supplied relative to an east-north
+coordinate system onto the local grid directions in the model ;
+\item the addition of a surface restoring term to observed SST and/or SSS (\np{ln\_ssr}\forcode{ = .true.}) ;
+\item the modification of fluxes below ice-covered areas (using observed ice-cover or a sea-ice model) (\np{nn\_ice}\forcode{ = 0..3}) ;
+\item the addition of river runoffs as surface freshwater fluxes or lateral inflow (\np{ln\_rnf}\forcode{ = .true.}) ;
+\item the addition of isf melting as lateral inflow (parameterisation) or as fluxes applied at the land-ice ocean interface (\np{ln\_isf}) ;
+\item the addition of a freshwater flux adjustment in order to avoid a mean sea-level drift (\np{nn\_fwb}\forcode{ = 0..2}) ;
+\item the transformation of the solar radiation (if provided as daily mean) into a diurnal cycle (\np{ln\_dm2dc}\forcode{ = .true.}) ;
+and a neutral drag coefficient can be read from an external wave model (\np{ln\_cdgw}\forcode{ = .true.}).
+\item
+  the rotation of vector components supplied relative to an east-north coordinate system onto
+  the local grid directions in the model;
+\item
+  the addition of a surface restoring term to observed SST and/or SSS (\np{ln\_ssr}\forcode{ = .true.});
+\item
+  the modification of fluxes below ice-covered areas (using observed ice-cover or a sea-ice model)
+  (\np{nn\_ice}\forcode{ = 0..3});
+\item
+  the addition of river runoffs as surface freshwater fluxes or lateral inflow (\np{ln\_rnf}\forcode{ = .true.});
+\item
+  the addition of isf melting as lateral inflow (parameterisation) or
+  as fluxes applied at the land-ice ocean interface (\np{ln\_isf}) ;
+\item
+  the addition of a freshwater flux adjustment in order to avoid a mean sea-level drift
+  (\np{nn\_fwb}\forcode{ = 0..2});
+\item
+  the transformation of the solar radiation (if provided as daily mean) into a diurnal cycle
+  (\np{ln\_dm2dc}\forcode{ = .true.});
+  and a neutral drag coefficient can be read from an external wave model (\np{ln\_cdgw}\forcode{ = .true.}).
 \end{itemize}
 The latter option is possible only in case core or mfs bulk formulas are selected.
 In this chapter, we first discuss where the surface boundary condition appears in the
 model equations. Then we present the five ways of providing the surface boundary condition,
+In this chapter, we first discuss where the surface boundary condition appears in the model equations.
+Then we present the five ways of providing the surface boundary condition,
 followed by the description of the atmospheric pressure and the river runoff.
 Next the scheme for interpolation on the fly is described.
 Finally, the different options that further modify the fluxes applied to the ocean are discussed.
+One of these is modification by icebergs (see \autoref{sec:ICB_icebergs}), which act as drifting sources of fresh water.
+One of these is modification by icebergs (see \autoref{sec:ICB_icebergs}),
+which act as drifting sources of fresh water.
 Another example of modification is that due to the ice shelf melting/freezing (see \autoref{sec:SBC_isf}),
 which provides additional sources of fresh water.
 …
 \label{sec:SBC_general}
+The surface ocean stress is the stress exerted by the wind and the sea-ice
+on the ocean. It is applied in \mdl{dynzdf} module as a surface boundary condition of the
+computation of the momentum vertical mixing trend (see \autoref{eq:dynzdf_sbc} in \autoref{sec:DYN_zdf}).
+As such, it has to be provided as a 2D vector interpolated
+onto the horizontal velocity ocean mesh, $i.e.$ resolved onto the model
+(\textbf{i},\textbf{j}) direction at $u$- and $v$-points.
+The surface heat flux is decomposed into two parts, a non solar and a solar heat
+flux, $Q_{ns}$ and $Q_{sr}$, respectively. The former is the non penetrative part
+of the heat flux ($i.e.$ the sum of sensible, latent and long wave heat fluxes
+plus the heat content of the mass exchange with the atmosphere and sea-ice).
+It is applied in \mdl{trasbc} module as a surface boundary condition trend of
+the first level temperature time evolution equation (see \autoref{eq:tra_sbc}
+and \autoref{eq:tra_sbc_lin} in \autoref{subsec:TRA_sbc}).
+The latter is the penetrative part of the heat flux. It is applied as a 3D
+trends of the temperature equation (\mdl{traqsr} module) when \np{ln\_traqsr}\forcode{ = .true.}.
+The way the light penetrates inside the water column is generally a sum of decreasing
+exponentials (see \autoref{subsec:TRA_qsr}).
+The surface ocean stress is the stress exerted by the wind and the sea-ice on the ocean.
+It is applied in \mdl{dynzdf} module as a surface boundary condition of the computation of
+the momentum vertical mixing trend (see \autoref{eq:dynzdf_sbc} in \autoref{sec:DYN_zdf}).
+As such, it has to be provided as a 2D vector interpolated onto the horizontal velocity ocean mesh,
+$i.e.$ resolved onto the model (\textbf{i},\textbf{j}) direction at $u$- and $v$-points.
+The surface heat flux is decomposed into two parts, a non solar and a solar heat flux,
+$Q_{ns}$ and $Q_{sr}$, respectively.
+The former is the non penetrative part of the heat flux
+($i.e.$ the sum of sensible, latent and long wave heat fluxes plus
+the heat content of the mass exchange with the atmosphere and sea-ice).
+It is applied in \mdl{trasbc} module as a surface boundary condition trend of
+the first level temperature time evolution equation
+(see \autoref{eq:tra_sbc} and \autoref{eq:tra_sbc_lin} in \autoref{subsec:TRA_sbc}).
+The latter is the penetrative part of the heat flux.
+It is applied as a 3D trends of the temperature equation (\mdl{traqsr} module) when
+\np{ln\_traqsr}\forcode{ = .true.}.
+The way the light penetrates inside the water column is generally a sum of decreasing exponentials
+(see \autoref{subsec:TRA_qsr}).
 The surface freshwater budget is provided by the \textit{emp} field.
 It represents the mass flux exchanged with the atmosphere (evaporation minus precipitation)
+and possibly with the sea-ice and ice shelves (freezing minus melting of ice).
 It affects both the ocean in two different ways:
 $(i)$   it changes the volume of the ocean and therefore appears in the sea surface height
 equation as a volume flux, and
 $(ii)$  it changes the surface temperature and salinity through the heat and salt contents
 of the mass exchanged with the atmosphere, the sea-ice and the ice shelves.
+It represents the mass flux exchanged with the atmosphere (evaporation minus precipitation) and
+possibly with the sea-ice and ice shelves (freezing minus melting of ice).
+It affects both the ocean in two different ways:
+$(i)$  it changes the volume of the ocean and therefore appears in the sea surface height equation as
+a volume flux, and
+$(ii)$ it changes the surface temperature and salinity through the heat and salt contents of
+the mass exchanged with the atmosphere, the sea-ice and the ice shelves.
 …
 %\colorbox{yellow}{End Miss }
 The ocean model provides, at each time step, to the surface module (\mdl{sbcmod})
+The ocean model provides, at each time step, to the surface module (\mdl{sbcmod})
 the surface currents, temperature and salinity.
+These variables are averaged over \np{nn\_fsbc} time-step (\autoref{tab:ssm}),
+and it is these averaged fields which are used to computes the surface fluxes
+at a frequency of \np{nn\_fsbc} time-step.
+These variables are averaged over \np{nn\_fsbc} time-step (\autoref{tab:ssm}), and
+it is these averaged fields which are used to computes the surface fluxes at a frequency of \np{nn\_fsbc} time-step.
 …
 Sea surface salinty              & sss\_m & $psu$        & T \\   \hline
 \end{tabular}
 \caption{  \protect\label{tab:ssm}
+Ocean variables provided by the ocean to the surface module (SBC).
+The variable are averaged over nn{\_}fsbc time step,
 $i.e.$ the frequency of computation of surface fluxes.}
+\caption{  \protect\label{tab:ssm}
+  Ocean variables provided by the ocean to the surface module (SBC).
+  The variable are averaged over nn{\_}fsbc time step,
+  $i.e.$ the frequency of computation of surface fluxes.}
 \end{center}   \end{table}
 %--------------------------------------------------------------------------------------------------------------
 …
 \label{sec:SBC_input}
+A generic interface has been introduced to manage the way input data (2D or 3D fields,
+like surface forcing or ocean T and S) are specify in \NEMO. This task is archieved by \mdl{fldread}.
+A generic interface has been introduced to manage the way input data
+(2D or 3D fields, like surface forcing or ocean T and S) are specify in \NEMO.
+This task is archieved by \mdl{fldread}.
 The module was design with four main objectives in mind:
+\begin{enumerate}
+\item optionally provide a time interpolation of the input data at model time-step,
+whatever their input frequency is, and according to the different calendars available in the model.
+\item optionally provide an on-the-fly space interpolation from the native input data grid to the model grid.
+\item make the run duration independent from the period cover by the input files.
+\item provide a simple user interface and a rather simple developer interface by limiting the
+ number of prerequisite information.
+\begin{enumerate}
+\item
+  optionally provide a time interpolation of the input data at model time-step, whatever their input frequency is,
+  and according to the different calendars available in the model.
+\item
+  optionally provide an on-the-fly space interpolation from the native input data grid to the model grid.
+\item
+  make the run duration independent from the period cover by the input files.
+\item
+  provide a simple user interface and a rather simple developer interface by
+  limiting the number of prerequisite information.
 \end{enumerate}
 As a results the user have only to fill in for each variable a structure in the namelist file
+to defined the input data file and variable names, the frequency of the data (in hours or months),
 whether its is climatological data or not, the period covered by the input file (one year, month, week or day),
 and three additional parameters for on-the-fly interpolation. When adding a new input variable,
+the developer has to add the associated structure in the namelist, read this information
+by mirroring the namelist read in \rou{sbc\_blk\_init} for example, and simply call \rou{fld\_read}
 to obtain the desired input field at the model time-step and grid points.
+As a results the user have only to fill in for each variable a structure in the namelist file to
+define the input data file and variable names, the frequency of the data (in hours or months),
+whether its is climatological data or not, the period covered by the input file (one year, month, week or day),
+and three additional parameters for on-the-fly interpolation.
+When adding a new input variable, the developer has to add the associated structure in the namelist,
+read this information by mirroring the namelist read in \rou{sbc\_blk\_init} for example,
+and simply call \rou{fld\_read} to obtain the desired input field at the model time-step and grid points.
 The only constraints are that the input file is a NetCDF file, the file name follows a nomenclature
+(see \autoref{subsec:SBC_fldread}), the period it cover is one year, month, week or day, and, if on-the-fly
+interpolation is used, a file of weights must be supplied (see \autoref{subsec:SBC_iof}).
+Note that when an input data is archived on a disc which is accessible directly
+from the workspace where the code is executed, then the use can set the \np{cn\_dir}
+to the pathway leading to the data. By default, the data are assumed to have been
+copied so that cn\_dir='./'.
+(see \autoref{subsec:SBC_fldread}), the period it cover is one year, month, week or day, and,
+if on-the-fly interpolation is used, a file of weights must be supplied (see \autoref{subsec:SBC_iof}).
+Note that when an input data is archived on a disc which is accessible directly from the workspace where
+the code is executed, then the use can set the \np{cn\_dir} to the pathway leading to the data.
+By default, the data are assumed to have been copied so that cn\_dir='./'.
 % -------------------------------------------------------------------------------------------------------------
 …
 where
 \begin{description}
+\item[File name]: the stem name of the NetCDF file to be open.
+This stem will be completed automatically by the model, with the addition of a '.nc' at its end
+and by date information and possibly a prefix (when using AGRIF).
+Tab.\autoref{tab:fldread} provides the resulting file name in all possible cases according to whether
+it is a climatological file or not, and to the open/close frequency (see below for definition).
+\item[File name]:
+  the stem name of the NetCDF file to be open.
+  This stem will be completed automatically by the model, with the addition of a '.nc' at its end and
+  by date information and possibly a prefix (when using AGRIF).
+  Tab.\autoref{tab:fldread} provides the resulting file name in all possible cases according to
+  whether it is a climatological file or not, and to the open/close frequency (see below for definition).
 %--------------------------------------------------TABLE--------------------------------------------------
 …
 \end{tabular}
 \end{center}
+\caption{ \protect\label{tab:fldread}   naming nomenclature for climatological or interannual input file,
+as a function of the Open/close frequency. The stem name is assumed to be 'fn'.
+For weekly files, the 'LLL' corresponds to the first three letters of the first day of the week ($i.e.$ 'sun','sat','fri','thu','wed','tue','mon'). The 'YYYY', 'MM' and 'DD' should be replaced by the
+actual year/month/day, always coded with 4 or 2 digits. Note that (1) in mpp, if the file is split
+over each subdomain, the suffix '.nc' is replaced by '\_PPPP.nc', where 'PPPP' is the
+process number coded with 4 digits; (2) when using AGRIF, the prefix
+'\_N' is added to files,
+where 'N'  is the child grid number.}
+\caption{ \protect\label{tab:fldread}
+  naming nomenclature for climatological or interannual input file, as a function of the Open/close frequency.
+  The stem name is assumed to be 'fn'.
+  For weekly files, the 'LLL' corresponds to the first three letters of the first day of the week
+  ($i.e.$ 'sun','sat','fri','thu','wed','tue','mon').
+  The 'YYYY', 'MM' and 'DD' should be replaced by the actual year/month/day, always coded with 4 or 2 digits.
+  Note that (1) in mpp, if the file is split over each subdomain, the suffix '.nc' is replaced by '\_PPPP.nc',
+  where 'PPPP' is the process number coded with 4 digits;
+  (2) when using AGRIF, the prefix '\_N' is added to files, where 'N'  is the child grid number.}
 \end{table}
 %--------------------------------------------------------------------------------------------------------------
+\item[Record frequency]: the frequency of the records contained in the input file.
+Its unit is in hours if it is positive (for example 24 for daily forcing) or in months if negative
+(for example -1 for monthly forcing or -12 for annual forcing).
+Note that this frequency must really be an integer and not a real.
+On some computers, seting it to '24.' can be interpreted as 240!
+\item[Variable name]: the name of the variable to be read in the input NetCDF file.
+\item[Time interpolation]: a logical to activate, or not, the time interpolation. If set to 'false',
+the forcing will have a steplike shape remaining constant during each forcing period.
+For example, when using a daily forcing without time interpolation, the forcing remaining
+constant from 00h00'00'' to 23h59'59". If set to 'true', the forcing will have a broken line shape.
+Records are assumed to be dated the middle of the forcing period.
+For example, when using a daily forcing with time interpolation, linear interpolation will
+be performed between mid-day of two consecutive days.
+\item[Climatological forcing]: a logical to specify if a input file contains climatological forcing
+which can be cycle in time, or an interannual forcing which will requires additional files
+if the period covered by the simulation exceed the one of the file. See the above the file
+naming strategy which impacts the expected name of the file to be opened.
+\item[Open/close frequency]: the frequency at which forcing files must be opened/closed.
+Four cases are coded: 'daily', 'weekLLL' (with 'LLL' the first 3 letters of the first day of the week),
+'monthly' and 'yearly' which means the forcing files will contain data for one day, one week,
+one month or one year. Files are assumed to contain data from the beginning of the open/close period.
+For example, the first record of a yearly file containing daily data is Jan 1st even if the experiment
+is not starting at the beginning of the year.
+\item[Others]: 'weights filename', 'pairing rotation' and 'land/sea mask' are associted with on-the-fly interpolation
+which is described in \autoref{subsec:SBC_iof}.
+\item[Record frequency]:
+  the frequency of the records contained in the input file.
+  Its unit is in hours if it is positive (for example 24 for daily forcing) or in months if negative
+  (for example -1 for monthly forcing or -12 for annual forcing).
+  Note that this frequency must really be an integer and not a real.
+  On some computers, seting it to '24.' can be interpreted as 240!
+\item[Variable name]:
+  the name of the variable to be read in the input NetCDF file.
+\item[Time interpolation]:
+  a logical to activate, or not, the time interpolation.
+  If set to 'false', the forcing will have a steplike shape remaining constant during each forcing period.
+  For example, when using a daily forcing without time interpolation, the forcing remaining constant from
+h00'00'' to 23h59'59".
+  If set to 'true', the forcing will have a broken line shape.
+  Records are assumed to be dated the middle of the forcing period.
+  For example, when using a daily forcing with time interpolation,
+  linear interpolation will be performed between mid-day of two consecutive days.
+\item[Climatological forcing]:
+  a logical to specify if a input file contains climatological forcing which can be cycle in time,
+  or an interannual forcing which will requires additional files if
+  the period covered by the simulation exceed the one of the file.
+  See the above the file naming strategy which impacts the expected name of the file to be opened.
+\item[Open/close frequency]:
+  the frequency at which forcing files must be opened/closed.
+  Four cases are coded:
+  'daily', 'weekLLL' (with 'LLL' the first 3 letters of the first day of the week), 'monthly' and 'yearly' which
+  means the forcing files will contain data for one day, one week, one month or one year.
+  Files are assumed to contain data from the beginning of the open/close period.
+  For example, the first record of a yearly file containing daily data is Jan 1st even if
+  the experiment is not starting at the beginning of the year.
+\item[Others]:
+  'weights filename', 'pairing rotation' and 'land/sea mask' are associated with
+  on-the-fly interpolation which is described in \autoref{subsec:SBC_iof}.
 \end{description}
 Additional remarks:\\
 (1) The time interpolation is a simple linear interpolation between two consecutive records of
+the input data. The only tricky point is therefore to specify the date at which we need to do
 the interpolation and the date of the records read in the input files.
 Following \citet{Leclair_Madec_OM09}, the date of a time step is set at the middle of the
+time step. For example, for an experiment starting at 0h00'00" with a one hour time-step,
+(1) The time interpolation is a simple linear interpolation between two consecutive records of the input data.
+The only tricky point is therefore to specify the date at which we need to do the interpolation and
+the date of the records read in the input files.
+Following \citet{Leclair_Madec_OM09}, the date of a time step is set at the middle of the time step.
+For example, for an experiment starting at 0h00'00" with a one hour time-step,
 a time interpolation will be performed at the following time: 0h30'00", 1h30'00", 2h30'00", etc.
 However, for forcing data related to the surface module, values are not needed at every
+time-step but at every \np{nn\_fsbc} time-step. For example with \np{nn\_fsbc}\forcode{ = 3},
+the surface module will be called at time-steps 1, 4, 7, etc. The date used for the time interpolation
+is thus redefined to be at the middle of \np{nn\_fsbc} time-step period. In the previous example,
 this leads to: 1h30'00", 4h30'00", 7h30'00", etc. \\
 (2) For code readablility and maintenance issues, we don't take into account the NetCDF input file
+calendar. The calendar associated with the forcing field is build according to the information
+provided by user in the record frequency, the open/close frequency and the type of temporal interpolation.
 For example, the first record of a yearly file containing daily data that will be interpolated in time
 is assumed to be start Jan 1st at 12h00'00" and end Dec 31st at 12h00'00". \\
 (3) If a time interpolation is requested, the code will pick up the needed data in the previous (next) file
+when interpolating data with the first (last) record of the open/close period.
+However, for forcing data related to the surface module,
+values are not needed at every time-step but at every \np{nn\_fsbc} time-step.
+For example with \np{nn\_fsbc}\forcode{ = 3}, the surface module will be called at time-steps 1, 4, 7, etc.
+The date used for the time interpolation is thus redefined to be at the middle of \np{nn\_fsbc} time-step period.
+In the previous example, this leads to: 1h30'00", 4h30'00", 7h30'00", etc. \\
+(2) For code readablility and maintenance issues, we don't take into account the NetCDF input file calendar.
+The calendar associated with the forcing field is build according to the information provided by
+user in the record frequency, the open/close frequency and the type of temporal interpolation.
+For example, the first record of a yearly file containing daily data that will be interpolated in time is assumed to
+be start Jan 1st at 12h00'00" and end Dec 31st at 12h00'00". \\
+(3) If a time interpolation is requested, the code will pick up the needed data in the previous (next) file when
+interpolating data with the first (last) record of the open/close period.
 For example, if the input file specifications are ''yearly, containing daily data to be interpolated in time'',
+the values given by the code between 00h00'00" and 11h59'59" on Jan 1st will be interpolated values
+between Dec 31st 12h00'00" and Jan 1st 12h00'00". If the forcing is climatological, Dec and Jan will
+be keep-up from the same year. However, if the forcing is not climatological, at the end of the
+open/close period the code will automatically close the current file and open the next one.
+Note that, if the experiment is starting (ending) at the beginning (end) of an open/close period
+we do accept that the previous (next) file is not existing. In this case, the time interpolation
+will be performed between two identical values. For example, when starting an experiment on
+Jan 1st of year Y with yearly files and daily data to be interpolated, we do accept that the file
+related to year Y-1 is not existing. The value of Jan 1st will be used as the missing one for
+Dec 31st of year Y-1. If the file of year Y-1 exists, the code will read its last record.
+Therefore, this file can contain only one record corresponding to Dec 31st, a useful feature for
+user considering that it is too heavy to manipulate the complete file for year Y-1.
+the values given by the code between 00h00'00" and 11h59'59" on Jan 1st will be interpolated values between
+Dec 31st 12h00'00" and Jan 1st 12h00'00".
+If the forcing is climatological, Dec and Jan will be keep-up from the same year.
+However, if the forcing is not climatological, at the end of
+the open/close period the code will automatically close the current file and open the next one.
+Note that, if the experiment is starting (ending) at the beginning (end) of
+an open/close period we do accept that the previous (next) file is not existing.
+In this case, the time interpolation will be performed between two identical values.
+For example, when starting an experiment on Jan 1st of year Y with yearly files and daily data to be interpolated,
+we do accept that the file related to year Y-1 is not existing.
+The value of Jan 1st will be used as the missing one for Dec 31st of year Y-1.
+If the file of year Y-1 exists, the code will read its last record.
+Therefore, this file can contain only one record corresponding to Dec 31st,
+a useful feature for user considering that it is too heavy to manipulate the complete file for year Y-1.
 …
 \label{subsec:SBC_iof}
+Interpolation on the Fly allows the user to supply input files required
+for the surface forcing on grids other than the model grid.
+To do this he or she must supply, in addition to the source data file,
+a file of weights to be used to interpolate from the data grid to the model grid.
+The original development of this code used the SCRIP package (freely available
+\href{http://climate.lanl.gov/Software/SCRIP}{here} under a copyright agreement).
+In principle, any package can be used to generate the weights, but the
+variables in the input weights file must have the same names and meanings as
+assumed by the model.
+Interpolation on the Fly allows the user to supply input files required for the surface forcing on
+grids other than the model grid.
+To do this he or she must supply, in addition to the source data file, a file of weights to be used to
+interpolate from the data grid to the model grid.
+The original development of this code used the SCRIP package
+(freely available \href{http://climate.lanl.gov/Software/SCRIP}{here} under a copyright agreement).
+In principle, any package can be used to generate the weights, but the variables in
+the input weights file must have the same names and meanings as assumed by the model.
 Two methods are currently available: bilinear and bicubic interpolation.
+Prior to the interpolation, providing a land/sea mask file, the user can decide to
+ remove land points from the input file and substitute the corresponding values
+with the average of the 8 neighbouring points in the native external grid.
+ Only "sea points" are considered for the averaging. The land/sea mask file must
+be provided in the structure associated with the input variable.
+ The netcdf land/sea mask variable name must be 'LSM' it must have the same
+horizontal and vertical dimensions of the associated variable and should
+be equal to 1 over land and 0 elsewhere.
+The procedure can be recursively applied setting nn\_lsm > 1 in namsbc namelist.
+Prior to the interpolation, providing a land/sea mask file, the user can decide to remove land points from
+the input file and substitute the corresponding values with the average of the 8 neighbouring points in
+the native external grid.
+Only "sea points" are considered for the averaging.
+The land/sea mask file must be provided in the structure associated with the input variable.
+The netcdf land/sea mask variable name must be 'LSM' it must have the same horizontal and vertical dimensions of
+the associated variable and should be equal to 1 over land and 0 elsewhere.
+The procedure can be recursively applied setting nn\_lsm > 1 in namsbc namelist.
 Note that nn\_lsm=0 forces the code to not apply the procedure even if a file for land/sea mask is supplied.
 …
 \label{subsec:SBC_iof_bilinear}
 The input weights file in this case has two sets of variables: src01, src02,
 src03, src04 and wgt01, wgt02, wgt03, wgt04.
 The "src" variables correspond to the point in the input grid to which the weight
+"wgt" is to be applied. Each src value is an integer corresponding to the index of a
+point in the input grid when written as a one dimensional array.  For example, for an input grid
 of size 5x10, point (3,2) is referenced as point 8, since (2-1)*5+3=8.
+The input weights file in this case has two sets of variables:
+src01, src02, src03, src04 and wgt01, wgt02, wgt03, wgt04.
+The "src" variables correspond to the point in the input grid to which the weight "wgt" is to be applied.
+Each src value is an integer corresponding to the index of a point in the input grid when
+written as a one dimensional array.
+For example, for an input grid of size 5x10, point (3,2) is referenced as point 8, since (2-1)*5+3=8.
 There are four of each variable because bilinear interpolation uses the four points defining
 the grid box containing the point to be interpolated.
 All of these arrays are on the model grid, so that values src01(i,j) and
 wgt01(i,j) are used to generate a value for point (i,j) in the model.
+All of these arrays are on the model grid, so that values src01(i,j) and wgt01(i,j) are used to
+generate a value for point (i,j) in the model.
 Symbolically, the algorithm used is:
 \begin{equation}
 f_{m}(i,j) = f_{m}(i,j) + \sum_{k=1}^{4} {wgt(k)f(idx(src(k)))}
 …
 \end{split}
 \end{equation*}
+The gradients here are taken with respect to the horizontal indices and not distances since the spatial dependency has been absorbed into the weights.
+The gradients here are taken with respect to the horizontal indices and not distances since
+the spatial dependency has been absorbed into the weights.
 \subsubsection{Implementation}
 \label{subsec:SBC_iof_imp}
+To activate this option, a non-empty string should be supplied in the weights filename column
+of the relevant namelist; if this is left as an empty string no action is taken.
+In the model, weights files are read in and stored in a structured type (WGT) in the fldread
+module, as and when they are first required.
+This initialisation procedure determines whether the input data grid should be treated
+as cyclical or not by inspecting a global attribute stored in the weights input file.
+To activate this option, a non-empty string should be supplied in
+the weights filename column of the relevant namelist;
+if this is left as an empty string no action is taken.
+In the model, weights files are read in and stored in a structured type (WGT) in the fldread module,
+as and when they are first required.
+This initialisation procedure determines whether the input data grid should be treated as cyclical or not by
+inspecting a global attribute stored in the weights input file.
 This attribute must be called "ew\_wrap" and be of integer type.
 If it is negative, the input non-model grid is assumed not to be cyclic.
 …
 if longitudes are 0.5, 2.5, .... , 358.5, 360.5, 362.5, ew\_wrap should be 2.
 If the model does not find attribute ew\_wrap, then a value of -999 is assumed.
 In this case the \rou{fld\_read} routine defaults ew\_wrap to value 0 and therefore the grid
 is assumed to be cyclic with no overlapping columns.
+In this case the \rou{fld\_read} routine defaults ew\_wrap to value 0 and
+therefore the grid is assumed to be cyclic with no overlapping columns.
 (In fact this only matters when bicubic interpolation is required.)
 Note that no testing is done to check the validity in the model, since there is no way
 of knowing the name used for the longitude variable,
+Note that no testing is done to check the validity in the model,
+since there is no way of knowing the name used for the longitude variable,
 so it is up to the user to make sure his or her data is correctly represented.
 Next the routine reads in the weights.
+Bicubic interpolation is assumed if it finds a variable with name "src05", otherwise
+bilinear interpolation is used. The WGT structure includes dynamic arrays both for
+the storage of the weights (on the model grid), and when required, for reading in
+the variable to be interpolated (on the input data grid).
+The size of the input data array is determined by examining the values in the "src"
+arrays to find the minimum and maximum i and j values required.
+Since bicubic interpolation requires the calculation of gradients at each point on the grid,
+Bicubic interpolation is assumed if it finds a variable with name "src05", otherwise bilinear interpolation is used.
+The WGT structure includes dynamic arrays both for the storage of the weights (on the model grid),
+and when required, for reading in the variable to be interpolated (on the input data grid).
+The size of the input data array is determined by examining the values in the "src" arrays to
+find the minimum and maximum i and j values required.
+Since bicubic interpolation requires the calculation of gradients at each point on the grid,
 the corresponding arrays are dimensioned with a halo of width one grid point all the way around.
 When the array of points from the data file is adjacent to an edge of the data grid,
 the halo is either a copy of the row/column next to it (non-cyclical case), or is a copy
 of one from the first few columns on the opposite side of the grid (cyclical case).
+When the array of points from the data file is adjacent to an edge of the data grid,
+the halo is either a copy of the row/column next to it (non-cyclical case),
+or is a copy of one from the first few columns on the opposite side of the grid (cyclical case).
 \subsubsection{Limitations}
 …
 \begin{enumerate}
+\item  The case where input data grids are not logically rectangular has not been tested.
+\item  This code is not guaranteed to produce positive definite answers from positive definite inputs
+          when a bicubic interpolation method is used.
+\item  The cyclic condition is only applied on left and right columns, and not to top and bottom rows.
+\item  The gradients across the ends of a cyclical grid assume that the grid spacing between
+          the two columns involved are consistent with the weights used.
+\item  Neither interpolation scheme is conservative. (There is a conservative scheme available
+          in SCRIP, but this has not been implemented.)
+\item
+  The case where input data grids are not logically rectangular has not been tested.
+\item
+  This code is not guaranteed to produce positive definite answers from positive definite inputs when
+  a bicubic interpolation method is used.
+\item
+  The cyclic condition is only applied on left and right columns, and not to top and bottom rows.
+\item
+  The gradients across the ends of a cyclical grid assume that the grid spacing between
+  the two columns involved are consistent with the weights used.
+\item
+  Neither interpolation scheme is conservative. (There is a conservative scheme available in SCRIP,
+  but this has not been implemented.)
 \end{enumerate}
 …
 % to be completed
 A set of utilities to create a weights file for a rectilinear input grid is available
+A set of utilities to create a weights file for a rectilinear input grid is available
 (see the directory NEMOGCM/TOOLS/WEIGHTS).
 …
 %--------------------------------------------------------------------------------------------------------------
 In some circumstances it may be useful to avoid calculating the 3D temperature, salinity and velocity fields
 and simply read them in from a previous run or receive them from OASIS.
+In some circumstances it may be useful to avoid calculating the 3D temperature,
+salinity and velocity fields and simply read them in from a previous run or receive them from OASIS.
 For example:
 \begin{itemize}
+\item  Multiple runs of the model are required in code development to see the effect of different algorithms in
+       the bulk formulae.
+\item  The effect of different parameter sets in the ice model is to be examined.
+\item  Development of sea-ice algorithms or parameterizations.
+\item  spinup of the iceberg floats
+\item  ocean/sea-ice simulation with both media running in parallel (\np{ln\_mixcpl}\forcode{ = .true.})
+\item
+  Multiple runs of the model are required in code development to
+  see the effect of different algorithms in the bulk formulae.
+\item
+  The effect of different parameter sets in the ice model is to be examined.
+\item
+  Development of sea-ice algorithms or parameterizations.
+\item
+  Spinup of the iceberg floats
+\item
+  Ocean/sea-ice simulation with both media running in parallel (\np{ln\_mixcpl}\forcode{ = .true.})
 \end{itemize}
 …
 Its options are defined through the \ngn{namsbc\_sas} namelist variables.
 A new copy of the model has to be compiled with a configuration based on ORCA2\_SAS\_LIM.
 However no namelist parameters need be changed from the settings of the previous run (except perhaps nn{\_}date0)
+However no namelist parameters need be changed from the settings of the previous run (except perhaps nn{\_}date0).
 In this configuration, a few routines in the standard model are overriden by new versions.
 Routines replaced are:
 \begin{itemize}
+\item \mdl{nemogcm} : This routine initialises the rest of the model and repeatedly calls the stp time stepping routine (\mdl{step})
+       Since the ocean state is not calculated all associated initialisations have been removed.
+\item  \mdl{step} : The main time stepping routine now only needs to call the sbc routine (and a few utility functions).
+\item  \mdl{sbcmod} : This has been cut down and now only calculates surface forcing and the ice model required.  New surface modules
+       that can function when only the surface level of the ocean state is defined can also be added (e.g. icebergs).
+\item  \mdl{daymod} : No ocean restarts are read or written (though the ice model restarts are retained), so calls to restart functions
+       have been removed.  This also means that the calendar cannot be controlled by time in a restart file, so the user
+       must make sure that nn{\_}date0 in the model namelist is correct for his or her purposes.
+\item  \mdl{stpctl} : Since there is no free surface solver, references to it have been removed from \rou{stp\_ctl} module.
+\item  \mdl{diawri} : All 3D data have been removed from the output.  The surface temperature, salinity and velocity components (which
+       have been read in) are written along with relevant forcing and ice data.
+\item
+  \mdl{nemogcm}:
+  This routine initialises the rest of the model and repeatedly calls the stp time stepping routine (\mdl{step}).
+  Since the ocean state is not calculated all associated initialisations have been removed.
+\item
+  \mdl{step}:
+  The main time stepping routine now only needs to call the sbc routine (and a few utility functions).
+\item
+  \mdl{sbcmod}:
+  This has been cut down and now only calculates surface forcing and the ice model required.
+  New surface modules that can function when only the surface level of the ocean state is defined can also be added
+  (e.g. icebergs).
+\item
+  \mdl{daymod}:
+  No ocean restarts are read or written (though the ice model restarts are retained),
+  so calls to restart functions have been removed.
+  This also means that the calendar cannot be controlled by time in a restart file,
+  so the user must make sure that nn{\_}date0 in the model namelist is correct for his or her purposes.
+\item
+  \mdl{stpctl}:
+  Since there is no free surface solver, references to it have been removed from \rou{stp\_ctl} module.
+\item
+  \mdl{diawri}:
+  All 3D data have been removed from the output.
+  The surface temperature, salinity and velocity components (which have been read in) are written along with
+  relevant forcing and ice data.
 \end{itemize}
 …
 \begin{itemize}
+\item  \mdl{sbcsas} : This module initialises the input files needed for reading temperature, salinity and velocity arrays at the surface.
+       These filenames are supplied in namelist namsbc{\_}sas.  Unfortunately because of limitations with the \mdl{iom} module,
+       the full 3D fields from the mean files have to be read in and interpolated in time, before using just the top level.
+       Since fldread is used to read in the data, Interpolation on the Fly may be used to change input data resolution.
+\item
+  \mdl{sbcsas}:
+  This module initialises the input files needed for reading temperature, salinity and
+  velocity arrays at the surface.
+  These filenames are supplied in namelist namsbc{\_}sas.
+  Unfortunately because of limitations with the \mdl{iom} module,
+  the full 3D fields from the mean files have to be read in and interpolated in time,
+  before using just the top level.
+  Since fldread is used to read in the data, Interpolation on the Fly may be used to change input data resolution.
 \end{itemize}
 …
 The analytical formulation of the surface boundary condition is the default scheme.
 In this case, all the six fluxes needed by the ocean are assumed to
+be uniform in space. They take constant values given in the namelist
+\ngn{namsbc{\_}ana} by the variables \np{rn\_utau0}, \np{rn\_vtau0}, \np{rn\_qns0},
+\np{rn\_qsr0}, and \np{rn\_emp0} ($\textit{emp}=\textit{emp}_S$). The runoff is set to zero.
+In addition, the wind is allowed to reach its nominal value within a given number
 of time steps (\np{nn\_tau000}).
 If a user wants to apply a different analytical forcing, the \mdl{sbcana}
+module can be modified to use another scheme. As an example,
+the \mdl{sbc\_ana\_gyre} routine provides the analytical forcing for the
 GYRE configuration (see GYRE configuration manual, in preparation).
+In this case, all the six fluxes needed by the ocean are assumed to be uniform in space.
+They take constant values given in the namelist \ngn{namsbc{\_}ana} by
+the variables \np{rn\_utau0}, \np{rn\_vtau0}, \np{rn\_qns0}, \np{rn\_qsr0}, and \np{rn\_emp0}
+($\textit{emp}=\textit{emp}_S$).
+The runoff is set to zero.
+In addition, the wind is allowed to reach its nominal value within a given number of time steps (\np{nn\_tau000}).
+If a user wants to apply a different analytical forcing,
+the \mdl{sbcana} module can be modified to use another scheme.
+As an example, the \mdl{sbc\_ana\_gyre} routine provides the analytical forcing for the GYRE configuration
+(see GYRE configuration manual, in preparation).
 …
 %-------------------------------------------------------------------------------------------------------------
+In the flux formulation (\np{ln\_flx}\forcode{ = .true.}), the surface boundary
+condition fields are directly read from input files. The user has to define
+in the namelist \ngn{namsbc{\_}flx} the name of the file, the name of the variable
+read in the file, the time frequency at which it is given (in hours), and a logical
+setting whether a time interpolation to the model time step is required
+for this field. See \autoref{subsec:SBC_fldread} for a more detailed description of the parameters.
+Note that in general, a flux formulation is used in associated with a
+restoring term to observed SST and/or SSS. See \autoref{subsec:SBC_ssr} for its
+specification.
+In the flux formulation (\np{ln\_flx}\forcode{ = .true.}),
+the surface boundary condition fields are directly read from input files.
+The user has to define in the namelist \ngn{namsbc{\_}flx} the name of the file,
+the name of the variable read in the file, the time frequency at which it is given (in hours),
+and a logical setting whether a time interpolation to the model time step is required for this field.
+See \autoref{subsec:SBC_fldread} for a more detailed description of the parameters.
+Note that in general, a flux formulation is used in associated with a restoring term to observed SST and/or SSS.
+See \autoref{subsec:SBC_ssr} for its specification.
 …
 \label{sec:SBC_blk}
+In the bulk formulation, the surface boundary condition fields are computed
+using bulk formulae and atmospheric fields and ocean (and ice) variables.
+The atmospheric fields used depend on the bulk formulae used. Three bulk formulations
+are available : the CORE, the CLIO and the MFS bulk formulea. The choice is made by setting to true
+one of the following namelist variable : \np{ln\_core} ; \np{ln\_clio} or  \np{ln\_mfs}.
+Note : in forced mode, when a sea-ice model is used, a bulk formulation (CLIO or CORE) have to be used.
+Therefore the two bulk (CLIO and CORE) formulea include the computation of the fluxes over both
+an ocean and an ice surface.
+In the bulk formulation, the surface boundary condition fields are computed using bulk formulae and atmospheric fields and ocean (and ice) variables.
+The atmospheric fields used depend on the bulk formulae used.
+Three bulk formulations are available:
+the CORE, the CLIO and the MFS bulk formulea.
+The choice is made by setting to true one of the following namelist variable:
+\np{ln\_core} ; \np{ln\_clio} or  \np{ln\_mfs}.
+Note:
+in forced mode, when a sea-ice model is used, a bulk formulation (CLIO or CORE) have to be used.
+Therefore the two bulk (CLIO and CORE) formulea include the computation of the fluxes over
+both an ocean and an ice surface.
 % -------------------------------------------------------------------------------------------------------------
 …
 %-------------------------------------------------------------------------------------------------------------
+The CORE bulk formulae have been developed by \citet{Large_Yeager_Rep04}.
+They have been designed to handle the CORE forcing, a mixture of NCEP
+reanalysis and satellite data. They use an inertial dissipative method to compute
+the turbulent transfer coefficients (momentum, sensible heat and evaporation)
+from the 10 metre wind speed, air temperature and specific humidity.
+This \citet{Large_Yeager_Rep04} dataset is available through the
+\href{http://nomads.gfdl.noaa.gov/nomads/forms/mom4/CORE.html}{GFDL web site}.
+Note that substituting ERA40 to NCEP reanalysis fields
+does not require changes in the bulk formulea themself.
+This is the so-called DRAKKAR Forcing Set (DFS) \citep{Brodeau_al_OM09}.
+The CORE bulk formulae have been developed by \citet{Large_Yeager_Rep04}.
+They have been designed to handle the CORE forcing, a mixture of NCEP reanalysis and satellite data.
+They use an inertial dissipative method to compute the turbulent transfer coefficients
+(momentum, sensible heat and evaporation) from the 10 metre wind speed, air temperature and specific humidity.
+This \citet{Large_Yeager_Rep04} dataset is available through
+the \href{http://nomads.gfdl.noaa.gov/nomads/forms/mom4/CORE.html}{GFDL web site}.
+Note that substituting ERA40 to NCEP reanalysis fields does not require changes in the bulk formulea themself.
+This is the so-called DRAKKAR Forcing Set (DFS) \citep{Brodeau_al_OM09}.
 Options are defined through the  \ngn{namsbc\_core} namelist variables.
 …
 %--------------------------------------------------------------------------------------------------------------
+Note that the air velocity is provided at a tracer ocean point, not at a velocity ocean
+point ($u$- and $v$-points). It is simpler and faster (less fields to be read),
+but it is not the recommended method when the ocean grid size is the same
+or larger than the one of the input atmospheric fields.
+The \np{sn\_wndi}, \np{sn\_wndj}, \np{sn\_qsr}, \np{sn\_qlw}, \np{sn\_tair}, \np{sn\_humi},
+\np{sn\_prec}, \np{sn\_snow}, \np{sn\_tdif} parameters describe the fields
+and the way they have to be used (spatial and temporal interpolations).
+Note that the air velocity is provided at a tracer ocean point, not at a velocity ocean point ($u$- and $v$-points).
+It is simpler and faster (less fields to be read), but it is not the recommended method when
+the ocean grid size is the same or larger than the one of the input atmospheric fields.
+The \np{sn\_wndi}, \np{sn\_wndj}, \np{sn\_qsr}, \np{sn\_qlw}, \np{sn\_tair}, \np{sn\_humi}, \np{sn\_prec},
+\np{sn\_snow}, \np{sn\_tdif} parameters describe the fields and the way they have to be used
+(spatial and temporal interpolations).
 \np{cn\_dir} is the directory of location of bulk files
 …
 \np{rn\_zu}: is the height of wind measurements (m)
 Three multiplicative factors are availables :
 \np{rn\_pfac} and \np{rn\_efac} allows to adjust (if necessary) the global freshwater budget
 by increasing/reducing the precipitations (total and snow) and or evaporation, respectively.
 The third one,\np{rn\_vfac}, control to which extend the ice/ocean velocities are taken into account
+in the calculation of surface wind stress. Its range should be between zero and one,
 and it is recommended to set it to 0.
+Three multiplicative factors are availables:
+\np{rn\_pfac} and \np{rn\_efac} allows to adjust (if necessary) the global freshwater budget by
+increasing/reducing the precipitations (total and snow) and or evaporation, respectively.
+The third one,\np{rn\_vfac}, control to which extend the ice/ocean velocities are taken into account in
+the calculation of surface wind stress.
+Its range should be between zero and one, and it is recommended to set it to 0.
 % -------------------------------------------------------------------------------------------------------------
 …
 %-------------------------------------------------------------------------------------------------------------
 The CLIO bulk formulae were developed several years ago for the
 Louvain-la-neuve coupled ice-ocean model (CLIO, \cite{Goosse_al_JGR99}).
 They are simpler bulk formulae. They assume the stress to be known and
 compute the radiative fluxes from a climatological cloud cover.
+The CLIO bulk formulae were developed several years ago for the Louvain-la-neuve coupled ice-ocean model
+(CLIO, \cite{Goosse_al_JGR99}).
+They are simpler bulk formulae.
+They assume the stress to be known and compute the radiative fluxes from a climatological cloud cover.
 Options are defined through the  \ngn{namsbc\_clio} namelist variables.
 …
 %--------------------------------------------------------------------------------------------------------------
+As for the flux formulation, information about the input data required by the
+model is provided in the namsbc\_blk\_core or namsbc\_blk\_clio
+namelist (see \autoref{subsec:SBC_fldread}).
+As for the flux formulation, information about the input data required by the model is provided in
+the namsbc\_blk\_core or namsbc\_blk\_clio namelist (see \autoref{subsec:SBC_fldread}).
 % -------------------------------------------------------------------------------------------------------------
 …
 %----------------------------------------------------------------------------------------------------------
+The MFS (Mediterranean Forecasting System) bulk formulae have been developed by
+ \citet{Castellari_al_JMS1998}.
+They have been designed to handle the ECMWF operational data and are currently
+in use in the MFS operational system \citep{Tonani_al_OS08}, \citep{Oddo_al_OS09}.
+The MFS (Mediterranean Forecasting System) bulk formulae have been developed by \citet{Castellari_al_JMS1998}.
+They have been designed to handle the ECMWF operational data and are currently in use in
+the MFS operational system \citep{Tonani_al_OS08}, \citep{Oddo_al_OS09}.
 The wind stress computation uses a drag coefficient computed according to \citet{Hellerman_Rosenstein_JPO83}.
+The surface boundary condition for temperature involves the balance between surface solar radiation,
+net long-wave radiation, the latent and sensible heat fluxes.
+Solar radiation is dependent on cloud cover and is computed by means of
+an astronomical formula \citep{Reed_JPO77}. Albedo monthly values are from \citet{Payne_JAS72}
+as means of the values at $40^{o}N$ and $30^{o}N$ for the Atlantic Ocean (hence the same latitudinal
+band of the Mediterranean Sea). The net long-wave radiation flux
+\citep{Bignami_al_JGR95} is a function of
+The surface boundary condition for temperature involves the balance between
+surface solar radiation, net long-wave radiation, the latent and sensible heat fluxes.
+Solar radiation is dependent on cloud cover and is computed by means of an astronomical formula \citep{Reed_JPO77}.
+Albedo monthly values are from \citet{Payne_JAS72} as means of the values at $40^{o}N$ and $30^{o}N$ for
+the Atlantic Ocean (hence the same latitudinal band of the Mediterranean Sea).
+The net long-wave radiation flux \citep{Bignami_al_JGR95} is a function of
 air temperature, sea-surface temperature, cloud cover and relative humidity.
 Sensible heat and latent heat fluxes are computed by classical
 bulk formulae parameterised according to \citet{Kondo1975}.
+Sensible heat and latent heat fluxes are computed by classical bulk formulae parameterised according to
+\citet{Kondo1975}.
 Details on the bulk formulae used can be found in \citet{Maggiore_al_PCE98} and \citet{Castellari_al_JMS1998}.
 Options are defined through the  \ngn{namsbc\_mfs} namelist variables.
 The required 7 input fields must be provided on the model Grid-T and  are:
+Options are defined through the \ngn{namsbc\_mfs} namelist variables.
+The required 7 input fields must be provided on the model Grid-T and are:
 \begin{itemize}
 \item          Zonal Component of the 10m wind ($ms^{-1}$)  (\np{sn\_windi})
 …
 %-------------------------------------------------------------------------------------------------------------
 In the coupled formulation of the surface boundary condition, the fluxes are
+provided by the OASIS coupler at a frequency which is defined in the OASIS coupler,
 while sea and ice surface temperature, ocean and ice albedo, and ocean currents
 are sent to the atmospheric component.
 A generalised coupled interface has been developed.
 It is currently interfaced with OASIS-3-MCT (\key{oasis3}).
 It has been successfully used to interface \NEMO to most of the European atmospheric
+GCM (ARPEGE, ECHAM, ECMWF, HadAM, HadGAM, LMDz),
 as well as to \href{http://wrf-model.org/}{WRF} (Weather Research and Forecasting Model).
 Note that in addition to the setting of \np{ln\_cpl} to true, the \key{coupled} have to be defined.
 The CPP key is mainly used in sea-ice to ensure that the atmospheric fluxes are
 actually recieved by the ice-ocean system (no calculation of ice sublimation in coupled mode).
+In the coupled formulation of the surface boundary condition,
+the fluxes are provided by the OASIS coupler at a frequency which is defined in the OASIS coupler,
+while sea and ice surface temperature, ocean and ice albedo, and ocean currents are sent to
+the atmospheric component.
+A generalised coupled interface has been developed.
+It is currently interfaced with OASIS-3-MCT (\key{oasis3}).
+It has been successfully used to interface \NEMO to most of the European atmospheric GCM
+(ARPEGE, ECHAM, ECMWF, HadAM, HadGAM, LMDz), as well as to \href{http://wrf-model.org/}{WRF}
+(Weather Research and Forecasting Model).
+Note that in addition to the setting of \np{ln\_cpl} to true, the \key{coupled} have to be defined.
+The CPP key is mainly used in sea-ice to ensure that the atmospheric fluxes are actually received by
+the ice-ocean system (no calculation of ice sublimation in coupled mode).
 When PISCES biogeochemical model (\key{top} and \key{pisces}) is also used in the coupled system,
+the whole carbon cycle is computed by defining \key{cpl\_carbon\_cycle}. In this case,
+CO$_2$ fluxes will be exchanged between the atmosphere and the ice-ocean system (and need to be activated in \ngn{namsbc{\_}cpl} ).
+The namelist above allows control of various aspects of the coupling fields (particularly for
+vectors) and now allows for any coupling fields to have multiple sea ice categories (as required by LIM3
+and CICE).  When indicating a multi-category coupling field in namsbc{\_}cpl the number of categories will be
+determined by the number used in the sea ice model.  In some limited cases it may be possible to specify
+single category coupling fields even when the sea ice model is running with multiple categories - in this
+case the user should examine the code to be sure the assumptions made are satisfactory.  In cases where
+this is definitely not possible the model should abort with an error message.  The new code has been tested using
+ECHAM with LIM2, and HadGAM3 with CICE but although it will compile with \key{lim3} additional minor code changes
+may be required to run using LIM3.
+the whole carbon cycle is computed by defining \key{cpl\_carbon\_cycle}.
+In this case, CO$_2$ fluxes will be exchanged between the atmosphere and the ice-ocean system
+(and need to be activated in \ngn{namsbc{\_}cpl} ).
+The namelist above allows control of various aspects of the coupling fields (particularly for vectors) and
+now allows for any coupling fields to have multiple sea ice categories (as required by LIM3 and CICE).
+When indicating a multi-category coupling field in namsbc{\_}cpl the number of categories will be determined by
+the number used in the sea ice model.
+In some limited cases it may be possible to specify single category coupling fields even when
+the sea ice model is running with multiple categories -
+in this case the user should examine the code to be sure the assumptions made are satisfactory.
+In cases where this is definitely not possible the model should abort with an error message.
+The new code has been tested using ECHAM with LIM2, and HadGAM3 with CICE but
+although it will compile with \key{lim3} additional minor code changes may be required to run using LIM3.
 …
 %-------------------------------------------------------------------------------------------------------------
+The optional atmospheric pressure can be used to force ocean and ice dynamics
+(\np{ln\_apr\_dyn}\forcode{ = .true.}, \textit{\ngn{namsbc}} namelist ).
+The input atmospheric forcing defined via \np{sn\_apr} structure (\textit{namsbc\_apr} namelist)
+can be interpolated in time to the model time step, and even in space when the
+interpolation on-the-fly is used. When used to force the dynamics, the atmospheric
+pressure is further transformed into an equivalent inverse barometer sea surface height,
+$\eta_{ib}$, using:
+The optional atmospheric pressure can be used to force ocean and ice dynamics
+(\np{ln\_apr\_dyn}\forcode{ = .true.}, \textit{\ngn{namsbc}} namelist).
+The input atmospheric forcing defined via \np{sn\_apr} structure (\textit{namsbc\_apr} namelist)
+can be interpolated in time to the model time step, and even in space when the interpolation on-the-fly is used.
+When used to force the dynamics, the atmospheric pressure is further transformed into
+an equivalent inverse barometer sea surface height, $\eta_{ib}$, using:
 \begin{equation} \label{eq:SBC_ssh_ib}
    \eta_{ib} = -  \frac{1}{g\,\rho_o}  \left( P_{atm} - P_o \right)
 \end{equation}
 where $P_{atm}$ is the atmospheric pressure and $P_o$ a reference atmospheric pressure.
+A value of $101,000~N/m^2$ is used unless \np{ln\_ref\_apr} is set to true. In this case $P_o$
+is set to the value of $P_{atm}$ averaged over the ocean domain, $i.e.$ the mean value of
+$\eta_{ib}$ is kept to zero at all time step.
+The gradient of $\eta_{ib}$ is added to the RHS of the ocean momentum equation
+(see \mdl{dynspg} for the ocean). For sea-ice, the sea surface height, $\eta_m$,
+which is provided to the sea ice model is set to $\eta - \eta_{ib}$ (see \mdl{sbcssr} module).
+$\eta_{ib}$ can be set in the output. This can simplify altimetry data and model comparison
+as inverse barometer sea surface height is usually removed from these date prior to their distribution.
+When using time-splitting and BDY package for open boundaries conditions, the equivalent
+inverse barometer sea surface height $\eta_{ib}$ can be added to BDY ssh data:
+A value of $101,000~N/m^2$ is used unless \np{ln\_ref\_apr} is set to true.
+In this case $P_o$ is set to the value of $P_{atm}$ averaged over the ocean domain,
+$i.e.$ the mean value of $\eta_{ib}$ is kept to zero at all time step.
+The gradient of $\eta_{ib}$ is added to the RHS of the ocean momentum equation (see \mdl{dynspg} for the ocean).
+For sea-ice, the sea surface height, $\eta_m$, which is provided to the sea ice model is set to $\eta - \eta_{ib}$
+(see \mdl{sbcssr} module).
+$\eta_{ib}$ can be set in the output.
+This can simplify altimetry data and model comparison as
+inverse barometer sea surface height is usually removed from these date prior to their distribution.
+When using time-splitting and BDY package for open boundaries conditions,
+the equivalent inverse barometer sea surface height $\eta_{ib}$ can be added to BDY ssh data:
 \np{ln\_apr\_obc}  might be set to true.
 …
 %-----------------------------------------------------------------------------------------
+The tidal forcing, generated by the gravity forces of the Earth-Moon and Earth-Sun sytems, is activated if \np{ln\_tide} and \np{ln\_tide\_pot} are both set to \np{.true.} in \ngn{nam\_tide}. This translates as an additional barotropic force in the momentum equations \ref{eq:PE_dyn} such that:
+The tidal forcing, generated by the gravity forces of the Earth-Moon and Earth-Sun sytems,
+is activated if \np{ln\_tide} and \np{ln\_tide\_pot} are both set to \np{.true.} in \ngn{nam\_tide}.
+This translates as an additional barotropic force in the momentum equations \ref{eq:PE_dyn} such that:
 \begin{equation}     \label{eq:PE_dyn_tides}
 \frac{\partial {\rm {\bf U}}_h }{\partial t}= ...
 …
 where $\Pi_{eq}$ stands for the equilibrium tidal forcing and $\Pi_{sal}$ a self-attraction and loading term (SAL).
+The equilibrium tidal forcing is expressed as a sum over the chosen constituents $l$ in \ngn{nam\_tide}. The constituents are defined such that \np{clname(1) = 'M2', clname(2)='S2', etc...}. For the three types of tidal frequencies it reads : \\
+The equilibrium tidal forcing is expressed as a sum over the chosen constituents $l$ in \ngn{nam\_tide}.
+The constituents are defined such that \np{clname(1) = 'M2', clname(2)='S2', etc...}.
+For the three types of tidal frequencies it reads: \\
 Long period tides :
 \begin{equation}
 …
 \Pi_{eq}(l)=A_{l}(1+k-h)(cos^{2}\phi)cos(\omega_{l}t+2\lambda+V_{l})
 \end{equation}
+Here $A_{l}$ is the amplitude, $\omega_{l}$ is the frequency, $\phi$ the latitude, $\lambda$ the longitude, $V_{0l}$ a phase shift with respect to Greenwich meridian and $t$ the time. The Love number factor $(1+k-h)$ is here taken as a constant (0.7).
+The SAL term should in principle be computed online as it depends on the model tidal prediction itself (see \citet{Arbic2004} for a discussion about the practical implementation of this term). Nevertheless, the complex calculations involved would make this computationally too expensive. Here, practical solutions are whether to read complex estimates $\Pi_{sal}(l)$ from an external model (\np{ln\_read\_load=.true.}) or use a ``scalar approximation'' (\np{ln\_scal\_load=.true.}). In the latter case, it reads:\\
+Here $A_{l}$ is the amplitude, $\omega_{l}$ is the frequency, $\phi$ the latitude, $\lambda$ the longitude,
+$V_{0l}$ a phase shift with respect to Greenwich meridian and $t$ the time.
+The Love number factor $(1+k-h)$ is here taken as a constant (0.7).
+The SAL term should in principle be computed online as it depends on the model tidal prediction itself
+(see \citet{Arbic2004} for a discussion about the practical implementation of this term).
+Nevertheless, the complex calculations involved would make this computationally too expensive.
+Here, practical solutions are whether to read complex estimates $\Pi_{sal}(l)$ from an external model
+(\np{ln\_read\_load=.true.}) or use a ``scalar approximation'' (\np{ln\_scal\_load=.true.}).
+In the latter case, it reads:\\
 \begin{equation}
 \Pi_{sal} = \beta \eta
 \end{equation}
+where $\beta$ (\np{rn\_scal\_load}, $\approx0.09$) is a spatially constant scalar, often chosen to minimize tidal prediction errors. Setting both \np{ln\_read\_load} and \np{ln\_scal\_load} to false removes the SAL contribution.
+where $\beta$ (\np{rn\_scal\_load}, $\approx0.09$) is a spatially constant scalar,
+often chosen to minimize tidal prediction errors.
+Setting both \np{ln\_read\_load} and \np{ln\_scal\_load} to false removes the SAL contribution.
 % ================================================================
 …
 River runoff generally enters the ocean at a nonzero depth rather than through the surface.
 Many models, however, have traditionally inserted river runoff to the top model cell.
+This was the case in \NEMO prior to the version 3.3, and was combined with an option
+to increase vertical mixing near the river mouth.
+However, with this method numerical and physical problems arise when the top grid cells are
+of the order of one meter. This situation is common in coastal modelling and is becoming
+more common in open ocean and climate modelling
+\footnote{At least a top cells thickness of 1~meter and a 3 hours forcing frequency are
+required to properly represent the diurnal cycle \citep{Bernie_al_JC05}. see also \autoref{fig:SBC_dcy}.}.
+As such from V~3.3 onwards it is possible to add river runoff through a non-zero depth, and for the
+temperature and salinity of the river to effect the surrounding ocean.
+The user is able to specify, in a NetCDF input file, the temperature and salinity of the river, along with the
+depth (in metres) which the river should be added to.
+Namelist variables in \ngn{namsbc\_rnf}, \np{ln\_rnf\_depth}, \np{ln\_rnf\_sal} and \np{ln\_rnf\_temp} control whether
+the river attributes (depth, salinity and temperature) are read in and used.  If these are set
+as false the river is added to the surface box only, assumed to be fresh (0~psu), and/or
+taken as surface temperature respectively.
+This was the case in \NEMO prior to the version 3.3,
+and was combined with an option to increase vertical mixing near the river mouth.
+However, with this method numerical and physical problems arise when the top grid cells are of the order of one meter.
+This situation is common in coastal modelling and is becoming more common in open ocean and climate modelling
+\footnote{
+  At least a top cells thickness of 1~meter and a 3 hours forcing frequency are required to
+  properly represent the diurnal cycle \citep{Bernie_al_JC05}.
+  see also \autoref{fig:SBC_dcy}.}.
+As such from V~3.3 onwards it is possible to add river runoff through a non-zero depth,
+and for the temperature and salinity of the river to effect the surrounding ocean.
+The user is able to specify, in a NetCDF input file, the temperature and salinity of the river,
+along with the depth (in metres) which the river should be added to.
+Namelist variables in \ngn{namsbc\_rnf}, \np{ln\_rnf\_depth}, \np{ln\_rnf\_sal} and
+\np{ln\_rnf\_temp} control whether the river attributes (depth, salinity and temperature) are read in and used.
+If these are set as false the river is added to the surface box only, assumed to be fresh (0~psu),
+and/or taken as surface temperature respectively.
 The runoff value and attributes are read in in sbcrnf.
 For temperature -999 is taken as missing data and the river temperature is taken to be the
 surface temperatue at the river point.
+For temperature -999 is taken as missing data and the river temperature is taken to
+be the surface temperatue at the river point.
 For the depth parameter a value of -1 means the river is added to the surface box only,
 and a value of -999 means the river is added through the entire water column.
+After being read in the temperature and salinity variables are multiplied by the amount of runoff (converted into m/s)
+to give the heat and salt content of the river runoff.
+After the user specified depth is read ini, the number of grid boxes this corresponds to is
+calculated and stored in the variable \np{nz\_rnf}.
+The variable \textit{h\_dep} is then calculated to be the depth (in metres) of the bottom of the
+lowest box the river water is being added to (i.e. the total depth that river water is being added to in the model).
+The mass/volume addition due to the river runoff is, at each relevant depth level, added to the horizontal divergence
+(\textit{hdivn}) in the subroutine \rou{sbc\_rnf\_div} (called from \mdl{divcur}).
+After being read in the temperature and salinity variables are multiplied by the amount of runoff
+(converted into m/s) to give the heat and salt content of the river runoff.
+After the user specified depth is read ini,
+the number of grid boxes this corresponds to is calculated and stored in the variable \np{nz\_rnf}.
+The variable \textit{h\_dep} is then calculated to be the depth (in metres) of
+the bottom of the lowest box the river water is being added to
+(i.e. the total depth that river water is being added to in the model).
+The mass/volume addition due to the river runoff is, at each relevant depth level, added to
+the horizontal divergence (\textit{hdivn}) in the subroutine \rou{sbc\_rnf\_div} (called from \mdl{divcur}).
 This increases the diffusion term in the vicinity of the river, thereby simulating a momentum flux.
 The sea surface height is calculated using the sum of the horizontal divergence terms, and so the
 river runoff indirectly forces an increase in sea surface height.
+The sea surface height is calculated using the sum of the horizontal divergence terms,
+and so the river runoff indirectly forces an increase in sea surface height.
 The \textit{hdivn} terms are used in the tracer advection modules to force vertical velocities.
+This causes a mass of water, equal to the amount of runoff, to be moved into the box above.
+The heat and salt content of the river runoff is not included in this step, and so the tracer
+concentrations are diluted as water of ocean temperature and salinity is moved upward out of the box
+and replaced by the same volume of river water with no corresponding heat and salt addition.
+For the linear free surface case, at the surface box the tracer advection causes a flux of water
+(of equal volume to the runoff) through the sea surface out of the domain, which causes a salt and heat flux out of the model.
+This causes a mass of water, equal to the amount of runoff, to be moved into the box above.
+The heat and salt content of the river runoff is not included in this step,
+and so the tracer concentrations are diluted as water of ocean temperature and salinity is moved upward out of
+the box and replaced by the same volume of river water with no corresponding heat and salt addition.
+For the linear free surface case, at the surface box the tracer advection causes a flux of water
+(of equal volume to the runoff) through the sea surface out of the domain,
+which causes a salt and heat flux out of the model.
 As such the volume of water does not change, but the water is diluted.
 For the non-linear free surface case (\key{vvl}), no flux is allowed through the surface.
 Instead in the surface box (as well as water moving up from the boxes below) a volume of runoff water
 is added with no corresponding heat and salt addition and so as happens in the lower boxes there is a dilution effect.
 (The runoff addition to the top box along with the water being moved up through boxes below means the surface box has a large
 increase in volume, whilst all other boxes remain the same size)
+Instead in the surface box (as well as water moving up from the boxes below) a volume of runoff water is added with
+no corresponding heat and salt addition and so as happens in the lower boxes there is a dilution effect.
+(The runoff addition to the top box along with the water being moved up through
+boxes below means the surface box has a large increase in volume, whilst all other boxes remain the same size)
 In trasbc the addition of heat and salt due to the river runoff is added.
 This is done in the same way for both vvl and non-vvl.
+The temperature and salinity are increased through the specified depth according to the heat and salt content of the river.
+In the non-linear free surface case (vvl), near the end of the time step the change in sea surface height is redistrubuted
+through the grid boxes, so that the original ratios of grid box heights are restored.
+In doing this water is moved into boxes below, throughout the water column, so the large volume addition to the surface box is spread between all the grid boxes.
+It is also possible for runnoff to be specified as a negative value for modelling flow through straits, i.e. modelling the Baltic flow in and out of the North Sea.
+When the flow is out of the domain there is no change in temperature and salinity, regardless of the namelist options used, as the ocean water leaving the domain removes heat and salt (at the same concentration) with it.
+The temperature and salinity are increased through the specified depth according to
+the heat and salt content of the river.
+In the non-linear free surface case (vvl),
+near the end of the time step the change in sea surface height is redistrubuted through the grid boxes,
+so that the original ratios of grid box heights are restored.
+In doing this water is moved into boxes below, throughout the water column,
+so the large volume addition to the surface box is spread between all the grid boxes.
+It is also possible for runnoff to be specified as a negative value for modelling flow through straits,
+i.e. modelling the Baltic flow in and out of the North Sea.
+When the flow is out of the domain there is no change in temperature and salinity,
+regardless of the namelist options used,
+as the ocean water leaving the domain removes heat and salt (at the same concentration) with it.
 …
 \begin{description}
 \item[\np{nn\_isf}\forcode{ = 1}]
+The ice shelf cavity is represented (\np{ln\_isfcav}\forcode{ = .true.} needed). The fwf and heat flux are computed.
+Two different bulk formula are available:
+  The ice shelf cavity is represented (\np{ln\_isfcav}\forcode{ = .true.} needed).
+  The fwf and heat flux are computed.
+  Two different bulk formula are available:
    \begin{description}
    \item[\np{nn\_isfblk}\forcode{ = 1}]
    The bulk formula used to compute the melt is based the one described in \citet{Hunter2006}.
         This formulation is based on a balance between the upward ocean heat flux and the latent heat flux at the ice shelf base.
    \item[\np{nn\_isfblk}\forcode{ = 2}]
    The bulk formula used to compute the melt is based the one described in \citet{Jenkins1991}.
         This formulation is based on a 3 equations formulation (a heat flux budget, a salt flux budget
          and a linearised freezing point temperature equation).
+     The bulk formula used to compute the melt is based the one described in \citet{Hunter2006}.
+     This formulation is based on a balance between the upward ocean heat flux and
+     the latent heat flux at the ice shelf base.
+   \item[\np{nn\_isfblk}\forcode{ = 2}]
+     The bulk formula used to compute the melt is based the one described in \citet{Jenkins1991}.
+     This formulation is based on a 3 equations formulation
+     (a heat flux budget, a salt flux budget and a linearised freezing point temperature equation).
    \end{description}
+For this 2 bulk formulations, there are 3 different ways to compute the exchange coeficient:
+   For this 2 bulk formulations, there are 3 different ways to compute the exchange coeficient:
    \begin{description}
+        \item[\np{nn\_gammablk}\forcode{ = 0}]
+   The salt and heat exchange coefficients are constant and defined by \np{rn\_gammas0} and \np{rn\_gammat0}
+   \item[\np{nn\_gammablk}\forcode{ = 0}]
+     The salt and heat exchange coefficients are constant and defined by \np{rn\_gammas0} and \np{rn\_gammat0}
    \item[\np{nn\_gammablk}\forcode{ = 1}]
+   The salt and heat exchange coefficients are velocity dependent and defined as \np{rn\_gammas0}$ \times u_{*}$ and \np{rn\_gammat0}$ \times u_{*}$
         where $u_{*}$ is the friction velocity in the top boundary layer (ie first \np{rn\_hisf\_tbl} meters).
         See \citet{Jenkins2010} for all the details on this formulation.
+     The salt and heat exchange coefficients are velocity dependent and defined as
+     \np{rn\_gammas0}$ \times u_{*}$ and \np{rn\_gammat0}$ \times u_{*}$ where
+     $u_{*}$ is the friction velocity in the top boundary layer (ie first \np{rn\_hisf\_tbl} meters).
+     See \citet{Jenkins2010} for all the details on this formulation.
    \item[\np{nn\_gammablk}\forcode{ = 2}]
+   The salt and heat exchange coefficients are velocity and stability dependent and defined as
         $\gamma_{T,S} = \frac{u_{*}}{\Gamma_{Turb} + \Gamma^{T,S}_{Mole}}$
         where $u_{*}$ is the friction velocity in the top boundary layer (ie first \np{rn\_hisf\_tbl} meters),
         $\Gamma_{Turb}$ the contribution of the ocean stability and
         $\Gamma^{T,S}_{Mole}$ the contribution of the molecular diffusion.
         See \citet{Holland1999} for all the details on this formulation.
         \end{description}
+\item[\np{nn\_isf}\forcode{ = 2}]
+A parameterisation of isf is used. The ice shelf cavity is not represented.
+The fwf is distributed along the ice shelf edge between the depth of the average grounding line (GL)
+(\np{sn\_depmax\_isf}) and the base of the ice shelf along the calving front (\np{sn\_depmin\_isf}) as in (\np{nn\_isf}\forcode{ = 3}).
+Furthermore the fwf and heat flux are computed using the \citet{Beckmann2003} parameterisation of isf melting.
 The effective melting length (\np{sn\_Leff\_isf}) is read from a file.
+\item[\np{nn\_isf}\forcode{ = 3}]
+A simple parameterisation of isf is used. The ice shelf cavity is not represented.
+The fwf (\np{sn\_rnfisf}) is prescribed and distributed along the ice shelf edge between the depth of the average grounding line (GL)
+(\np{sn\_depmax\_isf}) and the base of the ice shelf along the calving front (\np{sn\_depmin\_isf}).
 The heat flux ($Q_h$) is computed as $Q_h = fwf \times L_f$.
+\item[\np{nn\_isf}\forcode{ = 4}]
+The ice shelf cavity is opened (\np{ln\_isfcav}\forcode{ = .true.} needed). However, the fwf is not computed but specified from file \np{sn\_fwfisf}).
 The heat flux ($Q_h$) is computed as $Q_h = fwf \times L_f$.\\
+     The salt and heat exchange coefficients are velocity and stability dependent and defined as
+     $\gamma_{T,S} = \frac{u_{*}}{\Gamma_{Turb} + \Gamma^{T,S}_{Mole}}$ where
+     $u_{*}$ is the friction velocity in the top boundary layer (ie first \np{rn\_hisf\_tbl} meters),
+     $\Gamma_{Turb}$ the contribution of the ocean stability and
+     $\Gamma^{T,S}_{Mole}$ the contribution of the molecular diffusion.
+     See \citet{Holland1999} for all the details on this formulation.
+   \end{description}
+ \item[\np{nn\_isf}\forcode{ = 2}]
+   A parameterisation of isf is used. The ice shelf cavity is not represented.
+   The fwf is distributed along the ice shelf edge between the depth of the average grounding line (GL)
+   (\np{sn\_depmax\_isf}) and the base of the ice shelf along the calving front
+   (\np{sn\_depmin\_isf}) as in (\np{nn\_isf}\forcode{ = 3}).
+   Furthermore the fwf and heat flux are computed using the \citet{Beckmann2003} parameterisation of isf melting.
+   The effective melting length (\np{sn\_Leff\_isf}) is read from a file.
+ \item[\np{nn\_isf}\forcode{ = 3}]
+   A simple parameterisation of isf is used. The ice shelf cavity is not represented.
+   The fwf (\np{sn\_rnfisf}) is prescribed and distributed along the ice shelf edge between
+   the depth of the average grounding line (GL) (\np{sn\_depmax\_isf}) and
+   the base of the ice shelf along the calving front (\np{sn\_depmin\_isf}).
+   The heat flux ($Q_h$) is computed as $Q_h = fwf \times L_f$.
+ \item[\np{nn\_isf}\forcode{ = 4}]
+   The ice shelf cavity is opened (\np{ln\_isfcav}\forcode{ = .true.} needed).
+   However, the fwf is not computed but specified from file \np{sn\_fwfisf}).
+   The heat flux ($Q_h$) is computed as $Q_h = fwf \times L_f$.\\
 \end{description}
+$\bullet$ \np{nn\_isf}\forcode{ = 1} and \np{nn\_isf}\forcode{ = 2} compute a melt rate based on the water mass properties, ocean velocities and depth.
+ This flux is thus highly dependent of the model resolution (horizontal and vertical), realism of the water masses onto the shelf ...\\
+$\bullet$ \np{nn\_isf}\forcode{ = 3} and \np{nn\_isf}\forcode{ = 4} read the melt rate from a file. You have total control of the fwf forcing.
+This can be usefull if the water masses on the shelf are not realistic or the resolution (horizontal/vertical) are too
+coarse to have realistic melting or for studies where you need to control your heat and fw input.\\
+A namelist parameters control over how many meters the heat and fw fluxes are spread.
+\np{rn\_hisf\_tbl}] is the top boundary layer thickness as defined in \citet{Losch2008}.
+This parameter is only used if \np{nn\_isf}\forcode{ = 1} or \np{nn\_isf}\forcode{ = 4}
+$\bullet$ \np{nn\_isf}\forcode{ = 1} and \np{nn\_isf}\forcode{ = 2} compute a melt rate based on
+the water mass properties, ocean velocities and depth.
+This flux is thus highly dependent of the model resolution (horizontal and vertical),
+realism of the water masses onto the shelf ...\\
+$\bullet$ \np{nn\_isf}\forcode{ = 3} and \np{nn\_isf}\forcode{ = 4} read the melt rate from a file.
+You have total control of the fwf forcing.
+This can be useful if the water masses on the shelf are not realistic or
+the resolution (horizontal/vertical) are too coarse to have realistic melting or
+for studies where you need to control your heat and fw input.\\
+A namelist parameters control over how many meters the heat and fw fluxes are spread.
+\np{rn\_hisf\_tbl}] is the top boundary layer thickness as defined in \citet{Losch2008}.
+This parameter is only used if \np{nn\_isf}\forcode{ = 1} or \np{nn\_isf}\forcode{ = 4}.
 If \np{rn\_hisf\_tbl}\forcode{ = 0}., the fluxes are put in the top level whatever is its tickness.
+If \np{rn\_hisf\_tbl} $>$ 0., the fluxes are spread over the first \np{rn\_hisf\_tbl} m (ie over one or several cells).\\
+If \np{rn\_hisf\_tbl} $>$ 0., the fluxes are spread over the first \np{rn\_hisf\_tbl} m
+(ie over one or several cells).\\
 The ice shelf melt is implemented as a volume flux with in the same way as for the runoff.
 The fw addition due to the ice shelf melting is, at each relevant depth level, added to the horizontal divergence
+(\textit{hdivn}) in the subroutine \rou{sbc\_isf\_div}, called from \mdl{divcur}.
 See the runoff section \autoref{sec:SBC_rnf} for all the details about the divergence correction.
+The fw addition due to the ice shelf melting is, at each relevant depth level, added to
+the horizontal divergence (\textit{hdivn}) in the subroutine \rou{sbc\_isf\_div}, called from \mdl{divcur}.
+See the runoff section \autoref{sec:SBC_rnf} for all the details about the divergence correction.
 …
 \nlst{namsbc_iscpl}
 %--------------------------------------------------------------------------------------------------------
 Ice sheet/ocean coupling is done through file exchange at the restart step. NEMO, at each restart step,
+read the bathymetry and ice shelf draft variable in a netcdf file.
 If \np{ln\_iscpl}\forcode{ = .true.}, the isf draft is assume to be different at each restart step
 with potentially some new wet/dry cells due to the ice sheet dynamics/thermodynamics.
+Ice sheet/ocean coupling is done through file exchange at the restart step.
+NEMO, at each restart step, read the bathymetry and ice shelf draft variable in a netcdf file.
+If \np{ln\_iscpl}\forcode{ = .true.}, the isf draft is assume to be different at each restart step with
+potentially some new wet/dry cells due to the ice sheet dynamics/thermodynamics.
 The wetting and drying scheme applied on the restart is very simple and described below for the 6 different cases:
 \begin{description}
 \item[Thin a cell down:]
+   T/S/ssh are unchanged and U/V in the top cell are corrected to keep the barotropic transport (bt) constant ($bt_b=bt_n$).
+  T/S/ssh are unchanged and U/V in the top cell are corrected to keep the barotropic transport (bt) constant
+  ($bt_b=bt_n$).
 \item[Enlarge  a cell:]
    See case "Thin a cell down"
+  See case "Thin a cell down"
 \item[Dry a cell:]
+   mask, T/S, U/V and ssh are set to 0. Furthermore, U/V into the water column are modified to satisfy ($bt_b=bt_n$).
+  mask, T/S, U/V and ssh are set to 0.
+  Furthermore, U/V into the water column are modified to satisfy ($bt_b=bt_n$).
 \item[Wet a cell:]
+   mask is set to 1, T/S is extrapolated from neighbours, $ssh_n = ssh_b$ and U/V set to 0. If no neighbours along i,j and k, T/S/U/V and mask are set to 0.
+  mask is set to 1, T/S is extrapolated from neighbours, $ssh_n = ssh_b$ and U/V set to 0.
+  If no neighbours along i,j and k, T/S/U/V and mask are set to 0.
 \item[Dry a column:]
    mask, T/S, U/V are set to 0 everywhere in the column and ssh set to 0.
 \item[Wet a column:]
+   set mask to 1, T/S is extrapolated from neighbours, ssh is extrapolated from neighbours and U/V set to 0. If no neighbour, T/S/U/V and mask set to 0.
+  set mask to 1, T/S is extrapolated from neighbours, ssh is extrapolated from neighbours and U/V set to 0.
+  If no neighbour, T/S/U/V and mask set to 0.
 \end{description}
+The extrapolation is call \np{nn\_drown} times. It means that if the grounding line retreat by more than \np{nn\_drown} cells between 2 coupling steps,
+ the code will be unable to fill all the new wet cells properly. The default number is set up for the MISOMIP idealised experiments.\\
+This coupling procedure is able to take into account grounding line and calving front migration. However, it is a non-conservative processe.
+This could lead to a trend in heat/salt content and volume. In order to remove the trend and keep the conservation level as close to 0 as possible,
+ a simple conservation scheme is available with \np{ln\_hsb}\forcode{ = .true.}. The heat/salt/vol. gain/loss is diagnose, as well as the location.
+Based on what is done on sbcrnf to prescribed a source of heat/salt/vol., the heat/salt/vol. gain/loss is removed/added,
+ over a period of \np{rn\_fiscpl} time step, into the system.
+The extrapolation is call \np{nn\_drown} times.
+It means that if the grounding line retreat by more than \np{nn\_drown} cells between 2 coupling steps,
+the code will be unable to fill all the new wet cells properly.
+The default number is set up for the MISOMIP idealised experiments.\\
+This coupling procedure is able to take into account grounding line and calving front migration.
+However, it is a non-conservative processe.
+This could lead to a trend in heat/salt content and volume.
+In order to remove the trend and keep the conservation level as close to 0 as possible,
+a simple conservation scheme is available with \np{ln\_hsb}\forcode{ = .true.}.
+The heat/salt/vol. gain/loss is diagnose, as well as the location.
+Based on what is done on sbcrnf to prescribed a source of heat/salt/vol.,
+the heat/salt/vol. gain/loss is removed/added, over a period of \np{rn\_fiscpl} time step, into the system.
 So after \np{rn\_fiscpl} time step, all the heat/salt/vol. gain/loss due to extrapolation process is canceled.\\
+As the before and now fields are not compatible (modification of the geometry), the restart time step is prescribed to be an euler time step instead of a leap frog and $fields_b = fields_n$.
+As the before and now fields are not compatible (modification of the geometry),
+the restart time step is prescribed to be an euler time step instead of a leap frog and $fields_b = fields_n$.
+%
 % ================================================================
 …
 Their physical behaviour is controlled by equations as described in \citet{Martin_Adcroft_OM10} ).
 (Note that the authors kindly provided a copy of their code to act as a basis for implementation in NEMO).
 Icebergs are initially spawned into one of ten classes which have specific mass and thickness as described
+in the \ngn{namberg} namelist:
+\np{rn\_initial\_mass} and \np{rn\_initial\_thickness}.
+Each class has an associated scaling (\np{rn\_mass\_scaling}), which is an integer representing how many icebergs
 of this class are being described as one lagrangian point (this reduces the numerical problem of tracking every single iceberg).
+Icebergs are initially spawned into one of ten classes which have specific mass and thickness as
+described in the \ngn{namberg} namelist: \np{rn\_initial\_mass} and \np{rn\_initial\_thickness}.
+Each class has an associated scaling (\np{rn\_mass\_scaling}),
+which is an integer representing how many icebergs of this class are being described as one lagrangian point
+(this reduces the numerical problem of tracking every single iceberg).
 They are enabled by setting \np{ln\_icebergs}\forcode{ = .true.}.
 …
 \begin{description}
 \item[\np{nn\_test\_icebergs}~$>$~0]
+In this scheme, the value of \np{nn\_test\_icebergs} represents the class of iceberg to generate
+(so between 1 and 10), and \np{nn\_test\_icebergs} provides a lon/lat box in the domain at each
+grid point of which an iceberg is generated at the beginning of the run.
 (Note that this happens each time the timestep equals \np{nn\_nit000}.)
+\np{nn\_test\_icebergs} is defined by four numbers in \np{nn\_test\_box} representing the corners
 of the geographical box: lonmin,lonmax,latmin,latmax
+  In this scheme, the value of \np{nn\_test\_icebergs} represents the class of iceberg to generate
+  (so between 1 and 10), and \np{nn\_test\_icebergs} provides a lon/lat box in the domain at each grid point of
+  which an iceberg is generated at the beginning of the run.
+  (Note that this happens each time the timestep equals \np{nn\_nit000}.)
+  \np{nn\_test\_icebergs} is defined by four numbers in \np{nn\_test\_box} representing the corners of
+  the geographical box: lonmin,lonmax,latmin,latmax
 \item[\np{nn\_test\_icebergs}\forcode{ = -1}]
+In this scheme the model reads a calving file supplied in the \np{sn\_icb} parameter.
+This should be a file with a field on the configuration grid (typically ORCA) representing ice accumulation rate at each model point.
+These should be ocean points adjacent to land where icebergs are known to calve.
+Most points in this input grid are going to have value zero.
+When the model runs, ice is accumulated at each grid point which has a non-zero source term.
+At each time step, a test is performed to see if there is enough ice mass to calve an iceberg of each class in order (1 to 10).
+Note that this is the initial mass multiplied by the number each particle represents ($i.e.$ the scaling).
+If there is enough ice, a new iceberg is spawned and the total available ice reduced accordingly.
+  In this scheme the model reads a calving file supplied in the \np{sn\_icb} parameter.
+  This should be a file with a field on the configuration grid (typically ORCA)
+  representing ice accumulation rate at each model point.
+  These should be ocean points adjacent to land where icebergs are known to calve.
+  Most points in this input grid are going to have value zero.
+  When the model runs, ice is accumulated at each grid point which has a non-zero source term.
+  At each time step, a test is performed to see if there is enough ice mass to
+  calve an iceberg of each class in order (1 to 10).
+  Note that this is the initial mass multiplied by the number each particle represents ($i.e.$ the scaling).
+  If there is enough ice, a new iceberg is spawned and the total available ice reduced accordingly.
 \end{description}
 Icebergs are influenced by wind, waves and currents, bottom melt and erosion.
+The latter act to disintegrate the iceberg. This is either all melted freshwater, or
+(if \np{rn\_bits\_erosion\_fraction}~$>$~0) into melt and additionally small ice bits
+The latter act to disintegrate the iceberg.
+This is either all melted freshwater,
+or (if \np{rn\_bits\_erosion\_fraction}~$>$~0) into melt and additionally small ice bits
 which are assumed to propagate with their larger parent and thus delay fluxing into the ocean.
 Melt water (and other variables on the configuration grid) are written into the main NEMO model output files.
 …
 The amount of information is controlled by two integer parameters:
 \begin{description}
 \item[\np{nn\_verbose\_level}]  takes a value between one and four and represents
+an increasing number of points in the code at which variables are written, and an
 increasing level of obscurity.
+\item[\np{nn\_verbose\_level}] takes a value between one and four and
+  represents an increasing number of points in the code at which variables are written,
+  and an increasing level of obscurity.
 \item[\np{nn\_verbose\_write}] is the number of timesteps between writes
 \end{description}
 …
 When \key{mpp\_mpi} is defined, each output file contains only those icebergs in the corresponding processor.
 Trajectory points are written out in the order of their parent iceberg in the model's "linked list" of icebergs.
 So care is needed to recreate data for individual icebergs, since its trajectory data may be spread across
 multiple files.
+So care is needed to recreate data for individual icebergs,
+since its trajectory data may be spread across multiple files.
 …
 \begin{figure}[!t]    \begin{center}
 \includegraphics[width=0.8\textwidth]{Fig_SBC_diurnal}
+\caption{ \protect\label{fig:SBC_diurnal}
+Example of recontruction of the diurnal cycle variation of short wave flux
+from daily mean values. The reconstructed diurnal cycle (black line) is chosen
+as the mean value of the analytical cycle (blue line) over a time step, not
+as the mid time step value of the analytically cycle (red square). From \citet{Bernie_al_CD07}.}
+\caption{ \protect\label{fig:SBC_diurnal}
+  Example of recontruction of the diurnal cycle variation of short wave flux from daily mean values.
+  The reconstructed diurnal cycle (black line) is chosen as
+  the mean value of the analytical cycle (blue line) over a time step,
+  not as the mid time step value of the analytically cycle (red square).
+  From \citet{Bernie_al_CD07}.}
 \end{center}   \end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 \cite{Bernie_al_JC05} have shown that to capture 90$\%$ of the diurnal variability of
+SST requires a vertical resolution in upper ocean of 1~m or better and a temporal resolution
+of the surface fluxes of 3~h or less. Unfortunately high frequency forcing fields are rare,
+not to say inexistent. Nevertheless, it is possible to obtain a reasonable diurnal cycle
+of the SST knowning only short wave flux (SWF) at high frequency \citep{Bernie_al_CD07}.
+Furthermore, only the knowledge of daily mean value of SWF is needed,
 as higher frequency variations can be reconstructed from them, assuming that
+the diurnal cycle of SWF is a scaling of the top of the atmosphere diurnal cycle
+of incident SWF. The \cite{Bernie_al_CD07} reconstruction algorithm is available
+in \NEMO by setting \np{ln\_dm2dc}\forcode{ = .true.} (a \textit{\ngn{namsbc}} namelist variable) when using
+CORE bulk formulea (\np{ln\_blk\_core}\forcode{ = .true.}) or the flux formulation (\np{ln\_flx}\forcode{ = .true.}).
 The reconstruction is performed in the \mdl{sbcdcy} module. The detail of the algoritm used
+can be found in the appendix~A of \cite{Bernie_al_CD07}. The algorithm preserve the daily
+mean incomming SWF as the reconstructed SWF at a given time step is the mean value
+of the analytical cycle over this time step (\autoref{fig:SBC_diurnal}).
 The use of diurnal cycle reconstruction requires the input SWF to be daily
+\cite{Bernie_al_JC05} have shown that to capture 90$\%$ of the diurnal variability of SST requires a vertical resolution in upper ocean of 1~m or better and a temporal resolution of the surface fluxes of 3~h or less.
+Unfortunately high frequency forcing fields are rare, not to say inexistent.
+Nevertheless, it is possible to obtain a reasonable diurnal cycle of the SST knowning only short wave flux (SWF) at
+high frequency \citep{Bernie_al_CD07}.
+Furthermore, only the knowledge of daily mean value of SWF is needed,
+as higher frequency variations can be reconstructed from them,
+assuming that the diurnal cycle of SWF is a scaling of the top of the atmosphere diurnal cycle of incident SWF.
+The \cite{Bernie_al_CD07} reconstruction algorithm is available in \NEMO by
+setting \np{ln\_dm2dc}\forcode{ = .true.} (a \textit{\ngn{namsbc}} namelist variable) when
+using CORE bulk formulea (\np{ln\_blk\_core}\forcode{ = .true.}) or
+the flux formulation (\np{ln\_flx}\forcode{ = .true.}).
+The reconstruction is performed in the \mdl{sbcdcy} module.
+The detail of the algoritm used can be found in the appendix~A of \cite{Bernie_al_CD07}.
+The algorithm preserve the daily mean incoming SWF as the reconstructed SWF at
+a given time step is the mean value of the analytical cycle over this time step (\autoref{fig:SBC_diurnal}).
+The use of diurnal cycle reconstruction requires the input SWF to be daily
 ($i.e.$ a frequency of 24 and a time interpolation set to true in \np{sn\_qsr} namelist parameter).
 Furthermore, it is recommended to have a least 8 surface module time step per day,
 that is  $\rdt \ nn\_fsbc < 10,800~s = 3~h$. An example of recontructed SWF
+is given in \autoref{fig:SBC_dcy} for a 12 reconstructed diurnal cycle, one every 2~hours
 (from 1am to 11pm).
+that is  $\rdt \ nn\_fsbc < 10,800~s = 3~h$.
+An example of recontructed SWF is given in \autoref{fig:SBC_dcy} for a 12 reconstructed diurnal cycle,
+one every 2~hours (from 1am to 11pm).
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 \begin{figure}[!t]  \begin{center}
 \includegraphics[width=0.7\textwidth]{Fig_SBC_dcy}
 \caption{ \protect\label{fig:SBC_dcy}
+Example of recontruction of the diurnal cycle variation of short wave flux
+from daily mean values on an ORCA2 grid with a time sampling of 2~hours (from 1am to 11pm).
 The display is on (i,j) plane. }
+\caption{ \protect\label{fig:SBC_dcy}
+  Example of recontruction of the diurnal cycle variation of short wave flux from
+  daily mean values on an ORCA2 grid with a time sampling of 2~hours (from 1am to 11pm).
+  The display is on (i,j) plane. }
 \end{center}   \end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+Note also that the setting a diurnal cycle in SWF is highly recommended  when
+the top layer thickness approach 1~m or less, otherwise large error in SST can
+appear due to an inconsistency between the scale of the vertical resolution
+and the forcing acting on that scale.
+Note also that the setting a diurnal cycle in SWF is highly recommended when
+the top layer thickness approach 1~m or less, otherwise large error in SST can appear due to
+an inconsistency between the scale of the vertical resolution and the forcing acting on that scale.
 % -------------------------------------------------------------------------------------------------------------
 …
 \label{subsec:SBC_rotation}
+When using a flux (\np{ln\_flx}\forcode{ = .true.}) or bulk (\np{ln\_clio}\forcode{ = .true.} or \np{ln\_core}\forcode{ = .true.}) formulation,
+pairs of vector components can be rotated from east-north directions onto the local grid directions.
+This is particularly useful when interpolation on the fly is used since here any vectors are likely to be defined
+relative to a rectilinear grid.
+When using a flux (\np{ln\_flx}\forcode{ = .true.}) or
+bulk (\np{ln\_clio}\forcode{ = .true.} or \np{ln\_core}\forcode{ = .true.}) formulation,
+pairs of vector components can be rotated from east-north directions onto the local grid directions.
+This is particularly useful when interpolation on the fly is used since here any vectors are likely to
+be defined relative to a rectilinear grid.
 To activate this option a non-empty string is supplied in the rotation pair column of the relevant namelist.
 The eastward component must start with "U" and the northward component with "V".
 The remaining characters in the strings are used to identify which pair of components go together.
 So for example, strings "U1" and "V1" next to "utau" and "vtau" would pair the wind stress components together
+and rotate them on to the model grid directions; "U2" and "V2" could be used against a second pair of components,
 and so on.
+So for example, strings "U1" and "V1" next to "utau" and "vtau" would pair the wind stress components together and
+rotate them on to the model grid directions;
+"U2" and "V2" could be used against a second pair of components, and so on.
 The extra characters used in the strings are arbitrary.
 The rot\_rep routine from the \mdl{geo2ocean} module is used to perform the rotation.
 …
 %-------------------------------------------------------------------------------------------------------------
 IOptions are defined through the  \ngn{namsbc\_ssr} namelist variables.
+n forced mode using a flux formulation (\np{ln\_flx}\forcode{ = .true.}), a
 feedback term \emph{must} be added to the surface heat flux $Q_{ns}^o$:
+IOptions are defined through the \ngn{namsbc\_ssr} namelist variables.
+On forced mode using a flux formulation (\np{ln\_flx}\forcode{ = .true.}),
+a feedback term \emph{must} be added to the surface heat flux $Q_{ns}^o$:
 \begin{equation} \label{eq:sbc_dmp_q}
 Q_{ns} = Q_{ns}^o + \frac{dQ}{dT} \left( \left. T \right|_{k=1} - SST_{Obs} \right)
 \end{equation}
+where SST is a sea surface temperature field (observed or climatological), $T$ is
+the model surface layer temperature and $\frac{dQ}{dT}$ is a negative feedback
+coefficient usually taken equal to $-40~W/m^2/K$. For a $50~m$
+mixed-layer depth, this value corresponds to a relaxation time scale of two months.
+This term ensures that if $T$ perfectly matches the supplied SST, then $Q$ is
+equal to $Q_o$.
+In the fresh water budget, a feedback term can also be added. Converted into an
+equivalent freshwater flux, it takes the following expression :
+where SST is a sea surface temperature field (observed or climatological),
+$T$ is the model surface layer temperature and
+$\frac{dQ}{dT}$ is a negative feedback coefficient usually taken equal to $-40~W/m^2/K$.
+For a $50~m$ mixed-layer depth, this value corresponds to a relaxation time scale of two months.
+This term ensures that if $T$ perfectly matches the supplied SST, then $Q$ is equal to $Q_o$.
+In the fresh water budget, a feedback term can also be added.
+Converted into an equivalent freshwater flux, it takes the following expression :
 \begin{equation} \label{eq:sbc_dmp_emp}
 …
 \end{equation}
+where $\textit{emp}_{o }$ is a net surface fresh water flux (observed, climatological or an
+atmospheric model product), \textit{SSS}$_{Obs}$ is a sea surface salinity (usually a time
+interpolation of the monthly mean Polar Hydrographic Climatology \citep{Steele2001}),
+$\left.S\right|_{k=1}$ is the model surface layer salinity and $\gamma_s$ is a negative
+feedback coefficient which is provided as a namelist parameter. Unlike heat flux, there is no
+physical justification for the feedback term in \autoref{eq:sbc_dmp_emp} as the atmosphere
+does not care about ocean surface salinity \citep{Madec1997}. The SSS restoring
+term should be viewed as a flux correction on freshwater fluxes to reduce the
+uncertainties we have on the observed freshwater budget.
+where $\textit{emp}_{o }$ is a net surface fresh water flux
+(observed, climatological or an atmospheric model product),
+\textit{SSS}$_{Obs}$ is a sea surface salinity
+(usually a time interpolation of the monthly mean Polar Hydrographic Climatology \citep{Steele2001}),
+$\left.S\right|_{k=1}$ is the model surface layer salinity and
+$\gamma_s$ is a negative feedback coefficient which is provided as a namelist parameter.
+Unlike heat flux, there is no physical justification for the feedback term in \autoref{eq:sbc_dmp_emp} as
+the atmosphere does not care about ocean surface salinity \citep{Madec1997}.
+The SSS restoring term should be viewed as a flux correction on freshwater fluxes to
+reduce the uncertainties we have on the observed freshwater budget.
 % -------------------------------------------------------------------------------------------------------------
 …
 \label{subsec:SBC_ice-cover}
 The presence at the sea surface of an ice covered area modifies all the fluxes
+transmitted to the ocean. There are several way to handle sea-ice in the system
+depending on the value of the \np{nn\_ice} namelist parameter found in \ngn{namsbc} namelist.
+The presence at the sea surface of an ice covered area modifies all the fluxes transmitted to the ocean.
+There are several way to handle sea-ice in the system depending on
+the value of the \np{nn\_ice} namelist parameter found in \ngn{namsbc} namelist.
 \begin{description}
+\item[nn{\_}ice = 0]  there will never be sea-ice in the computational domain.
+This is a typical namelist value used for tropical ocean domain. The surface fluxes
+are simply specified for an ice-free ocean. No specific things is done for sea-ice.
+\item[nn{\_}ice = 1]  sea-ice can exist in the computational domain, but no sea-ice model
+is used. An observed ice covered area is read in a file. Below this area, the SST is
+restored to the freezing point and the heat fluxes are set to $-4~W/m^2$ ($-2~W/m^2$)
+in the northern (southern) hemisphere. The associated modification of the freshwater
+fluxes are done in such a way that the change in buoyancy fluxes remains zero.
+This prevents deep convection to occur when trying to reach the freezing point
+(and so ice covered area condition) while the SSS is too large. This manner of
+managing sea-ice area, just by using si IF case, is usually referred as the \textit{ice-if}
+model. It can be found in the \mdl{sbcice{\_}if} module.
+\item[nn{\_}ice = 2 or more]  A full sea ice model is used. This model computes the
+ice-ocean fluxes, that are combined with the air-sea fluxes using the ice fraction of
+each model cell to provide the surface ocean fluxes. Note that the activation of a
+sea-ice model is is done by defining a CPP key (\key{lim3} or \key{cice}).
+The activation automatically overwrites the read value of nn{\_}ice to its appropriate
+value ($i.e.$ $2$ for LIM-3 or $3$ for CICE).
+\item[nn{\_}ice = 0]
+  there will never be sea-ice in the computational domain.
+  This is a typical namelist value used for tropical ocean domain.
+  The surface fluxes are simply specified for an ice-free ocean.
+  No specific things is done for sea-ice.
+\item[nn{\_}ice = 1]
+  sea-ice can exist in the computational domain, but no sea-ice model is used.
+  An observed ice covered area is read in a file.
+  Below this area, the SST is restored to the freezing point and
+  the heat fluxes are set to $-4~W/m^2$ ($-2~W/m^2$) in the northern (southern) hemisphere.
+  The associated modification of the freshwater fluxes are done in such a way that
+  the change in buoyancy fluxes remains zero.
+  This prevents deep convection to occur when trying to reach the freezing point
+  (and so ice covered area condition) while the SSS is too large.
+  This manner of managing sea-ice area, just by using si IF case,
+  is usually referred as the \textit{ice-if} model.
+  It can be found in the \mdl{sbcice{\_}if} module.
+\item[nn{\_}ice = 2 or more]
+  A full sea ice model is used.
+  This model computes the ice-ocean fluxes,
+  that are combined with the air-sea fluxes using the ice fraction of each model cell to
+  provide the surface ocean fluxes.
+  Note that the activation of a sea-ice model is is done by defining a CPP key (\key{lim3} or \key{cice}).
+  The activation automatically overwrites the read value of nn{\_}ice to its appropriate value
+  ($i.e.$ $2$ for LIM-3 or $3$ for CICE).
 \end{description}
 …
 \label{subsec:SBC_cice}
+It is now possible to couple a regional or global NEMO configuration (without AGRIF) to the CICE sea-ice
+model by using \key{cice}.  The CICE code can be obtained from
+\href{http://oceans11.lanl.gov/trac/CICE/}{LANL} and the additional 'hadgem3' drivers will be required,
+even with the latest code release.  Input grid files consistent with those used in NEMO will also be needed,
+and CICE CPP keys \textbf{ORCA\_GRID}, \textbf{CICE\_IN\_NEMO} and \textbf{coupled} should be used (seek advice from UKMO
+if necessary).  Currently the code is only designed to work when using the CORE forcing option for NEMO (with
+\textit{calc\_strair}\forcode{ = .true.} and \textit{calc\_Tsfc}\forcode{ = .true.} in the CICE name-list), or alternatively when NEMO
+is coupled to the HadGAM3 atmosphere model (with \textit{calc\_strair}\forcode{ = .false.} and \textit{calc\_Tsfc}\forcode{ = false}).
+The code is intended to be used with \np{nn\_fsbc} set to 1 (although coupling ocean and ice less frequently
+should work, it is possible the calculation of some of the ocean-ice fluxes needs to be modified slightly - the
+user should check that results are not significantly different to the standard case).
+There are two options for the technical coupling between NEMO and CICE.  The standard version allows
+complete flexibility for the domain decompositions in the individual models, but this is at the expense of global
+gather and scatter operations in the coupling which become very expensive on larger numbers of processors. The
+alternative option (using \key{nemocice\_decomp} for both NEMO and CICE) ensures that the domain decomposition is
+identical in both models (provided domain parameters are set appropriately, and
+\textit{processor\_shape~=~square-ice} and \textit{distribution\_wght~=~block} in the CICE name-list) and allows
+much more efficient direct coupling on individual processors.  This solution scales much better although it is at
+the expense of having more idle CICE processors in areas where there is no sea ice.
+It is now possible to couple a regional or global NEMO configuration (without AGRIF)
+to the CICE sea-ice model by using \key{cice}.
+The CICE code can be obtained from \href{http://oceans11.lanl.gov/trac/CICE/}{LANL} and
+the additional 'hadgem3' drivers will be required, even with the latest code release.
+Input grid files consistent with those used in NEMO will also be needed,
+and CICE CPP keys \textbf{ORCA\_GRID}, \textbf{CICE\_IN\_NEMO} and \textbf{coupled} should be used
+(seek advice from UKMO if necessary).
+Currently the code is only designed to work when using the CORE forcing option for NEMO
+(with \textit{calc\_strair}\forcode{ = .true.} and \textit{calc\_Tsfc}\forcode{ = .true.} in the CICE name-list),
+or alternatively when NEMO is coupled to the HadGAM3 atmosphere model
+(with \textit{calc\_strair}\forcode{ = .false.} and \textit{calc\_Tsfc}\forcode{ = false}).
+The code is intended to be used with \np{nn\_fsbc} set to 1
+(although coupling ocean and ice less frequently should work,
+it is possible the calculation of some of the ocean-ice fluxes needs to be modified slightly -
+the user should check that results are not significantly different to the standard case).
+There are two options for the technical coupling between NEMO and CICE.
+The standard version allows complete flexibility for the domain decompositions in the individual models,
+but this is at the expense of global gather and scatter operations in the coupling which
+become very expensive on larger numbers of processors.
+The alternative option (using \key{nemocice\_decomp} for both NEMO and CICE) ensures that
+the domain decomposition is identical in both models (provided domain parameters are set appropriately,
+and \textit{processor\_shape~=~square-ice} and \textit{distribution\_wght~=~block} in the CICE name-list) and
+allows much more efficient direct coupling on individual processors.
+This solution scales much better although it is at the expense of having more idle CICE processors in areas where
+there is no sea ice.
 % -------------------------------------------------------------------------------------------------------------
 …
 \label{subsec:SBC_fwb}
 For global ocean simulation it can be useful to introduce a control of the mean sea
+level in order to prevent unrealistic drift of the sea surface height due to inaccuracy
 in the freshwater fluxes. In \NEMO, two way of controlling the the freshwater budget.
+For global ocean simulation it can be useful to introduce a control of the mean sea level in order to
+prevent unrealistic drift of the sea surface height due to inaccuracy in the freshwater fluxes.
+In \NEMO, two way of controlling the the freshwater budget.
 \begin{description}
+\item[\np{nn\_fwb}\forcode{ = 0}]  no control at all. The mean sea level is free to drift, and will
+certainly do so.
+\item[\np{nn\_fwb}\forcode{ = 1}]  global mean \textit{emp} set to zero at each model time step.
+\item[\np{nn\_fwb}\forcode{ = 0}]
+  no control at all.
+  The mean sea level is free to drift, and will certainly do so.
+\item[\np{nn\_fwb}\forcode{ = 1}]
+  global mean \textit{emp} set to zero at each model time step.
 %Note that with a sea-ice model, this technique only control the mean sea level with linear free surface (\key{vvl} not defined) and no mass flux between ocean and ice (as it is implemented in the current ice-ocean coupling).
 \item[\np{nn\_fwb}\forcode{ = 2}]  freshwater budget is adjusted from the previous year annual
+mean budget which is read in the \textit{EMPave\_old.dat} file. As the model uses the
+Boussinesq approximation, the annual mean fresh water budget is simply evaluated
+from the change in the mean sea level at January the first and saved in the
 \textit{EMPav.dat} file.
+\item[\np{nn\_fwb}\forcode{ = 2}]
+  freshwater budget is adjusted from the previous year annual mean budget which
+  is read in the \textit{EMPave\_old.dat} file.
+  As the model uses the Boussinesq approximation, the annual mean fresh water budget is simply evaluated from
+  the change in the mean sea level at January the first and saved in the \textit{EMPav.dat} file.
 \end{description}
 …
 %-------------------------------------------------------------------------------------------------------------
 In order to read a neutral drag coefficient, from an external data source ($i.e.$ a wave model), the
+logical variable \np{ln\_cdgw} in \ngn{namsbc} namelist must be set to \forcode{.true.}.
 The \mdl{sbcwave} module containing the routine \np{sbc\_wave} reads the
+namelist \ngn{namsbc\_wave} (for external data names, locations, frequency, interpolation and all
+the miscellanous options allowed by Input Data generic Interface see \autoref{sec:SBC_input})
 and a 2D field of neutral drag coefficient.
+In order to read a neutral drag coefficient, from an external data source ($i.e.$ a wave model),
+the logical variable \np{ln\_cdgw} in \ngn{namsbc} namelist must be set to \forcode{.true.}.
+The \mdl{sbcwave} module containing the routine \np{sbc\_wave} reads the namelist \ngn{namsbc\_wave}
+(for external data names, locations, frequency, interpolation and all the miscellanous options allowed by
+Input Data generic Interface see \autoref{sec:SBC_input}) and
+a 2D field of neutral drag coefficient.
 Then using the routine TURB\_CORE\_1Z or TURB\_CORE\_2Z, and starting from the neutral drag coefficent provided,
+the drag coefficient is computed according to stable/unstable conditions of the air-sea interface following \citet{Large_Yeager_Rep04}.
+the drag coefficient is computed according to stable/unstable conditions of the air-sea interface following
+\citet{Large_Yeager_Rep04}.

NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_STO.tex

-                      r10146
+                      r10368
 The stochastic parametrization module aims to explicitly simulate uncertainties in the model.
 More particularly, \cite{Brankart_OM2013} has shown that,
 because of the nonlinearity of the seawater equation of state, unresolved scales represent
+a major source of uncertainties in the computation of the large scale horizontal density gradient
+(from T/S large scale fields), and that the impact of these uncertainties can be simulated
 by random processes representing unresolved T/S fluctuations.
+The stochastic parametrization module aims to explicitly simulate uncertainties in the model.
+More particularly, \cite{Brankart_OM2013} has shown that,
+because of the nonlinearity of the seawater equation of state, unresolved scales represent a major source of
+uncertainties in the computation of the large scale horizontal density gradient (from T/S large scale fields),
+and that the impact of these uncertainties can be simulated by
+random processes representing unresolved T/S fluctuations.
 The stochastic formulation of the equation of state can be written as:
 …
   \rho = \frac{1}{2} \sum_{i=1}^m\{ \rho[T+\Delta T_i,S+\Delta S_i,p_o(z)] + \rho[T-\Delta T_i,S-\Delta S_i,p_o(z)] \}
 \end{equation}
 where $p_o(z)$ is the reference pressure depending on the depth and,
 $\Delta T_i$ and $\Delta S_i$ are a set of T/S perturbations defined as the scalar product
 of the respective local T/S gradients with random walks $\mathbf{\xi}$:
+where $p_o(z)$ is the reference pressure depending on the depth and,
+$\Delta T_i$ and $\Delta S_i$ are a set of T/S perturbations defined as
+the scalar product of the respective local T/S gradients with random walks $\mathbf{\xi}$:
 \begin{equation}
  \label{eq:sto_pert}
  \Delta T_i = \mathbf{\xi}_i \cdot \nabla T \qquad \hbox{and} \qquad \Delta S_i = \mathbf{\xi}_i \cdot \nabla S
 \end{equation}
 $\mathbf{\xi}_i$ are produced by a first-order autoregressive processes (AR-1) with
 a parametrized decorrelation time scale, and horizontal and vertical standard deviations $\sigma_s$.
+$\mathbf{\xi}_i$ are produced by a first-order autoregressive processes (AR-1) with
+a parametrized decorrelation time scale, and horizontal and vertical standard deviations $\sigma_s$.
 $\mathbf{\xi}$ are uncorrelated over the horizontal and fully correlated along the vertical.
 …
 \label{sec:STO_the_details}
+The starting point of our implementation of stochastic parameterizations
+in NEMO is to observe that many existing parameterizations are based
+on autoregressive processes, which are used as a basic source of randomness
+to transform a deterministic model into a probabilistic model.
+The starting point of our implementation of stochastic parameterizations in NEMO is to observe that
+many existing parameterizations are based on autoregressive processes,
+which are used as a basic source of randomness to transform a deterministic model into a probabilistic model.
 A generic approach is thus to add one single new module in NEMO,
+generating processes with appropriate statistics
+to simulate each kind of uncertainty in the model
+generating processes with appropriate statistics to simulate each kind of uncertainty in the model
 (see \cite{Brankart_al_GMD2015} for more details).
 In practice, at every model grid point, independent Gaussian autoregressive
+processes~$\xi^{(i)},\,i=1,\ldots,m$ are first generated
 using the same basic equation:
+In practice, at every model grid point,
+independent Gaussian autoregressive processes~$\xi^{(i)},\,i=1,\ldots,m$ are first generated using
+the same basic equation:
 \begin{equation}
 …
 \noindent
+where $k$ is the index of the model timestep; and
+$a^{(i)}$, $b^{(i)}$, $c^{(i)}$ are parameters defining
+the mean ($\mu^{(i)}$) standard deviation ($\sigma^{(i)}$)
+and correlation timescale ($\tau^{(i)}$) of each process:
+where $k$ is the index of the model timestep and
+$a^{(i)}$, $b^{(i)}$, $c^{(i)}$ are parameters defining the mean ($\mu^{(i)}$) standard deviation ($\sigma^{(i)}$) and
+correlation timescale ($\tau^{(i)}$) of each process:
 \begin{itemize}
 \item for order~1 processes, $w^{(i)}$ is a Gaussian white noise,
+with zero mean and standard deviation equal to~1, and the parameters
 $a^{(i)}$, $b^{(i)}$, $c^{(i)}$ are given by:
+\item
+  for order~1 processes, $w^{(i)}$ is a Gaussian white noise, with zero mean and standard deviation equal to~1,
+  and the parameters $a^{(i)}$, $b^{(i)}$, $c^{(i)}$ are given by:
 \begin{equation}
 …
 \end{equation}
+\item for order~$n>1$ processes, $w^{(i)}$ is an order~$n-1$ autoregressive process,
+with zero mean, standard deviation equal to~$\sigma^{(i)}$; correlation timescale
+equal to~$\tau^{(i)}$; and the parameters $a^{(i)}$, $b^{(i)}$, $c^{(i)}$ are given by:
+\item
+  for order~$n>1$ processes, $w^{(i)}$ is an order~$n-1$ autoregressive process, with zero mean,
+  standard deviation equal to~$\sigma^{(i)}$;
+  correlation timescale equal to~$\tau^{(i)}$;
+  and the parameters $a^{(i)}$, $b^{(i)}$, $c^{(i)}$ are given by:
 \begin{equation}
 …
 \noindent
+In this way, higher order processes can be easily generated recursively using
+the same piece of code implementing (\autoref{eq:autoreg}),
+and using succesively processes from order $0$ to~$n-1$ as~$w^{(i)}$.
+The parameters in (\autoref{eq:ord2}) are computed so that this recursive application
+of (\autoref{eq:autoreg}) leads to processes with the required standard deviation
+and correlation timescale, with the additional condition that
+the $n-1$ first derivatives of the autocorrelation function
+are equal to zero at~$t=0$, so that the resulting processes
+become smoother and smoother as $n$ is increased.
+In this way, higher order processes can be easily generated recursively using the same piece of code implementing
+(\autoref{eq:autoreg}), and using succesively processes from order $0$ to~$n-1$ as~$w^{(i)}$.
+The parameters in (\autoref{eq:ord2}) are computed so that this recursive application of
+(\autoref{eq:autoreg}) leads to processes with the required standard deviation and correlation timescale,
+with the additional condition that the $n-1$ first derivatives of the autocorrelation function are equal to
+zero at~$t=0$, so that the resulting processes become smoother and smoother as $n$ is increased.
+Overall, this method provides quite a simple and generic way of generating
+a wide class of stochastic processes.
+However, this also means that new model parameters are needed to specify each of
+these stochastic processes. As in any parameterization of lacking physics,
+a very important issues then to tune these new parameters using either first principles,
+model simulations, or real-world observations.
+Overall, this method provides quite a simple and generic way of generating a wide class of stochastic processes.
+However, this also means that new model parameters are needed to specify each of these stochastic processes.
+As in any parameterization of lacking physics, a very important issues then to tune these new parameters using
+either first principles, model simulations, or real-world observations.
 \section{Implementation details}
 …
 It involves three modules :
 \begin{description}
+\item[\mdl{stopar}] : define the Stochastic parameters and their time evolution.
+\item[\mdl{storng}] : a random number generator based on (and includes) the 64-bit KISS
+                      (Keep It Simple Stupid) random number generator distributed by George Marsaglia
+                      (see \href{https://groups.google.com/forum/#!searchin/comp.lang.fortran/64-bit$20KISS$20RNGs}{here})
+\item[\mdl{stopts}] : stochastic parametrisation associated with the non-linearity of the equation of seawater,
+ implementing \autoref{eq:sto_pert} and specific piece of code in the equation of state implementing \autoref{eq:eos_sto}.
+\item[\mdl{stopar}:]
+  define the Stochastic parameters and their time evolution.
+\item[\mdl{storng}:]
+  a random number generator based on (and includes) the 64-bit KISS (Keep It Simple Stupid) random number generator
+  distributed by George Marsaglia
+  (see \href{https://groups.google.com/forum/#!searchin/comp.lang.fortran/64-bit$20KISS$20RNGs}{here})
+\item[\mdl{stopts}:]
+  stochastic parametrisation associated with the non-linearity of the equation of seawater,
+  implementing \autoref{eq:sto_pert} and specific piece of code in
+  the equation of state implementing \autoref{eq:eos_sto}.
 \end{description}
 …
 The first routine (\rou{sto\_par}) is a direct implementation of (\autoref{eq:autoreg}),
+applied at each model grid point (in 2D or 3D),
+and called at each model time step ($k$) to update
+every autoregressive process ($i=1,\ldots,m$).
+applied at each model grid point (in 2D or 3D), and called at each model time step ($k$) to
+update every autoregressive process ($i=1,\ldots,m$).
 This routine also includes a filtering operator, applied to $w^{(i)}$,
 to introduce a spatial correlation between the stochastic processes.
+The second routine (\rou{sto\_par\_init}) is an initialization routine mainly dedicated
+to the computation of parameters $a^{(i)}, b^{(i)}, c^{(i)}$
+for each autoregressive process, as a function of the statistical properties
+required by the model user (mean, standard deviation, time correlation,
+order of the process,\ldots).
+The second routine (\rou{sto\_par\_init}) is an initialization routine mainly dedicated to
+the computation of parameters $a^{(i)}, b^{(i)}, c^{(i)}$ for each autoregressive process,
+as a function of the statistical properties required by the model user
+(mean, standard deviation, time correlation, order of the process,\ldots).
 Parameters for the processes can be specified through the following \ngn{namsto} namelist parameters:
 \begin{description}
+   \item[\np{nn\_sto\_eos}]   : number of independent random walks
    \item[\np{rn\_eos\_stdxy}] : random walk horz. standard deviation (in grid points)
    \item[\np{rn\_eos\_stdz}]  : random walk vert. standard deviation (in grid points)
    \item[\np{rn\_eos\_tcor}]  : random walk time correlation (in timesteps)
    \item[\np{nn\_eos\_ord}]   : order of autoregressive processes
    \item[\np{nn\_eos\_flt}]   : passes of Laplacian filter
    \item[\np{rn\_eos\_lim}]   : limitation factor (default = 3.0)
+\item[\np{nn\_sto\_eos}:]   number of independent random walks
+\item[\np{rn\_eos\_stdxy}:] random walk horz. standard deviation (in grid points)
+\item[\np{rn\_eos\_stdz}:]  random walk vert. standard deviation (in grid points)
+\item[\np{rn\_eos\_tcor}:]  random walk time correlation (in timesteps)
+\item[\np{nn\_eos\_ord}:]   order of autoregressive processes
+\item[\np{nn\_eos\_flt}:]   passes of Laplacian filter
+\item[\np{rn\_eos\_lim}:]   limitation factor (default = 3.0)
 \end{description}
 This routine also includes the initialization (seeding) of the random number generator.
 The third routine (\rou{sto\_rst\_write}) writes a restart file (which suffix name is
+given by \np{cn\_storst\_out} namelist parameter) containing the current value of
+The third routine (\rou{sto\_rst\_write}) writes a restart file
+(which suffix name is given by \np{cn\_storst\_out} namelist parameter) containing the current value of
 all autoregressive processes to allow restarting a simulation from where it has been interrupted.
 This file also contains the current state of the random number generator.
+When \np{ln\_rststo} is set to \forcode{.true.}), the restart file (which suffix name is
+given by \np{cn\_storst\_in} namelist parameter) is read by the initialization routine
+(\rou{sto\_par\_init}). The simulation will continue exactly as if it was not interrupted
+only  when \np{ln\_rstseed} is set to \forcode{.true.}, $i.e.$ when the state of
+the random number generator is read in the restart file.
+When \np{ln\_rststo} is set to \forcode{.true.}),
+the restart file (which suffix name is given by \np{cn\_storst\_in} namelist parameter) is read by
+the initialization routine (\rou{sto\_par\_init}).
+The simulation will continue exactly as if it was not interrupted only
+when \np{ln\_rstseed} is set to \forcode{.true.},
+$i.e.$ when the state of the random number generator is read in the restart file.

NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_TRA.tex

-                      r10146
+                      r10368
 %$\ $\newline    % force a new ligne
+Using the representation described in \autoref{chap:DOM}, several semi-discrete
+space forms of the tracer equations are available depending on the vertical
+coordinate used and on the physics used. In all the equations presented
+here, the masking has been omitted for simplicity. One must be aware that
+all the quantities are masked fields and that each time a mean or difference
+operator is used, the resulting field is multiplied by a mask.
+The two active tracers are potential temperature and salinity. Their prognostic
+equations can be summarized as follows:
+Using the representation described in \autoref{chap:DOM},
+several semi-discrete space forms of the tracer equations are available depending on
+the vertical coordinate used and on the physics used.
+In all the equations presented here, the masking has been omitted for simplicity.
+One must be aware that all the quantities are masked fields and
+that each time a mean or difference operator is used,
+the resulting field is multiplied by a mask.
+The two active tracers are potential temperature and salinity.
+Their prognostic equations can be summarized as follows:
 \begin{equation*}
 \text{NXT} = \text{ADV}+\text{LDF}+\text{ZDF}+\text{SBC}
 …
 \end{equation*}
+NXT stands for next, referring to the time-stepping. From left to right, the terms
+on the rhs of the tracer equations are the advection (ADV), the lateral diffusion
+(LDF), the vertical diffusion (ZDF), the contributions from the external forcings
+(SBC: Surface Boundary Condition, QSR: penetrative Solar Radiation, and BBC:
+Bottom Boundary Condition), the contribution from the bottom boundary Layer
+(BBL) parametrisation, and an internal damping (DMP) term. The terms QSR,
+BBC, BBL and DMP are optional. The external forcings and parameterisations
+require complex inputs and complex calculations ($e.g.$ bulk formulae, estimation
+of mixing coefficients) that are carried out in the SBC, LDF and ZDF modules and
+described in \autoref{chap:SBC}, \autoref{chap:LDF} and \autoref{chap:ZDF}, respectively.
+Note that \mdl{tranpc}, the non-penetrative convection module, although
+located in the NEMO/OPA/TRA directory as it directly modifies the tracer fields,
+is described with the model vertical physics (ZDF) together with other available
+parameterization of convection.
+In the present chapter we also describe the diagnostic equations used to compute
+the sea-water properties (density, Brunt-V\"{a}is\"{a}l\"{a} frequency, specific heat and
+freezing point with associated modules \mdl{eosbn2} and \mdl{phycst}).
+The different options available to the user are managed by namelist logicals or CPP keys.
+For each equation term  \textit{TTT}, the namelist logicals are \textit{ln\_traTTT\_xxx},
+where \textit{xxx} is a 3 or 4 letter acronym corresponding to each optional scheme.
+The CPP key (when it exists) is \key{traTTT}. The equivalent code can be
+found in the \textit{traTTT} or \textit{traTTT\_xxx} module, in the NEMO/OPA/TRA directory.
+The user has the option of extracting each tendency term on the RHS of the tracer
+equation for output (\np{ln\_tra\_trd} or \np{ln\_tra\_mxl}\forcode{ = .true.}), as described in \autoref{chap:DIA}.
+NXT stands for next, referring to the time-stepping.
+From left to right, the terms on the rhs of the tracer equations are the advection (ADV),
+the lateral diffusion (LDF), the vertical diffusion (ZDF), the contributions from the external forcings
+(SBC: Surface Boundary Condition, QSR: penetrative Solar Radiation, and BBC: Bottom Boundary Condition),
+the contribution from the bottom boundary Layer (BBL) parametrisation, and an internal damping (DMP) term.
+The terms QSR, BBC, BBL and DMP are optional.
+The external forcings and parameterisations require complex inputs and complex calculations
+($e.g.$ bulk formulae, estimation of mixing coefficients) that are carried out in the SBC, LDF and ZDF modules and
+described in \autoref{chap:SBC}, \autoref{chap:LDF} and \autoref{chap:ZDF}, respectively.
+Note that \mdl{tranpc}, the non-penetrative convection module, although located in the NEMO/OPA/TRA directory as
+it directly modifies the tracer fields, is described with the model vertical physics (ZDF) together with
+other available parameterization of convection.
+In the present chapter we also describe the diagnostic equations used to compute the sea-water properties
+(density, Brunt-V\"{a}is\"{a}l\"{a} frequency, specific heat and freezing point with
+associated modules \mdl{eosbn2} and \mdl{phycst}).
+The different options available to the user are managed by namelist logicals or CPP keys.
+For each equation term  \textit{TTT}, the namelist logicals are \textit{ln\_traTTT\_xxx},
+where \textit{xxx} is a 3 or 4 letter acronym corresponding to each optional scheme.
+The CPP key (when it exists) is \key{traTTT}.
+The equivalent code can be found in the \textit{traTTT} or \textit{traTTT\_xxx} module,
+in the NEMO/OPA/TRA directory.
+The user has the option of extracting each tendency term on the RHS of the tracer equation for output
+(\np{ln\_tra\_trd} or \np{ln\_tra\_mxl}\forcode{ = .true.}), as described in \autoref{chap:DIA}.
 $\ $\newline    % force a new ligne
 …
 %-------------------------------------------------------------------------------------------------------------
+When considered ($i.e.$ when \np{ln\_traadv\_NONE} is not set to \forcode{.true.}),
+the advection tendency of a tracer is expressed in flux form,
+$i.e.$ as the divergence of the advective fluxes. Its discrete expression is given by :
+When considered ($i.e.$ when \np{ln\_traadv\_NONE} is not set to \forcode{.true.}),
+the advection tendency of a tracer is expressed in flux form,
+$i.e.$ as the divergence of the advective fluxes.
+Its discrete expression is given by :
 \begin{equation} \label{eq:tra_adv}
 ADV_\tau =-\frac{1}{b_t} \left(
 …
 -\frac{1}{e_{3t}} \;\delta _k \left[ w\; \tau _w \right]
 \end{equation}
 where $\tau$ is either T or S, and $b_t= e_{1t}\,e_{2t}\,e_{3t}$ is the volume of $T$-cells.
 The flux form in \autoref{eq:tra_adv}
+implicitly requires the use of the continuity equation. Indeed, it is obtained
+by using the following equality : $\nabla \cdot \left( \vect{U}\,T \right)=\vect{U} \cdot \nabla T$
+which results from the use of the continuity equation,  $\partial _t e_3 + e_3\;\nabla \cdot \vect{U}=0$
 (which reduces to $\nabla \cdot \vect{U}=0$ in linear free surface, $i.e.$ \np{ln\_linssh}\forcode{ = .true.}).
 Therefore it is of paramount importance to design the discrete analogue of the
+advection tendency so that it is consistent with the continuity equation in order to
+enforce the conservation properties of the continuous equations. In other words,
+by setting $\tau = 1$ in (\autoref{eq:tra_adv}) we recover the discrete form of
+where $\tau$ is either T or S, and $b_t= e_{1t}\,e_{2t}\,e_{3t}$ is the volume of $T$-cells.
+The flux form in \autoref{eq:tra_adv} implicitly requires the use of the continuity equation.
+Indeed, it is obtained by using the following equality:
+$\nabla \cdot \left( \vect{U}\,T \right)=\vect{U} \cdot \nabla T$ which
+results from the use of the continuity equation,  $\partial _t e_3 + e_3\;\nabla \cdot \vect{U}=0$
+(which reduces to $\nabla \cdot \vect{U}=0$ in linear free surface, $i.e.$ \np{ln\_linssh}\forcode{ = .true.}).
+Therefore it is of paramount importance to design the discrete analogue of the advection tendency so that
+it is consistent with the continuity equation in order to enforce the conservation properties of
+the continuous equations.
+In other words, by setting $\tau = 1$ in (\autoref{eq:tra_adv}) we recover the discrete form of
 the continuity equation which is used to calculate the vertical velocity.
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!t]    \begin{center}
+\includegraphics[width=0.9\textwidth]{Fig_adv_scheme}
+\caption{   \protect\label{fig:adv_scheme}
+Schematic representation of some ways used to evaluate the tracer value
+at $u$-point and the amount of tracer exchanged between two neighbouring grid
+points. Upsteam biased scheme (ups): the upstream value is used and the black
+area is exchanged. Piecewise parabolic method (ppm): a parabolic interpolation
+is used and the black and dark grey areas are exchanged. Monotonic upstream
+scheme for conservative laws (muscl):  a parabolic interpolation is used and black,
+dark grey and grey areas are exchanged. Second order scheme (cen2): the mean
+value is used and black, dark grey, grey and light grey areas are exchanged. Note
+that this illustration does not include the flux limiter used in ppm and muscl schemes.}
+\end{center}   \end{figure}
+\begin{figure}[!t]
+  \begin{center}
+    \includegraphics[width=0.9\textwidth]{Fig_adv_scheme}
+    \caption{  \protect\label{fig:adv_scheme}
+      Schematic representation of some ways used to evaluate the tracer value at $u$-point and
+      the amount of tracer exchanged between two neighbouring grid points.
+      Upsteam biased scheme (ups):
+      the upstream value is used and the black area is exchanged.
+      Piecewise parabolic method (ppm):
+      a parabolic interpolation is used and the black and dark grey areas are exchanged.
+      Monotonic upstream scheme for conservative laws (muscl):
+      a parabolic interpolation is used and black, dark grey and grey areas are exchanged.
+      Second order scheme (cen2):
+      the mean value is used and black, dark grey, grey and light grey areas are exchanged.
+      Note that this illustration does not include the flux limiter used in ppm and muscl schemes.
+    }
+  \end{center}
+\end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 The key difference between the advection schemes available in \NEMO is the choice
+made in space and time interpolation to define the value of the tracer at the
+velocity points (\autoref{fig:adv_scheme}).
 Along solid lateral and bottom boundaries a zero tracer flux is automatically
 specified, since the normal velocity is zero there. At the sea surface the
+boundary condition depends on the type of sea surface chosen:
+The key difference between the advection schemes available in \NEMO is the choice made in space and
+time interpolation to define the value of the tracer at the velocity points
+(\autoref{fig:adv_scheme}).
+Along solid lateral and bottom boundaries a zero tracer flux is automatically specified,
+since the normal velocity is zero there.
+At the sea surface the boundary condition depends on the type of sea surface chosen:
 \begin{description}
+\item [linear free surface:] (\np{ln\_linssh}\forcode{ = .true.}) the first level thickness is constant in time:
+the vertical boundary condition is applied at the fixed surface $z=0$
+rather than on the moving surface $z=\eta$. There is a non-zero advective
+flux which is set for all advection schemes as
+$\left. {\tau _w } \right|_{k=1/2} =T_{k=1} $, $i.e.$
+the product of surface velocity (at $z=0$) by the first level tracer value.
+\item [non-linear free surface:] (\np{ln\_linssh}\forcode{ = .false.})
+convergence/divergence in the first ocean level moves the free surface
+up/down. There is no tracer advection through it so that the advective
+fluxes through the surface are also zero
+\item[linear free surface:]
+  (\np{ln\_linssh}\forcode{ = .true.})
+  the first level thickness is constant in time:
+  the vertical boundary condition is applied at the fixed surface $z=0$ rather than on the moving surface $z=\eta$.
+  There is a non-zero advective flux which is set for all advection schemes as
+  $\left. {\tau _w } \right|_{k=1/2} =T_{k=1} $,
+  $i.e.$ the product of surface velocity (at $z=0$) by the first level tracer value.
+\item[non-linear free surface:]
+  (\np{ln\_linssh}\forcode{ = .false.})
+  convergence/divergence in the first ocean level moves the free surface up/down.
+  There is no tracer advection through it so that the advective fluxes through the surface are also zero.
 \end{description}
+In all cases, this boundary condition retains local conservation of tracer.
+Global conservation is obtained in non-linear free surface case,
+but \textit{not} in the linear free surface case. Nevertheless, in the latter case,
+it is achieved to a good approximation since the non-conservative
+term is the product of the time derivative of the tracer and the free surface
+height, two quantities that are not correlated \citep{Roullet_Madec_JGR00,Griffies_al_MWR01,Campin2004}.
+The velocity field that appears in (\autoref{eq:tra_adv}) and (\autoref{eq:tra_adv_zco})
+is the centred (\textit{now}) \textit{effective} ocean velocity, $i.e.$ the \textit{eulerian} velocity
+(see \autoref{chap:DYN}) plus the eddy induced velocity (\textit{eiv})
+and/or the mixed layer eddy induced velocity (\textit{eiv})
+when those parameterisations are used (see \autoref{chap:LDF}).
+Several tracer advection scheme are proposed, namely
+a $2^{nd}$ or $4^{th}$ order centred schemes (CEN),
+In all cases, this boundary condition retains local conservation of tracer.
+Global conservation is obtained in non-linear free surface case, but \textit{not} in the linear free surface case.
+Nevertheless, in the latter case, it is achieved to a good approximation since
+the non-conservative term is the product of the time derivative of the tracer and the free surface height,
+two quantities that are not correlated \citep{Roullet_Madec_JGR00,Griffies_al_MWR01,Campin2004}.
+The velocity field that appears in (\autoref{eq:tra_adv} and \autoref{eq:tra_adv_zco})
+is the centred (\textit{now}) \textit{effective} ocean velocity,
+$i.e.$ the \textit{eulerian} velocity (see \autoref{chap:DYN}) plus
+the eddy induced velocity (\textit{eiv}) and/or
+the mixed layer eddy induced velocity (\textit{eiv}) when
+those parameterisations are used (see \autoref{chap:LDF}).
+Several tracer advection scheme are proposed, namely a $2^{nd}$ or $4^{th}$ order centred schemes (CEN),
 a $2^{nd}$ or $4^{th}$ order Flux Corrected Transport scheme (FCT),
 a Monotone Upstream Scheme for Conservative Laws scheme (MUSCL),
 a $3^{rd}$ Upstream Biased Scheme (UBS, also often called UP3), and
 a Quadratic Upstream Interpolation for Convective Kinematics with
+a $3^{rd}$ Upstream Biased Scheme (UBS, also often called UP3),
+and a Quadratic Upstream Interpolation for Convective Kinematics with
 Estimated Streaming Terms scheme (QUICKEST).
 The choice is made in the \textit{\ngn{namtra\_adv}} namelist, by
+setting to \forcode{.true.} one of the logicals \textit{ln\_traadv\_xxx}.
 The corresponding code can be found in the \mdl{traadv\_xxx} module,
 where \textit{xxx} is a 3 or 4 letter acronym corresponding to each scheme.
 By default ($i.e.$ in the reference namelist, \ngn{namelist\_ref}), all the logicals
+are set to \forcode{.false.}. If the user does not select an advection scheme
 in the configuration namelist (\ngn{namelist\_cfg}), the tracers will \textit{not} be advected !
 Details of the advection schemes are given below. The choosing an advection scheme
+is a complex matter which depends on the model physics, model resolution,
+The choice is made in the \textit{\ngn{namtra\_adv}} namelist,
+by setting to \forcode{.true.} one of the logicals \textit{ln\_traadv\_xxx}.
+The corresponding code can be found in the \mdl{traadv\_xxx} module,
+where \textit{xxx} is a 3 or 4 letter acronym corresponding to each scheme.
+By default ($i.e.$ in the reference namelist, \ngn{namelist\_ref}), all the logicals are set to \forcode{.false.}.
+If the user does not select an advection scheme in the configuration namelist (\ngn{namelist\_cfg}),
+the tracers will \textit{not} be advected!
+Details of the advection schemes are given below.
+The choosing an advection scheme is a complex matter which depends on the model physics, model resolution,
 type of tracer, as well as the issue of numerical cost. In particular, we note that
 (1) CEN and FCT schemes require an explicit diffusion operator
+while the other schemes are diffusive enough so that they do not necessarily need additional diffusion ;
+(1) CEN and FCT schemes require an explicit diffusion operator while the other schemes are diffusive enough so that
+they do not necessarily need additional diffusion;
 (2) CEN and UBS are not \textit{positive} schemes
 \footnote{negative values can appear in an initially strictly positive tracer field
+which is advected}
+, implying that false extrema are permitted. Their use is not recommended on passive tracers ;
 (3) It is recommended that the same advection-diffusion scheme is
+used on both active and passive tracers. Indeed, if a source or sink of a
+passive tracer depends on an active one, the difference of treatment of
 active and passive tracers can create very nice-looking frontal structures
+that are pure numerical artefacts. Nevertheless, most of our users set a different
 treatment on passive and active tracers, that's the reason why this possibility
+is offered. We strongly suggest them to perform a sensitivity experiment
 using a same treatment to assess the robustness of their results.
+\footnote{negative values can appear in an initially strictly positive tracer field which is advected},
+implying that false extrema are permitted.
+Their use is not recommended on passive tracers;
+(3) It is recommended that the same advection-diffusion scheme is used on both active and passive tracers.
+Indeed, if a source or sink of a passive tracer depends on an active one,
+the difference of treatment of active and passive tracers can create very nice-looking frontal structures that
+are pure numerical artefacts.
+Nevertheless, most of our users set a different treatment on passive and active tracers,
+that's the reason why this possibility is offered.
+We strongly suggest them to perform a sensitivity experiment using a same treatment to
+assess the robustness of their results.
 % -------------------------------------------------------------------------------------------------------------
 …
 %        2nd order centred scheme
 The centred advection scheme (CEN) is used when \np{ln\_traadv\_cen}\forcode{ = .true.}.
 Its order ($2^{nd}$ or $4^{th}$) can be chosen independently on horizontal (iso-level)
+and vertical direction by setting \np{nn\_cen\_h} and \np{nn\_cen\_v} to $2$ or $4$.
+The centred advection scheme (CEN) is used when \np{ln\_traadv\_cen}\forcode{ = .true.}.
+Its order ($2^{nd}$ or $4^{th}$) can be chosen independently on horizontal (iso-level) and vertical direction by
+setting \np{nn\_cen\_h} and \np{nn\_cen\_v} to $2$ or $4$.
 CEN implementation can be found in the \mdl{traadv\_cen} module.
 In the $2^{nd}$ order centred formulation (CEN2), the tracer at velocity points
+is evaluated as the mean of the two neighbouring $T$-point values.
+In the $2^{nd}$ order centred formulation (CEN2), the tracer at velocity points is evaluated as the mean of
+the two neighbouring $T$-point values.
 For example, in the $i$-direction :
 \begin{equation} \label{eq:tra_adv_cen2}
 …
 \end{equation}
+CEN2 is non diffusive ($i.e.$ it conserves the tracer variance, $\tau^2)$
+but dispersive ($i.e.$ it may create false extrema). It is therefore notoriously
+noisy and must be used in conjunction with an explicit diffusion operator to
+produce a sensible solution. The associated time-stepping is performed using
+a leapfrog scheme in conjunction with an Asselin time-filter, so $T$ in
+(\autoref{eq:tra_adv_cen2}) is the \textit{now} tracer value.
+Note that using the CEN2, the overall tracer advection is of second
+order accuracy since both (\autoref{eq:tra_adv}) and (\autoref{eq:tra_adv_cen2})
+have this order of accuracy.
+CEN2 is non diffusive ($i.e.$ it conserves the tracer variance, $\tau^2)$ but dispersive
+($i.e.$ it may create false extrema).
+It is therefore notoriously noisy and must be used in conjunction with an explicit diffusion operator to
+produce a sensible solution.
+The associated time-stepping is performed using a leapfrog scheme in conjunction with an Asselin time-filter,
+so $T$ in (\autoref{eq:tra_adv_cen2}) is the \textit{now} tracer value.
+Note that using the CEN2, the overall tracer advection is of second order accuracy since
+both (\autoref{eq:tra_adv}) and (\autoref{eq:tra_adv_cen2}) have this order of accuracy.
 %        4nd order centred scheme
 In the $4^{th}$ order formulation (CEN4), tracer values are evaluated at u- and v-points as
 a $4^{th}$ order interpolation, and thus depend on the four neighbouring $T$-points.
+In the $4^{th}$ order formulation (CEN4), tracer values are evaluated at u- and v-points as
+a $4^{th}$ order interpolation, and thus depend on the four neighbouring $T$-points.
 For example, in the $i$-direction:
 \begin{equation} \label{eq:tra_adv_cen4}
 …
 =\overline{   T - \frac{1}{6}\,\delta _i \left[ \delta_{i+1/2}[T] \,\right]   }^{\,i+1/2}
 \end{equation}
+In the vertical direction (\np{nn\_cen\_v}\forcode{ = 4}), a $4^{th}$ COMPACT interpolation
+has been prefered \citep{Demange_PhD2014}.
+In the COMPACT scheme, both the field and its derivative are interpolated,
+which leads, after a matrix inversion, spectral characteristics
+similar to schemes of higher order \citep{Lele_JCP1992}.
+In the vertical direction (\np{nn\_cen\_v}\forcode{ = 4}),
+a $4^{th}$ COMPACT interpolation has been prefered \citep{Demange_PhD2014}.
+In the COMPACT scheme, both the field and its derivative are interpolated, which leads, after a matrix inversion,
+spectral characteristics similar to schemes of higher order \citep{Lele_JCP1992}.
+Strictly speaking, the CEN4 scheme is not a $4^{th}$ order advection scheme
+but a $4^{th}$ order evaluation of advective fluxes, since the divergence of
+advective fluxes \autoref{eq:tra_adv} is kept at $2^{nd}$ order.
+The expression \textit{$4^{th}$ order scheme} used in oceanographic literature
+is usually associated with the scheme presented here.
+Introducing a \forcode{.true.} $4^{th}$ order advection scheme is feasible but,
+for consistency reasons, it requires changes in the discretisation of the tracer
+advection together with changes in the continuity equation,
+and the momentum advection and pressure terms.
+A direct consequence of the pseudo-fourth order nature of the scheme is that
+it is not non-diffusive, $i.e.$ the global variance of a tracer is not preserved using CEN4.
+Furthermore, it must be used in conjunction with an explicit diffusion operator
+to produce a sensible solution.
+As in CEN2 case, the time-stepping is performed using a leapfrog scheme in conjunction
+with an Asselin time-filter, so $T$ in (\autoref{eq:tra_adv_cen4}) is the \textit{now} tracer.
+At a $T$-grid cell adjacent to a boundary (coastline, bottom and surface),
+an additional hypothesis must be made to evaluate $\tau _u^{cen4}$.
+This hypothesis usually reduces the order of the scheme.
+Here we choose to set the gradient of $T$ across the boundary to zero.
+Alternative conditions can be specified, such as a reduction to a second order scheme
+for these near boundary grid points.
+Strictly speaking, the CEN4 scheme is not a $4^{th}$ order advection scheme but
+a $4^{th}$ order evaluation of advective fluxes,
+since the divergence of advective fluxes \autoref{eq:tra_adv} is kept at $2^{nd}$ order.
+The expression \textit{$4^{th}$ order scheme} used in oceanographic literature is usually associated with
+the scheme presented here.
+Introducing a \forcode{.true.} $4^{th}$ order advection scheme is feasible but, for consistency reasons,
+it requires changes in the discretisation of the tracer advection together with changes in the continuity equation,
+and the momentum advection and pressure terms.
+A direct consequence of the pseudo-fourth order nature of the scheme is that it is not non-diffusive,
+$i.e.$ the global variance of a tracer is not preserved using CEN4.
+Furthermore, it must be used in conjunction with an explicit diffusion operator to produce a sensible solution.
+As in CEN2 case, the time-stepping is performed using a leapfrog scheme in conjunction with an Asselin time-filter,
+so $T$ in (\autoref{eq:tra_adv_cen4}) is the \textit{now} tracer.
+At a $T$-grid cell adjacent to a boundary (coastline, bottom and surface),
+an additional hypothesis must be made to evaluate $\tau _u^{cen4}$.
+This hypothesis usually reduces the order of the scheme.
+Here we choose to set the gradient of $T$ across the boundary to zero.
+Alternative conditions can be specified, such as a reduction to a second order scheme for
+these near boundary grid points.
 % -------------------------------------------------------------------------------------------------------------
 …
 \label{subsec:TRA_adv_tvd}
 The Flux Corrected Transport schemes (FCT) is used when \np{ln\_traadv\_fct}\forcode{ = .true.}.
 Its order ($2^{nd}$ or $4^{th}$) can be chosen independently on horizontal (iso-level)
 and vertical direction by setting \np{nn\_fct\_h} and \np{nn\_fct\_v} to $2$ or $4$.
+The Flux Corrected Transport schemes (FCT) is used when \np{ln\_traadv\_fct}\forcode{ = .true.}.
+Its order ($2^{nd}$ or $4^{th}$) can be chosen independently on horizontal (iso-level) and vertical direction by
+setting \np{nn\_fct\_h} and \np{nn\_fct\_v} to $2$ or $4$.
 FCT implementation can be found in the \mdl{traadv\_fct} module.
+In FCT formulation, the tracer at velocity points is evaluated using a combination of
+an upstream and a centred scheme. For example, in the $i$-direction :
+In FCT formulation, the tracer at velocity points is evaluated using a combination of an upstream and
+a centred scheme.
+For example, in the $i$-direction :
 \begin{equation} \label{eq:tra_adv_fct}
 \begin{split}
 …
 \end{split}
 \end{equation}
+where $c_u$ is a flux limiter function taking values between 0 and 1.
+The FCT order is the one of the centred scheme used ($i.e.$ it depends on the setting of
+\np{nn\_fct\_h} and \np{nn\_fct\_v}.
+There exist many ways to define $c_u$, each corresponding to a different
+FCT scheme. The one chosen in \NEMO is described in \citet{Zalesak_JCP79}.
+$c_u$ only departs from $1$ when the advective term produces a local extremum in the tracer field.
+The resulting scheme is quite expensive but \emph{positive}.
+It can be used on both active and passive tracers.
+A comparison of FCT-2 with MUSCL and a MPDATA scheme can be found in \citet{Levy_al_GRL01}.
+An additional option has been added controlled by \np{nn\_fct\_zts}. By setting this integer to
+a value larger than zero, a $2^{nd}$ order FCT scheme is used on both horizontal and vertical direction,
+but on the latter, a split-explicit time stepping is used, with a number of sub-timestep equals
+to \np{nn\_fct\_zts}. This option can be useful when the size of the timestep is limited
+by vertical advection \citep{Lemarie_OM2015}. Note that in this case, a similar split-explicit
+time stepping should be used on vertical advection of momentum to insure a better stability
+(see \autoref{subsec:DYN_zad}).
+For stability reasons (see \autoref{chap:STP}), $\tau _u^{cen}$ is evaluated in (\autoref{eq:tra_adv_fct})
+using the \textit{now} tracer while $\tau _u^{ups}$ is evaluated using the \textit{before} tracer. In other words,
+the advective part of the scheme is time stepped with a leap-frog scheme
+where $c_u$ is a flux limiter function taking values between 0 and 1.
+The FCT order is the one of the centred scheme used
+($i.e.$ it depends on the setting of \np{nn\_fct\_h} and \np{nn\_fct\_v}).
+There exist many ways to define $c_u$, each corresponding to a different FCT scheme.
+The one chosen in \NEMO is described in \citet{Zalesak_JCP79}.
+$c_u$ only departs from $1$ when the advective term produces a local extremum in the tracer field.
+The resulting scheme is quite expensive but \emph{positive}.
+It can be used on both active and passive tracers.
+A comparison of FCT-2 with MUSCL and a MPDATA scheme can be found in \citet{Levy_al_GRL01}.
+An additional option has been added controlled by \np{nn\_fct\_zts}.
+By setting this integer to a value larger than zero,
+a $2^{nd}$ order FCT scheme is used on both horizontal and vertical direction, but on the latter,
+a split-explicit time stepping is used, with a number of sub-timestep equals to \np{nn\_fct\_zts}.
+This option can be useful when the size of the timestep is limited by vertical advection \citep{Lemarie_OM2015}.
+Note that in this case, a similar split-explicit time stepping should be used on vertical advection of momentum to
+insure a better stability (see \autoref{subsec:DYN_zad}).
+For stability reasons (see \autoref{chap:STP}),
+$\tau _u^{cen}$ is evaluated in (\autoref{eq:tra_adv_fct}) using the \textit{now} tracer while
+$\tau _u^{ups}$ is evaluated using the \textit{before} tracer.
+In other words, the advective part of the scheme is time stepped with a leap-frog scheme
 while a forward scheme is used for the diffusive part.
 …
 \label{subsec:TRA_adv_mus}
 The Monotone Upstream Scheme for Conservative Laws (MUSCL) is used when \np{ln\_traadv\_mus}\forcode{ = .true.}.
+The Monotone Upstream Scheme for Conservative Laws (MUSCL) is used when \np{ln\_traadv\_mus}\forcode{ = .true.}.
 MUSCL implementation can be found in the \mdl{traadv\_mus} module.
+MUSCL has been first implemented in \NEMO by \citet{Levy_al_GRL01}. In its formulation, the tracer at velocity points
+is evaluated assuming a linear tracer variation between two $T$-points
+(\autoref{fig:adv_scheme}). For example, in the $i$-direction :
+MUSCL has been first implemented in \NEMO by \citet{Levy_al_GRL01}.
+In its formulation, the tracer at velocity points is evaluated assuming a linear tracer variation between
+two $T$-points (\autoref{fig:adv_scheme}).
+For example, in the $i$-direction :
 \begin{equation} \label{eq:tra_adv_mus}
    \tau _u^{mus} = \left\{      \begin{aligned}
 …
    \end{aligned}    \right.
 \end{equation}
 where $\widetilde{\partial _i \tau}$ is the slope of the tracer on which a limitation
 is imposed to ensure the \textit{positive} character of the scheme.
 The time stepping is performed using a forward scheme, that is the \textit{before}
 tracer field is used to evaluate $\tau _u^{mus}$.
 For an ocean grid point adjacent to land and where the ocean velocity is
+directed toward land, an upstream flux is used. This choice ensure
+the \textit{positive} character of the scheme.
 In addition, fluxes round a grid-point where a runoff is applied can optionally be
 computed using upstream fluxes (\np{ln\_mus\_ups}\forcode{ = .true.}).
+where $\widetilde{\partial _i \tau}$ is the slope of the tracer on which a limitation is imposed to
+ensure the \textit{positive} character of the scheme.
+The time stepping is performed using a forward scheme,
+that is the \textit{before} tracer field is used to evaluate $\tau _u^{mus}$.
+For an ocean grid point adjacent to land and where the ocean velocity is directed toward land,
+an upstream flux is used.
+This choice ensure the \textit{positive} character of the scheme.
+In addition, fluxes round a grid-point where a runoff is applied can optionally be computed using upstream fluxes
+(\np{ln\_mus\_ups}\forcode{ = .true.}).
 % -------------------------------------------------------------------------------------------------------------
 …
 \label{subsec:TRA_adv_ubs}
 The Upstream-Biased Scheme (UBS) is used when \np{ln\_traadv\_ubs}\forcode{ = .true.}.
+The Upstream-Biased Scheme (UBS) is used when \np{ln\_traadv\_ubs}\forcode{ = .true.}.
 UBS implementation can be found in the \mdl{traadv\_mus} module.
 The UBS scheme, often called UP3, is also known as the Cell Averaged QUICK scheme
 (Quadratic Upstream Interpolation for Convective Kinematics). It is an upstream-biased
+third order scheme based on an upstream-biased parabolic interpolation.
 For example, in the $i$-direction :
+The UBS scheme, often called UP3, is also known as the Cell Averaged QUICK scheme
+(Quadratic Upstream Interpolation for Convective Kinematics).
+It is an upstream-biased third order scheme based on an upstream-biased parabolic interpolation.
+For example, in the $i$-direction:
 \begin{equation} \label{eq:tra_adv_ubs}
    \tau _u^{ubs} =\overline T ^{i+1/2}-\;\frac{1}{6} \left\{
 …
 where $\tau "_i =\delta _i \left[ {\delta _{i+1/2} \left[ \tau \right]} \right]$.
+This results in a dissipatively dominant (i.e. hyper-diffusive) truncation
+error \citep{Shchepetkin_McWilliams_OM05}. The overall performance of
+ the advection scheme is similar to that reported in \cite{Farrow1995}.
+It is a relatively good compromise between accuracy and smoothness.
+Nevertheless the scheme is not \emph{positive}, meaning that false extrema are permitted,
+but the amplitude of such are significantly reduced over the centred second
+or fourth order method. therefore it is not recommended that it should be
+applied to a passive tracer that requires positivity.
+The intrinsic diffusion of UBS makes its use risky in the vertical direction
+where the control of artificial diapycnal fluxes is of paramount importance \citep{Shchepetkin_McWilliams_OM05, Demange_PhD2014}.
+Therefore the vertical flux is evaluated using either a $2^nd$ order FCT scheme
+or a $4^th$ order COMPACT scheme (\np{nn\_cen\_v}\forcode{ = 2 or 4}).
+For stability reasons  (see \autoref{chap:STP}),
+the first term  in \autoref{eq:tra_adv_ubs} (which corresponds to a second order
+centred scheme) is evaluated using the \textit{now} tracer (centred in time)
+while the second term (which is the diffusive part of the scheme), is
+evaluated using the \textit{before} tracer (forward in time).
+This choice is discussed by \citet{Webb_al_JAOT98} in the context of the
+QUICK advection scheme. UBS and QUICK schemes only differ
+by one coefficient. Replacing 1/6 with 1/8 in \autoref{eq:tra_adv_ubs}
+leads to the QUICK advection scheme \citep{Webb_al_JAOT98}.
+This option is not available through a namelist parameter, since the
+/6 coefficient is hard coded. Nevertheless it is quite easy to make the
+substitution in the \mdl{traadv\_ubs} module and obtain a QUICK scheme.
+This results in a dissipatively dominant (i.e. hyper-diffusive) truncation error
+\citep{Shchepetkin_McWilliams_OM05}.
+The overall performance of the advection scheme is similar to that reported in \cite{Farrow1995}.
+It is a relatively good compromise between accuracy and smoothness.
+Nevertheless the scheme is not \emph{positive}, meaning that false extrema are permitted,
+but the amplitude of such are significantly reduced over the centred second or fourth order method.
+Therefore it is not recommended that it should be applied to a passive tracer that requires positivity.
+The intrinsic diffusion of UBS makes its use risky in the vertical direction where
+the control of artificial diapycnal fluxes is of paramount importance
+\citep{Shchepetkin_McWilliams_OM05, Demange_PhD2014}.
+Therefore the vertical flux is evaluated using either a $2^nd$ order FCT scheme or a $4^th$ order COMPACT scheme
+(\np{nn\_cen\_v}\forcode{ = 2 or 4}).
+For stability reasons (see \autoref{chap:STP}), the first term  in \autoref{eq:tra_adv_ubs}
+(which corresponds to a second order centred scheme)
+is evaluated using the \textit{now} tracer (centred in time) while the second term
+(which is the diffusive part of the scheme),
+is evaluated using the \textit{before} tracer (forward in time).
+This choice is discussed by \citet{Webb_al_JAOT98} in the context of the QUICK advection scheme.
+UBS and QUICK schemes only differ by one coefficient.
+Replacing 1/6 with 1/8 in \autoref{eq:tra_adv_ubs} leads to the QUICK advection scheme \citep{Webb_al_JAOT98}.
+This option is not available through a namelist parameter, since the 1/6 coefficient is hard coded.
+Nevertheless it is quite easy to make the substitution in the \mdl{traadv\_ubs} module and obtain a QUICK scheme.
 Note that it is straightforward to rewrite \autoref{eq:tra_adv_ubs} as follows:
 …
 \end{equation}
 \autoref{eq:traadv_ubs2} has several advantages. Firstly, it clearly reveals
+that the UBS scheme is based on the fourth order scheme to which an
+upstream-biased diffusion term is added. Secondly, this emphasises that the
+$4^{th}$ order part (as well as the $2^{nd}$ order part as stated above) has
+to be evaluated at the \emph{now} time step using \autoref{eq:tra_adv_ubs}.
 Thirdly, the diffusion term is in fact a biharmonic operator with an eddy
 coefficient which is simply proportional to the velocity:
+ $A_u^{lm}= \frac{1}{12}\,{e_{1u}}^3\,|u|$. Note the current version of NEMO uses
 the computationally more efficient formulation \autoref{eq:tra_adv_ubs}.
+\autoref{eq:traadv_ubs2} has several advantages.
+Firstly, it clearly reveals that the UBS scheme is based on the fourth order scheme to which
+an upstream-biased diffusion term is added.
+Secondly, this emphasises that the $4^{th}$ order part (as well as the $2^{nd}$ order part as stated above) has to
+be evaluated at the \emph{now} time step using \autoref{eq:tra_adv_ubs}.
+Thirdly, the diffusion term is in fact a biharmonic operator with an eddy coefficient which
+is simply proportional to the velocity:
+$A_u^{lm}= \frac{1}{12}\,{e_{1u}}^3\,|u|$.
+Note the current version of NEMO uses the computationally more efficient formulation \autoref{eq:tra_adv_ubs}.
 % -------------------------------------------------------------------------------------------------------------
 …
 \label{subsec:TRA_adv_qck}
+The Quadratic Upstream Interpolation for Convective Kinematics with
+Estimated Streaming Terms (QUICKEST) scheme proposed by \citet{Leonard1979}
+is used when \np{ln\_traadv\_qck}\forcode{ = .true.}.
+The Quadratic Upstream Interpolation for Convective Kinematics with Estimated Streaming Terms (QUICKEST) scheme
+proposed by \citet{Leonard1979} is used when \np{ln\_traadv\_qck}\forcode{ = .true.}.
 QUICKEST implementation can be found in the \mdl{traadv\_qck} module.
+QUICKEST is the third order Godunov scheme which is associated with the ULTIMATE QUICKEST
+limiter \citep{Leonard1991}. It has been implemented in NEMO by G. Reffray
+(MERCATOR-ocean) and can be found in the \mdl{traadv\_qck} module.
+The resulting scheme is quite expensive but \emph{positive}.
+It can be used on both active and passive tracers.
+However, the intrinsic diffusion of QCK makes its use risky in the vertical
+direction where the control of artificial diapycnal fluxes is of paramount importance.
+Therefore the vertical flux is evaluated using the CEN2 scheme.
+This no longer guarantees the positivity of the scheme.
+The use of FCT in the vertical direction (as for the UBS case) should be implemented
+to restore this property.
+QUICKEST is the third order Godunov scheme which is associated with the ULTIMATE QUICKEST limiter
+\citep{Leonard1991}.
+It has been implemented in NEMO by G. Reffray (MERCATOR-ocean) and can be found in the \mdl{traadv\_qck} module.
+The resulting scheme is quite expensive but \emph{positive}.
+It can be used on both active and passive tracers.
+However, the intrinsic diffusion of QCK makes its use risky in the vertical direction where
+the control of artificial diapycnal fluxes is of paramount importance.
+Therefore the vertical flux is evaluated using the CEN2 scheme.
+This no longer guarantees the positivity of the scheme.
+The use of FCT in the vertical direction (as for the UBS case) should be implemented to restore this property.
 %%%gmcomment   :  Cross term are missing in the current implementation....
 …
 Options are defined through the \ngn{namtra\_ldf} namelist variables.
 They are regrouped in four items, allowing to specify
 $(i)$   the type of operator used (none, laplacian, bilaplacian),
 $(ii)$  the direction along which the operator acts (iso-level, horizontal, iso-neutral),
 $(iii)$ some specific options related to the rotated operators ($i.e.$ non-iso-level operator), and
+$(i)$   the type of operator used (none, laplacian, bilaplacian),
+$(ii)$  the direction along which the operator acts (iso-level, horizontal, iso-neutral),
+$(iii)$ some specific options related to the rotated operators ($i.e.$ non-iso-level operator), and
 $(iv)$  the specification of eddy diffusivity coefficient (either constant or variable in space and time).
+Item $(iv)$ will be described in \autoref{chap:LDF} .
+The direction along which the operators act is defined through the slope between this direction and the iso-level surfaces.
+The slope is computed in the \mdl{ldfslp} module and will also be described in \autoref{chap:LDF}.
+The lateral diffusion of tracers is evaluated using a forward scheme,
+$i.e.$ the tracers appearing in its expression are the \textit{before} tracers in time,
+except for the pure vertical component that appears when a rotation tensor is used.
+This latter component is solved implicitly together with the vertical diffusion term (see \autoref{chap:STP}).
+When \np{ln\_traldf\_msc}\forcode{ = .true.}, a Method of Stabilizing Correction is used in which
+Item $(iv)$ will be described in \autoref{chap:LDF}.
+The direction along which the operators act is defined through the slope between
+this direction and the iso-level surfaces.
+The slope is computed in the \mdl{ldfslp} module and will also be described in \autoref{chap:LDF}.
+The lateral diffusion of tracers is evaluated using a forward scheme,
+$i.e.$ the tracers appearing in its expression are the \textit{before} tracers in time,
+except for the pure vertical component that appears when a rotation tensor is used.
+This latter component is solved implicitly together with the vertical diffusion term (see \autoref{chap:STP}).
+When \np{ln\_traldf\_msc}\forcode{ = .true.}, a Method of Stabilizing Correction is used in which
 the pure vertical component is split into an explicit and an implicit part \citep{Lemarie_OM2012}.
 …
 Three operator options are proposed and, one and only one of them must be selected:
 \begin{description}
+\item [\np{ln\_traldf\_NONE}\forcode{ = .true.}]: no operator selected, the lateral diffusive tendency will not be
+applied to the tracer equation. This option can be used when the selected advection scheme
+is diffusive enough (MUSCL scheme for example).
+\item [\np{ln\_traldf\_lap}\forcode{ = .true.}]: a laplacian operator is selected. This harmonic operator
+takes the following expression:  $\mathpzc{L}(T)=\nabla \cdot A_{ht}\;\nabla T $,
+where the gradient operates along the selected direction (see \autoref{subsec:TRA_ldf_dir}),
+and $A_{ht}$ is the eddy diffusivity coefficient expressed in $m^2/s$ (see \autoref{chap:LDF}).
+\item [\np{ln\_traldf\_blp}\forcode{ = .true.}]: a bilaplacian operator is selected. This biharmonic operator
+takes the following expression:
+$\mathpzc{B}=- \mathpzc{L}\left(\mathpzc{L}(T) \right) = -\nabla \cdot b\nabla \left( {\nabla \cdot b\nabla T} \right)$
+where the gradient operats along the selected direction,
+and $b^2=B_{ht}$ is the eddy diffusivity coefficient expressed in $m^4/s$  (see \autoref{chap:LDF}).
+In the code, the bilaplacian operator is obtained by calling the laplacian twice.
+\item[\np{ln\_traldf\_NONE}\forcode{ = .true.}:]
+  no operator selected, the lateral diffusive tendency will not be applied to the tracer equation.
+  This option can be used when the selected advection scheme is diffusive enough (MUSCL scheme for example).
+\item[\np{ln\_traldf\_lap}\forcode{ = .true.}:]
+  a laplacian operator is selected.
+  This harmonic operator takes the following expression:  $\mathpzc{L}(T)=\nabla \cdot A_{ht}\;\nabla T $,
+  where the gradient operates along the selected direction (see \autoref{subsec:TRA_ldf_dir}),
+  and $A_{ht}$ is the eddy diffusivity coefficient expressed in $m^2/s$ (see \autoref{chap:LDF}).
+\item[\np{ln\_traldf\_blp}\forcode{ = .true.}]:
+  a bilaplacian operator is selected.
+  This biharmonic operator takes the following expression:
+  $\mathpzc{B}=- \mathpzc{L}\left(\mathpzc{L}(T) \right) = -\nabla \cdot b\nabla \left( {\nabla \cdot b\nabla T} \right)$
+  where the gradient operats along the selected direction,
+  and $b^2=B_{ht}$ is the eddy diffusivity coefficient expressed in $m^4/s$  (see \autoref{chap:LDF}).
+  In the code, the bilaplacian operator is obtained by calling the laplacian twice.
 \end{description}
 Both laplacian and bilaplacian operators ensure the total tracer variance decrease.
 Their primary role is to provide strong dissipation at the smallest scale supported
+by the grid while minimizing the impact on the larger scale features.
 The main difference between the two operators is the scale selectiveness.
 The bilaplacian damping time ($i.e.$ its spin down time) scales like $\lambda^{-4}$
+for disturbances of wavelength $\lambda$ (so that short waves damped more rapidelly than long ones),
+Both laplacian and bilaplacian operators ensure the total tracer variance decrease.
+Their primary role is to provide strong dissipation at the smallest scale supported by the grid while
+minimizing the impact on the larger scale features.
+The main difference between the two operators is the scale selectiveness.
+The bilaplacian damping time ($i.e.$ its spin down time) scales like $\lambda^{-4}$ for
+disturbances of wavelength $\lambda$ (so that short waves damped more rapidelly than long ones),
 whereas the laplacian damping time scales only like $\lambda^{-2}$.
 …
 \label{subsec:TRA_ldf_dir}
 The choice of a direction of action determines the form of operator used.
 The operator is a simple (re-entrant) laplacian acting in the (\textbf{i},\textbf{j}) plane
+when iso-level option is used (\np{ln\_traldf\_lev}\forcode{ = .true.})
+or when a horizontal ($i.e.$ geopotential) operator is demanded in \textit{z}-coordinate
 (\np{ln\_traldf\_hor} and \np{ln\_zco} equal \forcode{.true.}).
+The choice of a direction of action determines the form of operator used.
+The operator is a simple (re-entrant) laplacian acting in the (\textbf{i},\textbf{j}) plane when
+iso-level option is used (\np{ln\_traldf\_lev}\forcode{ = .true.}) or
+when a horizontal ($i.e.$ geopotential) operator is demanded in \textit{z}-coordinate
+(\np{ln\_traldf\_hor} and \np{ln\_zco} equal \forcode{.true.}).
 The associated code can be found in the \mdl{traldf\_lap\_blp} module.
+The operator is a rotated (re-entrant) laplacian when the direction along which it acts
+does not coincide with the iso-level surfaces,
+that is when standard or triad iso-neutral option is used (\np{ln\_traldf\_iso} or
+ \np{ln\_traldf\_triad} equals \forcode{.true.}, see \mdl{traldf\_iso} or \mdl{traldf\_triad} module, resp.),
+or when a horizontal ($i.e.$ geopotential) operator is demanded in \textit{s}-coordinate
+The operator is a rotated (re-entrant) laplacian when
+the direction along which it acts does not coincide with the iso-level surfaces,
+that is when standard or triad iso-neutral option is used
+(\np{ln\_traldf\_iso} or \np{ln\_traldf\_triad} equals \forcode{.true.},
+see \mdl{traldf\_iso} or \mdl{traldf\_triad} module, resp.), or
+when a horizontal ($i.e.$ geopotential) operator is demanded in \textit{s}-coordinate
 (\np{ln\_traldf\_hor} and \np{ln\_sco} equal \forcode{.true.})
 \footnote{In this case, the standard iso-neutral operator will be automatically selected}.
 In that case, a rotation is applied to the gradient(s) that appears in the operator
 so that diffusive fluxes acts on the three spatial direction.
 The resulting discret form of the three operators (one iso-level and two rotated one)
 is given in the next two sub-sections.
+\footnote{In this case, the standard iso-neutral operator will be automatically selected}.
+In that case, a rotation is applied to the gradient(s) that appears in the operator so that
+diffusive fluxes acts on the three spatial direction.
+The resulting discret form of the three operators (one iso-level and two rotated one) is given in
+the next two sub-sections.
 …
 + \delta _{j}\left[ A_v^{lT} \;  \frac{e_{1v}\,e_{3v}}{e_{2v}} \;\delta _{j+1/2} [T] \right]  \;\right)
 \end{equation}
 where  $b_t$=$e_{1t}\,e_{2t}\,e_{3t}$  is the volume of $T$-cells
+and where zero diffusive fluxes is assumed across solid boundaries,
 first (and third in bilaplacian case) horizontal tracer derivative are masked.
 It is implemented in the \rou{traldf\_lap} subroutine found in the \mdl{traldf\_lap} module.
 The module also contains \rou{traldf\_blp}, the subroutine calling twice \rou{traldf\_lap}
 in order to compute the iso-level bilaplacian operator.
 It is a \emph{horizontal} operator ($i.e.$ acting along geopotential surfaces) in the $z$-coordinate
+with or without partial steps, but is simply an iso-level operator in the $s$-coordinate.
 It is thus used when, in addition to \np{ln\_traldf\_lap} or \np{ln\_traldf\_blp}\forcode{ = .true.},
 we have \np{ln\_traldf\_lev}\forcode{ = .true.} or \np{ln\_traldf\_hor}~=~\np{ln\_zco}\forcode{ = .true.}.
 In both cases, it significantly contributes to diapycnal mixing.
+where  $b_t$=$e_{1t}\,e_{2t}\,e_{3t}$  is the volume of $T$-cells and
+where zero diffusive fluxes is assumed across solid boundaries,
+first (and third in bilaplacian case) horizontal tracer derivative are masked.
+It is implemented in the \rou{traldf\_lap} subroutine found in the \mdl{traldf\_lap} module.
+The module also contains \rou{traldf\_blp}, the subroutine calling twice \rou{traldf\_lap} in order to
+compute the iso-level bilaplacian operator.
+It is a \emph{horizontal} operator ($i.e.$ acting along geopotential surfaces) in
+the $z$-coordinate with or without partial steps, but is simply an iso-level operator in the $s$-coordinate.
+It is thus used when, in addition to \np{ln\_traldf\_lap} or \np{ln\_traldf\_blp}\forcode{ = .true.},
+we have \np{ln\_traldf\_lev}\forcode{ = .true.} or \np{ln\_traldf\_hor}~=~\np{ln\_zco}\forcode{ = .true.}.
+In both cases, it significantly contributes to diapycnal mixing.
 It is therefore never recommended, even when using it in the bilaplacian case.
+Note that in the partial step $z$-coordinate (\np{ln\_zps}\forcode{ = .true.}), tracers in horizontally
+adjacent cells are located at different depths in the vicinity of the bottom.
+In this case, horizontal derivatives in (\autoref{eq:tra_ldf_lap}) at the bottom level
+require a specific treatment. They are calculated in the \mdl{zpshde} module,
+described in \autoref{sec:TRA_zpshde}.
+Note that in the partial step $z$-coordinate (\np{ln\_zps}\forcode{ = .true.}),
+tracers in horizontally adjacent cells are located at different depths in the vicinity of the bottom.
+In this case, horizontal derivatives in (\autoref{eq:tra_ldf_lap}) at the bottom level require a specific treatment.
+They are calculated in the \mdl{zpshde} module, described in \autoref{sec:TRA_zpshde}.
 …
 \subsubsection{Standard rotated (bi-)laplacian operator (\protect\mdl{traldf\_iso})}
 \label{subsec:TRA_ldf_iso}
 The general form of the second order lateral tracer subgrid scale physics
 (\autoref{eq:PE_zdf}) takes the following semi-discrete space form in $z$- and $s$-coordinates:
+The general form of the second order lateral tracer subgrid scale physics (\autoref{eq:PE_zdf})
+takes the following semi-discrete space form in $z$- and $s$-coordinates:
 \begin{equation} \label{eq:tra_ldf_iso}
 \begin{split}
 …
 & \left. {\left. {   \qquad \qquad \ \ \ \left. {
         +\;\frac{e_{1w}\,e_{2w}}{e_{3w}} \,\left( r_{1w}^2 + r_{2w}^2 \right)
+           \,\delta_{k+1/2} [T] } \right) } \right] \quad } \right\}
+ \end{split}
+ \end{equation}
+where $b_t$=$e_{1t}\,e_{2t}\,e_{3t}$  is the volume of $T$-cells,
+$r_1$ and $r_2$ are the slopes between the surface of computation
+($z$- or $s$-surfaces) and the surface along which the diffusion operator
+acts ($i.e.$ horizontal or iso-neutral surfaces).  It is thus used when,
+in addition to \np{ln\_traldf\_lap}\forcode{ = .true.}, we have \np{ln\_traldf\_iso}\forcode{ = .true.},
+or both \np{ln\_traldf\_hor}\forcode{ = .true.} and \np{ln\_zco}\forcode{ = .true.}. The way these
+slopes are evaluated is given in \autoref{sec:LDF_slp}. At the surface, bottom
+and lateral boundaries, the turbulent fluxes of heat and salt are set to zero
+using the mask technique (see \autoref{sec:LBC_coast}).
+The operator in \autoref{eq:tra_ldf_iso} involves both lateral and vertical
+derivatives. For numerical stability, the vertical second derivative must
+be solved using the same implicit time scheme as that used in the vertical
+physics (see \autoref{sec:TRA_zdf}). For computer efficiency reasons, this term
+is not computed in the \mdl{traldf\_iso} module, but in the \mdl{trazdf} module
+where, if iso-neutral mixing is used, the vertical mixing coefficient is simply
+increased by $\frac{e_{1w}\,e_{2w} }{e_{3w} }\ \left( {r_{1w} ^2+r_{2w} ^2} \right)$.
+This formulation conserves the tracer but does not ensure the decrease
+of the tracer variance. Nevertheless the treatment performed on the slopes
+(see \autoref{chap:LDF}) allows the model to run safely without any additional
+background horizontal diffusion \citep{Guilyardi_al_CD01}.
+Note that in the partial step $z$-coordinate (\np{ln\_zps}\forcode{ = .true.}), the horizontal derivatives
+at the bottom level in \autoref{eq:tra_ldf_iso} require a specific treatment.
+        \,\delta_{k+1/2} [T] } \right) } \right] \quad } \right\}
+\end{split}
+\end{equation}
+where $b_t$=$e_{1t}\,e_{2t}\,e_{3t}$  is the volume of $T$-cells,
+$r_1$ and $r_2$ are the slopes between the surface of computation ($z$- or $s$-surfaces) and
+the surface along which the diffusion operator acts ($i.e.$ horizontal or iso-neutral surfaces).
+It is thus used when, in addition to \np{ln\_traldf\_lap}\forcode{ = .true.},
+we have \np{ln\_traldf\_iso}\forcode{ = .true.},
+or both \np{ln\_traldf\_hor}\forcode{ = .true.} and \np{ln\_zco}\forcode{ = .true.}.
+The way these slopes are evaluated is given in \autoref{sec:LDF_slp}.
+At the surface, bottom and lateral boundaries, the turbulent fluxes of heat and salt are set to zero using
+the mask technique (see \autoref{sec:LBC_coast}).
+The operator in \autoref{eq:tra_ldf_iso} involves both lateral and vertical derivatives.
+For numerical stability, the vertical second derivative must be solved using the same implicit time scheme as that
+used in the vertical physics (see \autoref{sec:TRA_zdf}).
+For computer efficiency reasons, this term is not computed in the \mdl{traldf\_iso} module,
+but in the \mdl{trazdf} module where, if iso-neutral mixing is used,
+the vertical mixing coefficient is simply increased by
+$\frac{e_{1w}\,e_{2w} }{e_{3w} }\ \left( {r_{1w} ^2+r_{2w} ^2} \right)$.
+This formulation conserves the tracer but does not ensure the decrease of the tracer variance.
+Nevertheless the treatment performed on the slopes (see \autoref{chap:LDF}) allows the model to run safely without
+any additional background horizontal diffusion \citep{Guilyardi_al_CD01}.
+Note that in the partial step $z$-coordinate (\np{ln\_zps}\forcode{ = .true.}),
+the horizontal derivatives at the bottom level in \autoref{eq:tra_ldf_iso} require a specific treatment.
 They are calculated in module zpshde, described in \autoref{sec:TRA_zpshde}.
 …
 \label{subsec:TRA_ldf_triad}
+If the Griffies triad scheme is employed (\np{ln\_traldf\_triad}\forcode{ = .true.} ; see \autoref{apdx:triad})
+An alternative scheme developed by \cite{Griffies_al_JPO98} which ensures tracer variance decreases
+is also available in \NEMO (\np{ln\_traldf\_grif}\forcode{ = .true.}). A complete description of
+the algorithm is given in \autoref{apdx:triad}.
+The lateral fourth order bilaplacian operator on tracers is obtained by
+applying (\autoref{eq:tra_ldf_lap}) twice. The operator requires an additional assumption
+on boundary conditions: both first and third derivative terms normal to the
+coast are set to zero.
+The lateral fourth order operator formulation on tracers is obtained by
+applying (\autoref{eq:tra_ldf_iso}) twice. It requires an additional assumption
+on boundary conditions: first and third derivative terms normal to the
+coast, normal to the bottom and normal to the surface are set to zero.
+If the Griffies triad scheme is employed (\np{ln\_traldf\_triad}\forcode{ = .true.}; see \autoref{apdx:triad})
+An alternative scheme developed by \cite{Griffies_al_JPO98} which ensures tracer variance decreases
+is also available in \NEMO (\np{ln\_traldf\_grif}\forcode{ = .true.}).
+A complete description of the algorithm is given in \autoref{apdx:triad}.
+The lateral fourth order bilaplacian operator on tracers is obtained by applying (\autoref{eq:tra_ldf_lap}) twice.
+The operator requires an additional assumption on boundary conditions:
+both first and third derivative terms normal to the coast are set to zero.
+The lateral fourth order operator formulation on tracers is obtained by applying (\autoref{eq:tra_ldf_iso}) twice.
+It requires an additional assumption on boundary conditions:
+first and third derivative terms normal to the coast,
+normal to the bottom and normal to the surface are set to zero.
 %&&    Option for the rotated operators
 …
 \np{ln\_triad\_iso} = pure horizontal mixing in ML (triad only)
+\np{rn\_sw\_triad} =1 switching triad ; =0 all 4 triads used (triad only)
+\np{rn\_sw\_triad} =1 switching triad;
+                   =0 all 4 triads used (triad only)
 \np{ln\_botmix\_triad} = lateral mixing on bottom (triad only)
 …
 Options are defined through the \ngn{namzdf} namelist variables.
+The formulation of the vertical subgrid scale tracer physics is the same
+for all the vertical coordinates, and is based on a laplacian operator.
+The vertical diffusion operator given by (\autoref{eq:PE_zdf}) takes the
+following semi-discrete space form:
+The formulation of the vertical subgrid scale tracer physics is the same for all the vertical coordinates,
+and is based on a laplacian operator.
+The vertical diffusion operator given by (\autoref{eq:PE_zdf}) takes the following semi-discrete space form:
 \begin{equation} \label{eq:tra_zdf}
 \begin{split}
 …
 \end{split}
 \end{equation}
+where $A_w^{vT}$ and $A_w^{vS}$ are the vertical eddy diffusivity
+coefficients on temperature and salinity, respectively. Generally,
+$A_w^{vT}=A_w^{vS}$ except when double diffusive mixing is
+parameterised ($i.e.$ \key{zdfddm} is defined). The way these coefficients
+are evaluated is given in \autoref{chap:ZDF} (ZDF). Furthermore, when
+iso-neutral mixing is used, both mixing coefficients are increased
+by $\frac{e_{1w}\,e_{2w} }{e_{3w} }\ \left( {r_{1w} ^2+r_{2w} ^2} \right)$
+to account for the vertical second derivative of \autoref{eq:tra_ldf_iso}.
+At the surface and bottom boundaries, the turbulent fluxes of
+heat and salt must be specified. At the surface they are prescribed
+from the surface forcing and added in a dedicated routine (see \autoref{subsec:TRA_sbc}),
+whilst at the bottom they are set to zero for heat and salt unless
+a geothermal flux forcing is prescribed as a bottom boundary
+condition (see \autoref{subsec:TRA_bbc}).
+The large eddy coefficient found in the mixed layer together with high
+vertical resolution implies that in the case of explicit time stepping
+(\np{ln\_zdfexp}\forcode{ = .true.}) there would be too restrictive a constraint on
+the time step. Therefore, the default implicit time stepping is preferred
+for the vertical diffusion since it overcomes the stability constraint.
+A forward time differencing scheme (\np{ln\_zdfexp}\forcode{ = .true.}) using a time
+splitting technique (\np{nn\_zdfexp} $> 1$) is provided as an alternative.
+Namelist variables \np{ln\_zdfexp} and \np{nn\_zdfexp} apply to both
+tracers and dynamics.
+where $A_w^{vT}$ and $A_w^{vS}$ are the vertical eddy diffusivity coefficients on temperature and salinity,
+respectively.
+Generally, $A_w^{vT}=A_w^{vS}$ except when double diffusive mixing is parameterised ($i.e.$ \key{zdfddm} is defined).
+The way these coefficients are evaluated is given in \autoref{chap:ZDF} (ZDF).
+Furthermore, when iso-neutral mixing is used, both mixing coefficients are increased by
+$\frac{e_{1w}\,e_{2w} }{e_{3w} }\ \left( {r_{1w} ^2+r_{2w} ^2} \right)$ to account for
+the vertical second derivative of \autoref{eq:tra_ldf_iso}.
+At the surface and bottom boundaries, the turbulent fluxes of heat and salt must be specified.
+At the surface they are prescribed from the surface forcing and added in a dedicated routine
+(see \autoref{subsec:TRA_sbc}), whilst at the bottom they are set to zero for heat and salt unless
+a geothermal flux forcing is prescribed as a bottom boundary condition (see \autoref{subsec:TRA_bbc}).
+The large eddy coefficient found in the mixed layer together with high vertical resolution implies that
+in the case of explicit time stepping (\np{ln\_zdfexp}\forcode{ = .true.})
+there would be too restrictive a constraint on the time step.
+Therefore, the default implicit time stepping is preferred for the vertical diffusion since
+it overcomes the stability constraint.
+A forward time differencing scheme (\np{ln\_zdfexp}\forcode{ = .true.}) using
+a time splitting technique (\np{nn\_zdfexp} $> 1$) is provided as an alternative.
+Namelist variables \np{ln\_zdfexp} and \np{nn\_zdfexp} apply to both tracers and dynamics.
 % ================================================================
 …
 \label{subsec:TRA_sbc}
 The surface boundary condition for tracers is implemented in a separate
+module (\mdl{trasbc}) instead of entering as a boundary condition on the vertical
+diffusion operator (as in the case of momentum). This has been found to
+enhance readability of the code. The two formulations are completely
 equivalent; the forcing terms in trasbc are the surface fluxes divided by
+the thickness of the top model layer.
+Due to interactions and mass exchange of water ($F_{mass}$) with other Earth system components
+($i.e.$ atmosphere, sea-ice, land), the change in the heat and salt content of the surface layer
+of the ocean is due both to the heat and salt fluxes crossing the sea surface (not linked with $F_{mass}$)
+and to the heat and salt content of the mass exchange. They are both included directly in $Q_{ns}$,
 the surface heat flux, and $F_{salt}$, the surface salt flux (see \autoref{chap:SBC} for further details).
+The surface boundary condition for tracers is implemented in a separate module (\mdl{trasbc}) instead of
+entering as a boundary condition on the vertical diffusion operator (as in the case of momentum).
+This has been found to enhance readability of the code.
+The two formulations are completely equivalent;
+the forcing terms in trasbc are the surface fluxes divided by the thickness of the top model layer.
+Due to interactions and mass exchange of water ($F_{mass}$) with other Earth system components
+($i.e.$ atmosphere, sea-ice, land), the change in the heat and salt content of the surface layer of the ocean is due
+both to the heat and salt fluxes crossing the sea surface (not linked with $F_{mass}$) and
+to the heat and salt content of the mass exchange.
+They are both included directly in $Q_{ns}$, the surface heat flux,
+and $F_{salt}$, the surface salt flux (see \autoref{chap:SBC} for further details).
 By doing this, the forcing formulation is the same for any tracer (including temperature and salinity).
+The surface module (\mdl{sbcmod}, see \autoref{chap:SBC}) provides the following
+forcing fields (used on tracers):
+$\bullet$ $Q_{ns}$, the non-solar part of the net surface heat flux that crosses the sea surface
+The surface module (\mdl{sbcmod}, see \autoref{chap:SBC}) provides the following forcing fields (used on tracers):
+$\bullet$ $Q_{ns}$, the non-solar part of the net surface heat flux that crosses the sea surface
 (i.e. the difference between the total surface heat flux and the fraction of the short wave flux that
 penetrates into the water column, see \autoref{subsec:TRA_qsr}) plus the heat content associated with
 of the mass exchange with the atmosphere and lands.
+penetrates into the water column, see \autoref{subsec:TRA_qsr})
+plus the heat content associated with of the mass exchange with the atmosphere and lands.
 $\bullet$ $\textit{sfx}$, the salt flux resulting from ice-ocean mass exchange (freezing, melting, ridging...)
 $\bullet$ \textit{emp}, the mass flux exchanged with the atmosphere (evaporation minus precipitation)
  and possibly with the sea-ice and ice-shelves.
 $\bullet$ \textit{rnf}, the mass flux associated with runoff
+$\bullet$ \textit{emp}, the mass flux exchanged with the atmosphere (evaporation minus precipitation) and
+possibly with the sea-ice and ice-shelves.
+$\bullet$ \textit{rnf}, the mass flux associated with runoff
 (see \autoref{sec:SBC_rnf} for further detail of how it acts on temperature and salinity tendencies)
 $\bullet$ \textit{fwfisf}, the mass flux associated with ice shelf melt,
+$\bullet$ \textit{fwfisf}, the mass flux associated with ice shelf melt,
 (see \autoref{sec:SBC_isf} for further details on how the ice shelf melt is computed and applied).
 …
  \end{aligned}
 \end{equation}
+where $\overline{x }^t$ means that $x$ is averaged over two consecutive time steps
+($t-\rdt/2$ and $t+\rdt/2$). Such time averaging prevents the
+divergence of odd and even time step (see \autoref{chap:STP}).
+In the linear free surface case (\np{ln\_linssh}\forcode{ = .true.}),
+an additional term has to be added on both temperature and salinity.
+On temperature, this term remove the heat content associated with mass exchange
+that has been added to $Q_{ns}$. On salinity, this term mimics the concentration/dilution effect that
+would have resulted from a change in the volume of the first level.
+where $\overline{x }^t$ means that $x$ is averaged over two consecutive time steps ($t-\rdt/2$ and $t+\rdt/2$).
+Such time averaging prevents the divergence of odd and even time step (see \autoref{chap:STP}).
+In the linear free surface case (\np{ln\_linssh}\forcode{ = .true.}), an additional term has to be added on
+both temperature and salinity.
+On temperature, this term remove the heat content associated with mass exchange that has been added to $Q_{ns}$.
+On salinity, this term mimics the concentration/dilution effect that would have resulted from a change in
+the volume of the first level.
 The resulting surface boundary condition is applied as follows:
 \begin{equation} \label{eq:tra_sbc_lin}
 …
  \end{aligned}
 \end{equation}
 Note that an exact conservation of heat and salt content is only achieved with non-linear free surface.
 In the linear free surface case, there is a small imbalance. The imbalance is larger
+than the imbalance associated with the Asselin time filter \citep{Leclair_Madec_OM09}.
+Note that an exact conservation of heat and salt content is only achieved with non-linear free surface.
+In the linear free surface case, there is a small imbalance.
+The imbalance is larger than the imbalance associated with the Asselin time filter \citep{Leclair_Madec_OM09}.
 This is the reason why the modified filter is not applied in the linear free surface case (see \autoref{chap:STP}).
 …
 Options are defined through the \ngn{namtra\_qsr} namelist variables.
+When the penetrative solar radiation option is used (\np{ln\_flxqsr}\forcode{ = .true.}),
+the solar radiation penetrates the top few tens of meters of the ocean. If it is not used
+(\np{ln\_flxqsr}\forcode{ = .false.}) all the heat flux is absorbed in the first ocean level.
+Thus, in the former case a term is added to the time evolution equation of
+temperature \autoref{eq:PE_tra_T} and the surface boundary condition is
+modified to take into account only the non-penetrative part of the surface
+When the penetrative solar radiation option is used (\np{ln\_flxqsr}\forcode{ = .true.}),
+the solar radiation penetrates the top few tens of meters of the ocean.
+If it is not used (\np{ln\_flxqsr}\forcode{ = .false.}) all the heat flux is absorbed in the first ocean level.
+Thus, in the former case a term is added to the time evolution equation of temperature \autoref{eq:PE_tra_T} and
+the surface boundary condition is modified to take into account only the non-penetrative part of the surface
 heat flux:
 \begin{equation} \label{eq:PE_qsr}
 …
 \end{split}
 \end{equation}
 where $Q_{sr}$ is the penetrative part of the surface heat flux ($i.e.$ the shortwave radiation)
+and $I$ is the downward irradiance ($\left. I \right|_{z=\eta}=Q_{sr}$).
+where $Q_{sr}$ is the penetrative part of the surface heat flux ($i.e.$ the shortwave radiation) and
+$I$ is the downward irradiance ($\left. I \right|_{z=\eta}=Q_{sr}$).
 The additional term in \autoref{eq:PE_qsr} is discretized as follows:
 \begin{equation} \label{eq:tra_qsr}
 …
 \end{equation}
 The shortwave radiation, $Q_{sr}$, consists of energy distributed across a wide spectral range.
 The ocean is strongly absorbing for wavelengths longer than 700~nm and these
+wavelengths contribute to heating the upper few tens of centimetres. The fraction of $Q_{sr}$
+that resides in these almost non-penetrative wavebands, $R$, is $\sim 58\%$ (specified
+through namelist parameter \np{rn\_abs}). It is assumed to penetrate the ocean
+with a decreasing exponential profile, with an e-folding depth scale, $\xi_0$,
+The shortwave radiation, $Q_{sr}$, consists of energy distributed across a wide spectral range.
+The ocean is strongly absorbing for wavelengths longer than 700~nm and these wavelengths contribute to
+heating the upper few tens of centimetres.
+The fraction of $Q_{sr}$ that resides in these almost non-penetrative wavebands, $R$, is $\sim 58\%$
+(specified through namelist parameter \np{rn\_abs}).
+It is assumed to penetrate the ocean with a decreasing exponential profile, with an e-folding depth scale, $\xi_0$,
 of a few tens of centimetres (typically $\xi_0=0.35~m$ set as \np{rn\_si0} in the \ngn{namtra\_qsr} namelist).
+For shorter wavelengths (400-700~nm), the ocean is more transparent, and solar energy
+propagates to larger depths where it contributes to
+local heating.
+The way this second part of the solar energy penetrates into the ocean depends on
+which formulation is chosen. In the simple 2-waveband light penetration scheme (\np{ln\_qsr\_2bd}\forcode{ = .true.})
+a chlorophyll-independent monochromatic formulation is chosen for the shorter wavelengths,
+For shorter wavelengths (400-700~nm), the ocean is more transparent, and solar energy propagates to
+larger depths where it contributes to local heating.
+The way this second part of the solar energy penetrates into the ocean depends on which formulation is chosen.
+In the simple 2-waveband light penetration scheme (\np{ln\_qsr\_2bd}\forcode{ = .true.})
+a chlorophyll-independent monochromatic formulation is chosen for the shorter wavelengths,
 leading to the following expression \citep{Paulson1977}:
 \begin{equation} \label{eq:traqsr_iradiance}
 I(z) = Q_{sr} \left[Re^{-z / \xi_0} + \left( 1-R\right) e^{-z / \xi_1} \right]
 \end{equation}
+where $\xi_1$ is the second extinction length scale associated with the shorter wavelengths.
+It is usually chosen to be 23~m by setting the \np{rn\_si0} namelist parameter.
+The set of default values ($\xi_0$, $\xi_1$, $R$) corresponds to a Type I water in
+Jerlov's (1968) classification (oligotrophic waters).
+Such assumptions have been shown to provide a very crude and simplistic
+representation of observed light penetration profiles (\cite{Morel_JGR88}, see also
+\autoref{fig:traqsr_irradiance}). Light absorption in the ocean depends on
+particle concentration and is spectrally selective. \cite{Morel_JGR88} has shown
+that an accurate representation of light penetration can be provided by a 61 waveband
+formulation. Unfortunately, such a model is very computationally expensive.
+Thus, \cite{Lengaigne_al_CD07} have constructed a simplified version of this
+formulation in which visible light is split into three wavebands: blue (400-500 nm),
+green (500-600 nm) and red (600-700nm). For each wave-band, the chlorophyll-dependent
+attenuation coefficient is fitted to the coefficients computed from the full spectral model
+of \cite{Morel_JGR88} (as modified by \cite{Morel_Maritorena_JGR01}), assuming
+the same power-law relationship. As shown in \autoref{fig:traqsr_irradiance},
+this formulation, called RGB (Red-Green-Blue), reproduces quite closely
+the light penetration profiles predicted by the full spectal model, but with much greater
+computational efficiency. The 2-bands formulation does not reproduce the full model very well.
+The RGB formulation is used when \np{ln\_qsr\_rgb}\forcode{ = .true.}. The RGB attenuation coefficients
+($i.e.$ the inverses of the extinction length scales) are tabulated over 61 nonuniform
+chlorophyll classes ranging from 0.01 to 10 g.Chl/L (see the routine \rou{trc\_oce\_rgb}
+in \mdl{trc\_oce} module). Four types of chlorophyll can be chosen in the RGB formulation:
+where $\xi_1$ is the second extinction length scale associated with the shorter wavelengths.
+It is usually chosen to be 23~m by setting the \np{rn\_si0} namelist parameter.
+The set of default values ($\xi_0$, $\xi_1$, $R$) corresponds to a Type I water in Jerlov's (1968) classification
+(oligotrophic waters).
+Such assumptions have been shown to provide a very crude and simplistic representation of
+observed light penetration profiles (\cite{Morel_JGR88}, see also \autoref{fig:traqsr_irradiance}).
+Light absorption in the ocean depends on particle concentration and is spectrally selective.
+\cite{Morel_JGR88} has shown that an accurate representation of light penetration can be provided by
+a 61 waveband formulation.
+Unfortunately, such a model is very computationally expensive.
+Thus, \cite{Lengaigne_al_CD07} have constructed a simplified version of this formulation in which
+visible light is split into three wavebands: blue (400-500 nm), green (500-600 nm) and red (600-700nm).
+For each wave-band, the chlorophyll-dependent attenuation coefficient is fitted to the coefficients computed from
+the full spectral model of \cite{Morel_JGR88} (as modified by \cite{Morel_Maritorena_JGR01}),
+assuming the same power-law relationship.
+As shown in \autoref{fig:traqsr_irradiance}, this formulation, called RGB (Red-Green-Blue),
+reproduces quite closely the light penetration profiles predicted by the full spectal model,
+but with much greater computational efficiency.
+The 2-bands formulation does not reproduce the full model very well.
+The RGB formulation is used when \np{ln\_qsr\_rgb}\forcode{ = .true.}.
+The RGB attenuation coefficients ($i.e.$ the inverses of the extinction length scales) are tabulated over
+nonuniform chlorophyll classes ranging from 0.01 to 10 g.Chl/L
+(see the routine \rou{trc\_oce\_rgb} in \mdl{trc\_oce} module).
+Four types of chlorophyll can be chosen in the RGB formulation:
 \begin{description}
 \item[\np{nn\_chdta}\forcode{ = 0}]
 a constant 0.05 g.Chl/L value everywhere ;
 \item[\np{nn\_chdta}\forcode{ = 1}]
+an observed time varying chlorophyll deduced from satellite surface ocean color measurement
+spread uniformly in the vertical direction ;
 \item[\np{nn\_chdta}\forcode{ = 2}]
+same as previous case except that a vertical profile of chlorophyl is used.
 Following \cite{Morel_Berthon_LO89}, the profile is computed from the local surface chlorophyll value ;
 \item[\np{ln\_qsr\_bio}\forcode{ = .true.}]
+simulated time varying chlorophyll by TOP biogeochemical model.
+In this case, the RGB formulation is used to calculate both the phytoplankton
 light limitation in PISCES or LOBSTER and the oceanic heating rate.
+\item[\np{nn\_chdta}\forcode{ = 0}]
+  a constant 0.05 g.Chl/L value everywhere ;
+\item[\np{nn\_chdta}\forcode{ = 1}]
+  an observed time varying chlorophyll deduced from satellite surface ocean color measurement spread uniformly in
+  the vertical direction;
+\item[\np{nn\_chdta}\forcode{ = 2}]
+  same as previous case except that a vertical profile of chlorophyl is used.
+  Following \cite{Morel_Berthon_LO89}, the profile is computed from the local surface chlorophyll value;
+\item[\np{ln\_qsr\_bio}\forcode{ = .true.}]
+  simulated time varying chlorophyll by TOP biogeochemical model.
+  In this case, the RGB formulation is used to calculate both the phytoplankton light limitation in
+  PISCES or LOBSTER and the oceanic heating rate.
 \end{description}
 The trend in \autoref{eq:tra_qsr} associated with the penetration of the solar radiation
+is added to the temperature trend, and the surface heat flux is modified in routine \mdl{traqsr}.
 When the $z$-coordinate is preferred to the $s$-coordinate, the depth of $w-$levels does
+not significantly vary with location. The level at which the light has been totally
+absorbed ($i.e.$ it is less than the computer precision) is computed once,
+and the trend associated with the penetration of the solar radiation is only added down to that level.
+Finally, note that when the ocean is shallow ($<$ 200~m), part of the
+solar radiation can reach the ocean floor. In this case, we have
+chosen that all remaining radiation is absorbed in the last ocean
 level ($i.e.$ $I$ is masked).
+The trend in \autoref{eq:tra_qsr} associated with the penetration of the solar radiation is added to
+the temperature trend, and the surface heat flux is modified in routine \mdl{traqsr}.
+When the $z$-coordinate is preferred to the $s$-coordinate,
+the depth of $w-$levels does not significantly vary with location.
+The level at which the light has been totally absorbed
+($i.e.$ it is less than the computer precision) is computed once,
+and the trend associated with the penetration of the solar radiation is only added down to that level.
+Finally, note that when the ocean is shallow ($<$ 200~m), part of the solar radiation can reach the ocean floor.
+In this case, we have chosen that all remaining radiation is absorbed in the last ocean level
+($i.e.$ $I$ is masked).
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!t]     \begin{center}
+\includegraphics[width=1.0\textwidth]{Fig_TRA_Irradiance}
+\caption{    \protect\label{fig:traqsr_irradiance}
+Penetration profile of the downward solar irradiance calculated by four models.
+Two waveband chlorophyll-independent formulation (blue), a chlorophyll-dependent
+monochromatic formulation (green), 4 waveband RGB formulation (red),
+waveband Morel (1988) formulation (black) for a chlorophyll concentration of
+(a) Chl=0.05 mg/m$^3$ and (b) Chl=0.5 mg/m$^3$. From \citet{Lengaigne_al_CD07}.}
+\end{center}   \end{figure}
+\begin{figure}[!t]
+  \begin{center}
+    \includegraphics[width=1.0\textwidth]{Fig_TRA_Irradiance}
+    \caption{   \protect\label{fig:traqsr_irradiance}
+      Penetration profile of the downward solar irradiance calculated by four models.
+      Two waveband chlorophyll-independent formulation (blue),
+      a chlorophyll-dependent monochromatic formulation (green),
+waveband RGB formulation (red),
+waveband Morel (1988) formulation (black) for a chlorophyll concentration of
+      (a) Chl=0.05 mg/m$^3$ and (b) Chl=0.5 mg/m$^3$.
+      From \citet{Lengaigne_al_CD07}.
+    }
+  \end{center}
+\end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 …
 %--------------------------------------------------------------------------------------------------------------
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!t]     \begin{center}
+\includegraphics[width=1.0\textwidth]{Fig_TRA_geoth}
+\caption{   \protect\label{fig:geothermal}
+Geothermal Heat flux (in $mW.m^{-2}$) used by \cite{Emile-Geay_Madec_OS09}.
+It is inferred from the age of the sea floor and the formulae of \citet{Stein_Stein_Nat92}.}
+\end{center}   \end{figure}
+\begin{figure}[!t]
+  \begin{center}
+    \includegraphics[width=1.0\textwidth]{Fig_TRA_geoth}
+    \caption{  \protect\label{fig:geothermal}
+      Geothermal Heat flux (in $mW.m^{-2}$) used by \cite{Emile-Geay_Madec_OS09}.
+      It is inferred from the age of the sea floor and the formulae of \citet{Stein_Stein_Nat92}.
+    }
+  \end{center}
+\end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+Usually it is assumed that there is no exchange of heat or salt through
+the ocean bottom, $i.e.$ a no flux boundary condition is applied on active
+tracers at the bottom. This is the default option in \NEMO, and it is
+implemented using the masking technique. However, there is a
+non-zero heat flux across the seafloor that is associated with solid
+earth cooling. This flux is weak compared to surface fluxes (a mean
+global value of $\sim0.1\;W/m^2$ \citep{Stein_Stein_Nat92}), but it warms
+systematically the ocean and acts on the densest water masses.
+Taking this flux into account in a global ocean model increases
+the deepest overturning cell ($i.e.$ the one associated with the Antarctic
+Bottom Water) by a few Sverdrups  \citep{Emile-Geay_Madec_OS09}.
+Usually it is assumed that there is no exchange of heat or salt through the ocean bottom,
+$i.e.$ a no flux boundary condition is applied on active tracers at the bottom.
+This is the default option in \NEMO, and it is implemented using the masking technique.
+However, there is a non-zero heat flux across the seafloor that is associated with solid earth cooling.
+This flux is weak compared to surface fluxes (a mean global value of $\sim0.1\;W/m^2$ \citep{Stein_Stein_Nat92}),
+but it warms systematically the ocean and acts on the densest water masses.
+Taking this flux into account in a global ocean model increases the deepest overturning cell
+($i.e.$ the one associated with the Antarctic Bottom Water) by a few Sverdrups  \citep{Emile-Geay_Madec_OS09}.
 Options are defined through the  \ngn{namtra\_bbc} namelist variables.
+The presence of geothermal heating is controlled by setting the namelist
+parameter  \np{ln\_trabbc} to true. Then, when \np{nn\_geoflx} is set to 1,
+a constant geothermal heating is introduced whose value is given by the
+\np{nn\_geoflx\_cst}, which is also a namelist parameter.
+When  \np{nn\_geoflx} is set to 2, a spatially varying geothermal heat flux is
+introduced which is provided in the \ifile{geothermal\_heating} NetCDF file
+(\autoref{fig:geothermal}) \citep{Emile-Geay_Madec_OS09}.
+The presence of geothermal heating is controlled by setting the namelist parameter \np{ln\_trabbc} to true.
+Then, when \np{nn\_geoflx} is set to 1, a constant geothermal heating is introduced whose value is given by
+the \np{nn\_geoflx\_cst}, which is also a namelist parameter.
+When \np{nn\_geoflx} is set to 2, a spatially varying geothermal heat flux is introduced which is provided in
+the \ifile{geothermal\_heating} NetCDF file (\autoref{fig:geothermal}) \citep{Emile-Geay_Madec_OS09}.
 % ================================================================
 …
 Options are defined through the  \ngn{nambbl} namelist variables.
+In a $z$-coordinate configuration, the bottom topography is represented by a
+series of discrete steps. This is not adequate to represent gravity driven
+downslope flows. Such flows arise either downstream of sills such as the Strait of
+Gibraltar or Denmark Strait, where dense water formed in marginal seas flows
+into a basin filled with less dense water, or along the continental slope when dense
+water masses are formed on a continental shelf. The amount of entrainment
+that occurs in these gravity plumes is critical in determining the density
+and volume flux of the densest waters of the ocean, such as Antarctic Bottom Water,
+or North Atlantic Deep Water. $z$-coordinate models tend to overestimate the
+entrainment, because the gravity flow is mixed vertically by convection
+as it goes ''downstairs'' following the step topography, sometimes over a thickness
+much larger than the thickness of the observed gravity plume. A similar problem
+occurs in the $s$-coordinate when the thickness of the bottom level varies rapidly
+downstream of a sill \citep{Willebrand_al_PO01}, and the thickness
+of the plume is not resolved.
+The idea of the bottom boundary layer (BBL) parameterisation, first introduced by
+\citet{Beckmann_Doscher1997}, is to allow a direct communication between
+two adjacent bottom cells at different levels, whenever the densest water is
+located above the less dense water. The communication can be by a diffusive flux
+(diffusive BBL), an advective flux (advective BBL), or both. In the current
+implementation of the BBL, only the tracers are modified, not the velocities.
+Furthermore, it only connects ocean bottom cells, and therefore does not include
+all the improvements introduced by \citet{Campin_Goosse_Tel99}.
+In a $z$-coordinate configuration, the bottom topography is represented by a series of discrete steps.
+This is not adequate to represent gravity driven downslope flows.
+Such flows arise either downstream of sills such as the Strait of Gibraltar or Denmark Strait,
+where dense water formed in marginal seas flows into a basin filled with less dense water,
+or along the continental slope when dense water masses are formed on a continental shelf.
+The amount of entrainment that occurs in these gravity plumes is critical in determining the density and
+volume flux of the densest waters of the ocean, such as Antarctic Bottom Water, or North Atlantic Deep Water.
+$z$-coordinate models tend to overestimate the entrainment,
+because the gravity flow is mixed vertically by convection as it goes ''downstairs'' following the step topography,
+sometimes over a thickness much larger than the thickness of the observed gravity plume.
+A similar problem occurs in the $s$-coordinate when the thickness of the bottom level varies rapidly downstream of
+a sill \citep{Willebrand_al_PO01}, and the thickness of the plume is not resolved.
+The idea of the bottom boundary layer (BBL) parameterisation, first introduced by \citet{Beckmann_Doscher1997},
+is to allow a direct communication between two adjacent bottom cells at different levels,
+whenever the densest water is located above the less dense water.
+The communication can be by a diffusive flux (diffusive BBL), an advective flux (advective BBL), or both.
+In the current implementation of the BBL, only the tracers are modified, not the velocities.
+Furthermore, it only connects ocean bottom cells, and therefore does not include all the improvements introduced by
+\citet{Campin_Goosse_Tel99}.
 % -------------------------------------------------------------------------------------------------------------
 …
 \label{subsec:TRA_bbl_diff}
 When applying sigma-diffusion (\key{trabbl} defined and \np{nn\_bbl\_ldf} set to 1),
+When applying sigma-diffusion (\key{trabbl} defined and \np{nn\_bbl\_ldf} set to 1),
 the diffusive flux between two adjacent cells at the ocean floor is given by
 \begin{equation} \label{eq:tra_bbl_diff}
 {\rm {\bf F}}_\sigma=A_l^\sigma \; \nabla_\sigma T
 \end{equation}
+with $\nabla_\sigma$ the lateral gradient operator taken between bottom cells,
+and  $A_l^\sigma$ the lateral diffusivity in the BBL. Following \citet{Beckmann_Doscher1997},
+the latter is prescribed with a spatial dependence, $i.e.$ in the conditional form
+with $\nabla_\sigma$ the lateral gradient operator taken between bottom cells,
+and  $A_l^\sigma$ the lateral diffusivity in the BBL.
+Following \citet{Beckmann_Doscher1997}, the latter is prescribed with a spatial dependence,
+$i.e.$ in the conditional form
 \begin{equation} \label{eq:tra_bbl_coef}
 A_l^\sigma (i,j,t)=\left\{ {\begin{array}{l}
 …
  \end{array}} \right.
 \end{equation}
 where $A_{bbl}$ is the BBL diffusivity coefficient, given by the namelist
+parameter \np{rn\_ahtbbl} and usually set to a value much larger
+than the one used for lateral mixing in the open ocean. The constraint in \autoref{eq:tra_bbl_coef}
+implies that sigma-like diffusion only occurs when the density above the sea floor, at the top of
+the slope, is larger than in the deeper ocean (see green arrow in \autoref{fig:bbl}).
 In practice, this constraint is applied separately in the two horizontal directions,
+where $A_{bbl}$ is the BBL diffusivity coefficient, given by the namelist parameter \np{rn\_ahtbbl} and
+usually set to a value much larger than the one used for lateral mixing in the open ocean.
+The constraint in \autoref{eq:tra_bbl_coef} implies that sigma-like diffusion only occurs when
+the density above the sea floor, at the top of the slope, is larger than in the deeper ocean
+(see green arrow in \autoref{fig:bbl}).
+In practice, this constraint is applied separately in the two horizontal directions,
 and the density gradient in \autoref{eq:tra_bbl_coef} is evaluated with the log gradient formulation:
 \begin{equation} \label{eq:tra_bbl_Drho}
    \nabla_\sigma \rho / \rho = \alpha \,\nabla_\sigma T + \beta   \,\nabla_\sigma S
 \end{equation}
+where $\rho$, $\alpha$ and $\beta$ are functions of $\overline{T}^\sigma$,
+$\overline{S}^\sigma$ and $\overline{H}^\sigma$, the along bottom mean temperature,
+salinity and depth, respectively.
+where $\rho$, $\alpha$ and $\beta$ are functions of $\overline{T}^\sigma$,
+$\overline{S}^\sigma$ and $\overline{H}^\sigma$, the along bottom mean temperature, salinity and depth, respectively.
 % -------------------------------------------------------------------------------------------------------------
 …
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!t]   \begin{center}
+\includegraphics[width=0.7\textwidth]{Fig_BBL_adv}
+\caption{   \protect\label{fig:bbl}
+Advective/diffusive Bottom Boundary Layer. The BBL parameterisation is
+activated when $\rho^i_{kup}$ is larger than $\rho^{i+1}_{kdnw}$.
+Red arrows indicate the additional overturning circulation due to the advective BBL.
+The transport of the downslope flow is defined either as the transport of the bottom
+ocean cell (black arrow), or as a function of the along slope density gradient.
+The green arrow indicates the diffusive BBL flux directly connecting $kup$ and $kdwn$
+ocean bottom cells.
+connection}
+\end{center}   \end{figure}
+\begin{figure}[!t]
+  \begin{center}
+    \includegraphics[width=0.7\textwidth]{Fig_BBL_adv}
+    \caption{  \protect\label{fig:bbl}
+      Advective/diffusive Bottom Boundary Layer.
+      The BBL parameterisation is activated when $\rho^i_{kup}$ is larger than $\rho^{i+1}_{kdnw}$.
+      Red arrows indicate the additional overturning circulation due to the advective BBL.
+      The transport of the downslope flow is defined either as the transport of the bottom ocean cell (black arrow),
+      or as a function of the along slope density gradient.
+      The green arrow indicates the diffusive BBL flux directly connecting $kup$ and $kdwn$ ocean bottom cells.
+    }
+  \end{center}
+\end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 …
 %%%gmcomment   :  this section has to be really written
 When applying an advective BBL (\np{nn\_bbl\_adv}\forcode{ = 1..2}), an overturning
 circulation is added which connects two adjacent bottom grid-points only if dense
 water overlies less dense water on the slope. The density difference causes dense
+water to move down the slope.
+\np{nn\_bbl\_adv}\forcode{ = 1} : the downslope velocity is chosen to be the Eulerian
+ocean velocity just above the topographic step (see black arrow in \autoref{fig:bbl})
+\citep{Beckmann_Doscher1997}. It is a \textit{conditional advection}, that is, advection
 is allowed only if dense water overlies less dense water on the slope ($i.e.$
+$\nabla_\sigma \rho  \cdot  \nabla H<0$) and if the velocity is directed towards
+greater depth ($i.e.$ $\vect{U}  \cdot  \nabla H>0$).
 \np{nn\_bbl\_adv}\forcode{ = 2} : the downslope velocity is chosen to be proportional to $\Delta \rho$,
+When applying an advective BBL (\np{nn\_bbl\_adv}\forcode{ = 1..2}), an overturning circulation is added which
+connects two adjacent bottom grid-points only if dense water overlies less dense water on the slope.
+The density difference causes dense water to move down the slope.
+\np{nn\_bbl\_adv}\forcode{ = 1}:
+the downslope velocity is chosen to be the Eulerian ocean velocity just above the topographic step
+(see black arrow in \autoref{fig:bbl}) \citep{Beckmann_Doscher1997}.
+It is a \textit{conditional advection}, that is, advection is allowed only
+if dense water overlies less dense water on the slope ($i.e.$ $\nabla_\sigma \rho  \cdot  \nabla H<0$) and
+if the velocity is directed towards greater depth ($i.e.$ $\vect{U}  \cdot  \nabla H>0$).
+\np{nn\_bbl\_adv}\forcode{ = 2}:
+the downslope velocity is chosen to be proportional to $\Delta \rho$,
 the density difference between the higher cell and lower cell densities \citep{Campin_Goosse_Tel99}.
 The advection is allowed only  if dense water overlies less dense water on the slope ($i.e.$
+$\nabla_\sigma \rho  \cdot  \nabla H<0$). For example, the resulting transport of the
+downslope flow, here in the $i$-direction (\autoref{fig:bbl}), is simply given by the
 following expression:
+The advection is allowed only  if dense water overlies less dense water on the slope
+($i.e.$ $\nabla_\sigma \rho  \cdot  \nabla H<0$).
+For example, the resulting transport of the downslope flow, here in the $i$-direction (\autoref{fig:bbl}),
+is simply given by the following expression:
 \begin{equation} \label{eq:bbl_Utr}
  u^{tr}_{bbl} = \gamma \, g \frac{\Delta \rho}{\rho_o}  e_{1u} \; min \left( {e_{3u}}_{kup},{e_{3u}}_{kdwn} \right)
 \end{equation}
+where $\gamma$, expressed in seconds, is the coefficient of proportionality
+provided as \np{rn\_gambbl}, a namelist parameter, and \textit{kup} and \textit{kdwn}
+are the vertical index of the higher and lower cells, respectively.
+The parameter $\gamma$ should take a different value for each bathymetric
+step, but for simplicity, and because no direct estimation of this parameter is
+available, a uniform value has been assumed. The possible values for $\gamma$
+range between 1 and $10~s$ \citep{Campin_Goosse_Tel99}.
+Scalar properties are advected by this additional transport $( u^{tr}_{bbl}, v^{tr}_{bbl} )$
+using the upwind scheme. Such a diffusive advective scheme has been chosen
+to mimic the entrainment between the downslope plume and the surrounding
+water at intermediate depths. The entrainment is replaced by the vertical mixing
+implicit in the advection scheme. Let us consider as an example the
+case displayed in \autoref{fig:bbl} where the density at level $(i,kup)$ is
+larger than the one at level $(i,kdwn)$. The advective BBL scheme
+modifies the tracer time tendency of the ocean cells near the
+topographic step by the downslope flow \autoref{eq:bbl_dw},
+the horizontal \autoref{eq:bbl_hor} and the upward \autoref{eq:bbl_up}
+return flows as follows:
+where $\gamma$, expressed in seconds, is the coefficient of proportionality provided as \np{rn\_gambbl},
+a namelist parameter, and \textit{kup} and \textit{kdwn} are the vertical index of the higher and lower cells,
+respectively.
+The parameter $\gamma$ should take a different value for each bathymetric step, but for simplicity,
+and because no direct estimation of this parameter is available, a uniform value has been assumed.
+The possible values for $\gamma$ range between 1 and $10~s$ \citep{Campin_Goosse_Tel99}.
+Scalar properties are advected by this additional transport $( u^{tr}_{bbl}, v^{tr}_{bbl} )$ using the upwind scheme.
+Such a diffusive advective scheme has been chosen to mimic the entrainment between the downslope plume and
+the surrounding water at intermediate depths.
+The entrainment is replaced by the vertical mixing implicit in the advection scheme.
+Let us consider as an example the case displayed in \autoref{fig:bbl} where
+the density at level $(i,kup)$ is larger than the one at level $(i,kdwn)$.
+The advective BBL scheme modifies the tracer time tendency of the ocean cells near the topographic step by
+the downslope flow \autoref{eq:bbl_dw}, the horizontal \autoref{eq:bbl_hor} and
+the upward \autoref{eq:bbl_up} return flows as follows:
 \begin{align}
 \partial_t T^{do}_{kdw} &\equiv \partial_t T^{do}_{kdw}
 …
 where $b_t$ is the $T$-cell volume.
+Note that the BBL transport, $( u^{tr}_{bbl}, v^{tr}_{bbl} )$, is available in
+the model outputs. It has to be used to compute the effective velocity
+as well as the effective overturning circulation.
+Note that the BBL transport, $( u^{tr}_{bbl}, v^{tr}_{bbl} )$, is available in the model outputs.
+It has to be used to compute the effective velocity as well as the effective overturning circulation.
 % ================================================================
 …
 %--------------------------------------------------------------------------------------------------------------
+In some applications it can be useful to add a Newtonian damping term
+into the temperature and salinity equations:
+In some applications it can be useful to add a Newtonian damping term into the temperature and salinity equations:
 \begin{equation} \label{eq:tra_dmp}
 \begin{split}
  \frac{\partial T}{\partial t}=\;\cdots \;-\gamma \,\left( {T-T_o } \right)  \\
  \frac{\partial S}{\partial t}=\;\cdots \;-\gamma \,\left( {S-S_o } \right)
  \end{split}
  \end{equation}
 where $\gamma$ is the inverse of a time scale, and $T_o$ and $S_o$
+are given temperature and salinity fields (usually a climatology).
+ \frac{\partial S}{\partial t}=\;\cdots \;-\gamma \,\left( {S-S_o } \right)
+\end{split}
+\end{equation}
+where $\gamma$ is the inverse of a time scale, and $T_o$ and $S_o$ are given temperature and salinity fields
+(usually a climatology).
 Options are defined through the  \ngn{namtra\_dmp} namelist variables.
+The restoring term is added when the namelist parameter \np{ln\_tradmp} is set to true.
+It also requires that both \np{ln\_tsd\_init} and \np{ln\_tsd\_tradmp} are set to true
+in \textit{namtsd} namelist as well as \np{sn\_tem} and \np{sn\_sal} structures are
+correctly set  ($i.e.$ that $T_o$ and $S_o$ are provided in input files and read
+using \mdl{fldread}, see \autoref{subsec:SBC_fldread}).
+The restoring coefficient $\gamma$ is a three-dimensional array read in during the \rou{tra\_dmp\_init} routine. The file name is specified by the namelist variable \np{cn\_resto}. The DMP\_TOOLS tool is provided to allow users to generate the netcdf file.
+The two main cases in which \autoref{eq:tra_dmp} is used are \textit{(a)}
+the specification of the boundary conditions along artificial walls of a
+limited domain basin and \textit{(b)} the computation of the velocity
+field associated with a given $T$-$S$ field (for example to build the
+initial state of a prognostic simulation, or to use the resulting velocity
+field for a passive tracer study). The first case applies to regional
+models that have artificial walls instead of open boundaries.
+In the vicinity of these walls, $\gamma$ takes large values (equivalent to
+a time scale of a few days) whereas it is zero in the interior of the
+model domain. The second case corresponds to the use of the robust
+diagnostic method \citep{Sarmiento1982}. It allows us to find the velocity
+field consistent with the model dynamics whilst having a $T$, $S$ field
+close to a given climatological field ($T_o$, $S_o$).
+The robust diagnostic method is very efficient in preventing temperature
+drift in intermediate waters but it produces artificial sources of heat and salt
+within the ocean. It also has undesirable effects on the ocean convection.
+It tends to prevent deep convection and subsequent deep-water formation,
+by stabilising the water column too much.
+The namelist parameter \np{nn\_zdmp} sets whether the damping should be applied in the whole water column or only below the mixed layer (defined either on a density or $S_o$ criterion). It is common to set the damping to zero in the mixed layer as the adjustment time scale is short here \citep{Madec_al_JPO96}.
+The restoring term is added when the namelist parameter \np{ln\_tradmp} is set to true.
+It also requires that both \np{ln\_tsd\_init} and \np{ln\_tsd\_tradmp} are set to true in
+\textit{namtsd} namelist as well as \np{sn\_tem} and \np{sn\_sal} structures are correctly set
+($i.e.$ that $T_o$ and $S_o$ are provided in input files and read using \mdl{fldread},
+see \autoref{subsec:SBC_fldread}).
+The restoring coefficient $\gamma$ is a three-dimensional array read in during the \rou{tra\_dmp\_init} routine.
+The file name is specified by the namelist variable \np{cn\_resto}.
+The DMP\_TOOLS tool is provided to allow users to generate the netcdf file.
+The two main cases in which \autoref{eq:tra_dmp} is used are
+\textit{(a)} the specification of the boundary conditions along artificial walls of a limited domain basin and
+\textit{(b)} the computation of the velocity field associated with a given $T$-$S$ field
+(for example to build the initial state of a prognostic simulation,
+or to use the resulting velocity field for a passive tracer study).
+The first case applies to regional models that have artificial walls instead of open boundaries.
+In the vicinity of these walls, $\gamma$ takes large values (equivalent to a time scale of a few days) whereas
+it is zero in the interior of the model domain.
+The second case corresponds to the use of the robust diagnostic method \citep{Sarmiento1982}.
+It allows us to find the velocity field consistent with the model dynamics whilst
+having a $T$, $S$ field close to a given climatological field ($T_o$, $S_o$).
+The robust diagnostic method is very efficient in preventing temperature drift in intermediate waters but
+it produces artificial sources of heat and salt within the ocean.
+It also has undesirable effects on the ocean convection.
+It tends to prevent deep convection and subsequent deep-water formation, by stabilising the water column too much.
+The namelist parameter \np{nn\_zdmp} sets whether the damping should be applied in the whole water column or
+only below the mixed layer (defined either on a density or $S_o$ criterion).
+It is common to set the damping to zero in the mixed layer as the adjustment time scale is short here
+\citep{Madec_al_JPO96}.
 \subsection{Generating \ifile{resto} using DMP\_TOOLS}
+DMP\_TOOLS can be used to generate a netcdf file containing the restoration coefficient $\gamma$.
+Note that in order to maintain bit comparison with previous NEMO versions DMP\_TOOLS must be compiled
+and run on the same machine as the NEMO model. A \ifile{mesh\_mask} file for the model configuration is required as an input.
+This can be generated by carrying out a short model run with the namelist parameter \np{nn\_msh} set to 1.
+The namelist parameter \np{ln\_tradmp} will also need to be set to .false. for this to work.
+The \nl{nam\_dmp\_create} namelist in the DMP\_TOOLS directory is used to specify options for the restoration coefficient.
+DMP\_TOOLS can be used to generate a netcdf file containing the restoration coefficient $\gamma$.
+Note that in order to maintain bit comparison with previous NEMO versions DMP\_TOOLS must be compiled and
+run on the same machine as the NEMO model.
+A \ifile{mesh\_mask} file for the model configuration is required as an input.
+This can be generated by carrying out a short model run with the namelist parameter \np{nn\_msh} set to 1.
+The namelist parameter \np{ln\_tradmp} will also need to be set to .false. for this to work.
+The \nl{nam\_dmp\_create} namelist in the DMP\_TOOLS directory is used to specify options for
+the restoration coefficient.
 %--------------------------------------------nam_dmp_create-------------------------------------------------
 …
 %-------------------------------------------------------------------------------------------------------
+\np{cp\_cfg}, \np{cp\_cpz}, \np{jp\_cfg} and \np{jperio} specify the model configuration being used and should be the same as specified in \nl{namcfg}. The variable \nl{lzoom} is used to specify that the damping is being used as in case \textit{a} above to provide boundary conditions to a zoom configuration. In the case of the arctic or antarctic zoom configurations this includes some specific treatment. Otherwise damping is applied to the 6 grid points along the ocean boundaries. The open boundaries are specified by the variables \np{lzoom\_n}, \np{lzoom\_e}, \np{lzoom\_s}, \np{lzoom\_w} in the \nl{nam\_zoom\_dmp} name list.
+The remaining switch namelist variables determine the spatial variation of the restoration coefficient in non-zoom configurations.
+\np{ln\_full\_field} specifies that newtonian damping should be applied to the whole model domain.
+\np{ln\_med\_red\_seas} specifies grid specific restoration coefficients in the Mediterranean Sea
+for the ORCA4, ORCA2 and ORCA05 configurations.
+If \np{ln\_old\_31\_lev\_code} is set then the depth variation of the coeffients will be specified as
+a function of the model number. This option is included to allow backwards compatability of the ORCA2 reference
+configurations with previous model versions.
+\np{ln\_coast} specifies that the restoration coefficient should be reduced near to coastlines.
+This option only has an effect if \np{ln\_full\_field} is true.
+\np{ln\_zero\_top\_layer} specifies that the restoration coefficient should be zero in the surface layer.
+Finally \np{ln\_custom} specifies that the custom module will be called.
+This module is contained in the file \mdl{custom} and can be edited by users. For example damping could be applied in a specific region.
+The restoration coefficient can be set to zero in equatorial regions by specifying a positive value of \np{nn\_hdmp}.
+\np{cp\_cfg}, \np{cp\_cpz}, \np{jp\_cfg} and \np{jperio} specify the model configuration being used and
+should be the same as specified in \nl{namcfg}.
+The variable \nl{lzoom} is used to specify that the damping is being used as in case \textit{a} above to
+provide boundary conditions to a zoom configuration.
+In the case of the arctic or antarctic zoom configurations this includes some specific treatment.
+Otherwise damping is applied to the 6 grid points along the ocean boundaries.
+The open boundaries are specified by the variables \np{lzoom\_n}, \np{lzoom\_e}, \np{lzoom\_s}, \np{lzoom\_w} in
+the \nl{nam\_zoom\_dmp} name list.
+The remaining switch namelist variables determine the spatial variation of the restoration coefficient in
+non-zoom configurations.
+\np{ln\_full\_field} specifies that newtonian damping should be applied to the whole model domain.
+\np{ln\_med\_red\_seas} specifies grid specific restoration coefficients in the Mediterranean Sea for
+the ORCA4, ORCA2 and ORCA05 configurations.
+If \np{ln\_old\_31\_lev\_code} is set then the depth variation of the coeffients will be specified as
+a function of the model number.
+This option is included to allow backwards compatability of the ORCA2 reference configurations with
+previous model versions.
+\np{ln\_coast} specifies that the restoration coefficient should be reduced near to coastlines.
+This option only has an effect if \np{ln\_full\_field} is true.
+\np{ln\_zero\_top\_layer} specifies that the restoration coefficient should be zero in the surface layer.
+Finally \np{ln\_custom} specifies that the custom module will be called.
+This module is contained in the file \mdl{custom} and can be edited by users.
+For example damping could be applied in a specific region.
+The restoration coefficient can be set to zero in equatorial regions by
+specifying a positive value of \np{nn\_hdmp}.
 Equatorward of this latitude the restoration coefficient will be zero with a smooth transition to
 the full values of a 10\deg latitud band.
+This is often used because of the short adjustment time scale in the equatorial region
+\citep{Reverdin1991, Fujio1991, Marti_PhD92}. The time scale associated with the damping depends on the depth as a
+hyperbolic tangent, with \np{rn\_surf} as surface value, \np{rn\_bot} as bottom value and a transition depth of \np{rn\_dep}.
+This is often used because of the short adjustment time scale in the equatorial region
+\citep{Reverdin1991, Fujio1991, Marti_PhD92}.
+The time scale associated with the damping depends on the depth as a hyperbolic tangent,
+with \np{rn\_surf} as surface value, \np{rn\_bot} as bottom value and a transition depth of \np{rn\_dep}.
 % ================================================================
 …
 Options are defined through the  \ngn{namdom} namelist variables.
+The general framework for tracer time stepping is a modified leap-frog scheme
+\citep{Leclair_Madec_OM09}, $i.e.$ a three level centred time scheme associated
+with a Asselin time filter (cf. \autoref{sec:STP_mLF}):
+The general framework for tracer time stepping is a modified leap-frog scheme \citep{Leclair_Madec_OM09},
+$i.e.$ a three level centred time scheme associated with a Asselin time filter (cf. \autoref{sec:STP_mLF}):
 \begin{equation} \label{eq:tra_nxt}
 \begin{aligned}
 …
 \end{aligned}
 \end{equation}
 where RHS is the right hand side of the temperature equation,
+the subscript $f$ denotes filtered values, $\gamma$ is the Asselin coefficient,
+and $S$ is the total forcing applied on $T$ ($i.e.$ fluxes plus content in mass exchanges).
 $\gamma$ is initialized as \np{rn\_atfp} (\textbf{namelist} parameter).
 Its default value is \np{rn\_atfp}\forcode{ = 10.e-3}. Note that the forcing correction term in the filter
+is not applied in linear free surface (\jp{lk\_vvl}\forcode{ = .false.}) (see \autoref{subsec:TRA_sbc}.
+Not also that in constant volume case, the time stepping is performed on $T$,
 not on its content, $e_{3t}T$.
 When the vertical mixing is solved implicitly, the update of the \textit{next} tracer
+fields is done in module \mdl{trazdf}. In this case only the swapping of arrays
 and the Asselin filtering is done in the \mdl{tranxt} module.
 In order to prepare for the computation of the \textit{next} time step,
 a swap of tracer arrays is performed: $T^{t-\rdt} = T^t$ and $T^t = T_f$.
+where RHS is the right hand side of the temperature equation, the subscript $f$ denotes filtered values,
+$\gamma$ is the Asselin coefficient, and $S$ is the total forcing applied on $T$
+($i.e.$ fluxes plus content in mass exchanges).
+$\gamma$ is initialized as \np{rn\_atfp} (\textbf{namelist} parameter).
+Its default value is \np{rn\_atfp}\forcode{ = 10.e-3}.
+Note that the forcing correction term in the filter is not applied in linear free surface
+(\jp{lk\_vvl}\forcode{ = .false.}) (see \autoref{subsec:TRA_sbc}.
+Not also that in constant volume case, the time stepping is performed on $T$, not on its content, $e_{3t}T$.
+When the vertical mixing is solved implicitly,
+the update of the \textit{next} tracer fields is done in module \mdl{trazdf}.
+In this case only the swapping of arrays and the Asselin filtering is done in the \mdl{tranxt} module.
+In order to prepare for the computation of the \textit{next} time step, a swap of tracer arrays is performed:
+$T^{t-\rdt} = T^t$ and $T^t = T_f$.
 % ================================================================
 …
 \label{subsec:TRA_eos}
 The Equation Of Seawater (EOS) is an empirical nonlinear thermodynamic relationship
+linking seawater density, $\rho$, to a number of state variables,
+most typically temperature, salinity and pressure.
+Because density gradients control the pressure gradient force through the hydrostatic balance,
 the equation of state provides a fundamental bridge between the distribution of active tracers
+and the fluid dynamics. Nonlinearities of the EOS are of major importance, in particular
+influencing the circulation through determination of the static stability below the mixed layer,
 thus controlling rates of exchange between the atmosphere  and the ocean interior \citep{Roquet_JPO2015}.
 Therefore an accurate EOS based on either the 1980 equation of state (EOS-80, \cite{UNESCO1983})
+or TEOS-10 \citep{TEOS10} standards should be used anytime a simulation of the real
+ocean circulation is attempted \citep{Roquet_JPO2015}.
 The use of TEOS-10 is highly recommended because
 \textit{(i)} it is the new official EOS,
 \textit{(ii)} it is more accurate, being based on an updated database of laboratory measurements, and
 \textit{(iii)} it uses Conservative Temperature and Absolute Salinity (instead of potential temperature
+and practical salinity for EOS-980, both variables being more suitable for use as model variables
 \citep{TEOS10, Graham_McDougall_JPO13}.
+The Equation Of Seawater (EOS) is an empirical nonlinear thermodynamic relationship linking seawater density,
+$\rho$, to a number of state variables, most typically temperature, salinity and pressure.
+Because density gradients control the pressure gradient force through the hydrostatic balance,
+the equation of state provides a fundamental bridge between the distribution of active tracers and
+the fluid dynamics.
+Nonlinearities of the EOS are of major importance, in particular influencing the circulation through
+determination of the static stability below the mixed layer,
+thus controlling rates of exchange between the atmosphere and the ocean interior \citep{Roquet_JPO2015}.
+Therefore an accurate EOS based on either the 1980 equation of state (EOS-80, \cite{UNESCO1983}) or
+TEOS-10 \citep{TEOS10} standards should be used anytime a simulation of the real ocean circulation is attempted
+\citep{Roquet_JPO2015}.
+The use of TEOS-10 is highly recommended because
+\textit{(i)}   it is the new official EOS,
+\textit{(ii)}  it is more accurate, being based on an updated database of laboratory measurements, and
+\textit{(iii)} it uses Conservative Temperature and Absolute Salinity (instead of potential temperature and
+practical salinity for EOS-980, both variables being more suitable for use as model variables
+\citep{TEOS10, Graham_McDougall_JPO13}.
 EOS-80 is an obsolescent feature of the NEMO system, kept only for backward compatibility.
+For process studies, it is often convenient to use an approximation of the EOS. To that purposed,
+a simplified EOS (S-EOS) inspired by \citet{Vallis06} is also available.
+In the computer code, a density anomaly, $d_a= \rho / \rho_o - 1$,
+is computed, with $\rho_o$ a reference density. Called \textit{rau0}
+in the code, $\rho_o$ is set in \mdl{phycst} to a value of $1,026~Kg/m^3$.
+This is a sensible choice for the reference density used in a Boussinesq ocean
+climate model, as, with the exception of only a small percentage of the ocean,
+For process studies, it is often convenient to use an approximation of the EOS.
+To that purposed, a simplified EOS (S-EOS) inspired by \citet{Vallis06} is also available.
+In the computer code, a density anomaly, $d_a= \rho / \rho_o - 1$, is computed, with $\rho_o$ a reference density.
+Called \textit{rau0} in the code, $\rho_o$ is set in \mdl{phycst} to a value of $1,026~Kg/m^3$.
+This is a sensible choice for the reference density used in a Boussinesq ocean climate model, as,
+with the exception of only a small percentage of the ocean,
 density in the World Ocean varies by no more than 2$\%$ from that value \citep{Gill1982}.
 Options are defined through the  \ngn{nameos} namelist variables, and in particular \np{nn\_eos}
 which controls the EOS used (\forcode{= -1} for TEOS10 ; \forcode{= 0} for EOS-80 ; \forcode{= 1} for S-EOS).
+Options are defined through the  \ngn{nameos} namelist variables, and in particular \np{nn\_eos} which
+controls the EOS used (\forcode{= -1} for TEOS10 ; \forcode{= 0} for EOS-80 ; \forcode{= 1} for S-EOS).
 \begin{description}
+\item[\np{nn\_eos}\forcode{ = -1}] the polyTEOS10-bsq equation of seawater \citep{Roquet_OM2015} is used.
+The accuracy of this approximation is comparable to the TEOS-10 rational function approximation,
+but it is optimized for a boussinesq fluid and the polynomial expressions have simpler
+and more computationally efficient expressions for their derived quantities
+which make them more adapted for use in ocean models.
+Note that a slightly higher precision polynomial form is now used replacement of the TEOS-10
+rational function approximation for hydrographic data analysis  \citep{TEOS10}.
+A key point is that conservative state variables are used:
+Absolute Salinity (unit: g/kg, notation: $S_A$) and Conservative Temperature (unit: \degC, notation: $\Theta$).
+The pressure in decibars is approximated by the depth in meters.
+With TEOS10, the specific heat capacity of sea water, $C_p$, is a constant. It is set to
+$C_p=3991.86795711963~J\,Kg^{-1}\,^{\circ}K^{-1}$, according to \citet{TEOS10}.
+Choosing polyTEOS10-bsq implies that the state variables used by the model are
+$\Theta$ and $S_A$. In particular, the initial state deined by the user have to be given as
+\textit{Conservative} Temperature and \textit{Absolute} Salinity.
+In addition, setting \np{ln\_useCT} to \forcode{.true.} convert the Conservative SST to potential SST
+prior to either computing the air-sea and ice-sea fluxes (forced mode)
+or sending the SST field to the atmosphere (coupled mode).
+\item[\np{nn\_eos}\forcode{ = 0}] the polyEOS80-bsq equation of seawater is used.
+It takes the same polynomial form as the polyTEOS10, but the coefficients have been optimized
+to accurately fit EOS80 (Roquet, personal comm.). The state variables used in both the EOS80
+and the ocean model are:
+the Practical Salinity ((unit: psu, notation: $S_p$)) and Potential Temperature (unit: $^{\circ}C$, notation: $\theta$).
+The pressure in decibars is approximated by the depth in meters.
+With thsi EOS, the specific heat capacity of sea water, $C_p$, is a function of temperature,
+salinity and pressure \citep{UNESCO1983}. Nevertheless, a severe assumption is made in order to
+have a heat content ($C_p T_p$) which is conserved by the model: $C_p$ is set to a constant
+value, the TEOS10 value.
+\item[\np{nn\_eos}\forcode{ = -1}]
+  the polyTEOS10-bsq equation of seawater \citep{Roquet_OM2015} is used.
+  The accuracy of this approximation is comparable to the TEOS-10 rational function approximation,
+  but it is optimized for a boussinesq fluid and the polynomial expressions have simpler and
+  more computationally efficient expressions for their derived quantities which make them more adapted for
+  use in ocean models.
+  Note that a slightly higher precision polynomial form is now used replacement of
+  the TEOS-10 rational function approximation for hydrographic data analysis \citep{TEOS10}.
+  A key point is that conservative state variables are used:
+  Absolute Salinity (unit: g/kg, notation: $S_A$) and Conservative Temperature (unit: \degC, notation: $\Theta$).
+  The pressure in decibars is approximated by the depth in meters.
+  With TEOS10, the specific heat capacity of sea water, $C_p$, is a constant.
+  It is set to $C_p=3991.86795711963~J\,Kg^{-1}\,^{\circ}K^{-1}$, according to \citet{TEOS10}.
+  Choosing polyTEOS10-bsq implies that the state variables used by the model are $\Theta$ and $S_A$.
+  In particular, the initial state deined by the user have to be given as \textit{Conservative} Temperature and
+  \textit{Absolute} Salinity.
+  In addition, setting \np{ln\_useCT} to \forcode{.true.} convert the Conservative SST to potential SST prior to
+  either computing the air-sea and ice-sea fluxes (forced mode) or
+  sending the SST field to the atmosphere (coupled mode).
+\item[\np{nn\_eos}\forcode{ = 0}]
+  the polyEOS80-bsq equation of seawater is used.
+  It takes the same polynomial form as the polyTEOS10, but the coefficients have been optimized to
+  accurately fit EOS80 (Roquet, personal comm.).
+  The state variables used in both the EOS80 and the ocean model are:
+  the Practical Salinity ((unit: psu, notation: $S_p$)) and
+  Potential Temperature (unit: $^{\circ}C$, notation: $\theta$).
+  The pressure in decibars is approximated by the depth in meters.
+  With thsi EOS, the specific heat capacity of sea water, $C_p$, is a function of temperature, salinity and
+  pressure \citep{UNESCO1983}.
+  Nevertheless, a severe assumption is made in order to have a heat content ($C_p T_p$) which
+  is conserved by the model: $C_p$ is set to a constant value, the TEOS10 value.
+\item[\np{nn\_eos}\forcode{ = 1}] a simplified EOS (S-EOS) inspired by \citet{Vallis06} is chosen,
+the coefficients of which has been optimized to fit the behavior of TEOS10 (Roquet, personal comm.)
+(see also \citet{Roquet_JPO2015}). It provides a simplistic linear representation of both
+cabbeling and thermobaricity effects which is enough for a proper treatment of the EOS
+in theoretical studies \citep{Roquet_JPO2015}.
+With such an equation of state there is no longer a distinction between
+\textit{conservative} and \textit{potential} temperature, as well as between \textit{absolute}
+and \textit{practical} salinity.
+S-EOS takes the following expression:
+\begin{equation} \label{eq:tra_S-EOS}
+\begin{split}
+  d_a(T,S,z)  =  ( & - a_0 \; ( 1 + 0.5 \; \lambda_1 \; T_a + \mu_1 \; z ) * T_a  \\
+                                & + b_0 \; ( 1 - 0.5 \; \lambda_2 \; S_a - \mu_2 \; z ) * S_a  \\
+                                & - \nu \; T_a \; S_a \;  ) \; / \; \rho_o                     \\
+  with \ \  T_a = T-10  \; ;  & \;  S_a = S-35  \; ;\;  \rho_o = 1026~Kg/m^3
+\end{split}
+\end{equation}
+where the computer name of the coefficients as well as their standard value are given in \autoref{tab:SEOS}.
+In fact, when choosing S-EOS, various approximation of EOS can be specified simply by changing
+the associated coefficients.
+Setting to zero the two thermobaric coefficients ($\mu_1$, $\mu_2$) remove thermobaric effect from S-EOS.
+setting to zero the three cabbeling coefficients ($\lambda_1$, $\lambda_2$, $\nu$) remove cabbeling effect from S-EOS.
+Keeping non-zero value to $a_0$ and $b_0$ provide a linear EOS function of T and S.
+\item[\np{nn\_eos}\forcode{ = 1}]
+  a simplified EOS (S-EOS) inspired by \citet{Vallis06} is chosen,
+  the coefficients of which has been optimized to fit the behavior of TEOS10
+  (Roquet, personal comm.) (see also \citet{Roquet_JPO2015}).
+  It provides a simplistic linear representation of both cabbeling and thermobaricity effects which
+  is enough for a proper treatment of the EOS in theoretical studies \citep{Roquet_JPO2015}.
+  With such an equation of state there is no longer a distinction between
+  \textit{conservative} and \textit{potential} temperature,
+  as well as between \textit{absolute} and \textit{practical} salinity.
+  S-EOS takes the following expression:
+  \begin{equation} \label{eq:tra_S-EOS}
+    \begin{split}
+      d_a(T,S,z)  =  ( & - a_0 \; ( 1 + 0.5 \; \lambda_1 \; T_a + \mu_1 \; z ) * T_a  \\
+      & + b_0 \; ( 1 - 0.5 \; \lambda_2 \; S_a - \mu_2 \; z ) * S_a  \\
+      & - \nu \; T_a \; S_a \;  ) \; / \; \rho_o                     \\
+      with \ \  T_a = T-10  \; ;  & \;  S_a = S-35  \; ;\;  \rho_o = 1026~Kg/m^3
+    \end{split}
+  \end{equation}
+  where the computer name of the coefficients as well as their standard value are given in \autoref{tab:SEOS}.
+  In fact, when choosing S-EOS, various approximation of EOS can be specified simply by changing the associated coefficients.
+  Setting to zero the two thermobaric coefficients ($\mu_1$, $\mu_2$) remove thermobaric effect from S-EOS.
+  setting to zero the three cabbeling coefficients ($\lambda_1$, $\lambda_2$, $\nu$) remove cabbeling effect from S-EOS.
+  Keeping non-zero value to $a_0$ and $b_0$ provide a linear EOS function of T and S.
 \end{description}
 …
 \end{tabular}
 \caption{ \protect\label{tab:SEOS}
+Standard value of S-EOS coefficients. }
+  Standard value of S-EOS coefficients.
+}
 \end{center}
 \end{table}
 …
 \label{subsec:TRA_bn2}
 An accurate computation of the ocean stability (i.e. of $N$, the brunt-V\"{a}is\"{a}l\"{a}
+ frequency) is of paramount importance as determine the ocean stratification and
+ is used in several ocean parameterisations (namely TKE, GLS, Richardson number dependent
+ vertical diffusion, enhanced vertical diffusion, non-penetrative convection, tidal mixing
+ parameterisation, iso-neutral diffusion). In particular, $N^2$ has to be computed at the local pressure
+ (pressure in decibar being approximated by the depth in meters). The expression for $N^2$
  is given by:
+An accurate computation of the ocean stability (i.e. of $N$, the brunt-V\"{a}is\"{a}l\"{a} frequency) is of
+paramount importance as determine the ocean stratification and is used in several ocean parameterisations
+(namely TKE, GLS, Richardson number dependent vertical diffusion, enhanced vertical diffusion,
+non-penetrative convection, tidal mixing  parameterisation, iso-neutral diffusion).
+In particular, $N^2$ has to be computed at the local pressure
+(pressure in decibar being approximated by the depth in meters).
+The expression for $N^2$  is given by:
 \begin{equation} \label{eq:tra_bn2}
 N^2 = \frac{g}{e_{3w}} \left(   \beta \;\delta_{k+1/2}[S] - \alpha \;\delta_{k+1/2}[T]   \right)
 \end{equation}
 where $(T,S) = (\Theta, S_A)$ for TEOS10, $= (\theta, S_p)$ for TEOS-80, or $=(T,S)$ for S-EOS,
 and, $\alpha$ and $\beta$ are the thermal and haline expansion coefficients.
 The coefficients are a polynomial function of temperature, salinity and depth which expression
+depends on the chosen EOS. They are computed through \textit{eos\_rab}, a \textsc{Fortran}
 function that can be found in \mdl{eosbn2}.
+where $(T,S) = (\Theta, S_A)$ for TEOS10, $= (\theta, S_p)$ for TEOS-80, or $=(T,S)$ for S-EOS,
+and, $\alpha$ and $\beta$ are the thermal and haline expansion coefficients.
+The coefficients are a polynomial function of temperature, salinity and depth which
+expression depends on the chosen EOS.
+They are computed through \textit{eos\_rab}, a \textsc{Fortran} function that can be found in \mdl{eosbn2}.
 % -------------------------------------------------------------------------------------------------------------
 …
 \end{equation}
 \autoref{eq:tra_eos_fzp} is only used to compute the potential freezing point of
+sea water ($i.e.$ referenced to the surface $p=0$), thus the pressure dependent
 terms in \autoref{eq:tra_eos_fzp} (last term) have been dropped. The freezing
+point is computed through \textit{eos\_fzp}, a \textsc{Fortran} function that can be found
 in \mdl{eosbn2}.
+\autoref{eq:tra_eos_fzp} is only used to compute the potential freezing point of sea water
+($i.e.$ referenced to the surface $p=0$),
+thus the pressure dependent terms in \autoref{eq:tra_eos_fzp} (last term) have been dropped.
+The freezing point is computed through \textit{eos\_fzp},
+a \textsc{Fortran} function that can be found in \mdl{eosbn2}.
 …
 \gmcomment{STEVEN: to be consistent with earlier discussion of differencing and averaging operators,
+                   I've changed "derivative" to "difference" and "mean" to "average"}
+With partial cells (\np{ln\_zps}\forcode{ = .true.}) at bottom and top (\np{ln\_isfcav}\forcode{ = .true.}), in general,
+tracers in horizontally adjacent cells live at different depths.
+Horizontal gradients of tracers are needed for horizontal diffusion (\mdl{traldf} module)
+and the hydrostatic pressure gradient calculations (\mdl{dynhpg} module).
+The partial cell properties at the top (\np{ln\_isfcav}\forcode{ = .true.}) are computed in the same way as for the bottom.
+I've changed "derivative" to "difference" and "mean" to "average"}
+With partial cells (\np{ln\_zps}\forcode{ = .true.}) at bottom and top (\np{ln\_isfcav}\forcode{ = .true.}),
+in general, tracers in horizontally adjacent cells live at different depths.
+Horizontal gradients of tracers are needed for horizontal diffusion (\mdl{traldf} module) and
+the hydrostatic pressure gradient calculations (\mdl{dynhpg} module).
+The partial cell properties at the top (\np{ln\_isfcav}\forcode{ = .true.}) are computed in the same way as
+for the bottom.
 So, only the bottom interpolation is explained below.
+Before taking horizontal gradients between the tracers next to the bottom, a linear
+interpolation in the vertical is used to approximate the deeper tracer as if it actually
+lived at the depth of the shallower tracer point (\autoref{fig:Partial_step_scheme}).
+For example, for temperature in the $i$-direction the needed interpolated
+temperature, $\widetilde{T}$, is:
+Before taking horizontal gradients between the tracers next to the bottom,
+a linear interpolation in the vertical is used to approximate the deeper tracer as if
+it actually lived at the depth of the shallower tracer point (\autoref{fig:Partial_step_scheme}).
+For example, for temperature in the $i$-direction the needed interpolated temperature, $\widetilde{T}$, is:
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!p]    \begin{center}
+\includegraphics[width=0.9\textwidth]{Partial_step_scheme}
+\caption{   \protect\label{fig:Partial_step_scheme}
+Discretisation of the horizontal difference and average of tracers in the $z$-partial
+step coordinate (\protect\np{ln\_zps}\forcode{ = .true.}) in the case $( e3w_k^{i+1} - e3w_k^i  )>0$.
+A linear interpolation is used to estimate $\widetilde{T}_k^{i+1}$, the tracer value
+at the depth of the shallower tracer point of the two adjacent bottom $T$-points.
+The horizontal difference is then given by: $\delta _{i+1/2} T_k=  \widetilde{T}_k^{\,i+1} -T_k^{\,i}$
+and the average by: $\overline{T}_k^{\,i+1/2}= ( \widetilde{T}_k^{\,i+1/2} - T_k^{\,i} ) / 2$.  }
+\end{center}   \end{figure}
+\begin{figure}[!p]
+  \begin{center}
+    \includegraphics[width=0.9\textwidth]{Fig_partial_step_scheme}
+    \caption{  \protect\label{fig:Partial_step_scheme}
+      Discretisation of the horizontal difference and average of tracers in the $z$-partial step coordinate
+      (\protect\np{ln\_zps}\forcode{ = .true.}) in the case $( e3w_k^{i+1} - e3w_k^i  )>0$.
+      A linear interpolation is used to estimate $\widetilde{T}_k^{i+1}$,
+      the tracer value at the depth of the shallower tracer point of the two adjacent bottom $T$-points.
+      The horizontal difference is then given by: $\delta _{i+1/2} T_k=  \widetilde{T}_k^{\,i+1} -T_k^{\,i}$ and
+      the average by: $\overline{T}_k^{\,i+1/2}= ( \widetilde{T}_k^{\,i+1/2} - T_k^{\,i} ) / 2$.
+    }
+  \end{center}
+\end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 \begin{equation*}
 …
             \end{aligned}   \right.
 \end{equation*}
+and the resulting forms for the horizontal difference and the horizontal average
+value of $T$ at a $U$-point are:
+and the resulting forms for the horizontal difference and the horizontal average value of $T$ at a $U$-point are:
 \begin{equation} \label{eq:zps_hde}
 \begin{aligned}
 …
 \end{equation}
+The computation of horizontal derivative of tracers as well as of density is
+performed once for all at each time step in \mdl{zpshde} module and stored
+in shared arrays to be used when needed. It has to be emphasized that the
+procedure used to compute the interpolated density, $\widetilde{\rho}$, is not
+the same as that used for $T$ and $S$. Instead of forming a linear approximation
+of density, we compute $\widetilde{\rho }$ from the interpolated values of $T$
+and $S$, and the pressure at a $u$-point (in the equation of state pressure is
+approximated by depth, see \autoref{subsec:TRA_eos} ) :
+The computation of horizontal derivative of tracers as well as of density is performed once for all at
+each time step in \mdl{zpshde} module and stored in shared arrays to be used when needed.
+It has to be emphasized that the procedure used to compute the interpolated density, $\widetilde{\rho}$,
+is not the same as that used for $T$ and $S$.
+Instead of forming a linear approximation of density, we compute $\widetilde{\rho }$ from the interpolated values of
+$T$ and $S$, and the pressure at a $u$-point
+(in the equation of state pressure is approximated by depth, see \autoref{subsec:TRA_eos} ):
 \begin{equation} \label{eq:zps_hde_rho}
 \widetilde{\rho } = \rho ( {\widetilde{T},\widetilde {S},z_u })
 …
 \end{equation}
 This is a much better approximation as the variation of $\rho$ with depth (and
+thus pressure) is highly non-linear with a true equation of state and thus is badly
+approximated with a linear interpolation. This approximation is used to compute
+both the horizontal pressure gradient (\autoref{sec:DYN_hpg}) and the slopes of neutral
+surfaces (\autoref{sec:LDF_slp})
+Note that in almost all the advection schemes presented in this Chapter, both
+averaging and differencing operators appear. Yet \autoref{eq:zps_hde} has not
+been used in these schemes: in contrast to diffusion and pressure gradient
+computations, no correction for partial steps is applied for advection. The main
+motivation is to preserve the domain averaged mean variance of the advected
+field when using the $2^{nd}$ order centred scheme. Sensitivity of the advection
+schemes to the way horizontal averages are performed in the vicinity of partial
 cells should be further investigated in the near future.
+This is a much better approximation as the variation of $\rho$ with depth (and thus pressure)
+is highly non-linear with a true equation of state and thus is badly approximated with a linear interpolation.
+This approximation is used to compute both the horizontal pressure gradient (\autoref{sec:DYN_hpg}) and
+the slopes of neutral surfaces (\autoref{sec:LDF_slp}).
+Note that in almost all the advection schemes presented in this Chapter,
+both averaging and differencing operators appear.
+Yet \autoref{eq:zps_hde} has not been used in these schemes:
+in contrast to diffusion and pressure gradient computations,
+no correction for partial steps is applied for advection.
+The main motivation is to preserve the domain averaged mean variance of the advected field when
+using the $2^{nd}$ order centred scheme.
+Sensitivity of the advection schemes to the way horizontal averages are performed in the vicinity of
+partial cells should be further investigated in the near future.
 %%%
 \gmcomment{gm :   this last remark has to be done}

NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_ZDF.tex

-                      r10146
+                      r10368
 \label{sec:ZDF_zdf}
+The discrete form of the ocean subgrid scale physics has been presented in
+\autoref{sec:TRA_zdf} and \autoref{sec:DYN_zdf}. At the surface and bottom boundaries,
+the turbulent fluxes of momentum, heat and salt have to be defined. At the
+surface they are prescribed from the surface forcing (see \autoref{chap:SBC}),
+while at the bottom they are set to zero for heat and salt, unless a geothermal
+flux forcing is prescribed as a bottom boundary condition ($i.e.$ \key{trabbl}
+defined, see \autoref{subsec:TRA_bbc}), and specified through a bottom friction
+parameterisation for momentum (see \autoref{sec:ZDF_bfr}).
+In this section we briefly discuss the various choices offered to compute
+the vertical eddy viscosity and diffusivity coefficients, $A_u^{vm}$ ,
+$A_v^{vm}$ and $A^{vT}$ ($A^{vS}$), defined at $uw$-, $vw$- and $w$-
+points, respectively (see \autoref{sec:TRA_zdf} and \autoref{sec:DYN_zdf}). These
+coefficients can be assumed to be either constant, or a function of the local
+Richardson number, or computed from a turbulent closure model (either
+TKE or GLS formulation). The computation of these coefficients is initialized
+in the \mdl{zdfini} module and performed in the \mdl{zdfric}, \mdl{zdftke} or
+\mdl{zdfgls} modules. The trends due to the vertical momentum and tracer
+diffusion, including the surface forcing, are computed and added to the
+general trend in the \mdl{dynzdf} and \mdl{trazdf} modules, respectively.
+These trends can be computed using either a forward time stepping scheme
+(namelist parameter \np{ln\_zdfexp}\forcode{ = .true.}) or a backward time stepping
+scheme (\np{ln\_zdfexp}\forcode{ = .false.}) depending on the magnitude of the mixing
+coefficients, and thus of the formulation used (see \autoref{chap:STP}).
+The discrete form of the ocean subgrid scale physics has been presented in
+\autoref{sec:TRA_zdf} and \autoref{sec:DYN_zdf}.
+At the surface and bottom boundaries, the turbulent fluxes of momentum, heat and salt have to be defined.
+At the surface they are prescribed from the surface forcing (see \autoref{chap:SBC}),
+while at the bottom they are set to zero for heat and salt,
+unless a geothermal flux forcing is prescribed as a bottom boundary condition ($i.e.$ \key{trabbl} defined,
+see \autoref{subsec:TRA_bbc}), and specified through a bottom friction parameterisation for momentum
+(see \autoref{sec:ZDF_bfr}).
+In this section we briefly discuss the various choices offered to compute the vertical eddy viscosity and
+diffusivity coefficients, $A_u^{vm}$ , $A_v^{vm}$ and $A^{vT}$ ($A^{vS}$), defined at $uw$-, $vw$- and $w$- points,
+respectively (see \autoref{sec:TRA_zdf} and \autoref{sec:DYN_zdf}).
+These coefficients can be assumed to be either constant, or a function of the local Richardson number,
+or computed from a turbulent closure model (either TKE or GLS formulation).
+The computation of these coefficients is initialized in the \mdl{zdfini} module and performed in
+the \mdl{zdfric}, \mdl{zdftke} or \mdl{zdfgls} modules.
+The trends due to the vertical momentum and tracer diffusion, including the surface forcing,
+are computed and added to the general trend in the \mdl{dynzdf} and \mdl{trazdf} modules, respectively.
+These trends can be computed using either a forward time stepping scheme
+(namelist parameter \np{ln\_zdfexp}\forcode{ = .true.}) or a backward time stepping scheme
+(\np{ln\_zdfexp}\forcode{ = .false.}) depending on the magnitude of the mixing coefficients,
+and thus of the formulation used (see \autoref{chap:STP}).
 % -------------------------------------------------------------------------------------------------------------
 …
 %--------------------------------------------------------------------------------------------------------------
+Options are defined through the  \ngn{namzdf} namelist variables.
+When \key{zdfcst} is defined, the momentum and tracer vertical eddy coefficients
+are set to constant values over the whole ocean. This is the crudest way to define
+the vertical ocean physics. It is recommended that this option is only used in
+process studies, not in basin scale simulations. Typical values used in this case are:
+Options are defined through the \ngn{namzdf} namelist variables.
+When \key{zdfcst} is defined, the momentum and tracer vertical eddy coefficients are set to
+constant values over the whole ocean.
+This is the crudest way to define the vertical ocean physics.
+It is recommended that this option is only used in process studies, not in basin scale simulations.
+Typical values used in this case are:
 \begin{align*}
 A_u^{vm} = A_v^{vm} &= 1.2\ 10^{-4}~m^2.s^{-1}  \\
 …
 These values are set through the \np{rn\_avm0} and \np{rn\_avt0} namelist parameters.
 In all cases, do not use values smaller that those associated with the molecular
+viscosity and diffusivity, that is $\sim10^{-6}~m^2.s^{-1}$ for momentum,
 $\sim10^{-7}~m^2.s^{-1}$ for temperature and $\sim10^{-9}~m^2.s^{-1}$ for salinity.
+In all cases, do not use values smaller that those associated with the molecular viscosity and diffusivity,
+that is $\sim10^{-6}~m^2.s^{-1}$ for momentum, $\sim10^{-7}~m^2.s^{-1}$ for temperature and
+$\sim10^{-9}~m^2.s^{-1}$ for salinity.
 …
 %--------------------------------------------------------------------------------------------------------------
+When \key{zdfric} is defined, a local Richardson number dependent formulation
+for the vertical momentum and tracer eddy coefficients is set through the  \ngn{namzdf\_ric}
+namelist variables.The vertical mixing
+coefficients are diagnosed from the large scale variables computed by the model.
+\textit{In situ} measurements have been used to link vertical turbulent activity to
+large scale ocean structures. The hypothesis of a mixing mainly maintained by the
+growth of Kelvin-Helmholtz like instabilities leads to a dependency between the
+vertical eddy coefficients and the local Richardson number ($i.e.$ the
+ratio of stratification to vertical shear). Following \citet{Pacanowski_Philander_JPO81}, the following
+formulation has been implemented:
+When \key{zdfric} is defined, a local Richardson number dependent formulation for the vertical momentum and
+tracer eddy coefficients is set through the \ngn{namzdf\_ric} namelist variables.
+The vertical mixing coefficients are diagnosed from the large scale variables computed by the model.
+\textit{In situ} measurements have been used to link vertical turbulent activity to large scale ocean structures.
+The hypothesis of a mixing mainly maintained by the growth of Kelvin-Helmholtz like instabilities leads to
+a dependency between the vertical eddy coefficients and the local Richardson number
+($i.e.$ the ratio of stratification to vertical shear).
+Following \citet{Pacanowski_Philander_JPO81}, the following formulation has been implemented:
 \begin{equation} \label{eq:zdfric}
    \left\{      \begin{aligned}
 …
    \end{aligned}    \right.
 \end{equation}
+where $Ri = N^2 / \left(\partial_z \textbf{U}_h \right)^2$ is the local Richardson
+number, $N$ is the local Brunt-Vais\"{a}l\"{a} frequency (see \autoref{subsec:TRA_bn2}),
+$A_b^{vT} $ and $A_b^{vm}$ are the constant background values set as in the
+constant case (see \autoref{subsec:ZDF_cst}), and $A_{ric}^{vT} = 10^{-4}~m^2.s^{-1}$
+is the maximum value that can be reached by the coefficient when $Ri\leq 0$,
+$a=5$ and $n=2$. The last three values can be modified by setting the
+\np{rn\_avmri}, \np{rn\_alp} and \np{nn\_ric} namelist parameters, respectively.
+A simple mixing-layer model to transfer and dissipate the atmospheric
+ forcings (wind-stress and buoyancy fluxes) can be activated setting
+the \np{ln\_mldw}\forcode{ = .true.} in the namelist.
+In this case, the local depth of turbulent wind-mixing or "Ekman depth"
+ $h_{e}(x,y,t)$ is evaluated and the vertical eddy coefficients prescribed within this layer.
+where $Ri = N^2 / \left(\partial_z \textbf{U}_h \right)^2$ is the local Richardson number,
+$N$ is the local Brunt-Vais\"{a}l\"{a} frequency (see \autoref{subsec:TRA_bn2}),
+$A_b^{vT} $ and $A_b^{vm}$ are the constant background values set as in the constant case
+(see \autoref{subsec:ZDF_cst}), and $A_{ric}^{vT} = 10^{-4}~m^2.s^{-1}$ is the maximum value that
+can be reached by the coefficient when $Ri\leq 0$, $a=5$ and $n=2$.
+The last three values can be modified by setting the \np{rn\_avmri}, \np{rn\_alp} and
+\np{nn\_ric} namelist parameters, respectively.
+A simple mixing-layer model to transfer and dissipate the atmospheric forcings
+(wind-stress and buoyancy fluxes) can be activated setting the \np{ln\_mldw}\forcode{ = .true.} in the namelist.
+In this case, the local depth of turbulent wind-mixing or "Ekman depth" $h_{e}(x,y,t)$ is evaluated and
+the vertical eddy coefficients prescribed within this layer.
 This depth is assumed proportional to the "depth of frictional influence" that is limited by rotation:
 \begin{equation}
+         h_{e} = Ek \frac {u^{*}} {f_{0}}    \\
+\end{equation}
+where, $Ek$ is an empirical parameter, $u^{*}$ is the friction velocity and $f_{0}$ is the Coriolis
+parameter.
+h_{e} = Ek \frac {u^{*}} {f_{0}}
+\end{equation}
+where, $Ek$ is an empirical parameter, $u^{*}$ is the friction velocity and $f_{0}$ is the Coriolis parameter.
 In this similarity height relationship, the turbulent friction velocity:
 \begin{equation}
+         u^{*} = \sqrt \frac {|\tau|} {\rho_o}     \\
+\end{equation}
+u^{*} = \sqrt \frac {|\tau|} {\rho_o}
+\end{equation}
 is computed from the wind stress vector $|\tau|$ and the reference density $ \rho_o$.
 The final $h_{e}$ is further constrained by the adjustable bounds \np{rn\_mldmin} and \np{rn\_mldmax}.
 Once $h_{e}$ is computed, the vertical eddy coefficients within $h_{e}$ are set to
+Once $h_{e}$ is computed, the vertical eddy coefficients within $h_{e}$ are set to
 the empirical values \np{rn\_wtmix} and \np{rn\_wvmix} \citep{Lermusiaux2001}.
 …
 %--------------------------------------------------------------------------------------------------------------
+The vertical eddy viscosity and diffusivity coefficients are computed from a TKE
+turbulent closure model based on a prognostic equation for $\bar{e}$, the turbulent
+kinetic energy, and a closure assumption for the turbulent length scales. This
+turbulent closure model has been developed by \citet{Bougeault1989} in the
+atmospheric case, adapted by \citet{Gaspar1990} for the oceanic case, and
+embedded in OPA, the ancestor of NEMO, by \citet{Blanke1993} for equatorial Atlantic
+simulations. Since then, significant modifications have been introduced by
+\citet{Madec1998} in both the implementation and the formulation of the mixing
+length scale. The time evolution of $\bar{e}$ is the result of the production of
+$\bar{e}$ through vertical shear, its destruction through stratification, its vertical
+diffusion, and its dissipation of \citet{Kolmogorov1942} type:
+The vertical eddy viscosity and diffusivity coefficients are computed from a TKE turbulent closure model based on
+a prognostic equation for $\bar{e}$, the turbulent kinetic energy,
+and a closure assumption for the turbulent length scales.
+This turbulent closure model has been developed by \citet{Bougeault1989} in the atmospheric case,
+adapted by \citet{Gaspar1990} for the oceanic case, and embedded in OPA, the ancestor of NEMO,
+by \citet{Blanke1993} for equatorial Atlantic simulations.
+Since then, significant modifications have been introduced by \citet{Madec1998} in both the implementation and
+the formulation of the mixing length scale.
+The time evolution of $\bar{e}$ is the result of the production of $\bar{e}$ through vertical shear,
+its destruction through stratification, its vertical diffusion, and its dissipation of \citet{Kolmogorov1942} type:
 \begin{equation} \label{eq:zdftke_e}
 \frac{\partial \bar{e}}{\partial t} =
 …
 where $N$ is the local Brunt-Vais\"{a}l\"{a} frequency (see \autoref{subsec:TRA_bn2}),
 $l_{\epsilon }$ and $l_{\kappa }$ are the dissipation and mixing length scales,
+$P_{rt}$ is the Prandtl number, $K_m$ and $K_\rho$ are the vertical eddy viscosity
+and diffusivity coefficients. The constants $C_k =  0.1$ and $C_\epsilon = \sqrt {2} /2$
+$\approx 0.7$ are designed to deal with vertical mixing at any depth \citep{Gaspar1990}.
+They are set through namelist parameters \np{nn\_ediff} and \np{nn\_ediss}.
+$P_{rt}$ can be set to unity or, following \citet{Blanke1993}, be a function
+of the local Richardson number, $R_i$:
+$P_{rt}$ is the Prandtl number, $K_m$ and $K_\rho$ are the vertical eddy viscosity and diffusivity coefficients.
+The constants $C_k =  0.1$ and $C_\epsilon = \sqrt {2} /2$ $\approx 0.7$ are designed to deal with
+vertical mixing at any depth \citep{Gaspar1990}.
+They are set through namelist parameters \np{nn\_ediff} and \np{nn\_ediss}.
+$P_{rt}$ can be set to unity or, following \citet{Blanke1993}, be a function of the local Richardson number, $R_i$:
 \begin{align*} \label{eq:prt}
 P_{rt} = \begin{cases}
 …
 The choice of $P_{rt}$ is controlled by the \np{nn\_pdl} namelist variable.
+At the sea surface, the value of $\bar{e}$ is prescribed from the wind
+stress field as $\bar{e}_o = e_{bb} |\tau| / \rho_o$, with $e_{bb}$ the \np{rn\_ebb}
+namelist parameter. The default value of $e_{bb}$ is 3.75. \citep{Gaspar1990}),
+however a much larger value can be used when taking into account the
+surface wave breaking (see below Eq. \autoref{eq:ZDF_Esbc}).
+The bottom value of TKE is assumed to be equal to the value of the level just above.
+The time integration of the $\bar{e}$ equation may formally lead to negative values
+because the numerical scheme does not ensure its positivity. To overcome this
+problem, a cut-off in the minimum value of $\bar{e}$ is used (\np{rn\_emin}
+namelist parameter). Following \citet{Gaspar1990}, the cut-off value is set
+to $\sqrt{2}/2~10^{-6}~m^2.s^{-2}$. This allows the subsequent formulations
+to match that of \citet{Gargett1984} for the diffusion in the thermocline and
+deep ocean :  $K_\rho = 10^{-3} / N$.
+In addition, a cut-off is applied on $K_m$ and $K_\rho$ to avoid numerical
+instabilities associated with too weak vertical diffusion. They must be
+specified at least larger than the molecular values, and are set through
+\np{rn\_avm0} and \np{rn\_avt0} (namzdf namelist, see \autoref{subsec:ZDF_cst}).
+At the sea surface, the value of $\bar{e}$ is prescribed from the wind stress field as
+$\bar{e}_o = e_{bb} |\tau| / \rho_o$, with $e_{bb}$ the \np{rn\_ebb} namelist parameter.
+The default value of $e_{bb}$ is 3.75. \citep{Gaspar1990}), however a much larger value can be used when
+taking into account the surface wave breaking (see below Eq. \autoref{eq:ZDF_Esbc}).
+The bottom value of TKE is assumed to be equal to the value of the level just above.
+The time integration of the $\bar{e}$ equation may formally lead to negative values because
+the numerical scheme does not ensure its positivity.
+To overcome this problem, a cut-off in the minimum value of $\bar{e}$ is used (\np{rn\_emin} namelist parameter).
+Following \citet{Gaspar1990}, the cut-off value is set to $\sqrt{2}/2~10^{-6}~m^2.s^{-2}$.
+This allows the subsequent formulations to match that of \citet{Gargett1984} for the diffusion in
+the thermocline and deep ocean :  $K_\rho = 10^{-3} / N$.
+In addition, a cut-off is applied on $K_m$ and $K_\rho$ to avoid numerical instabilities associated with
+too weak vertical diffusion.
+They must be specified at least larger than the molecular values, and are set through \np{rn\_avm0} and
+\np{rn\_avt0} (namzdf namelist, see \autoref{subsec:ZDF_cst}).
 \subsubsection{Turbulent length scale}
+For computational efficiency, the original formulation of the turbulent length
+scales proposed by \citet{Gaspar1990} has been simplified. Four formulations
+are proposed, the choice of which is controlled by the \np{nn\_mxl} namelist
+parameter. The first two are based on the following first order approximation
+\citep{Blanke1993}:
+For computational efficiency, the original formulation of the turbulent length scales proposed by
+\citet{Gaspar1990} has been simplified.
+Four formulations are proposed, the choice of which is controlled by the \np{nn\_mxl} namelist parameter.
+The first two are based on the following first order approximation \citep{Blanke1993}:
 \begin{equation} \label{eq:tke_mxl0_1}
 l_k = l_\epsilon = \sqrt {2 \bar{e}\; } / N
 \end{equation}
+which is valid in a stable stratified region with constant values of the Brunt-
+Vais\"{a}l\"{a} frequency. The resulting length scale is bounded by the distance
+to the surface or to the bottom (\np{nn\_mxl}\forcode{ = 0}) or by the local vertical scale factor
+(\np{nn\_mxl}\forcode{ = 1}). \citet{Blanke1993} notice that this simplification has two major
+drawbacks: it makes no sense for locally unstable stratification and the
+computation no longer uses all the information contained in the vertical density
+profile. To overcome these drawbacks, \citet{Madec1998} introduces the
+\np{nn\_mxl}\forcode{ = 2..3} cases, which add an extra assumption concerning the vertical
+gradient of the computed length scale. So, the length scales are first evaluated
+as in \autoref{eq:tke_mxl0_1} and then bounded such that:
+which is valid in a stable stratified region with constant values of the Brunt-Vais\"{a}l\"{a} frequency.
+The resulting length scale is bounded by the distance to the surface or to the bottom
+(\np{nn\_mxl}\forcode{ = 0}) or by the local vertical scale factor (\np{nn\_mxl}\forcode{ = 1}).
+\citet{Blanke1993} notice that this simplification has two major drawbacks:
+it makes no sense for locally unstable stratification and the computation no longer uses all
+the information contained in the vertical density profile.
+To overcome these drawbacks, \citet{Madec1998} introduces the \np{nn\_mxl}\forcode{ = 2..3} cases,
+which add an extra assumption concerning the vertical gradient of the computed length scale.
+So, the length scales are first evaluated as in \autoref{eq:tke_mxl0_1} and then bounded such that:
 \begin{equation} \label{eq:tke_mxl_constraint}
 \frac{1}{e_3 }\left| {\frac{\partial l}{\partial k}} \right| \leq 1
 \qquad \text{with }\  l =  l_k = l_\epsilon
 \end{equation}
+\autoref{eq:tke_mxl_constraint} means that the vertical variations of the length
+scale cannot be larger than the variations of depth. It provides a better
+approximation of the \citet{Gaspar1990} formulation while being much less
+time consuming. In particular, it allows the length scale to be limited not only
+by the distance to the surface or to the ocean bottom but also by the distance
+to a strongly stratified portion of the water column such as the thermocline
+(\autoref{fig:mixing_length}). In order to impose the \autoref{eq:tke_mxl_constraint}
+constraint, we introduce two additional length scales: $l_{up}$ and $l_{dwn}$,
+the upward and downward length scales, and evaluate the dissipation and
+mixing length scales as (and note that here we use numerical indexing):
+\autoref{eq:tke_mxl_constraint} means that the vertical variations of the length scale cannot be larger than
+the variations of depth.
+It provides a better approximation of the \citet{Gaspar1990} formulation while being much less
+time consuming.
+In particular, it allows the length scale to be limited not only by the distance to the surface or
+to the ocean bottom but also by the distance to a strongly stratified portion of the water column such as
+the thermocline (\autoref{fig:mixing_length}).
+In order to impose the \autoref{eq:tke_mxl_constraint} constraint, we introduce two additional length scales:
+$l_{up}$ and $l_{dwn}$, the upward and downward length scales, and
+evaluate the dissipation and mixing length scales as
+(and note that here we use numerical indexing):
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 \begin{figure}[!t] \begin{center}
 …
 \end{aligned}
 \end{equation}
+where $l^{(k)}$ is computed using \autoref{eq:tke_mxl0_1},
+$i.e.$ $l^{(k)} = \sqrt {2 {\bar e}^{(k)} / {N^2}^{(k)} }$.
+In the \np{nn\_mxl}\forcode{ = 2} case, the dissipation and mixing length scales take the same
+value: $ l_k=  l_\epsilon = \min \left(\ l_{up} \;,\;  l_{dwn}\ \right)$, while in the
+\np{nn\_mxl}\forcode{ = 3} case, the dissipation and mixing turbulent length scales are give
+as in \citet{Gaspar1990}:
+where $l^{(k)}$ is computed using \autoref{eq:tke_mxl0_1}, $i.e.$ $l^{(k)} = \sqrt {2 {\bar e}^{(k)} / {N^2}^{(k)} }$.
+In the \np{nn\_mxl}\forcode{ = 2} case, the dissipation and mixing length scales take the same value:
+$ l_k=  l_\epsilon = \min \left(\ l_{up} \;,\;  l_{dwn}\ \right)$, while in the \np{nn\_mxl}\forcode{ = 3} case,
+the dissipation and mixing turbulent length scales are give as in \citet{Gaspar1990}:
 \begin{equation} \label{eq:tke_mxl_gaspar}
 \begin{aligned}
 …
 \end{equation}
+At the ocean surface, a non zero length scale is set through the  \np{rn\_mxl0} namelist
+parameter. Usually the surface scale is given by $l_o = \kappa \,z_o$
+where $\kappa = 0.4$ is von Karman's constant and $z_o$ the roughness
+parameter of the surface. Assuming $z_o=0.1$~m \citep{Craig_Banner_JPO94}
+leads to a 0.04~m, the default value of \np{rn\_mxl0}. In the ocean interior
+a minimum length scale is set to recover the molecular viscosity when $\bar{e}$
+reach its minimum value ($1.10^{-6}= C_k\, l_{min} \,\sqrt{\bar{e}_{min}}$ ).
+At the ocean surface, a non zero length scale is set through the  \np{rn\_mxl0} namelist parameter.
+Usually the surface scale is given by $l_o = \kappa \,z_o$ where $\kappa = 0.4$ is von Karman's constant and
+$z_o$ the roughness parameter of the surface.
+Assuming $z_o=0.1$~m \citep{Craig_Banner_JPO94} leads to a 0.04~m, the default value of \np{rn\_mxl0}.
+In the ocean interior a minimum length scale is set to recover the molecular viscosity when
+$\bar{e}$ reach its minimum value ($1.10^{-6}= C_k\, l_{min} \,\sqrt{\bar{e}_{min}}$ ).
 \subsubsection{Surface wave breaking parameterization}
 %-----------------------------------------------------------------------%
+Following \citet{Mellor_Blumberg_JPO04}, the TKE turbulence closure model has been modified
+to include the effect of surface wave breaking energetics. This results in a reduction of summertime
+surface temperature when the mixed layer is relatively shallow. The \citet{Mellor_Blumberg_JPO04}
+modifications acts on surface length scale and TKE values and air-sea drag coefficient.
+Following \citet{Mellor_Blumberg_JPO04}, the TKE turbulence closure model has been modified to
+include the effect of surface wave breaking energetics.
+This results in a reduction of summertime surface temperature when the mixed layer is relatively shallow.
+The \citet{Mellor_Blumberg_JPO04} modifications acts on surface length scale and TKE values and
+air-sea drag coefficient.
 The latter concerns the bulk formulea and is not discussed here.
 …
 \bar{e}_o = \frac{1}{2}\,\left(  15.8\,\alpha_{CB} \right)^{2/3} \,\frac{|\tau|}{\rho_o}
 \end{equation}
+where $\alpha_{CB}$ is the \citet{Craig_Banner_JPO94} constant of proportionality
+which depends on the ''wave age'', ranging from 57 for mature waves to 146 for
+younger waves \citep{Mellor_Blumberg_JPO04}.
+where $\alpha_{CB}$ is the \citet{Craig_Banner_JPO94} constant of proportionality which depends on the ''wave age'',
+ranging from 57 for mature waves to 146 for younger waves \citep{Mellor_Blumberg_JPO04}.
 The boundary condition on the turbulent length scale follows the Charnock's relation:
 \begin{equation} \label{eq:ZDF_Lsbc}
 …
 \end{equation}
 where $\kappa=0.40$ is the von Karman constant, and $\beta$ is the Charnock's constant.
+\citet{Mellor_Blumberg_JPO04} suggest $\beta = 2.10^{5}$ the value chosen by \citet{Stacey_JPO99}
+citing observation evidence, and $\alpha_{CB} = 100$ the Craig and Banner's value.
+As the surface boundary condition on TKE is prescribed through $\bar{e}_o = e_{bb} |\tau| / \rho_o$,
+\citet{Mellor_Blumberg_JPO04} suggest $\beta = 2.10^{5}$ the value chosen by
+\citet{Stacey_JPO99} citing observation evidence, and
+$\alpha_{CB} = 100$ the Craig and Banner's value.
+As the surface boundary condition on TKE is prescribed through $\bar{e}_o = e_{bb} |\tau| / \rho_o$,
 with $e_{bb}$ the \np{rn\_ebb} namelist parameter, setting \np{rn\_ebb}\forcode{ = 67.83} corresponds
+to $\alpha_{CB} = 100$. Further setting  \np{ln\_mxl0} to true applies \autoref{eq:ZDF_Lsbc}
+as surface boundary condition on length scale, with $\beta$ hard coded to the Stacey's value.
+Note that a minimal threshold of \np{rn\_emin0}$=10^{-4}~m^2.s^{-2}$ (namelist parameters)
+is applied on surface $\bar{e}$ value.
+to $\alpha_{CB} = 100$.
+Further setting  \np{ln\_mxl0} to true applies \autoref{eq:ZDF_Lsbc} as surface boundary condition on length scale,
+with $\beta$ hard coded to the Stacey's value.
+Note that a minimal threshold of \np{rn\_emin0}$=10^{-4}~m^2.s^{-2}$ (namelist parameters) is applied on
+surface $\bar{e}$ value.
 \subsubsection{Langmuir cells}
 %--------------------------------------%
+Langmuir circulations (LC) can be described as ordered large-scale vertical motions
+in the surface layer of the oceans. Although LC have nothing to do with convection,
+the circulation pattern is rather similar to so-called convective rolls in the atmospheric
+boundary layer. The detailed physics behind LC is described in, for example,
+\citet{Craik_Leibovich_JFM76}. The prevailing explanation is that LC arise from
+a nonlinear interaction between the Stokes drift and wind drift currents.
+Here we introduced in the TKE turbulent closure the simple parameterization of
+Langmuir circulations proposed by \citep{Axell_JGR02} for a $k-\epsilon$ turbulent closure.
+The parameterization, tuned against large-eddy simulation, includes the whole effect
+of LC in an extra source terms of TKE, $P_{LC}$.
+The presence of $P_{LC}$ in \autoref{eq:zdftke_e}, the TKE equation, is controlled
+by setting \np{ln\_lc} to \forcode{.true.} in the namtke namelist.
+Langmuir circulations (LC) can be described as ordered large-scale vertical motions in
+the surface layer of the oceans.
+Although LC have nothing to do with convection, the circulation pattern is rather similar to
+so-called convective rolls in the atmospheric boundary layer.
+The detailed physics behind LC is described in, for example, \citet{Craik_Leibovich_JFM76}.
+The prevailing explanation is that LC arise from a nonlinear interaction between the Stokes drift and
+wind drift currents.
+Here we introduced in the TKE turbulent closure the simple parameterization of Langmuir circulations proposed by
+\citep{Axell_JGR02} for a $k-\epsilon$ turbulent closure.
+The parameterization, tuned against large-eddy simulation, includes the whole effect of LC in
+an extra source terms of TKE, $P_{LC}$.
+The presence of $P_{LC}$ in \autoref{eq:zdftke_e}, the TKE equation, is controlled by setting \np{ln\_lc} to
+\forcode{.true.} in the namtke namelist.
 By making an analogy with the characteristic convective velocity scale
 ($e.g.$, \citet{D'Alessio_al_JPO98}), $P_{LC}$ is assumed to be :
+By making an analogy with the characteristic convective velocity scale ($e.g.$, \citet{D'Alessio_al_JPO98}),
+$P_{LC}$ is assumed to be :
 \begin{equation}
 P_{LC}(z) = \frac{w_{LC}^3(z)}{H_{LC}}
 …
 With no information about the wave field, $w_{LC}$ is assumed to be proportional to
 the Stokes drift $u_s = 0.377\,\,|\tau|^{1/2}$, where $|\tau|$ is the surface wind stress module
 \footnote{Following \citet{Li_Garrett_JMR93}, the surface Stoke drift velocity
+may be expressed as $u_s =  0.016 \,|U_{10m}|$. Assuming an air density of
+$\rho_a=1.22 \,Kg/m^3$ and a drag coefficient of $1.5~10^{-3}$ give the expression
+used of $u_s$ as a function of the module of surface stress}.
 For the vertical variation, $w_{LC}$ is assumed to be zero at the surface as well as
 at a finite depth $H_{LC}$ (which is often close to the mixed layer depth), and simply
 varies as a sine function in between (a first-order profile for the Langmuir cell structures).
+\footnote{Following \citet{Li_Garrett_JMR93}, the surface Stoke drift velocity may be expressed as
+  $u_s =  0.016 \,|U_{10m}|$.
+  Assuming an air density of $\rho_a=1.22 \,Kg/m^3$ and a drag coefficient of
+  $1.5~10^{-3}$ give the expression used of $u_s$ as a function of the module of surface stress}.
+For the vertical variation, $w_{LC}$ is assumed to be zero at the surface as well as at
+a finite depth $H_{LC}$ (which is often close to the mixed layer depth),
+and simply varies as a sine function in between (a first-order profile for the Langmuir cell structures).
 The resulting expression for $w_{LC}$ is :
 \begin{equation}
 …
                  \end{cases}
 \end{equation}
 where $c_{LC} = 0.15$ has been chosen by \citep{Axell_JGR02} as a good compromise
+to fit LES data. The chosen value yields maximum vertical velocities $w_{LC}$ of the order
+of a few centimeters per second. The value of $c_{LC}$ is set through the \np{rn\_lc}
 namelist parameter, having in mind that it should stay between 0.15 and 0.54 \citep{Axell_JGR02}.
+where $c_{LC} = 0.15$ has been chosen by \citep{Axell_JGR02} as a good compromise to fit LES data.
+The chosen value yields maximum vertical velocities $w_{LC}$ of the order of a few centimeters per second.
+The value of $c_{LC}$ is set through the \np{rn\_lc} namelist parameter,
+having in mind that it should stay between 0.15 and 0.54 \citep{Axell_JGR02}.
 The $H_{LC}$ is estimated in a similar way as the turbulent length scale of TKE equations:
 $H_{LC}$ is depth to which a water parcel with kinetic energy due to Stoke drift
 can reach on its own by converting its kinetic energy to potential energy, according to
+$H_{LC}$ is depth to which a water parcel with kinetic energy due to Stoke drift can reach on its own by
+converting its kinetic energy to potential energy, according to
 \begin{equation}
 - \int_{-H_{LC}}^0 { N^2\;z  \;dz} = \frac{1}{2} u_s^2
 …
 %--------------------------------------------------------------%
 Vertical mixing parameterizations commonly used in ocean general circulation models
 tend to produce mixed-layer depths that are too shallow during summer months and windy conditions.
 This bias is particularly acute over the Southern Ocean.
 To overcome this systematic bias, an ad hoc parameterization is introduced into the TKE scheme  \cite{Rodgers_2014}.
 The parameterization is an empirical one, $i.e.$ not derived from theoretical considerations,
+Vertical mixing parameterizations commonly used in ocean general circulation models tend to
+produce mixed-layer depths that are too shallow during summer months and windy conditions.
+This bias is particularly acute over the Southern Ocean.
+To overcome this systematic bias, an ad hoc parameterization is introduced into the TKE scheme \cite{Rodgers_2014}.
+The parameterization is an empirical one, $i.e.$ not derived from theoretical considerations,
 but rather is meant to account for observed processes that affect the density structure of
 the ocean’s planetary boundary layer that are not explicitly captured by default in the TKE scheme
 ($i.e.$ near-inertial oscillations and ocean swells and waves).
+When using this parameterization ($i.e.$ when \np{nn\_etau}\forcode{ = 1}), the TKE input to the ocean ($S$)
+imposed by the winds in the form of near-inertial oscillations, swell and waves is parameterized
+by \autoref{eq:ZDF_Esbc} the standard TKE surface boundary condition, plus a depth depend one given by:
+When using this parameterization ($i.e.$ when \np{nn\_etau}\forcode{ = 1}),
+the TKE input to the ocean ($S$) imposed by the winds in the form of near-inertial oscillations,
+swell and waves is parameterized by \autoref{eq:ZDF_Esbc} the standard TKE surface boundary condition,
+plus a depth depend one given by:
 \begin{equation}  \label{eq:ZDF_Ehtau}
 S = (1-f_i) \; f_r \; e_s \; e^{-z / h_\tau}
 \end{equation}
+where
+$z$ is the depth,
+$e_s$ is TKE surface boundary condition,
+$f_r$ is the fraction of the surface TKE that penetrate in the ocean,
+$h_\tau$ is a vertical mixing length scale that controls exponential shape of the penetration,
+and $f_i$ is the ice concentration (no penetration if $f_i=1$, that is if the ocean is entirely
+covered by sea-ice).
+The value of $f_r$, usually a few percents, is specified through \np{rn\_efr} namelist parameter.
+The vertical mixing length scale, $h_\tau$, can be set as a 10~m uniform value (\np{nn\_etau}\forcode{ = 0})
+or a latitude dependent value (varying from 0.5~m at the Equator to a maximum value of 30~m
+at high latitudes (\np{nn\_etau}\forcode{ = 1}).
+Note that two other option existe, \np{nn\_etau}\forcode{ = 2..3}. They correspond to applying
+\autoref{eq:ZDF_Ehtau} only at the base of the mixed layer, or to using the high frequency part
+of the stress to evaluate the fraction of TKE that penetrate the ocean.
+where $z$ is the depth, $e_s$ is TKE surface boundary condition, $f_r$ is the fraction of the surface TKE that
+penetrate in the ocean, $h_\tau$ is a vertical mixing length scale that controls exponential shape of
+the penetration, and $f_i$ is the ice concentration
+(no penetration if $f_i=1$, that is if the ocean is entirely covered by sea-ice).
+The value of $f_r$, usually a few percents, is specified through \np{rn\_efr} namelist parameter.
+The vertical mixing length scale, $h_\tau$, can be set as a 10~m uniform value (\np{nn\_etau}\forcode{ = 0}) or
+a latitude dependent value (varying from 0.5~m at the Equator to a maximum value of 30~m at high latitudes
+(\np{nn\_etau}\forcode{ = 1}).
+Note that two other option existe, \np{nn\_etau}\forcode{ = 2..3}.
+They correspond to applying \autoref{eq:ZDF_Ehtau} only at the base of the mixed layer,
+or to using the high frequency part of the stress to evaluate the fraction of TKE that penetrate the ocean.
 Those two options are obsolescent features introduced for test purposes.
 They will be removed in the next release.
 …
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+The production of turbulence by vertical shear (the first term of the right hand side
+of \autoref{eq:zdftke_e}) should balance the loss of kinetic energy associated with
+the vertical momentum diffusion (first line in \autoref{eq:PE_zdf}). To do so a special care
+have to be taken for both the time and space discretization of the TKE equation
+\citep{Burchard_OM02,Marsaleix_al_OM08}.
+Let us first address the time stepping issue. \autoref{fig:TKE_time_scheme} shows
+how the two-level Leap-Frog time stepping of the momentum and tracer equations interplays
+with the one-level forward time stepping of TKE equation. With this framework, the total loss
+of kinetic energy (in 1D for the demonstration) due to the vertical momentum diffusion is
+obtained by multiplying this quantity by $u^t$ and summing the result vertically:
+The production of turbulence by vertical shear (the first term of the right hand side of
+\autoref{eq:zdftke_e}) should balance the loss of kinetic energy associated with the vertical momentum diffusion
+(first line in \autoref{eq:PE_zdf}).
+To do so a special care have to be taken for both the time and space discretization of
+the TKE equation \citep{Burchard_OM02,Marsaleix_al_OM08}.
+Let us first address the time stepping issue. \autoref{fig:TKE_time_scheme} shows how
+the two-level Leap-Frog time stepping of the momentum and tracer equations interplays with
+the one-level forward time stepping of TKE equation.
+With this framework, the total loss of kinetic energy (in 1D for the demonstration) due to
+the vertical momentum diffusion is obtained by multiplying this quantity by $u^t$ and
+summing the result vertically:
 \begin{equation} \label{eq:energ1}
 \begin{split}
 …
 \end{split}
 \end{equation}
+Here, the vertical diffusion of momentum is discretized backward in time
+with a coefficient, $K_m$, known at time $t$ (\autoref{fig:TKE_time_scheme}),
+as it is required when using the TKE scheme (see \autoref{sec:STP_forward_imp}).
+The first term of the right hand side of \autoref{eq:energ1} represents the kinetic energy
+transfer at the surface (atmospheric forcing) and at the bottom (friction effect).
+The second term is always negative. It is the dissipation rate of kinetic energy,
+and thus minus the shear production rate of $\bar{e}$. \autoref{eq:energ1}
+implies that, to be energetically consistent, the production rate of $\bar{e}$
+used to compute $(\bar{e})^t$ (and thus ${K_m}^t$) should be expressed as
+${K_m}^{t-\rdt}\,(\partial_z u)^{t-\rdt} \,(\partial_z u)^t$ (and not by the more straightforward
+$K_m \left( \partial_z u \right)^2$ expression taken at time $t$ or $t-\rdt$).
+A similar consideration applies on the destruction rate of $\bar{e}$ due to stratification
+(second term of the right hand side of \autoref{eq:zdftke_e}). This term
+must balance the input of potential energy resulting from vertical mixing.
+The rate of change of potential energy (in 1D for the demonstration) due vertical
+mixing is obtained by multiplying vertical density diffusion
+tendency by $g\,z$ and and summing the result vertically:
+Here, the vertical diffusion of momentum is discretized backward in time with a coefficient, $K_m$,
+known at time $t$ (\autoref{fig:TKE_time_scheme}), as it is required when using the TKE scheme
+(see \autoref{sec:STP_forward_imp}).
+The first term of the right hand side of \autoref{eq:energ1} represents the kinetic energy transfer at
+the surface (atmospheric forcing) and at the bottom (friction effect).
+The second term is always negative.
+It is the dissipation rate of kinetic energy, and thus minus the shear production rate of $\bar{e}$.
+\autoref{eq:energ1} implies that, to be energetically consistent,
+the production rate of $\bar{e}$ used to compute $(\bar{e})^t$ (and thus ${K_m}^t$) should be expressed as
+${K_m}^{t-\rdt}\,(\partial_z u)^{t-\rdt} \,(\partial_z u)^t$
+(and not by the more straightforward $K_m \left( \partial_z u \right)^2$ expression taken at time $t$ or $t-\rdt$).
+A similar consideration applies on the destruction rate of $\bar{e}$ due to stratification
+(second term of the right hand side of \autoref{eq:zdftke_e}).
+This term must balance the input of potential energy resulting from vertical mixing.
+The rate of change of potential energy (in 1D for the demonstration) due vertical mixing is obtained by
+multiplying vertical density diffusion tendency by $g\,z$ and and summing the result vertically:
 \begin{equation} \label{eq:energ2}
 \begin{split}
 …
 \end{equation}
 where we use $N^2 = -g \,\partial_k \rho / (e_3 \rho)$.
+The first term of the right hand side of \autoref{eq:energ2}  is always zero
+because there is no diffusive flux through the ocean surface and bottom).
+The second term is minus the destruction rate of  $\bar{e}$ due to stratification.
+Therefore \autoref{eq:energ1} implies that, to be energetically consistent, the product
+${K_\rho}^{t-\rdt}\,(N^2)^t$ should be used in \autoref{eq:zdftke_e}, the TKE equation.
+Let us now address the space discretization issue.
+The vertical eddy coefficients are defined at $w$-point whereas the horizontal velocity
+components are in the centre of the side faces of a $t$-box in staggered C-grid
+(\autoref{fig:cell}). A space averaging is thus required to obtain the shear TKE production term.
+By redoing the \autoref{eq:energ1} in the 3D case, it can be shown that the product of
+eddy coefficient by the shear at $t$ and $t-\rdt$ must be performed prior to the averaging.
+Furthermore, the possible time variation of $e_3$ (\key{vvl} case) have to be taken into
+account.
+The above energetic considerations leads to
+the following final discrete form for the TKE equation:
+The first term of the right hand side of \autoref{eq:energ2} is always zero because
+there is no diffusive flux through the ocean surface and bottom).
+The second term is minus the destruction rate of  $\bar{e}$ due to stratification.
+Therefore \autoref{eq:energ1} implies that, to be energetically consistent,
+the product ${K_\rho}^{t-\rdt}\,(N^2)^t$ should be used in \autoref{eq:zdftke_e}, the TKE equation.
+Let us now address the space discretization issue.
+The vertical eddy coefficients are defined at $w$-point whereas the horizontal velocity components are in
+the centre of the side faces of a $t$-box in staggered C-grid (\autoref{fig:cell}).
+A space averaging is thus required to obtain the shear TKE production term.
+By redoing the \autoref{eq:energ1} in the 3D case, it can be shown that the product of eddy coefficient by
+the shear at $t$ and $t-\rdt$ must be performed prior to the averaging.
+Furthermore, the possible time variation of $e_3$ (\key{vvl} case) have to be taken into account.
+The above energetic considerations leads to the following final discrete form for the TKE equation:
 \begin{equation} \label{eq:zdftke_ene}
 \begin{split}
 …
 \end{split}
 \end{equation}
+where the last two terms in \autoref{eq:zdftke_ene} (vertical diffusion and Kolmogorov dissipation)
+are time stepped using a backward scheme (see\autoref{sec:STP_forward_imp}).
+Note that the Kolmogorov term has been linearized in time in order to render
+the implicit computation possible. The restart of the TKE scheme
+requires the storage of $\bar {e}$, $K_m$, $K_\rho$ and $l_\epsilon$ as they all appear in
+the right hand side of \autoref{eq:zdftke_ene}. For the latter, it is in fact
+the ratio $\sqrt{\bar{e}}/l_\epsilon$ which is stored.
+where the last two terms in \autoref{eq:zdftke_ene} (vertical diffusion and Kolmogorov dissipation)
+are time stepped using a backward scheme (see\autoref{sec:STP_forward_imp}).
+Note that the Kolmogorov term has been linearized in time in order to render the implicit computation possible.
+The restart of the TKE scheme requires the storage of $\bar {e}$, $K_m$, $K_\rho$ and $l_\epsilon$ as
+they all appear in the right hand side of \autoref{eq:zdftke_ene}.
+For the latter, it is in fact the ratio $\sqrt{\bar{e}}/l_\epsilon$ which is stored.
 % -------------------------------------------------------------------------------------------------------------
 …
 %--------------------------------------------------------------------------------------------------------------
+The Generic Length Scale (GLS) scheme is a turbulent closure scheme based on
+two prognostic equations: one for the turbulent kinetic energy $\bar {e}$, and another
+for the generic length scale, $\psi$ \citep{Umlauf_Burchard_JMS03, Umlauf_Burchard_CSR05}.
+This later variable is defined as : $\psi = {C_{0\mu}}^{p} \ {\bar{e}}^{m} \ l^{n}$,
+where the triplet $(p, m, n)$ value given in Tab.\autoref{tab:GLS} allows to recover
+a number of well-known turbulent closures ($k$-$kl$ \citep{Mellor_Yamada_1982},
+$k$-$\epsilon$ \citep{Rodi_1987}, $k$-$\omega$ \citep{Wilcox_1988}
+among others \citep{Umlauf_Burchard_JMS03,Kantha_Carniel_CSR05}).
+The Generic Length Scale (GLS) scheme is a turbulent closure scheme based on two prognostic equations:
+one for the turbulent kinetic energy $\bar {e}$, and another for the generic length scale,
+$\psi$ \citep{Umlauf_Burchard_JMS03, Umlauf_Burchard_CSR05}.
+This later variable is defined as: $\psi = {C_{0\mu}}^{p} \ {\bar{e}}^{m} \ l^{n}$,
+where the triplet $(p, m, n)$ value given in Tab.\autoref{tab:GLS} allows to recover a number of
+well-known turbulent closures ($k$-$kl$ \citep{Mellor_Yamada_1982}, $k$-$\epsilon$ \citep{Rodi_1987},
+$k$-$\omega$ \citep{Wilcox_1988} among others \citep{Umlauf_Burchard_JMS03,Kantha_Carniel_CSR05}).
 The GLS scheme is given by the following set of equations:
 \begin{equation} \label{eq:zdfgls_e}
 …
 {\epsilon} = C_{0\mu} \,\frac{\bar {e}^{3/2}}{l} \;
 \end{equation}
+where $N$ is the local Brunt-Vais\"{a}l\"{a} frequency (see \autoref{subsec:TRA_bn2})
+and $\epsilon$ the dissipation rate.
+The constants $C_1$, $C_2$, $C_3$, ${\sigma_e}$, ${\sigma_{\psi}}$ and the wall function ($Fw$)
+depends of the choice of the turbulence model. Four different turbulent models are pre-defined
+(Tab.\autoref{tab:GLS}). They are made available through the \np{nn\_clo} namelist parameter.
+where $N$ is the local Brunt-Vais\"{a}l\"{a} frequency (see \autoref{subsec:TRA_bn2}) and
+$\epsilon$ the dissipation rate.
+The constants $C_1$, $C_2$, $C_3$, ${\sigma_e}$, ${\sigma_{\psi}}$ and the wall function ($Fw$) depends of
+the choice of the turbulence model.
+Four different turbulent models are pre-defined (Tab.\autoref{tab:GLS}).
+They are made available through the \np{nn\_clo} namelist parameter.
 %--------------------------------------------------TABLE--------------------------------------------------
 …
 \end{tabular}
 \caption{   \protect\label{tab:GLS}
+Set of predefined GLS parameters, or equivalently predefined turbulence models available
 with \protect\key{zdfgls} and controlled by the \protect\np{nn\_clos} namelist variable in \protect\ngn{namzdf\_gls} .}
+  Set of predefined GLS parameters, or equivalently predefined turbulence models available with
+  \protect\key{zdfgls} and controlled by the \protect\np{nn\_clos} namelist variable in \protect\ngn{namzdf\_gls}.}
 \end{center}   \end{table}
 %--------------------------------------------------------------------------------------------------------------
+In the Mellor-Yamada model, the negativity of $n$ allows to use a wall function to force
+the convergence of the mixing length towards $K z_b$ ($K$: Kappa and $z_b$: rugosity length)
+value near physical boundaries (logarithmic boundary layer law). $C_{\mu}$ and $C_{\mu'}$
+are calculated from stability function proposed by \citet{Galperin_al_JAS88}, or by \citet{Kantha_Clayson_1994}
+or one of the two functions suggested by \citet{Canuto_2001}  (\np{nn\_stab\_func}\forcode{ = 0..3}, resp.).
+In the Mellor-Yamada model, the negativity of $n$ allows to use a wall function to force the convergence of
+the mixing length towards $K z_b$ ($K$: Kappa and $z_b$: rugosity length) value near physical boundaries
+(logarithmic boundary layer law).
+$C_{\mu}$ and $C_{\mu'}$ are calculated from stability function proposed by \citet{Galperin_al_JAS88},
+or by \citet{Kantha_Clayson_1994} or one of the two functions suggested by \citet{Canuto_2001}
+(\np{nn\_stab\_func}\forcode{ = 0..3}, resp.).
 The value of $C_{0\mu}$ depends of the choice of the stability function.
+The surface and bottom boundary condition on both $\bar{e}$ and $\psi$ can be calculated
+thanks to Dirichlet or Neumann condition through \np{nn\_tkebc\_surf} and \np{nn\_tkebc\_bot}, resp.
+As for TKE closure , the wave effect on the mixing is considered when \np{ln\_crban}\forcode{ = .true.}
+\citep{Craig_Banner_JPO94, Mellor_Blumberg_JPO04}. The \np{rn\_crban} namelist parameter
+is $\alpha_{CB}$ in \autoref{eq:ZDF_Esbc} and \np{rn\_charn} provides the value of $\beta$ in \autoref{eq:ZDF_Lsbc}.
+The $\psi$ equation is known to fail in stably stratified flows, and for this reason
+almost all authors apply a clipping of the length scale as an \textit{ad hoc} remedy.
+With this clipping, the maximum permissible length scale is determined by
+$l_{max} = c_{lim} \sqrt{2\bar{e}}/ N$. A value of $c_{lim} = 0.53$ is often used
+\citep{Galperin_al_JAS88}. \cite{Umlauf_Burchard_CSR05} show that the value of
+the clipping factor is of crucial importance for the entrainment depth predicted in
+stably stratified situations, and that its value has to be chosen in accordance
+with the algebraic model for the turbulent fluxes. The clipping is only activated
+if \np{ln\_length\_lim}\forcode{ = .true.}, and the $c_{lim}$ is set to the \np{rn\_clim\_galp} value.
+The time and space discretization of the GLS equations follows the same energetic
+consideration as for the TKE case described in \autoref{subsec:ZDF_tke_ene}  \citep{Burchard_OM02}.
+The surface and bottom boundary condition on both $\bar{e}$ and $\psi$ can be calculated thanks to Dirichlet or
+Neumann condition through \np{nn\_tkebc\_surf} and \np{nn\_tkebc\_bot}, resp.
+As for TKE closure, the wave effect on the mixing is considered when
+\np{ln\_crban}\forcode{ = .true.} \citep{Craig_Banner_JPO94, Mellor_Blumberg_JPO04}.
+The \np{rn\_crban} namelist parameter is $\alpha_{CB}$ in \autoref{eq:ZDF_Esbc} and
+\np{rn\_charn} provides the value of $\beta$ in \autoref{eq:ZDF_Lsbc}.
+The $\psi$ equation is known to fail in stably stratified flows, and for this reason
+almost all authors apply a clipping of the length scale as an \textit{ad hoc} remedy.
+With this clipping, the maximum permissible length scale is determined by $l_{max} = c_{lim} \sqrt{2\bar{e}}/ N$.
+A value of $c_{lim} = 0.53$ is often used \citep{Galperin_al_JAS88}.
+\cite{Umlauf_Burchard_CSR05} show that the value of the clipping factor is of crucial importance for
+the entrainment depth predicted in stably stratified situations,
+and that its value has to be chosen in accordance with the algebraic model for the turbulent fluxes.
+The clipping is only activated if \np{ln\_length\_lim}\forcode{ = .true.},
+and the $c_{lim}$ is set to the \np{rn\_clim\_galp} value.
+The time and space discretization of the GLS equations follows the same energetic consideration as for
+the TKE case described in \autoref{subsec:ZDF_tke_ene} \citep{Burchard_OM02}.
 Examples of performance of the 4 turbulent closure scheme can be found in \citet{Warner_al_OM05}.
 …
 %--------------------------------------------------------------------------------------------------------------
+Static instabilities (i.e. light potential densities under heavy ones) may
+occur at particular ocean grid points. In nature, convective processes
+quickly re-establish the static stability of the water column. These
+processes have been removed from the model via the hydrostatic
+assumption so they must be parameterized. Three parameterisations
+are available to deal with convective processes: a non-penetrative
+convective adjustment or an enhanced vertical diffusion, or/and the
+use of a turbulent closure scheme.
+Static instabilities (i.e. light potential densities under heavy ones) may occur at particular ocean grid points.
+In nature, convective processes quickly re-establish the static stability of the water column.
+These processes have been removed from the model via the hydrostatic assumption so they must be parameterized.
+Three parameterisations are available to deal with convective processes:
+a non-penetrative convective adjustment or an enhanced vertical diffusion,
+or/and the use of a turbulent closure scheme.
 % -------------------------------------------------------------------------------------------------------------
 …
 \includegraphics[width=0.90\textwidth]{Fig_npc}
 \caption{  \protect\label{fig:npc}
+Example of an unstable density profile treated by the non penetrative
+convective adjustment algorithm. $1^{st}$ step: the initial profile is checked from
+the surface to the bottom. It is found to be unstable between levels 3 and 4.
+They are mixed. The resulting $\rho$ is still larger than $\rho$(5): levels 3 to 5
+are mixed. The resulting $\rho$ is still larger than $\rho$(6): levels 3 to 6 are
+mixed. The $1^{st}$ step ends since the density profile is then stable below
+the level 3. $2^{nd}$ step: the new $\rho$ profile is checked following the same
+procedure as in $1^{st}$ step: levels 2 to 5 are mixed. The new density profile
+is checked. It is found stable: end of algorithm.}
+  Example of an unstable density profile treated by the non penetrative convective adjustment algorithm.
+  $1^{st}$ step: the initial profile is checked from the surface to the bottom.
+  It is found to be unstable between levels 3 and 4.
+  They are mixed.
+  The resulting $\rho$ is still larger than $\rho$(5): levels 3 to 5 are mixed.
+  The resulting $\rho$ is still larger than $\rho$(6): levels 3 to 6 are mixed.
+  The $1^{st}$ step ends since the density profile is then stable below the level 3.
+  $2^{nd}$ step: the new $\rho$ profile is checked following the same procedure as in $1^{st}$ step:
+  levels 2 to 5 are mixed.
+  The new density profile is checked.
+  It is found stable: end of algorithm.}
 \end{center}   \end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+Options are defined through the  \ngn{namzdf} namelist variables.
+The non-penetrative convective adjustment is used when \np{ln\_zdfnpc}\forcode{ = .true.}.
+It is applied at each \np{nn\_npc} time step and mixes downwards instantaneously
+the statically unstable portion of the water column, but only until the density
+structure becomes neutrally stable ($i.e.$ until the mixed portion of the water
+column has \textit{exactly} the density of the water just below) \citep{Madec_al_JPO91}.
+The associated algorithm is an iterative process used in the following way
+(\autoref{fig:npc}): starting from the top of the ocean, the first instability is
+found. Assume in the following that the instability is located between levels
+$k$ and $k+1$. The temperature and salinity in the two levels are
+vertically mixed, conserving the heat and salt contents of the water column.
+The new density is then computed by a linear approximation. If the new
+density profile is still unstable between levels $k+1$ and $k+2$, levels $k$,
+$k+1$ and $k+2$ are then mixed. This process is repeated until stability is
+established below the level $k$ (the mixing process can go down to the
+ocean bottom). The algorithm is repeated to check if the density profile
+between level $k-1$ and $k$ is unstable and/or if there is no deeper instability.
+This algorithm is significantly different from mixing statically unstable levels
+two by two. The latter procedure cannot converge with a finite number
+of iterations for some vertical profiles while the algorithm used in \NEMO
+converges for any profile in a number of iterations which is less than the
+number of vertical levels. This property is of paramount importance as
+pointed out by \citet{Killworth1989}: it avoids the existence of permanent
+and unrealistic static instabilities at the sea surface. This non-penetrative
+convective algorithm has been proved successful in studies of the deep
+water formation in the north-western Mediterranean Sea
+\citep{Madec_al_JPO91, Madec_al_DAO91, Madec_Crepon_Bk91}.
+The current implementation has been modified in order to deal with any non linear
+equation of seawater (L. Brodeau, personnal communication).
+Two main differences have been introduced compared to the original algorithm:
+Options are defined through the \ngn{namzdf} namelist variables.
+The non-penetrative convective adjustment is used when \np{ln\_zdfnpc}\forcode{ = .true.}.
+It is applied at each \np{nn\_npc} time step and mixes downwards instantaneously the statically unstable portion of
+the water column, but only until the density structure becomes neutrally stable
+($i.e.$ until the mixed portion of the water column has \textit{exactly} the density of the water just below)
+\citep{Madec_al_JPO91}.
+The associated algorithm is an iterative process used in the following way (\autoref{fig:npc}):
+starting from the top of the ocean, the first instability is found.
+Assume in the following that the instability is located between levels $k$ and $k+1$.
+The temperature and salinity in the two levels are vertically mixed, conserving the heat and salt contents of
+the water column.
+The new density is then computed by a linear approximation.
+If the new density profile is still unstable between levels $k+1$ and $k+2$,
+levels $k$, $k+1$ and $k+2$ are then mixed.
+This process is repeated until stability is established below the level $k$
+(the mixing process can go down to the ocean bottom).
+The algorithm is repeated to check if the density profile between level $k-1$ and $k$ is unstable and/or
+if there is no deeper instability.
+This algorithm is significantly different from mixing statically unstable levels two by two.
+The latter procedure cannot converge with a finite number of iterations for some vertical profiles while
+the algorithm used in \NEMO converges for any profile in a number of iterations which is less than
+the number of vertical levels.
+This property is of paramount importance as pointed out by \citet{Killworth1989}:
+it avoids the existence of permanent and unrealistic static instabilities at the sea surface.
+This non-penetrative convective algorithm has been proved successful in studies of the deep water formation in
+the north-western Mediterranean Sea \citep{Madec_al_JPO91, Madec_al_DAO91, Madec_Crepon_Bk91}.
+The current implementation has been modified in order to deal with any non linear equation of seawater
+(L. Brodeau, personnal communication).
+Two main differences have been introduced compared to the original algorithm:
 $(i)$ the stability is now checked using the Brunt-V\"{a}is\"{a}l\"{a} frequency
+(not the the difference in potential density) ;
+$(ii)$ when two levels are found unstable, their thermal and haline expansion coefficients
+are vertically mixed in the same way their temperature and salinity has been mixed.
+These two modifications allow the algorithm to perform properly and accurately
+with TEOS10 or EOS-80 without having to recompute the expansion coefficients at each
+mixing iteration.
+(not the the difference in potential density);
+$(ii)$ when two levels are found unstable, their thermal and haline expansion coefficients are vertically mixed in
+the same way their temperature and salinity has been mixed.
+These two modifications allow the algorithm to perform properly and accurately with TEOS10 or EOS-80 without
+having to recompute the expansion coefficients at each mixing iteration.
 % -------------------------------------------------------------------------------------------------------------
 …
 Options are defined through the  \ngn{namzdf} namelist variables.
+The enhanced vertical diffusion parameterisation is used when \np{ln\_zdfevd}\forcode{ = .true.}.
+In this case, the vertical eddy mixing coefficients are assigned very large values
+(a typical value is $10\;m^2s^{-1})$ in regions where the stratification is unstable
+($i.e.$ when $N^2$ the Brunt-Vais\"{a}l\"{a} frequency is negative)
+\citep{Lazar_PhD97, Lazar_al_JPO99}. This is done either on tracers only
+(\np{nn\_evdm}\forcode{ = 0}) or on both momentum and tracers (\np{nn\_evdm}\forcode{ = 1}).
+In practice, where $N^2\leq 10^{-12}$, $A_T^{vT}$ and $A_T^{vS}$, and
+if \np{nn\_evdm}\forcode{ = 1}, the four neighbouring $A_u^{vm} \;\mbox{and}\;A_v^{vm}$
+values also, are set equal to the namelist parameter \np{rn\_avevd}. A typical value
+for $rn\_avevd$ is between 1 and $100~m^2.s^{-1}$. This parameterisation of
+convective processes is less time consuming than the convective adjustment
+algorithm presented above when mixing both tracers and momentum in the
+case of static instabilities. It requires the use of an implicit time stepping on
+vertical diffusion terms (i.e. \np{ln\_zdfexp}\forcode{ = .false.}).
+Note that the stability test is performed on both \textit{before} and \textit{now}
+values of $N^2$. This removes a potential source of divergence of odd and
+even time step in a leapfrog environment \citep{Leclair_PhD2010} (see \autoref{sec:STP_mLF}).
+The enhanced vertical diffusion parameterisation is used when \np{ln\_zdfevd}\forcode{ = .true.}.
+In this case, the vertical eddy mixing coefficients are assigned very large values
+(a typical value is $10\;m^2s^{-1})$ in regions where the stratification is unstable
+($i.e.$ when $N^2$ the Brunt-Vais\"{a}l\"{a} frequency is negative) \citep{Lazar_PhD97, Lazar_al_JPO99}.
+This is done either on tracers only (\np{nn\_evdm}\forcode{ = 0}) or
+on both momentum and tracers (\np{nn\_evdm}\forcode{ = 1}).
+In practice, where $N^2\leq 10^{-12}$, $A_T^{vT}$ and $A_T^{vS}$, and if \np{nn\_evdm}\forcode{ = 1},
+the four neighbouring $A_u^{vm} \;\mbox{and}\;A_v^{vm}$ values also, are set equal to
+the namelist parameter \np{rn\_avevd}.
+A typical value for $rn\_avevd$ is between 1 and $100~m^2.s^{-1}$.
+This parameterisation of convective processes is less time consuming than
+the convective adjustment algorithm presented above when mixing both tracers and
+momentum in the case of static instabilities.
+It requires the use of an implicit time stepping on vertical diffusion terms
+(i.e. \np{ln\_zdfexp}\forcode{ = .false.}).
+Note that the stability test is performed on both \textit{before} and \textit{now} values of $N^2$.
+This removes a potential source of divergence of odd and even time step in
+a leapfrog environment \citep{Leclair_PhD2010} (see \autoref{sec:STP_mLF}).
 % -------------------------------------------------------------------------------------------------------------
 …
 \label{subsec:ZDF_tcs}
+The turbulent closure scheme presented in \autoref{subsec:ZDF_tke} and \autoref{subsec:ZDF_gls}
+(\key{zdftke} or \key{zdftke} is defined) in theory solves the problem of statically
+unstable density profiles. In such a case, the term corresponding to the
+destruction of turbulent kinetic energy through stratification in \autoref{eq:zdftke_e}
+or \autoref{eq:zdfgls_e} becomes a source term, since $N^2$ is negative.
+It results in large values of $A_T^{vT}$ and  $A_T^{vT}$, and also the four neighbouring
+$A_u^{vm} {and}\;A_v^{vm}$ (up to $1\;m^2s^{-1})$. These large values
+restore the static stability of the water column in a way similar to that of the
+enhanced vertical diffusion parameterisation (\autoref{subsec:ZDF_evd}). However,
+in the vicinity of the sea surface (first ocean layer), the eddy coefficients
+computed by the turbulent closure scheme do not usually exceed $10^{-2}m.s^{-1}$,
+because the mixing length scale is bounded by the distance to the sea surface.
+It can thus be useful to combine the enhanced vertical
+diffusion with the turbulent closure scheme, $i.e.$ setting the \np{ln\_zdfnpc}
+namelist parameter to true and defining the turbulent closure CPP key all together.
+The KPP turbulent closure scheme already includes enhanced vertical diffusion
+in the case of convection, as governed by the variables $bvsqcon$ and $difcon$
+found in \mdl{zdfkpp}, therefore \np{ln\_zdfevd}\forcode{ = .false.} should be used with the KPP
+scheme. %gm%  + one word on non local flux with KPP scheme trakpp.F90 module...
+The turbulent closure scheme presented in \autoref{subsec:ZDF_tke} and \autoref{subsec:ZDF_gls}
+(\key{zdftke} or \key{zdftke} is defined) in theory solves the problem of statically unstable density profiles.
+In such a case, the term corresponding to the destruction of turbulent kinetic energy through stratification in
+\autoref{eq:zdftke_e} or \autoref{eq:zdfgls_e} becomes a source term, since $N^2$ is negative.
+It results in large values of $A_T^{vT}$ and  $A_T^{vT}$, and also the four neighbouring $A_u^{vm} {and}\;A_v^{vm}$
+(up to $1\;m^2s^{-1}$).
+These large values restore the static stability of the water column in a way similar to that of
+the enhanced vertical diffusion parameterisation (\autoref{subsec:ZDF_evd}).
+However, in the vicinity of the sea surface (first ocean layer), the eddy coefficients computed by
+the turbulent closure scheme do not usually exceed $10^{-2}m.s^{-1}$,
+because the mixing length scale is bounded by the distance to the sea surface.
+It can thus be useful to combine the enhanced vertical diffusion with the turbulent closure scheme,
+$i.e.$ setting the \np{ln\_zdfnpc} namelist parameter to true and
+defining the turbulent closure CPP key all together.
+The KPP turbulent closure scheme already includes enhanced vertical diffusion in the case of convection,
+as governed by the variables $bvsqcon$ and $difcon$ found in \mdl{zdfkpp},
+therefore \np{ln\_zdfevd}\forcode{ = .false.} should be used with the KPP scheme.
+% gm%  + one word on non local flux with KPP scheme trakpp.F90 module...
 % ================================================================
 …
 Options are defined through the  \ngn{namzdf\_ddm} namelist variables.
+Double diffusion occurs when relatively warm, salty water overlies cooler, fresher
+water, or vice versa. The former condition leads to salt fingering and the latter
+to diffusive convection. Double-diffusive phenomena contribute to diapycnal
+mixing in extensive regions of the ocean.  \citet{Merryfield1999} include a
+parameterisation of such phenomena in a global ocean model and show that
+it leads to relatively minor changes in circulation but exerts significant regional
+influences on temperature and salinity. This parameterisation has been
+introduced in \mdl{zdfddm} module and is controlled by the \key{zdfddm} CPP key.
+Double diffusion occurs when relatively warm, salty water overlies cooler, fresher water, or vice versa.
+The former condition leads to salt fingering and the latter to diffusive convection.
+Double-diffusive phenomena contribute to diapycnal mixing in extensive regions of the ocean.
+\citet{Merryfield1999} include a parameterisation of such phenomena in a global ocean model and show that
+it leads to relatively minor changes in circulation but exerts significant regional influences on
+temperature and salinity.
+This parameterisation has been introduced in \mdl{zdfddm} module and is controlled by the \key{zdfddm} CPP key.
 Diapycnal mixing of S and T are described by diapycnal diffusion coefficients
 …
     &A^{vS} = A_o^{vS}+A_f^{vS}+A_d^{vS}
 \end{align*}
+where subscript $f$ represents mixing by salt fingering, $d$ by diffusive convection,
+and $o$ by processes other than double diffusion. The rates of double-diffusive
+mixing depend on the buoyancy ratio $R_\rho = \alpha \partial_z T / \beta \partial_z S$,
+where $\alpha$ and $\beta$ are coefficients of thermal expansion and saline
+contraction (see \autoref{subsec:TRA_eos}). To represent mixing of $S$ and $T$ by salt
+fingering, we adopt the diapycnal diffusivities suggested by Schmitt (1981):
+where subscript $f$ represents mixing by salt fingering, $d$ by diffusive convection,
+and $o$ by processes other than double diffusion.
+The rates of double-diffusive mixing depend on the buoyancy ratio
+$R_\rho = \alpha \partial_z T / \beta \partial_z S$, where $\alpha$ and $\beta$ are coefficients of
+thermal expansion and saline contraction (see \autoref{subsec:TRA_eos}).
+To represent mixing of $S$ and $T$ by salt fingering, we adopt the diapycnal diffusivities suggested by Schmitt
+(1981):
 \begin{align} \label{eq:zdfddm_f}
 A_f^{vS} &=    \begin{cases}
 …
 \includegraphics[width=0.99\textwidth]{Fig_zdfddm}
 \caption{  \protect\label{fig:zdfddm}
+From \citet{Merryfield1999} : (a) Diapycnal diffusivities $A_f^{vT}$
+and $A_f^{vS}$ for temperature and salt in regions of salt fingering. Heavy
+curves denote $A^{\ast v} = 10^{-3}~m^2.s^{-1}$ and thin curves
+$A^{\ast v} = 10^{-4}~m^2.s^{-1}$ ; (b) diapycnal diffusivities $A_d^{vT}$ and
+$A_d^{vS}$ for temperature and salt in regions of diffusive convection. Heavy
+curves denote the Federov parameterisation and thin curves the Kelley
+parameterisation. The latter is not implemented in \NEMO. }
+  From \citet{Merryfield1999} :
+  (a) Diapycnal diffusivities $A_f^{vT}$ and $A_f^{vS}$ for temperature and salt in regions of salt fingering.
+  Heavy curves denote $A^{\ast v} = 10^{-3}~m^2.s^{-1}$ and thin curves $A^{\ast v} = 10^{-4}~m^2.s^{-1}$;
+  (b) diapycnal diffusivities $A_d^{vT}$ and $A_d^{vS}$ for temperature and salt in regions of diffusive convection.
+  Heavy curves denote the Federov parameterisation and thin curves the Kelley parameterisation.
+  The latter is not implemented in \NEMO. }
 \end{center}    \end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 The factor 0.7 in \autoref{eq:zdfddm_f_T} reflects the measured ratio
+$\alpha F_T /\beta F_S \approx  0.7$ of buoyancy flux of heat to buoyancy
+flux of salt ($e.g.$, \citet{McDougall_Taylor_JMR84}). Following  \citet{Merryfield1999},
+we adopt $R_c = 1.6$, $n = 6$, and $A^{\ast v} = 10^{-4}~m^2.s^{-1}$.
 To represent mixing of S and T by diffusive layering,  the diapycnal diffusivities suggested by Federov (1988) is used:
+The factor 0.7 in \autoref{eq:zdfddm_f_T} reflects the measured ratio $\alpha F_T /\beta F_S \approx  0.7$ of
+buoyancy flux of heat to buoyancy flux of salt ($e.g.$, \citet{McDougall_Taylor_JMR84}).
+Following  \citet{Merryfield1999}, we adopt $R_c = 1.6$, $n = 6$, and $A^{\ast v} = 10^{-4}~m^2.s^{-1}$.
+To represent mixing of S and T by diffusive layering,  the diapycnal diffusivities suggested by
+Federov (1988) is used:
 \begin{align}  \label{eq:zdfddm_d}
 A_d^{vT} &=    \begin{cases}
 …
 \end{align}
 The dependencies of \autoref{eq:zdfddm_f} to \autoref{eq:zdfddm_d_S} on $R_\rho$
+are illustrated in \autoref{fig:zdfddm}. Implementing this requires computing
+$R_\rho$ at each grid point on every time step. This is done in \mdl{eosbn2} at the
+same time as $N^2$ is computed. This avoids duplication in the computation of
 $\alpha$ and $\beta$ (which is usually quite expensive).
+The dependencies of \autoref{eq:zdfddm_f} to \autoref{eq:zdfddm_d_S} on $R_\rho$ are illustrated in
+\autoref{fig:zdfddm}.
+Implementing this requires computing $R_\rho$ at each grid point on every time step.
+This is done in \mdl{eosbn2} at the same time as $N^2$ is computed.
+This avoids duplication in the computation of $\alpha$ and $\beta$ (which is usually quite expensive).
 % ================================================================
 …
 %--------------------------------------------------------------------------------------------------------------
 Options to define the top and bottom friction are defined through the  \ngn{nambfr} namelist variables.
 The bottom friction represents the friction generated by the bathymetry.
 The top friction represents the friction generated by the ice shelf/ocean interface.
 As the friction processes at the top and bottom are treated in similar way,
+Options to define the top and bottom friction are defined through the \ngn{nambfr} namelist variables.
+The bottom friction represents the friction generated by the bathymetry.
+The top friction represents the friction generated by the ice shelf/ocean interface.
+As the friction processes at the top and bottom are treated in similar way,
 only the bottom friction is described in detail below.
 Both the surface momentum flux (wind stress) and the bottom momentum
+flux (bottom friction) enter the equations as a condition on the vertical
 diffusive flux. For the bottom boundary layer, one has:
+Both the surface momentum flux (wind stress) and the bottom momentum flux (bottom friction) enter the equations as
+a condition on the vertical diffusive flux.
+For the bottom boundary layer, one has:
 \begin{equation} \label{eq:zdfbfr_flux}
 A^{vm} \left( \partial {\textbf U}_h / \partial z \right) = {{\cal F}}_h^{\textbf U}
 \end{equation}
+where ${\cal F}_h^{\textbf U}$ is represents the downward flux of horizontal momentum
+outside the logarithmic turbulent boundary layer (thickness of the order of
+~m in the ocean). How ${\cal F}_h^{\textbf U}$ influences the interior depends on the
+vertical resolution of the model near the bottom relative to the Ekman layer
+depth. For example, in order to obtain an Ekman layer depth
+$d = \sqrt{2\;A^{vm}} / f = 50$~m, one needs a vertical diffusion coefficient
+$A^{vm} = 0.125$~m$^2$s$^{-1}$ (for a Coriolis frequency
+$f = 10^{-4}$~m$^2$s$^{-1}$). With a background diffusion coefficient
+$A^{vm} = 10^{-4}$~m$^2$s$^{-1}$, the Ekman layer depth is only 1.4~m.
+When the vertical mixing coefficient is this small, using a flux condition is
+equivalent to entering the viscous forces (either wind stress or bottom friction)
+as a body force over the depth of the top or bottom model layer. To illustrate
+this, consider the equation for $u$ at $k$, the last ocean level:
+where ${\cal F}_h^{\textbf U}$ is represents the downward flux of horizontal momentum outside
+the logarithmic turbulent boundary layer (thickness of the order of 1~m in the ocean).
+How ${\cal F}_h^{\textbf U}$ influences the interior depends on the vertical resolution of the model near
+the bottom relative to the Ekman layer depth.
+For example, in order to obtain an Ekman layer depth $d = \sqrt{2\;A^{vm}} / f = 50$~m,
+one needs a vertical diffusion coefficient $A^{vm} = 0.125$~m$^2$s$^{-1}$
+(for a Coriolis frequency $f = 10^{-4}$~m$^2$s$^{-1}$).
+With a background diffusion coefficient $A^{vm} = 10^{-4}$~m$^2$s$^{-1}$, the Ekman layer depth is only 1.4~m.
+When the vertical mixing coefficient is this small, using a flux condition is equivalent to
+entering the viscous forces (either wind stress or bottom friction) as a body force over the depth of the top or
+bottom model layer.
+To illustrate this, consider the equation for $u$ at $k$, the last ocean level:
 \begin{equation} \label{eq:zdfbfr_flux2}
 \frac{\partial u_k}{\partial t} = \frac{1}{e_{3u}} \left[ \frac{A_{uw}^{vm}}{e_{3uw}} \delta_{k+1/2}\;[u] - {\cal F}^u_h \right] \approx - \frac{{\cal F}^u_{h}}{e_{3u}}
 \end{equation}
+If the bottom layer thickness is 200~m, the Ekman transport will
+be distributed over that depth. On the other hand, if the vertical resolution
+is high (1~m or less) and a turbulent closure model is used, the turbulent
+Ekman layer will be represented explicitly by the model. However, the
+logarithmic layer is never represented in current primitive equation model
+applications: it is \emph{necessary} to parameterize the flux ${\cal F}^u_h $.
+Two choices are available in \NEMO: a linear and a quadratic bottom friction.
+Note that in both cases, the rotation between the interior velocity and the
+bottom friction is neglected in the present release of \NEMO.
+In the code, the bottom friction is imposed by adding the trend due to the bottom
+friction to the general momentum trend in \mdl{dynbfr}. For the time-split surface
+pressure gradient algorithm, the momentum trend due to the barotropic component
+needs to be handled separately. For this purpose it is convenient to compute and
+store coefficients which can be simply combined with bottom velocities and geometric
+values to provide the momentum trend due to bottom friction.
+These coefficients are computed in \mdl{zdfbfr} and generally take the form
+$c_b^{\textbf U}$ where:
+If the bottom layer thickness is 200~m, the Ekman transport will be distributed over that depth.
+On the other hand, if the vertical resolution is high (1~m or less) and a turbulent closure model is used,
+the turbulent Ekman layer will be represented explicitly by the model.
+However, the logarithmic layer is never represented in current primitive equation model applications:
+it is \emph{necessary} to parameterize the flux ${\cal F}^u_h $.
+Two choices are available in \NEMO: a linear and a quadratic bottom friction.
+Note that in both cases, the rotation between the interior velocity and the bottom friction is neglected in
+the present release of \NEMO.
+In the code, the bottom friction is imposed by adding the trend due to the bottom friction to
+the general momentum trend in \mdl{dynbfr}.
+For the time-split surface pressure gradient algorithm, the momentum trend due to
+the barotropic component needs to be handled separately.
+For this purpose it is convenient to compute and store coefficients which can be simply combined with
+bottom velocities and geometric values to provide the momentum trend due to bottom friction.
+These coefficients are computed in \mdl{zdfbfr} and generally take the form $c_b^{\textbf U}$ where:
 \begin{equation} \label{eq:zdfbfr_bdef}
 \frac{\partial {\textbf U_h}}{\partial t} =
 …
 \label{subsec:ZDF_bfr_linear}
+The linear bottom friction parameterisation (including the special case
+of a free-slip condition) assumes that the bottom friction
+is proportional to the interior velocity (i.e. the velocity of the last
+model level):
+The linear bottom friction parameterisation (including the special case of a free-slip condition) assumes that
+the bottom friction is proportional to the interior velocity (i.e. the velocity of the last model level):
 \begin{equation} \label{eq:zdfbfr_linear}
 {\cal F}_h^\textbf{U} = \frac{A^{vm}}{e_3} \; \frac{\partial \textbf{U}_h}{\partial k} = r \; \textbf{U}_h^b
 \end{equation}
+where $r$ is a friction coefficient expressed in ms$^{-1}$.
+This coefficient is generally estimated by setting a typical decay time
+$\tau$ in the deep ocean,
+and setting $r = H / \tau$, where $H$ is the ocean depth. Commonly accepted
+values of $\tau$ are of the order of 100 to 200 days \citep{Weatherly_JMR84}.
+A value $\tau^{-1} = 10^{-7}$~s$^{-1}$ equivalent to 115 days, is usually used
+in quasi-geostrophic models. One may consider the linear friction as an
+approximation of quadratic friction, $r \approx 2\;C_D\;U_{av}$ (\citet{Gill1982},
+Eq. 9.6.6). For example, with a drag coefficient $C_D = 0.002$, a typical speed
+of tidal currents of $U_{av} =0.1$~m\;s$^{-1}$, and assuming an ocean depth
+$H = 4000$~m, the resulting friction coefficient is $r = 4\;10^{-4}$~m\;s$^{-1}$.
+This is the default value used in \NEMO. It corresponds to a decay time scale
+of 115~days. It can be changed by specifying \np{rn\_bfri1} (namelist parameter).
+For the linear friction case the coefficients defined in the general
+expression \autoref{eq:zdfbfr_bdef} are:
+where $r$ is a friction coefficient expressed in ms$^{-1}$.
+This coefficient is generally estimated by setting a typical decay time $\tau$ in the deep ocean,
+and setting $r = H / \tau$, where $H$ is the ocean depth.
+Commonly accepted values of $\tau$ are of the order of 100 to 200 days \citep{Weatherly_JMR84}.
+A value $\tau^{-1} = 10^{-7}$~s$^{-1}$ equivalent to 115 days, is usually used in quasi-geostrophic models.
+One may consider the linear friction as an approximation of quadratic friction, $r \approx 2\;C_D\;U_{av}$
+(\citet{Gill1982}, Eq. 9.6.6).
+For example, with a drag coefficient $C_D = 0.002$, a typical speed of tidal currents of $U_{av} =0.1$~m\;s$^{-1}$,
+and assuming an ocean depth $H = 4000$~m, the resulting friction coefficient is $r = 4\;10^{-4}$~m\;s$^{-1}$.
+This is the default value used in \NEMO. It corresponds to a decay time scale of 115~days.
+It can be changed by specifying \np{rn\_bfri1} (namelist parameter).
+For the linear friction case the coefficients defined in the general expression \autoref{eq:zdfbfr_bdef} are:
 \begin{equation} \label{eq:zdfbfr_linbfr_b}
 \begin{split}
 …
 \end{split}
 \end{equation}
+When \np{nn\_botfr}\forcode{ = 1}, the value of $r$ used is \np{rn\_bfri1}.
+Setting \np{nn\_botfr}\forcode{ = 0} is equivalent to setting $r=0$ and leads to a free-slip
+bottom boundary condition. These values are assigned in \mdl{zdfbfr}.
+From v3.2 onwards there is support for local enhancement of these values
+via an externally defined 2D mask array (\np{ln\_bfr2d}\forcode{ = .true.}) given
+in the \ifile{bfr\_coef} input NetCDF file. The mask values should vary from 0 to 1.
+Locations with a non-zero mask value will have the friction coefficient increased
+by $mask\_value$*\np{rn\_bfrien}*\np{rn\_bfri1}.
+When \np{nn\_botfr}\forcode{ = 1}, the value of $r$ used is \np{rn\_bfri1}.
+Setting \np{nn\_botfr}\forcode{ = 0} is equivalent to setting $r=0$ and
+leads to a free-slip bottom boundary condition.
+These values are assigned in \mdl{zdfbfr}.
+From v3.2 onwards there is support for local enhancement of these values via an externally defined 2D mask array
+(\np{ln\_bfr2d}\forcode{ = .true.}) given in the \ifile{bfr\_coef} input NetCDF file.
+The mask values should vary from 0 to 1.
+Locations with a non-zero mask value will have the friction coefficient increased by
+$mask\_value$*\np{rn\_bfrien}*\np{rn\_bfri1}.
 % -------------------------------------------------------------------------------------------------------------
 …
 \label{subsec:ZDF_bfr_nonlinear}
+The non-linear bottom friction parameterisation assumes that the bottom
+friction is quadratic:
+The non-linear bottom friction parameterisation assumes that the bottom friction is quadratic:
 \begin{equation} \label{eq:zdfbfr_nonlinear}
 {\cal F}_h^\textbf{U} = \frac{A^{vm}}{e_3 }\frac{\partial \textbf {U}_h
 }{\partial k}=C_D \;\sqrt {u_b ^2+v_b ^2+e_b } \;\; \textbf {U}_h^b
 \end{equation}
+where $C_D$ is a drag coefficient, and $e_b $ a bottom turbulent kinetic energy
+due to tides, internal waves breaking and other short time scale currents.
+A typical value of the drag coefficient is $C_D = 10^{-3} $. As an example,
+the CME experiment \citep{Treguier_JGR92} uses $C_D = 10^{-3}$ and
+$e_b = 2.5\;10^{-3}$m$^2$\;s$^{-2}$, while the FRAM experiment \citep{Killworth1992}
+uses $C_D = 1.4\;10^{-3}$ and $e_b =2.5\;\;10^{-3}$m$^2$\;s$^{-2}$.
+The CME choices have been set as default values (\np{rn\_bfri2} and \np{rn\_bfeb2}
+namelist parameters).
+As for the linear case, the bottom friction is imposed in the code by
+adding the trend due to the bottom friction to the general momentum trend
+in \mdl{dynbfr}.
+For the non-linear friction case the terms
+computed in \mdl{zdfbfr}  are:
+where $C_D$ is a drag coefficient, and $e_b $ a bottom turbulent kinetic energy due to tides,
+internal waves breaking and other short time scale currents.
+A typical value of the drag coefficient is $C_D = 10^{-3} $.
+As an example, the CME experiment \citep{Treguier_JGR92} uses $C_D = 10^{-3}$ and
+$e_b = 2.5\;10^{-3}$m$^2$\;s$^{-2}$, while the FRAM experiment \citep{Killworth1992} uses $C_D = 1.4\;10^{-3}$ and
+$e_b =2.5\;\;10^{-3}$m$^2$\;s$^{-2}$.
+The CME choices have been set as default values (\np{rn\_bfri2} and \np{rn\_bfeb2} namelist parameters).
+As for the linear case, the bottom friction is imposed in the code by adding the trend due to
+the bottom friction to the general momentum trend in \mdl{dynbfr}.
+For the non-linear friction case the terms computed in \mdl{zdfbfr} are:
 \begin{equation} \label{eq:zdfbfr_nonlinbfr}
 \begin{split}
 …
 \end{equation}
 The coefficients that control the strength of the non-linear bottom friction are
 initialised as namelist parameters: $C_D$= \np{rn\_bfri2}, and $e_b$ =\np{rn\_bfeb2}.
 Note for applications which treat tides explicitly a low or even zero value of
+\np{rn\_bfeb2} is recommended. From v3.2 onwards a local enhancement of $C_D$ is possible
+via an externally defined 2D mask array (\np{ln\_bfr2d}\forcode{ = .true.}).  This works in the same way
 as for the linear bottom friction case with non-zero masked locations increased by
+The coefficients that control the strength of the non-linear bottom friction are initialised as namelist parameters:
+$C_D$= \np{rn\_bfri2}, and $e_b$ =\np{rn\_bfeb2}.
+Note for applications which treat tides explicitly a low or even zero value of \np{rn\_bfeb2} is recommended.
+From v3.2 onwards a local enhancement of $C_D$ is possible via an externally defined 2D mask array
+(\np{ln\_bfr2d}\forcode{ = .true.}).
+This works in the same way as for the linear bottom friction case with non-zero masked locations increased by
 $mask\_value$*\np{rn\_bfrien}*\np{rn\_bfri2}.
 …
 \label{subsec:ZDF_bfr_loglayer}
 In the non-linear bottom friction case, the drag coefficient, $C_D$, can be optionally
+enhanced using a "law of the wall" scaling. If  \np{ln\_loglayer} = .true., $C_D$ is no
+longer constant but is related to the thickness of the last wet layer in each column by:
+In the non-linear bottom friction case, the drag coefficient, $C_D$, can be optionally enhanced using
+a "law of the wall" scaling.
+If  \np{ln\_loglayer} = .true., $C_D$ is no longer constant but is related to the thickness of
+the last wet layer in each column by:
 \begin{equation}
 C_D = \left ( {\kappa \over {\rm log}\left ( 0.5e_{3t}/rn\_bfrz0 \right ) } \right )^2
 \end{equation}
 \noindent where $\kappa$ is the von-Karman constant and \np{rn\_bfrz0} is a roughness
 length provided via the namelist.
+\noindent where $\kappa$ is the von-Karman constant and \np{rn\_bfrz0} is a roughness length provided via
+the namelist.
 For stability, the drag coefficient is bounded such that it is kept greater or equal to
+the base \np{rn\_bfri2} value and it is not allowed to exceed the value of an additional
+namelist parameter: \np{rn\_bfri2\_max}, i.e.:
+the base \np{rn\_bfri2} value and it is not allowed to exceed the value of an additional namelist parameter:
+\np{rn\_bfri2\_max}, i.e.:
 \begin{equation}
 rn\_bfri2 \leq C_D \leq rn\_bfri2\_max
 \end{equation}
+\noindent Note also that a log-layer enhancement can also be applied to the top boundary
+friction if under ice-shelf cavities are in use (\np{ln\_isfcav}\forcode{ = .true.}).  In this case, the
+relevant namelist parameters are \np{rn\_tfrz0}, \np{rn\_tfri2}
+and \np{rn\_tfri2\_max}.
+\noindent Note also that a log-layer enhancement can also be applied to the top boundary friction if
+under ice-shelf cavities are in use (\np{ln\_isfcav}\forcode{ = .true.}).
+In this case, the relevant namelist parameters are \np{rn\_tfrz0}, \np{rn\_tfri2} and \np{rn\_tfri2\_max}.
 % -------------------------------------------------------------------------------------------------------------
 …
 \label{subsec:ZDF_bfr_stability}
+Some care needs to exercised over the choice of parameters to ensure that the
+implementation of bottom friction does not induce numerical instability. For
+the purposes of stability analysis, an approximation to \autoref{eq:zdfbfr_flux2}
+is:
+Some care needs to exercised over the choice of parameters to ensure that the implementation of
+bottom friction does not induce numerical instability.
+For the purposes of stability analysis, an approximation to \autoref{eq:zdfbfr_flux2} is:
 \begin{equation} \label{eq:Eqn_bfrstab}
 \begin{split}
 …
 \end{split}
 \end{equation}
 \noindent where linear bottom friction and a leapfrog timestep have been assumed.
+\noindent where linear bottom friction and a leapfrog timestep have been assumed.
 To ensure that the bottom friction cannot reverse the direction of flow it is necessary to have:
 \begin{equation}
 …
 r\frac{2\rdt}{e_{3u}} < 1 \qquad  \Rightarrow \qquad r < \frac{e_{3u}}{2\rdt}\\
 \end{equation}
 This same inequality can also be derived in the non-linear bottom friction case
+if a velocity of 1 m.s$^{-1}$ is assumed. Alternatively, this criterion can be
 rearranged to suggest a minimum bottom box thickness to ensure stability:
+This same inequality can also be derived in the non-linear bottom friction case if
+a velocity of 1 m.s$^{-1}$ is assumed.
+Alternatively, this criterion can be rearranged to suggest a minimum bottom box thickness to ensure stability:
 \begin{equation}
 e_{3u} > 2\;r\;\rdt
 \end{equation}
+\noindent which it may be necessary to impose if partial steps are being used.
+For example, if $|u| = 1$ m.s$^{-1}$, $rdt = 1800$ s, $r = 10^{-3}$ then
+$e_{3u}$ should be greater than 3.6 m. For most applications, with physically
+sensible parameters these restrictions should not be of concern. But
+caution may be necessary if attempts are made to locally enhance the bottom
+friction parameters.
+To ensure stability limits are imposed on the bottom friction coefficients both during
+initialisation and at each time step. Checks at initialisation are made in \mdl{zdfbfr}
+(assuming a 1 m.s$^{-1}$ velocity in the non-linear case).
+The number of breaches of the stability criterion are reported as well as the minimum
+and maximum values that have been set. The criterion is also checked at each time step,
+using the actual velocity, in \mdl{dynbfr}. Values of the bottom friction coefficient are
+reduced as necessary to ensure stability; these changes are not reported.
+\noindent which it may be necessary to impose if partial steps are being used.
+For example, if $|u| = 1$ m.s$^{-1}$, $rdt = 1800$ s, $r = 10^{-3}$ then $e_{3u}$ should be greater than 3.6 m.
+For most applications, with physically sensible parameters these restrictions should not be of concern.
+But caution may be necessary if attempts are made to locally enhance the bottom friction parameters.
+To ensure stability limits are imposed on the bottom friction coefficients both
+during initialisation and at each time step.
+Checks at initialisation are made in \mdl{zdfbfr} (assuming a 1 m.s$^{-1}$ velocity in the non-linear case).
+The number of breaches of the stability criterion are reported as well as
+the minimum and maximum values that have been set.
+The criterion is also checked at each time step, using the actual velocity, in \mdl{dynbfr}.
+Values of the bottom friction coefficient are reduced as necessary to ensure stability;
+these changes are not reported.
 Limits on the bottom friction coefficient are not imposed if the user has elected to
 handle the bottom friction implicitly (see \autoref{subsec:ZDF_bfr_imp}). The number of potential
 breaches of the explicit stability criterion are still reported for information purposes.
+handle the bottom friction implicitly (see \autoref{subsec:ZDF_bfr_imp}).
+The number of potential breaches of the explicit stability criterion are still reported for information purposes.
 % -------------------------------------------------------------------------------------------------------------
 …
 \label{subsec:ZDF_bfr_imp}
+An optional implicit form of bottom friction has been implemented to improve
+model stability. We recommend this option for shelf sea and coastal ocean applications, especially
+for split-explicit time splitting. This option can be invoked by setting \np{ln\_bfrimp}
+to \forcode{.true.} in the \textit{nambfr} namelist. This option requires \np{ln\_zdfexp} to be \forcode{.false.}
+in the \textit{namzdf} namelist.
+This implementation is realised in \mdl{dynzdf\_imp} and \mdl{dynspg\_ts}. In \mdl{dynzdf\_imp}, the
+bottom boundary condition is implemented implicitly.
+An optional implicit form of bottom friction has been implemented to improve model stability.
+We recommend this option for shelf sea and coastal ocean applications, especially for split-explicit time splitting.
+This option can be invoked by setting \np{ln\_bfrimp} to \forcode{.true.} in the \textit{nambfr} namelist.
+This option requires \np{ln\_zdfexp} to be \forcode{.false.} in the \textit{namzdf} namelist.
+This implementation is realised in \mdl{dynzdf\_imp} and \mdl{dynspg\_ts}. In \mdl{dynzdf\_imp},
+the bottom boundary condition is implemented implicitly.
 \begin{equation} \label{eq:dynzdf_bfr}
 …
 \end{equation}
 where $mbk$ is the layer number of the bottom wet layer. superscript $n+1$ means the velocity used in the
 friction formula is to be calculated, so, it is implicit.
 If split-explicit time splitting is used, care must be taken to avoid the double counting of
 the bottom friction in the 2-D barotropic momentum equations. As NEMO only updates the barotropic
+pressure gradient and Coriolis' forcing terms in the 2-D barotropic calculation, we need to remove
 the bottom friction induced by these two terms which has been included in the 3-D momentum trend
 and update it with the latest value. On the other hand, the bottom friction contributed by the
 other terms (e.g. the advection term, viscosity term) has been included in the 3-D momentum equations
+and should not be added in the 2-D barotropic mode.
+The implementation of the implicit bottom friction in \mdl{dynspg\_ts} is done in two steps as the
 following:
+where $mbk$ is the layer number of the bottom wet layer.
+Superscript $n+1$ means the velocity used in the friction formula is to be calculated, so, it is implicit.
+If split-explicit time splitting is used, care must be taken to avoid the double counting of the bottom friction in
+the 2-D barotropic momentum equations.
+As NEMO only updates the barotropic pressure gradient and Coriolis' forcing terms in the 2-D barotropic calculation,
+we need to remove the bottom friction induced by these two terms which has been included in the 3-D momentum trend
+and update it with the latest value.
+On the other hand, the bottom friction contributed by the other terms
+(e.g. the advection term, viscosity term) has been included in the 3-D momentum equations and
+should not be added in the 2-D barotropic mode.
+The implementation of the implicit bottom friction in \mdl{dynspg\_ts} is done in two steps as the following:
 \begin{equation} \label{eq:dynspg_ts_bfr1}
 …
 \end{equation}
+where $\textbf{T}$ is the vertical integrated 3-D momentum trend. We assume the leap-frog time-stepping
+is used here. $\Delta t$ is the barotropic mode time step and $\Delta t_{bc}$ is the baroclinic mode time step.
+ $c_{b}$ is the friction coefficient. $\eta$ is the sea surface level calculated in the barotropic loops
+while $\eta^{'}$ is the sea surface level used in the 3-D baroclinic mode. $\textbf{u}_{b}$ is the bottom
+layer horizontal velocity.
+where $\textbf{T}$ is the vertical integrated 3-D momentum trend.
+We assume the leap-frog time-stepping is used here.
+$\Delta t$ is the barotropic mode time step and $\Delta t_{bc}$ is the baroclinic mode time step.
+$c_{b}$ is the friction coefficient.
+$\eta$ is the sea surface level calculated in the barotropic loops while $\eta^{'}$ is the sea surface level used in
+the 3-D baroclinic mode.
+$\textbf{u}_{b}$ is the bottom layer horizontal velocity.
 …
 \label{subsec:ZDF_bfr_ts}
+When calculating the momentum trend due to bottom friction in \mdl{dynbfr}, the
+bottom velocity at the before time step is used. This velocity includes both the
+baroclinic and barotropic components which is appropriate when using either the
+explicit or filtered surface pressure gradient algorithms (\key{dynspg\_exp} or
+\key{dynspg\_flt}). Extra attention is required, however, when using
+split-explicit time stepping (\key{dynspg\_ts}). In this case the free surface
+equation is solved with a small time step \np{rn\_rdt}/\np{nn\_baro}, while the three
+dimensional prognostic variables are solved with the longer time step
+of \np{rn\_rdt} seconds. The trend in the barotropic momentum due to bottom
+friction appropriate to this method is that given by the selected parameterisation
+($i.e.$ linear or non-linear bottom friction) computed with the evolving velocities
+at each barotropic timestep.
+In the case of non-linear bottom friction, we have elected to partially linearise
+the problem by keeping the coefficients fixed throughout the barotropic
+time-stepping to those computed in \mdl{zdfbfr} using the now timestep.
+When calculating the momentum trend due to bottom friction in \mdl{dynbfr},
+the bottom velocity at the before time step is used.
+This velocity includes both the baroclinic and barotropic components which is appropriate when
+using either the explicit or filtered surface pressure gradient algorithms
+(\key{dynspg\_exp} or \key{dynspg\_flt}).
+Extra attention is required, however, when using split-explicit time stepping (\key{dynspg\_ts}).
+In this case the free surface equation is solved with a small time step \np{rn\_rdt}/\np{nn\_baro},
+while the three dimensional prognostic variables are solved with the longer time step of \np{rn\_rdt} seconds.
+The trend in the barotropic momentum due to bottom friction appropriate to this method is that given by
+the selected parameterisation ($i.e.$ linear or non-linear bottom friction) computed with
+the evolving velocities at each barotropic timestep.
+In the case of non-linear bottom friction, we have elected to partially linearise the problem by
+keeping the coefficients fixed throughout the barotropic time-stepping to those computed in
+\mdl{zdfbfr} using the now timestep.
 This decision allows an efficient use of the $c_b^{\vect{U}}$ coefficients to:
 \begin{enumerate}
+\item On entry to \rou{dyn\_spg\_ts}, remove the contribution of the before
+barotropic velocity to the bottom friction component of the vertically
+integrated momentum trend. Note the same stability check that is carried out
+on the bottom friction coefficient in \mdl{dynbfr} has to be applied here to
+ensure that the trend removed matches that which was added in \mdl{dynbfr}.
+\item At each barotropic step, compute the contribution of the current barotropic
+velocity to the trend due to bottom friction. Add this contribution to the
+vertically integrated momentum trend. This contribution is handled implicitly which
+eliminates the need to impose a stability criteria on the values of the bottom friction
+coefficient within the barotropic loop.
+\item On entry to \rou{dyn\_spg\_ts}, remove the contribution of the before barotropic velocity to
+  the bottom friction component of the vertically integrated momentum trend.
+  Note the same stability check that is carried out on the bottom friction coefficient in \mdl{dynbfr} has to
+  be applied here to ensure that the trend removed matches that which was added in \mdl{dynbfr}.
+\item At each barotropic step, compute the contribution of the current barotropic velocity to
+  the trend due to bottom friction.
+  Add this contribution to the vertically integrated momentum trend.
+  This contribution is handled implicitly which eliminates the need to impose a stability criteria on
+  the values of the bottom friction coefficient within the barotropic loop.
 \end{enumerate}
+Note that the use of an implicit formulation within the barotropic loop
+for the bottom friction trend means that any limiting of the bottom friction coefficient
+in \mdl{dynbfr} does not adversely affect the solution when using split-explicit time
+splitting. This is because the major contribution to bottom friction is likely to come from
+the barotropic component which uses the unrestricted value of the coefficient. However, if the
+limiting is thought to be having a major effect (a more likely prospect in coastal and shelf seas
+applications) then the fully implicit form of the bottom friction should be used (see \autoref{subsec:ZDF_bfr_imp} )
+Note that the use of an implicit formulation within the barotropic loop for the bottom friction trend means that
+any limiting of the bottom friction coefficient in \mdl{dynbfr} does not adversely affect the solution when
+using split-explicit time splitting.
+This is because the major contribution to bottom friction is likely to come from the barotropic component which
+uses the unrestricted value of the coefficient.
+However, if the limiting is thought to be having a major effect
+(a more likely prospect in coastal and shelf seas applications) then
+the fully implicit form of the bottom friction should be used (see \autoref{subsec:ZDF_bfr_imp})
 which can be selected by setting \np{ln\_bfrimp} $=$ \forcode{.true.}.
 …
 \end{equation}
 where $\bar U$ is the barotropic velocity, $H_e$ is the full depth (including sea surface height),
+$c_b^u$ is the bottom friction coefficient as calculated in \rou{zdf\_bfr} and $RHS$ represents
+all the components to the vertically integrated momentum trend except for that due to bottom friction.
+$c_b^u$ is the bottom friction coefficient as calculated in \rou{zdf\_bfr} and
+$RHS$ represents all the components to the vertically integrated momentum trend except for
+that due to bottom friction.
 …
 Options are defined through the  \ngn{namzdf\_tmx} namelist variables.
+The parameterization of tidal mixing follows the general formulation for
+the vertical eddy diffusivity proposed by \citet{St_Laurent_al_GRL02} and
+first introduced in an OGCM by \citep{Simmons_al_OM04}.
+In this formulation an additional vertical diffusivity resulting from internal tide breaking,
+$A^{vT}_{tides}$ is expressed as a function of $E(x,y)$, the energy transfer from barotropic
+tides to baroclinic tides :
+The parameterization of tidal mixing follows the general formulation for the vertical eddy diffusivity proposed by
+\citet{St_Laurent_al_GRL02} and first introduced in an OGCM by \citep{Simmons_al_OM04}.
+In this formulation an additional vertical diffusivity resulting from internal tide breaking,
+$A^{vT}_{tides}$ is expressed as a function of $E(x,y)$,
+the energy transfer from barotropic tides to baroclinic tides:
 \begin{equation} \label{eq:Ktides}
 A^{vT}_{tides} =  q \,\Gamma \,\frac{ E(x,y) \, F(z) }{ \rho \, N^2 }
 \end{equation}
 where $\Gamma$ is the mixing efficiency, $N$ the Brunt-Vais\"{a}l\"{a} frequency
 (see \autoref{subsec:TRA_bn2}), $\rho$ the density, $q$ the tidal dissipation efficiency,
+and $F(z)$ the vertical structure function.
+The mixing efficiency of turbulence is set by $\Gamma$ (\np{rn\_me} namelist parameter)
+and is usually taken to be the canonical value of $\Gamma = 0.2$ (Osborn 1980).
+The tidal dissipation efficiency is given by the parameter $q$ (\np{rn\_tfe} namelist parameter)
+represents the part of the internal wave energy flux $E(x, y)$ that is dissipated locally,
+with the remaining $1-q$ radiating away as low mode internal waves and
+contributing to the background internal wave field. A value of $q=1/3$ is typically used
 \citet{St_Laurent_al_GRL02}.
+The vertical structure function $F(z)$ models the distribution of the turbulent mixing in the vertical.
+It is implemented as a simple exponential decaying upward away from the bottom,
 with a vertical scale of $h_o$ (\np{rn\_htmx} namelist parameter, with a typical value of $500\,m$) \citep{St_Laurent_Nash_DSR04},
+where $\Gamma$ is the mixing efficiency, $N$ the Brunt-Vais\"{a}l\"{a} frequency (see \autoref{subsec:TRA_bn2}),
+$\rho$ the density, $q$ the tidal dissipation efficiency, and $F(z)$ the vertical structure function.
+The mixing efficiency of turbulence is set by $\Gamma$ (\np{rn\_me} namelist parameter) and
+is usually taken to be the canonical value of $\Gamma = 0.2$ (Osborn 1980).
+The tidal dissipation efficiency is given by the parameter $q$ (\np{rn\_tfe} namelist parameter)
+represents the part of the internal wave energy flux $E(x, y)$ that is dissipated locally,
+with the remaining $1-q$ radiating away as low mode internal waves and
+contributing to the background internal wave field.
+A value of $q=1/3$ is typically used \citet{St_Laurent_al_GRL02}.
+The vertical structure function $F(z)$ models the distribution of the turbulent mixing in the vertical.
+It is implemented as a simple exponential decaying upward away from the bottom,
+with a vertical scale of $h_o$ (\np{rn\_htmx} namelist parameter,
+with a typical value of $500\,m$) \citep{St_Laurent_Nash_DSR04},
 \begin{equation} \label{eq:Fz}
 F(i,j,k) = \frac{ e^{ -\frac{H+z}{h_o} } }{ h_o \left( 1- e^{ -\frac{H}{h_o} } \right) }
 …
 and is normalized so that vertical integral over the water column is unity.
+The associated vertical viscosity is calculated from the vertical
+diffusivity assuming a Prandtl number of 1, $i.e.$ $A^{vm}_{tides}=A^{vT}_{tides}$.
+In the limit of $N \rightarrow 0$ (or becoming negative), the vertical diffusivity
+is capped at $300\,cm^2/s$ and impose a lower limit on $N^2$ of \np{rn\_n2min}
+usually set to $10^{-8} s^{-2}$. These bounds are usually rarely encountered.
+The internal wave energy map, $E(x, y)$ in \autoref{eq:Ktides}, is derived
+from a barotropic model of the tides utilizing a parameterization of the
+conversion of barotropic tidal energy into internal waves.
+The essential goal of the parameterization is to represent the momentum
+exchange between the barotropic tides and the unrepresented internal waves
+induced by the tidal flow over rough topography in a stratified ocean.
+In the current version of \NEMO, the map is built from the output of
+The associated vertical viscosity is calculated from the vertical diffusivity assuming a Prandtl number of 1,
+$i.e.$ $A^{vm}_{tides}=A^{vT}_{tides}$.
+In the limit of $N \rightarrow 0$ (or becoming negative), the vertical diffusivity is capped at $300\,cm^2/s$ and
+impose a lower limit on $N^2$ of \np{rn\_n2min} usually set to $10^{-8} s^{-2}$.
+These bounds are usually rarely encountered.
+The internal wave energy map, $E(x, y)$ in \autoref{eq:Ktides}, is derived from a barotropic model of
+the tides utilizing a parameterization of the conversion of barotropic tidal energy into internal waves.
+The essential goal of the parameterization is to represent the momentum exchange between the barotropic tides and
+the unrepresented internal waves induced by the tidal flow over rough topography in a stratified ocean.
+In the current version of \NEMO, the map is built from the output of
 the barotropic global ocean tide model MOG2D-G \citep{Carrere_Lyard_GRL03}.
+This model provides the dissipation associated with internal wave energy for the M2 and K1
+tides component (\autoref{fig:ZDF_M2_K1_tmx}). The S2 dissipation is simply approximated
+as being $1/4$ of the M2 one. The internal wave energy is thus : $E(x, y) = 1.25 E_{M2} + E_{K1}$.
+Its global mean value is $1.1$ TW, in agreement with independent estimates
+\citep{Egbert_Ray_Nat00, Egbert_Ray_JGR01}.
+This model provides the dissipation associated with internal wave energy for the M2 and K1 tides component
+(\autoref{fig:ZDF_M2_K1_tmx}).
+The S2 dissipation is simply approximated as being $1/4$ of the M2 one.
+The internal wave energy is thus : $E(x, y) = 1.25 E_{M2} + E_{K1}$.
+Its global mean value is $1.1$ TW,
+in agreement with independent estimates \citep{Egbert_Ray_Nat00, Egbert_Ray_JGR01}.
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 …
 When the Indonesian Through Flow (ITF) area is included in the model domain,
+a specific treatment of tidal induced mixing in this area can be used.
+It is activated through the namelist logical \np{ln\_tmx\_itf}, and the user must provide
+an input NetCDF file, \ifile{mask\_itf}, which contains a mask array defining the ITF area
+where the specific treatment is applied.
+When \np{ln\_tmx\_itf}\forcode{ = .true.}, the two key parameters $q$ and $F(z)$ are adjusted following
+a specific treatment of tidal induced mixing in this area can be used.
+It is activated through the namelist logical \np{ln\_tmx\_itf}, and the user must provide an input NetCDF file,
+\ifile{mask\_itf}, which contains a mask array defining the ITF area where the specific treatment is applied.
+When \np{ln\_tmx\_itf}\forcode{ = .true.}, the two key parameters $q$ and $F(z)$ are adjusted following
 the parameterisation developed by \citet{Koch-Larrouy_al_GRL07}:
+First, the Indonesian archipelago is a complex geographic region
+with a series of large, deep, semi-enclosed basins connected via
+numerous narrow straits. Once generated, internal tides remain
+confined within this semi-enclosed area and hardly radiate away.
+Therefore all the internal tides energy is consumed within this area.
+First, the Indonesian archipelago is a complex geographic region with a series of
+large, deep, semi-enclosed basins connected via numerous narrow straits.
+Once generated, internal tides remain confined within this semi-enclosed area and hardly radiate away.
+Therefore all the internal tides energy is consumed within this area.
 So it is assumed that $q = 1$, $i.e.$ all the energy generated is available for mixing.
+Note that for test purposed, the ITF tidal dissipation efficiency is a
+namelist parameter (\np{rn\_tfe\_itf}). A value of $1$ or close to is
+this recommended for this parameter.
+Second, the vertical structure function, $F(z)$, is no more associated
+with a bottom intensification of the mixing, but with a maximum of
+energy available within the thermocline. \citet{Koch-Larrouy_al_GRL07}
+have suggested that the vertical distribution of the energy dissipation
+proportional to $N^2$ below the core of the thermocline and to $N$ above.
+Note that for test purposed, the ITF tidal dissipation efficiency is a namelist parameter (\np{rn\_tfe\_itf}).
+A value of $1$ or close to is this recommended for this parameter.
+Second, the vertical structure function, $F(z)$, is no more associated with a bottom intensification of the mixing,
+but with a maximum of energy available within the thermocline.
+\citet{Koch-Larrouy_al_GRL07} have suggested that the vertical distribution of
+the energy dissipation proportional to $N^2$ below the core of the thermocline and to $N$ above.
 The resulting $F(z)$ is:
 \begin{equation} \label{eq:Fz_itf}
 …
 Averaged over the ITF area, the resulting tidal mixing coefficient is $1.5\,cm^2/s$,
+which agrees with the independent estimates inferred from observations.
+Introduced in a regional OGCM, the parameterization improves the water mass
+characteristics in the different Indonesian seas, suggesting that the horizontal
+and vertical distributions of the mixing are adequately prescribed
+\citep{Koch-Larrouy_al_GRL07, Koch-Larrouy_al_OD08a, Koch-Larrouy_al_OD08b}.
+Note also that such a parameterisation has a significant impact on the behaviour
+of global coupled GCMs \citep{Koch-Larrouy_al_CD10}.
+which agrees with the independent estimates inferred from observations.
+Introduced in a regional OGCM, the parameterization improves the water mass characteristics in
+the different Indonesian seas, suggesting that the horizontal and vertical distributions of
+the mixing are adequately prescribed \citep{Koch-Larrouy_al_GRL07, Koch-Larrouy_al_OD08a, Koch-Larrouy_al_OD08b}.
+Note also that such a parameterisation has a significant impact on the behaviour of
+global coupled GCMs \citep{Koch-Larrouy_al_CD10}.
 …
 %--------------------------------------------------------------------------------------------------------------
 The parameterization of mixing induced by breaking internal waves is a generalization
+of the approach originally proposed by \citet{St_Laurent_al_GRL02}.
 A three-dimensional field of internal wave energy dissipation $\epsilon(x,y,z)$ is first constructed,
+The parameterization of mixing induced by breaking internal waves is a generalization of
+the approach originally proposed by \citet{St_Laurent_al_GRL02}.
+A three-dimensional field of internal wave energy dissipation $\epsilon(x,y,z)$ is first constructed,
 and the resulting diffusivity is obtained as
 \begin{equation} \label{eq:Kwave}
 A^{vT}_{wave} =  R_f \,\frac{ \epsilon }{ \rho \, N^2 }
 \end{equation}
+where $R_f$ is the mixing efficiency and $\epsilon$ is a specified three dimensional distribution
+of the energy available for mixing. If the \np{ln\_mevar} namelist parameter is set to false,
+the mixing efficiency is taken as constant and equal to 1/6 \citep{Osborn_JPO80}.
+In the opposite (recommended) case, $R_f$ is instead a function of the turbulence intensity parameter
+$Re_b = \frac{ \epsilon}{\nu \, N^2}$, with $\nu$ the molecular viscosity of seawater,
+following the model of \cite{Bouffard_Boegman_DAO2013}
+and the implementation of \cite{de_lavergne_JPO2016_efficiency}.
+Note that $A^{vT}_{wave}$ is bounded by $10^{-2}\,m^2/s$, a limit that is often reached when the mixing efficiency is constant.
+where $R_f$ is the mixing efficiency and $\epsilon$ is a specified three dimensional distribution of
+the energy available for mixing.
+If the \np{ln\_mevar} namelist parameter is set to false, the mixing efficiency is taken as constant and
+equal to 1/6 \citep{Osborn_JPO80}.
+In the opposite (recommended) case, $R_f$ is instead a function of
+the turbulence intensity parameter $Re_b = \frac{ \epsilon}{\nu \, N^2}$,
+with $\nu$ the molecular viscosity of seawater, following the model of \cite{Bouffard_Boegman_DAO2013} and
+the implementation of \cite{de_lavergne_JPO2016_efficiency}.
+Note that $A^{vT}_{wave}$ is bounded by $10^{-2}\,m^2/s$, a limit that is often reached when
+the mixing efficiency is constant.
 In addition to the mixing efficiency, the ratio of salt to heat diffusivities can chosen to vary
 as a function of $Re_b$ by setting the \np{ln\_tsdiff} parameter to true, a recommended choice).
 This parameterization of differential mixing, due to \cite{Jackson_Rehmann_JPO2014},
+as a function of $Re_b$ by setting the \np{ln\_tsdiff} parameter to true, a recommended choice.
+This parameterization of differential mixing, due to \cite{Jackson_Rehmann_JPO2014},
 is implemented as in \cite{de_lavergne_JPO2016_efficiency}.
 The three-dimensional distribution of the energy available for mixing, $\epsilon(i,j,k)$, is constructed
+from three static maps of column-integrated internal wave energy dissipation, $E_{cri}(i,j)$,
 $E_{pyc}(i,j)$, and $E_{bot}(i,j)$, combined to three corresponding vertical structures
+The three-dimensional distribution of the energy available for mixing, $\epsilon(i,j,k)$,
+is constructed from three static maps of column-integrated internal wave energy dissipation,
+$E_{cri}(i,j)$, $E_{pyc}(i,j)$, and $E_{bot}(i,j)$, combined to three corresponding vertical structures
 (de Lavergne et al., in prep):
 \begin{align*}
 …
 F_{bot}(i,j,k) &\propto N^2 \, e^{- h_{wkb} / h_{bot} }
 \end{align*}
 In the above formula, $h_{ab}$ denotes the height above bottom,
+In the above formula, $h_{ab}$ denotes the height above bottom,
 $h_{wkb}$ denotes the WKB-stretched height above bottom, defined by
 \begin{equation*}
 h_{wkb} = H \, \frac{ \int_{-H}^{z} N \, dz' } { \int_{-H}^{\eta} N \, dz'  } \; ,
 \end{equation*}
+The $n_p$ parameter (given by \np{nn\_zpyc} in \ngn{namzdf\_tmx\_new} namelist)  controls the stratification-dependence of the pycnocline-intensified dissipation.
+The $n_p$ parameter (given by \np{nn\_zpyc} in \ngn{namzdf\_tmx\_new} namelist)
+controls the stratification-dependence of the pycnocline-intensified dissipation.
 It can take values of 1 (recommended) or 2.
 Finally, the vertical structures $F_{cri}$ and $F_{bot}$ require the specification of
 the decay scales $h_{cri}(i,j)$ and $h_{bot}(i,j)$, which are defined by two additional input maps.
 $h_{cri}$ is related to the large-scale topography of the ocean (etopo2)
+and $h_{bot}$ is a function of the energy flux $E_{bot}$, the characteristic horizontal scale of
+Finally, the vertical structures $F_{cri}$ and $F_{bot}$ require the specification of
+the decay scales $h_{cri}(i,j)$ and $h_{bot}(i,j)$, which are defined by two additional input maps.
+$h_{cri}$ is related to the large-scale topography of the ocean (etopo2) and
+$h_{bot}$ is a function of the energy flux $E_{bot}$, the characteristic horizontal scale of
 the abyssal hill topography \citep{Goff_JGR2010} and the latitude.

NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_conservation.tex

-                      r9407
+                      r10368
 \minitoc
+The continuous equations of motion have many analytic properties. Many
+quantities (total mass, energy, enstrophy, etc.) are strictly conserved in
+the inviscid and unforced limit, while ocean physics conserve the total
+quantities on which they act (momentum, temperature, salinity) but dissipate
+their total variance (energy, enstrophy, etc.). Unfortunately, the finite
+difference form of these equations is not guaranteed to retain all these
+important properties. In constructing the finite differencing schemes, we
+wish to ensure that certain integral constraints will be maintained. In
+particular, it is desirable to construct the finite difference equations so
+that horizontal kinetic energy and/or potential enstrophy of horizontally
+non-divergent flow, and variance of temperature and salinity will be
+conserved in the absence of dissipative effects and forcing. \citet{Arakawa1966}
+has first pointed out the advantage of this approach. He showed that if
+integral constraints on energy are maintained, the computation will be free
+of the troublesome "non linear" instability originally pointed out by
+\citet{Phillips1959}. A consistent formulation of the energetic properties is
+also extremely important in carrying out long-term numerical simulations for
+an oceanographic model. Such a formulation avoids systematic errors that
+accumulate with time \citep{Bryan1997}.
+The general philosophy of OPA which has led to the discrete formulation
+presented in {\S}II.2 and II.3 is to choose second order non-diffusive
+scheme for advective terms for both dynamical and tracer equations. At this
+level of complexity, the resulting schemes are dispersive schemes.
+Therefore, they require the addition of a diffusive operator to be stable.
+The alternative is to use diffusive schemes such as upstream or flux
+corrected schemes. This last option was rejected because we prefer a
+complete handling of the model diffusion, i.e. of the model physics rather
+than letting the advective scheme produces its own implicit diffusion
+without controlling the space and time structure of this implicit diffusion.
+Note that in some very specific cases as passive tracer studies, the
+positivity of the advective scheme is required. In that case, and in that
+case only, the advective scheme used for passive tracer is a flux correction
+scheme \citep{Marti1992, Levy1996, Levy1998}.
+The continuous equations of motion have many analytic properties.
+Many quantities (total mass, energy, enstrophy, etc.) are strictly conserved in the inviscid and unforced limit,
+while ocean physics conserve the total quantities on which they act (momentum, temperature, salinity) but
+dissipate their total variance (energy, enstrophy, etc.).
+Unfortunately, the finite difference form of these equations is not guaranteed to
+retain all these important properties.
+In constructing the finite differencing schemes, we wish to ensure that
+certain integral constraints will be maintained.
+In particular, it is desirable to construct the finite difference equations so that
+horizontal kinetic energy and/or potential enstrophy of horizontally non-divergent flow,
+and variance of temperature and salinity will be conserved in the absence of dissipative effects and forcing.
+\citet{Arakawa1966} has first pointed out the advantage of this approach.
+He showed that if integral constraints on energy are maintained,
+the computation will be free of the troublesome "non linear" instability originally pointed out by
+\citet{Phillips1959}.
+A consistent formulation of the energetic properties is also extremely important in carrying out
+long-term numerical simulations for an oceanographic model.
+Such a formulation avoids systematic errors that accumulate with time \citep{Bryan1997}.
+The general philosophy of OPA which has led to the discrete formulation presented in {\S}II.2 and II.3 is to
+choose second order non-diffusive scheme for advective terms for both dynamical and tracer equations.
+At this level of complexity, the resulting schemes are dispersive schemes.
+Therefore, they require the addition of a diffusive operator to be stable.
+The alternative is to use diffusive schemes such as upstream or flux corrected schemes.
+This last option was rejected because we prefer a complete handling of the model diffusion,
+i.e. of the model physics rather than letting the advective scheme produces its own implicit diffusion without
+controlling the space and time structure of this implicit diffusion.
+Note that in some very specific cases as passive tracer studies, the positivity of the advective scheme is required.
+In that case, and in that case only, the advective scheme used for passive tracer is a flux correction scheme
+\citep{Marti1992, Levy1996, Levy1998}.
 % -------------------------------------------------------------------------------------------------------------
 …
 \label{sec:Invariant_dyn}
+The non linear term of the momentum equations has been split into a
+vorticity term, a gradient of horizontal kinetic energy and a vertical
+advection term. Three schemes are available for the former (see {\S}~II.2)
+according to the CPP variable defined (default option\textbf{
+}or \textbf{key{\_}vorenergy } or \textbf{key{\_}vorcombined
+} defined). They differ in their conservative
+properties (energy or enstrophy conserving scheme). The two latter terms
+preserve the total kinetic energy: the large scale kinetic energy is also
+preserved in practice. The remaining non-diffusive terms of the momentum
+equation (namely the hydrostatic and surface pressure gradient terms) also
+preserve the total kinetic energy and have no effect on the vorticity of the
+flow.
+The non linear term of the momentum equations has been split into a vorticity term,
+a gradient of horizontal kinetic energy and a vertical advection term.
+Three schemes are available for the former (see {\S}~II.2) according to the CPP variable defined
+(default option\textbf{?}or \textbf{key{\_}vorenergy} or \textbf{key{\_}vorcombined} defined).
+They differ in their conservative properties (energy or enstrophy conserving scheme).
+The two latter terms preserve the total kinetic energy:
+the large scale kinetic energy is also preserved in practice.
+The remaining non-diffusive terms of the momentum equation
+(namely the hydrostatic and surface pressure gradient terms) also preserve the total kinetic energy and
+have no effect on the vorticity of the flow.
 \textbf{* relative, planetary and total vorticity term:}
+Let us define as either the relative, planetary and total potential
+vorticity, i.e. , , and , respectively. The continuous formulation of the
+vorticity term satisfies following integral constraints:
+Let us define as either the relative, planetary and total potential vorticity, i.e. ?, ?, and ?, respectively.
+The continuous formulation of the vorticity term satisfies following integral constraints:
 \begin{equation} \label{eq:vor_vorticity}
 \int_D {{\textbf {k}}\cdot \frac{1}{e_3 }\nabla \times \left( {\varsigma
 …
 \int_D {{\textbf{U}}_h \times \left( {\varsigma \;{\textbf{k}}\times {\textbf{U}}_h } \right)\;dv} =0
 \end{equation}
+where $dv = e_1\, e_2\, e_3\, di\, dj\, dk$ is the volume element.
+(II.4.1a) means that $\varsigma $ is conserved. (II.4.1b) is obtained by an
+integration by part. It means that $\varsigma^2$ is conserved for a horizontally
+non-divergent flow.
+(II.4.1c) is even satisfied locally since the vorticity term is orthogonal
+to the horizontal velocity. It means that the vorticity term has no
+contribution to the evolution of the total kinetic energy. (II.4.1a) is
+obviously always satisfied, but (II.4.1b) and (II.4.1c) cannot be satisfied
+simultaneously with a second order scheme. Using the symmetry or
+anti-symmetry properties of the operators (Eqs II.1.10 and 11), it can be
+shown that the scheme (II.2.11) satisfies (II.4.1b) but not (II.4.1c), while
+scheme (II.2.12) satisfies (II.4.1c) but not (II.4.1b) (see appendix C).
+Note that the enstrophy conserving scheme on total vorticity has been chosen
+as the standard discrete form of the vorticity term.
+where $dv = e_1\, e_2\, e_3\, di\, dj\, dk$ is the volume element.
+(II.4.1a) means that $\varsigma $ is conserved. (II.4.1b) is obtained by an integration by part.
+It means that $\varsigma^2$ is conserved for a horizontally non-divergent flow.
+(II.4.1c) is even satisfied locally since the vorticity term is orthogonal to the horizontal velocity.
+It means that the vorticity term has no contribution to the evolution of the total kinetic energy.
+(II.4.1a) is obviously always satisfied, but (II.4.1b) and (II.4.1c) cannot be satisfied simultaneously with
+a second order scheme.
+Using the symmetry or anti-symmetry properties of the operators (Eqs II.1.10 and 11),
+it can be shown that the scheme (II.2.11) satisfies (II.4.1b) but not (II.4.1c),
+while scheme (II.2.12) satisfies (II.4.1c) but not (II.4.1b) (see appendix C).
+Note that the enstrophy conserving scheme on total vorticity has been chosen as the standard discrete form of
+the vorticity term.
 \textbf{* Gradient of kinetic energy / vertical advection}
 In continuous formulation, the gradient of horizontal kinetic energy has no
+contribution to the evolution of the vorticity as the curl of a gradient is
+zero. This property is satisfied locally with the discrete form of both the
+gradient and the curl operator we have made (property (II.1.9)~). Another
+continuous property is that the change of horizontal kinetic energy due to
 vertical advection is exactly balanced by the change of horizontal kinetic
 energy due to the horizontal gradient of horizontal kinetic energy:
+In continuous formulation, the gradient of horizontal kinetic energy has no contribution to the evolution of
+the vorticity as the curl of a gradient is zero.
+This property is satisfied locally with the discrete form of both the gradient and the curl operator we have made
+(property (II.1.9)~).
+Another continuous property is that the change of horizontal kinetic energy due to
+vertical advection is exactly balanced by the change of horizontal kinetic energy due to
+the horizontal gradient of horizontal kinetic energy:
 \begin{equation} \label{eq:keg_zad}
 …
 \end{equation}
 Using the discrete form given in {\S}II.2-a and the symmetry or
+anti-symmetry properties of the mean and difference operators, \autoref{eq:keg_zad} is
+demonstrated in the Appendix C. The main point here is that satisfying
+\autoref{eq:keg_zad} links the choice of the discrete forms of the vertical advection
+and of the horizontal gradient of horizontal kinetic energy. Choosing one
+imposes the other. The discrete form of the vertical advection given in
+{\S}II.2-a is a direct consequence of formulating the horizontal kinetic
 energy as $1/2 \left( \overline{u^2}^i + \overline{v^2}^j \right) $ in the gradient term.
+Using the discrete form given in {\S}II.2-a and the symmetry or anti-symmetry properties of
+the mean and difference operators, \autoref{eq:keg_zad} is demonstrated in the Appendix C.
+The main point here is that satisfying \autoref{eq:keg_zad} links the choice of the discrete forms of
+the vertical advection and of the horizontal gradient of horizontal kinetic energy.
+Choosing one imposes the other.
+The discrete form of the vertical advection given in {\S}II.2-a is a direct consequence of
+formulating the horizontal kinetic energy as $1/2 \left( \overline{u^2}^i + \overline{v^2}^j \right) $ in
+the gradient term.
 \textbf{* hydrostatic pressure gradient term}
+In continuous formulation, a pressure gradient has no contribution to the
+evolution of the vorticity as the curl of a gradient is zero. This
+properties is satisfied locally with the choice of discretization we have
+made (property (II.1.9)~). In addition, when the equation of state is linear
+(i.e. when an advective-diffusive equation for density can be derived from
+those of temperature and salinity) the change of horizontal kinetic energy
+due to the work of pressure forces is balanced by the change of potential
+energy due to buoyancy forces:
+In continuous formulation, a pressure gradient has no contribution to the evolution of the vorticity as
+the curl of a gradient is zero.
+This properties is satisfied locally with the choice of discretization we have made (property (II.1.9)~).
+In addition, when the equation of state is linear
+(i.e. when an advective-diffusive equation for density can be derived from those of temperature and salinity)
+the change of horizontal kinetic energy due to the work of pressure forces is balanced by the change of
+potential energy due to buoyancy forces:
 \begin{equation} \label{eq:hpg_pe}
 …
 \end{equation}
+Using the discrete form given in {\S}~II.2-a and the symmetry or
+anti-symmetry properties of the mean and difference operators, (II.4.3) is
+demonstrated in the Appendix C. The main point here is that satisfying
+(II.4.3) strongly constraints the discrete expression of the depth of
+$T$-points and of the term added to the pressure gradient in $s-$coordinates: the
+depth of a $T$-point, $z_T$, is defined as the sum the vertical scale
+factors at $w$-points starting from the surface.
+Using the discrete form given in {\S}~II.2-a and the symmetry or anti-symmetry properties of
+the mean and difference operators, (II.4.3) is demonstrated in the Appendix C.
+The main point here is that satisfying (II.4.3) strongly constraints the discrete expression of the depth of
+$T$-points and of the term added to the pressure gradient in $s-$coordinates: the depth of a $T$-point, $z_T$,
+is defined as the sum the vertical scale factors at $w$-points starting from the surface.
 \textbf{* surface pressure gradient term}
+In continuous formulation, the surface pressure gradient has no contribution
+to the evolution of vorticity. This properties is trivially satisfied
+locally as (II.2.3) (the equation verified by $\psi$ has been
+derived from the discrete formulation of the momentum equations, vertical
+sum and curl. Nevertheless, the $\psi$-equation is solved numerically by an
+iterative solver (see {\S}~III.5), thus the property is only satisfied with
+the accuracy required on the solver. In addition, with the rigid-lid
+approximation, the change of horizontal kinetic energy due to the work of
+In continuous formulation, the surface pressure gradient has no contribution to the evolution of vorticity.
+This properties is trivially satisfied locally as (II.2.3)
+(the equation verified by $\psi$ has been derived from the discrete formulation of the momentum equations,
+vertical sum and curl).
+Nevertheless, the $\psi$-equation is solved numerically by an iterative solver (see {\S}~III.5),
+thus the property is only satisfied with the accuracy required on the solver.
+In addition, with the rigid-lid approximation, the change of horizontal kinetic energy due to the work of
 surface pressure forces is exactly zero:
 \begin{equation} \label{eq:spg}
 …
 \end{equation}
+(II.4.4) is satisfied in discrete form only if the discrete barotropic
+streamfunction time evolution equation is given by (II.2.3) (see appendix
+C). This shows that (II.2.3) is the only way to compute the streamfunction,
+otherwise there is no guarantee that the surface pressure force work
+vanishes.
+(II.4.4) is satisfied in discrete form only if
+the discrete barotropic streamfunction time evolution equation is given by (II.2.3) (see appendix C).
+This shows that (II.2.3) is the only way to compute the streamfunction,
+otherwise there is no guarantee that the surface pressure force work vanishes.
 % -------------------------------------------------------------------------------------------------------------
 …
 \label{sec:Invariant_tra}
 In continuous formulation, the advective terms of the tracer equations
+conserve the tracer content and the quadratic form of the tracer, i.e.
+In continuous formulation, the advective terms of the tracer equations conserve the tracer content and
+the quadratic form of the tracer, $i.e.$
 \begin{equation} \label{eq:tra_tra2}
 \int_D {\nabla .\left( {T\;{\textbf{U}}} \right)\;dv} =0
 …
 \end{equation}
 The numerical scheme used ({\S}II.2-b) (equations in flux form, second order
+centred finite differences) satisfies (II.4.5) (see appendix C). Note that
+in both continuous and discrete formulations, there is generally no strict
+conservation of mass, since the equation of state is non linear with respect
 to $T$ and $S$. In practice, the mass is conserved with a very good accuracy.
+The numerical scheme used ({\S}II.2-b) (equations in flux form, second order centred finite differences) satisfies
+(II.4.5) (see appendix C).
+Note that in both continuous and discrete formulations, there is generally no strict conservation of mass,
+since the equation of state is non linear with respect to $T$ and $S$.
+In practice, the mass is conserved with a very good accuracy.
 % -------------------------------------------------------------------------------------------------------------
 …
 \textbf{* lateral momentum diffusion term}
+The continuous formulation of the horizontal diffusion of momentum satisfies
+the following integral constraints~:
+The continuous formulation of the horizontal diffusion of momentum satisfies the following integral constraints~:
 \begin{equation} \label{eq:dynldf_dyn}
 \int\limits_D {\frac{1}{e_3 }{\rm {\bf k}}\cdot \nabla \times \left[ {\nabla
 …
+(II.4.6a) and (II.4.6b) means that the horizontal diffusion of momentum
+conserve both the potential vorticity and the divergence of the flow, while
+Eqs (II.4.6c) to (II.4.6e) mean that it dissipates the energy, the enstrophy
+and the square of the divergence. The two latter properties are only
+satisfied when the eddy coefficients are horizontally uniform.
+Using (II.1.8) and (II.1.9), and the symmetry or anti-symmetry properties of
+the mean and difference operators, it is shown that the discrete form of the
+lateral momentum diffusion given in {\S}II.2-c satisfies all the integral
+constraints (II.4.6) (see appendix C). In particular, when the eddy
+coefficients are horizontally uniform, a complete separation of vorticity
+and horizontal divergence fields is ensured, so that diffusion (dissipation)
+of vorticity (enstrophy) does not generate horizontal divergence (variance
+of the horizontal divergence) and \textit{vice versa}. When the vertical curl of the horizontal
+diffusion of momentum (discrete sense) is taken, the term associated to the
+horizontal gradient of the divergence is zero locally. When the horizontal
+divergence of the horizontal diffusion of momentum (discrete sense) is
+taken, the term associated to the vertical curl of the vorticity is zero
+locally. The resulting term conserves $\chi$ and dissipates
+$\chi^2$ when the
+eddy coefficient is horizontally uniform.
+(II.4.6a) and (II.4.6b) means that the horizontal diffusion of momentum conserve both the potential vorticity and
+the divergence of the flow, while Eqs (II.4.6c) to (II.4.6e) mean that it dissipates the energy, the enstrophy and
+the square of the divergence.
+The two latter properties are only satisfied when the eddy coefficients are horizontally uniform.
+Using (II.1.8) and (II.1.9), and the symmetry or anti-symmetry properties of the mean and difference operators,
+it is shown that the discrete form of the lateral momentum diffusion given in
+{\S}II.2-c satisfies all the integral constraints (II.4.6) (see appendix C).
+In particular, when the eddy coefficients are horizontally uniform,
+a complete separation of vorticity and horizontal divergence fields is ensured,
+so that diffusion (dissipation) of vorticity (enstrophy) does not generate horizontal divergence
+(variance of the horizontal divergence) and \textit{vice versa}.
+When the vertical curl of the horizontal diffusion of momentum (discrete sense) is taken,
+the term associated to the horizontal gradient of the divergence is zero locally.
+When the horizontal divergence of the horizontal diffusion of momentum (discrete sense) is taken,
+the term associated to the vertical curl of the vorticity is zero locally.
+The resulting term conserves $\chi$ and dissipates $\chi^2$ when the eddy coefficient is horizontally uniform.
 \textbf{* vertical momentum diffusion term}
 As for the lateral momentum physics, the continuous form of the vertical
 diffusion of momentum satisfies following integral constraints~:
+As for the lateral momentum physics, the continuous form of the vertical diffusion of
+momentum satisfies following integral constraints~:
 conservation of momentum, dissipation of horizontal kinetic energy
 …
 \end{aligned}
 \end{equation}
+conservation of horizontal divergence, dissipation of square of the
+horizontal divergence
+conservation of horizontal divergence, dissipation of square of the horizontal divergence
 \begin{equation} \label{eq:dynzdf_div}
 \begin{aligned}
 …
 \end{equation}
+In discrete form, all these properties are satisfied in $z$-coordinate (see
+Appendix C). In $s$-coordinates, only first order properties can be
+demonstrated, i.e. the vertical momentum physics conserve momentum,
+potential vorticity, and horizontal divergence.
+In discrete form, all these properties are satisfied in $z$-coordinate (see Appendix C).
+In $s$-coordinates, only first order properties can be demonstrated,
+$i.e.$ the vertical momentum physics conserve momentum, potential vorticity, and horizontal divergence.
 % -------------------------------------------------------------------------------------------------------------
 …
 \label{subsec:Invariant_tra_physics}
+The numerical schemes used for tracer subgridscale physics are written in
+such a way that the heat and salt contents are conserved (equations in flux
+form, second order centred finite differences). As a form flux is used to
+compute the temperature and salinity, the quadratic form of these quantities
+(i.e. their variance) globally tends to diminish. As for the advective term,
+there is generally no strict conservation of mass even if, in practice, the
+mass is conserved with a very good accuracy.
+The numerical schemes used for tracer subgridscale physics are written in such a way that
+the heat and salt contents are conserved (equations in flux form, second order centred finite differences).
+As a form flux is used to compute the temperature and salinity,
+the quadratic form of these quantities (i.e. their variance) globally tends to diminish.
+As for the advective term, there is generally no strict conservation of mass even if,
+in practice, the mass is conserved with a very good accuracy.
 \textbf{* lateral physics: }conservation of tracer, dissipation of tracer
 …
 \end{equation}
+\textbf{* vertical physics: }conservation of tracer, dissipation of tracer
+variance, i.e.
+\textbf{* vertical physics: }conservation of tracer, dissipation of tracer variance, $i.e.$
 \begin{equation} \label{eq:trazdf_t_t2}
 …
 \end{equation}
+Using the symmetry or anti-symmetry properties of the mean and difference
+operators, it is shown that the discrete form of tracer physics given in
+{\S}~II.2-c satisfies all the integral constraints (II.4.8) and (II.4.9)
+except the dissipation of the square of the tracer when non-geopotential
+diffusion is used (see appendix C). A discrete form of the lateral tracer
+physics can be derived which satisfies these last properties. Nevertheless,
+it requires a horizontal averaging of the vertical component of the lateral
+physics that prevents the use of implicit resolution in the vertical. It has
+not been implemented.
+Using the symmetry or anti-symmetry properties of the mean and difference operators,
+it is shown that the discrete form of tracer physics given in {\S}~II.2-c satisfies all the integral constraints
+(II.4.8) and (II.4.9) except the dissipation of the square of the tracer when non-geopotential diffusion is used
+(see appendix C).
+A discrete form of the lateral tracer physics can be derived which satisfies these last properties.
+Nevertheless, it requires a horizontal averaging of the vertical component of the lateral physics that
+prevents the use of implicit resolution in the vertical.
+It has not been implemented.
 \end{document}

NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_misc.tex

-                      r10146
+                      r10368
 \label{sec:MISC_strait}
 In climate modeling, it often occurs that a crucial connections between water masses
+is broken as the grid mesh is too coarse to resolve narrow straits. For example, coarse
+grid spacing typically closes off the Mediterranean from the Atlantic at the Strait of
+Gibraltar. In this case, it is important for climate models to include the effects of salty
+water entering the Atlantic from the Mediterranean. Likewise, it is important for the
+Mediterranean to replenish its supply of water from the Atlantic to balance the net
+evaporation occurring over the Mediterranean region. This problem occurs even in
+eddy permitting simulations. For example, in ORCA 1/4\deg several straits of the Indonesian
 archipelago (Ombai, Lombok...) are much narrow than even a single ocean grid-point.
+We describe briefly here the three methods that can be used in \NEMO to handle
+such improperly resolved straits. The first two consist of opening the strait by hand
+while ensuring that the mass exchanges through the strait are not too large by
+either artificially reducing the surface of the strait grid-cells or, locally increasing
 the lateral friction. In the third one, the strait is closed but exchanges of mass,
 heat and salt across the land are allowed.
+Note that such modifications are so specific to a given configuration that no attempt
+has been made to set them in a generic way. However, examples of how
+they can be set up is given in the ORCA 2\deg and 0.5\deg configurations. For example,
+for details of implementation in ORCA2, search:
 \texttt{ IF( cp\_cfg == "orca" .AND. jp\_cfg == 2 ) }
+In climate modeling, it often occurs that a crucial connections between water masses is broken as
+the grid mesh is too coarse to resolve narrow straits.
+For example, coarse grid spacing typically closes off the Mediterranean from the Atlantic at
+the Strait of Gibraltar.
+In this case, it is important for climate models to include the effects of salty water entering the Atlantic from
+the Mediterranean.
+Likewise, it is important for the Mediterranean to replenish its supply of water from the Atlantic to
+balance the net evaporation occurring over the Mediterranean region.
+This problem occurs even in eddy permitting simulations.
+For example, in ORCA 1/4\deg several straits of the Indonesian archipelago (Ombai, Lombok...)
+are much narrow than even a single ocean grid-point.
+We describe briefly here the three methods that can be used in \NEMO to handle such improperly resolved straits.
+The first two consist of opening the strait by hand while ensuring that the mass exchanges through
+the strait are not too large by either artificially reducing the surface of the strait grid-cells or,
+locally increasing the lateral friction.
+In the third one, the strait is closed but exchanges of mass, heat and salt across the land are allowed.
+Note that such modifications are so specific to a given configuration that no attempt has been made to
+set them in a generic way.
+However, examples of how they can be set up is given in the ORCA 2\deg and 0.5\deg configurations.
+For example, for details of implementation in ORCA2, search: \texttt{IF( cp\_cfg == "orca" .AND. jp\_cfg == 2 )}
 % -------------------------------------------------------------------------------------------------------------
 …
 \label{subsec:MISC_strait_hand}
 $\bullet$ reduced scale factor in the cross-strait direction to a value in better agreement
 with the true mean width of the strait. (\autoref{fig:MISC_strait_hand}).
+$\bullet$ reduced scale factor in the cross-strait direction to a value in better agreement with
+the true mean width of the strait (\autoref{fig:MISC_strait_hand}).
 This technique is sometime called "partially open face" or "partially closed cells".
 The key issue here is only to reduce the faces of $T$-cell ($i.e.$ change the value
 of the horizontal scale factors at $u$- or $v$-point) but not the volume of the $T$-cell.
 Indeed, reducing the volume of strait $T$-cell can easily produce a numerical
 instability at that grid point that would require a reduction of the model time step.
 The changes associated with strait management are done in \mdl{domhgr},
+The key issue here is only to reduce the faces of $T$-cell
+($i.e.$ change the value of the horizontal scale factors at $u$- or $v$-point) but not the volume of the $T$-cell.
+Indeed, reducing the volume of strait $T$-cell can easily produce a numerical instability at
+that grid point that would require a reduction of the model time step.
+The changes associated with strait management are done in \mdl{domhgr},
 just after the definition or reading of the horizontal scale factors.
+$\bullet$ increase of the viscous boundary layer thickness by local increase of the
+fmask value at the coast (\autoref{fig:MISC_strait_hand}). This is done in
+\mdl{dommsk} together with the setting of the coastal value of fmask
+(see  \autoref{sec:LBC_coast})
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!tbp]     \begin{center}
+\includegraphics[width=0.80\textwidth]{Fig_Gibraltar}
+\includegraphics[width=0.80\textwidth]{Fig_Gibraltar2}
+\caption{   \protect\label{fig:MISC_strait_hand}
+Example of the Gibraltar strait defined in a $1^{\circ} \times 1^{\circ}$ mesh.
+\textit{Top}: using partially open cells. The meridional scale factor at $v$-point
+is reduced on both sides of the strait to account for the real width of the strait
+(about 20 km). Note that the scale factors of the strait $T$-point remains unchanged.
+\textit{Bottom}: using viscous boundary layers. The four fmask parameters
+along the strait coastlines are set to a value larger than 4, $i.e.$ "strong" no-slip
+case (see \autoref{fig:LBC_shlat}) creating a large viscous boundary layer
+that allows a reduced transport through the strait.}
+\end{center}   \end{figure}
+$\bullet$ increase of the viscous boundary layer thickness by local increase of the fmask value at the coast
+(\autoref{fig:MISC_strait_hand}).
+This is done in \mdl{dommsk} together with the setting of the coastal value of fmask (see  \autoref{sec:LBC_coast}).
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!tbp]
+  \begin{center}
+    \includegraphics[width=0.80\textwidth]{Fig_Gibraltar}
+    \includegraphics[width=0.80\textwidth]{Fig_Gibraltar2}
+    \caption{  \protect\label{fig:MISC_strait_hand}
+      Example of the Gibraltar strait defined in a $1^{\circ} \times 1^{\circ}$ mesh.
+      \textit{Top}: using partially open cells.
+      The meridional scale factor at $v$-point is reduced on both sides of the strait to account for
+      the real width of the strait (about 20 km).
+      Note that the scale factors of the strait $T$-point remains unchanged.
+      \textit{Bottom}: using viscous boundary layers.
+      The four fmask parameters along the strait coastlines are set to a value larger than 4,
+      $i.e.$ "strong" no-slip case (see \autoref{fig:LBC_shlat}) creating a large viscous boundary layer that
+      allows a reduced transport through the strait.}
+  \end{center}
+\end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 …
 \subsection{Simple subsetting of input files via NetCDF attributes}
+The extended grids for use with the under-shelf ice cavities will result in redundant rows
+around Antarctica if the ice cavities are not active. A simple mechanism for subsetting
+input files associated with the extended domains has been implemented to avoid the need to
+maintain different sets of input fields for use with or without active ice cavities. The
+existing 'zoom' options are overly complex for this task and marked for deletion anyway.
+This alternative subsetting operates for the j-direction only and works by optionally
+looking for and using a global file attribute (named: \np{open\_ocean\_jstart}) to
+determine the starting j-row for input. The use of this option is best explained with an
+example: Consider an ORCA1 configuration using the extended grid bathymetry and coordinate
+files:
+The extended grids for use with the under-shelf ice cavities will result in redundant rows around Antarctica if
+the ice cavities are not active.
+A simple mechanism for subsetting input files associated with the extended domains has been implemented to
+avoid the need to maintain different sets of input fields for use with or without active ice cavities.
+The existing 'zoom' options are overly complex for this task and marked for deletion anyway.
+This alternative subsetting operates for the j-direction only and works by optionally looking for and
+using a global file attribute (named: \np{open\_ocean\_jstart}) to determine the starting j-row for input.
+The use of this option is best explained with an example:
+consider an ORCA1 configuration using the extended grid bathymetry and coordinate files:
 \vspace{-10pt}
 \ifile{eORCA1\_bathymetry\_v2}
 \ifile{eORCA1\_coordinates}
+\noindent These files define a horizontal domain of 362x332. Assuming the first row with
+open ocean wet points in the non-isf bathymetry for this set is row 42 (Fortran indexing)
+then the formally correct setting for \np{open\_ocean\_jstart} is 41. Using this value as the
+first row to be read will result in a 362x292 domain which is the same size as the original
+ORCA1 domain. Thus the extended coordinates and bathymetry files can be used with all the
+original input files for ORCA1 if the ice cavities are not active (\np{ln\_isfcav =
+.false.}). Full instructions for achieving this are:
+\noindent These files define a horizontal domain of 362x332.
+Assuming the first row with open ocean wet points in the non-isf bathymetry for this set is row 42
+(Fortran indexing) then the formally correct setting for \np{open\_ocean\_jstart} is 41.
+Using this value as the first row to be read will result in a 362x292 domain which is the same size as
+the original ORCA1 domain.
+Thus the extended coordinates and bathymetry files can be used with all the original input files for ORCA1 if
+the ice cavities are not active (\np{ln\_isfcav = .false.}).
+Full instructions for achieving this are:
 \noindent Add the new attribute to any input files requiring a j-row offset, i.e:
 …
 %--------------------------------------------------------------------------------------------------------------
+\noindent Note the j-size of the global domain is the (extended j-size minus
+\np{open\_ocean\_jstart} + 1 ) and this must match the size of all datasets other than
+bathymetry and coordinates currently. However the option can be extended to any global, 2D
+and 3D, netcdf, input field by adding the:
+\noindent Note the j-size of the global domain is the (extended j-size minus \np{open\_ocean\_jstart} + 1 ) and
+this must match the size of all datasets other than bathymetry and coordinates currently.
+However the option can be extended to any global, 2D and 3D, netcdf, input field by adding the:
 \vspace{-10pt}
 \begin{forlines}
 lrowattr=ln_use_jattr
 \end{forlines}
+optional argument to the appropriate \np{iom\_get} call and the \np{open\_ocean\_jstart} attribute to the corresponding input files. It remains the users responsibility to set \np{jpjdta} and \np{jpjglo} values in the \np{namelist\_cfg} file according to their needs.
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!ht]    \begin{center}
+\includegraphics[width=0.90\textwidth]{Fig_LBC_zoom}
+\caption{   \protect\label{fig:LBC_zoom}
+Position of a model domain compared to the data input domain when the zoom functionality is used.}
+\end{center}   \end{figure}
+optional argument to the appropriate \np{iom\_get} call and the \np{open\_ocean\_jstart} attribute to
+the corresponding input files.
+It remains the users responsibility to set \np{jpjdta} and \np{jpjglo} values in
+the \np{namelist\_cfg} file according to their needs.
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!ht]
+  \begin{center}
+    \includegraphics[width=0.90\textwidth]{Fig_LBC_zoom}
+    \caption{  \protect\label{fig:LBC_zoom}
+      Position of a model domain compared to the data input domain when the zoom functionality is used.
+    }
+  \end{center}
+\end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 …
 \label{subsec:MISC_sign}
 The SIGN(A, B) is the \textsc {Fortran} intrinsic function delivers the magnitude
 of A with the sign of B. For example, SIGN(-3.0,2.0) has the value 3.0.
 The problematic case is when the second argument is zero, because, on platforms
+that support IEEE arithmetic, zero is actually a signed number.
+The SIGN(A, B) is the \textsc {Fortran} intrinsic function delivers the magnitude of A with the sign of B.
+For example, SIGN(-3.0,2.0) has the value 3.0.
+The problematic case is when the second argument is zero, because, on platforms that support IEEE arithmetic,
+zero is actually a signed number.
 There is a positive zero and a negative zero.
 In \textsc{Fortran}~90, the processor was required always to deliver a positive result for SIGN(A, B)
+if B was zero. Nevertheless, in \textsc{Fortran}~95, the processor is allowed to do the correct thing
+and deliver ABS(A) when B is a positive zero and -ABS(A) when B is a negative zero.
 This change in the specification becomes apparent only when B is of type real, and is zero,
 and the processor is capable of distinguishing between positive and negative zero,
 and B is negative real zero. Then SIGN delivers a negative result where, under \textsc{Fortran}~90
+rules,  it used to return a positive result.
 This change may be especially sensitive for the ice model, so we overwrite the intrinsinc
 function with our own function simply performing :   \\
+In \textsc{Fortran}~90, the processor was required always to deliver a positive result for SIGN(A, B) if B was zero.
+Nevertheless, in \textsc{Fortran}~95, the processor is allowed to do the correct thing and deliver ABS(A) when
+B is a positive zero and -ABS(A) when B is a negative zero.
+This change in the specification becomes apparent only when B is of type real, and is zero,
+and the processor is capable of distinguishing between positive and negative zero,
+and B is negative real zero.
+Then SIGN delivers a negative result where, under \textsc{Fortran}~90 rules, it used to return a positive result.
+This change may be especially sensitive for the ice model,
+so we overwrite the intrinsinc function with our own function simply performing :   \\
 \verb?   IF( B >= 0.e0 ) THEN   ;   SIGN(A,B) = ABS(A)  ?    \\
 \verb?   ELSE                   ;   SIGN(A,B) =-ABS(A)     ?  \\
 \verb?   ENDIF    ? \\
 This feature can be found in \mdl{lib\_fortran} module and is effective when \key{nosignedzero}
+is defined. We use a CPP key as the overwritting of a intrinsic function can present
 performance issues with some computers/compilers.
+This feature can be found in \mdl{lib\_fortran} module and is effective when \key{nosignedzero} is defined.
+We use a CPP key as the overwritting of a intrinsic function can present performance issues with
+some computers/compilers.
 …
 \label{subsec:MISC_glosum}
+The numerical reproducibility of simulations on distributed memory parallel computers
+is a critical issue. In particular, within NEMO global summation of distributed arrays
+is most susceptible to rounding errors, and their propagation and accumulation cause
+uncertainty in final simulation reproducibility on different numbers of processors.
+To avoid so, based on \citet{He_Ding_JSC01} review of different technics,
+we use a so called self-compensated summation method. The idea is to estimate
+the roundoff error, store it in a buffer, and then add it back in the next addition.
+Suppose we need to calculate $b = a_1 + a_2 + a_3$. The following algorithm
+will allow to split the sum in two ($sum_1 = a_{1} + a_{2}$ and $b = sum_2 = sum_1 + a_3$)
+with exactly the same rounding errors as the sum performed all at once.
+The numerical reproducibility of simulations on distributed memory parallel computers is a critical issue.
+In particular, within NEMO global summation of distributed arrays is most susceptible to rounding errors,
+and their propagation and accumulation cause uncertainty in final simulation reproducibility on
+different numbers of processors.
+To avoid so, based on \citet{He_Ding_JSC01} review of different technics,
+we use a so called self-compensated summation method.
+The idea is to estimate the roundoff error, store it in a buffer, and then add it back in the next addition.
+Suppose we need to calculate $b = a_1 + a_2 + a_3$.
+The following algorithm will allow to split the sum in two
+($sum_1 = a_{1} + a_{2}$ and $b = sum_2 = sum_1 + a_3$) with exactly the same rounding errors as
+the sum performed all at once.
 \begin{align*}
    sum_1 \ \  &= a_1 + a_2 \\
 …
 \end{align*}
 An example of this feature can be found in \mdl{lib\_fortran} module.
 It is systematicallt used in glob\_sum function (summation over the entire basin excluding
+duplicated rows and columns due to cyclic or north fold boundary condition as well as
+overlap MPP areas). The self-compensated summation method should be used in all summation
 in i- and/or j-direction. See \mdl{closea} module for an example.
+It is systematicallt used in glob\_sum function (summation over the entire basin excluding duplicated rows and
+columns due to cyclic or north fold boundary condition as well as overlap MPP areas).
+The self-compensated summation method should be used in all summation in i- and/or j-direction.
+See \mdl{closea} module for an example.
 Note also that this implementation may be sensitive to the optimization level.
 …
 \label{subsec:MISC_mppsca}
+The default method of communicating values across the north-fold in distributed memory applications
+(\key{mpp\_mpi}) uses a \textsc{MPI\_ALLGATHER} function to exchange values from each processing
+region in the northern row with every other processing region in the northern row. This enables a
+global width array containing the top 4 rows to be collated on every northern row processor and then
+folded with a simple algorithm. Although conceptually simple, this "All to All" communication will
+hamper performance scalability for large numbers of northern row processors. From version 3.4
+onwards an alternative method is available which only performs direct "Peer to Peer" communications
+between each processor and its immediate "neighbours" across the fold line. This is achieved by
+using the default \textsc{MPI\_ALLGATHER} method during initialisation to help identify the "active"
+neighbours. Stored lists of these neighbours are then used in all subsequent north-fold exchanges to
+restrict exchanges to those between associated regions. The collated global width array for each
+region is thus only partially filled but is guaranteed to be set at all the locations actually
+required by each individual for the fold operation. This alternative method should give identical
+results to the default \textsc{ALLGATHER} method and is recommended for large values of \np{jpni}.
+The new method is activated by setting \np{ln\_nnogather} to be true ({\bf nammpp}). The
+reproducibility of results using the two methods should be confirmed for each new, non-reference
+configuration.
+The default method of communicating values across the north-fold in distributed memory applications (\key{mpp\_mpi})
+uses a \textsc{MPI\_ALLGATHER} function to exchange values from each processing region in
+the northern row with every other processing region in the northern row.
+This enables a global width array containing the top 4 rows to be collated on every northern row processor and then
+folded with a simple algorithm.
+Although conceptually simple, this "All to All" communication will hamper performance scalability for
+large numbers of northern row processors.
+From version 3.4 onwards an alternative method is available which only performs direct "Peer to Peer" communications
+between each processor and its immediate "neighbours" across the fold line.
+This is achieved by using the default \textsc{MPI\_ALLGATHER} method during initialisation to
+help identify the "active" neighbours.
+Stored lists of these neighbours are then used in all subsequent north-fold exchanges to
+restrict exchanges to those between associated regions.
+The collated global width array for each region is thus only partially filled but is guaranteed to
+be set at all the locations actually required by each individual for the fold operation.
+This alternative method should give identical results to the default \textsc{ALLGATHER} method and
+is recommended for large values of \np{jpni}.
+The new method is activated by setting \np{ln\_nnogather} to be true ({\bf nammpp}).
+The reproducibility of results using the two methods should be confirmed for each new,
+non-reference configuration.
 % ================================================================
 …
 $\bullet$ Vector optimisation:
 \key{vectopt\_loop} enables the internal loops to collapse. This is very
+a very efficient way to increase the length of vector calculations and thus
+\key{vectopt\_loop} enables the internal loops to collapse.
+This is very a very efficient way to increase the length of vector calculations and thus
 to speed up the model on vector computers.
 …
 $\bullet$ Control print %: describe here 4 things:
 - \np{ln\_ctl} : compute and print the trends averaged over the interior domain
+in all TRA, DYN, LDF and ZDF modules. This option is very helpful when
 diagnosing the origin of an undesired change in model results.
 - also \np{ln\_ctl} but using the nictl and njctl namelist parameters to check
 the source of differences between mono and multi processor runs.
+- \np{ln\_ctl}: compute and print the trends averaged over the interior domain in all TRA, DYN, LDF and
+ZDF modules.
+This option is very helpful when diagnosing the origin of an undesired change in model results.
+- also \np{ln\_ctl} but using the nictl and njctl namelist parameters to check the source of differences between
+mono and multi processor runs.
 %%gm   to be removed both here and in the code
+- last digit comparison (\np{nn\_bit\_cmp}). In an MPP simulation, the computation of
+a sum over the whole domain is performed as the summation over all processors of
+each of their sums over their interior domains. This double sum never gives exactly
+the same result as a single sum over the whole domain, due to truncation differences.
+The "bit comparison" option has been introduced in order to be able to check that
+mono-processor and multi-processor runs give exactly the same results.
+%THIS is to be updated with the mpp_sum_glo  introduced in v3.3
+- last digit comparison (\np{nn\_bit\_cmp}).
+In an MPP simulation, the computation of a sum over the whole domain is performed as the summation over
+all processors of each of their sums over their interior domains.
+This double sum never gives exactly the same result as a single sum over the whole domain,
+due to truncation differences.
+The "bit comparison" option has been introduced in order to be able to check that
+mono-processor and multi-processor runs give exactly the same results.
+% THIS is to be updated with the mpp_sum_glo  introduced in v3.3
 % nn_bit_cmp  today only check that the nn_cla = 0 (no cross land advection)
 %%gm end
 $\bullet$  Benchmark (\np{nn\_bench}). This option defines a benchmark run based on
+a GYRE configuration (see \autoref{sec:CFG_gyre}) in which the resolution remains the same
+whatever the domain size. This allows a very large model domain to be used, just by
+changing the domain size (\jp{jpiglo}, \jp{jpjglo}) and without adjusting either the time-step
 or the physical parameterisations.
+$\bullet$  Benchmark (\np{nn\_bench}).
+This option defines a benchmark run based on a GYRE configuration (see \autoref{sec:CFG_gyre}) in which
+the resolution remains the same whatever the domain size.
+This allows a very large model domain to be used, just by changing the domain size (\jp{jpiglo}, \jp{jpjglo}) and
+without adjusting either the time-step or the physical parameterisations.
 % ================================================================

NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_model_basics.tex

-                      r9414
+                      r10368
+The ocean is a fluid that can be described to a good approximation by the primitive
+equations, $i.e.$ the Navier-Stokes equations along with a nonlinear equation of
+state which couples the two active tracers (temperature and salinity) to the fluid
+velocity, plus the following additional assumptions made from scale considerations:
+\textit{(1) spherical earth approximation: }the geopotential surfaces are assumed to
+be spheres so that gravity (local vertical) is parallel to the earth's radius
+\textit{(2) thin-shell approximation: }the ocean depth is neglected compared to the earth's radius
+\textit{(3) turbulent closure hypothesis: }the turbulent fluxes (which represent the effect
+of small scale processes on the large-scale) are expressed in terms of large-scale features
+\textit{(4) Boussinesq hypothesis:} density variations are neglected except in their
+contribution to the buoyancy force
+\textit{(5) Hydrostatic hypothesis: }the vertical momentum equation is reduced to a
+balance between the vertical pressure gradient and the buoyancy force (this removes
+convective processes from the initial Navier-Stokes equations and so convective processes
+must be parameterized instead)
+\textit{(6) Incompressibility hypothesis: }the three dimensional divergence of the velocity
+vector is assumed to be zero.
+Because the gravitational force is so dominant in the equations of large-scale motions,
+it is useful to choose an orthogonal set of unit vectors (\textbf{i},\textbf{j},\textbf{k}) linked
+to the earth such that \textbf{k} is the local upward vector and (\textbf{i},\textbf{j}) are two
+vectors orthogonal to \textbf{k}, $i.e.$ tangent to the geopotential surfaces. Let us define
+the following variables: \textbf{U} the vector velocity, $\textbf{U}=\textbf{U}_h + w\, \textbf{k}$
+The ocean is a fluid that can be described to a good approximation by the primitive equations,
+$i.e.$ the Navier-Stokes equations along with a nonlinear equation of state which
+couples the two active tracers (temperature and salinity) to the fluid velocity,
+plus the following additional assumptions made from scale considerations:
+\textit{(1) spherical earth approximation:} the geopotential surfaces are assumed to be spheres so that
+gravity (local vertical) is parallel to the earth's radius
+\textit{(2) thin-shell approximation:} the ocean depth is neglected compared to the earth's radius
+\textit{(3) turbulent closure hypothesis:} the turbulent fluxes
+(which represent the effect of small scale processes on the large-scale) are expressed in terms of
+large-scale features
+\textit{(4) Boussinesq hypothesis:} density variations are neglected except in their contribution to
+the buoyancy force
+\textit{(5) Hydrostatic hypothesis:} the vertical momentum equation is reduced to a balance between
+the vertical pressure gradient and the buoyancy force
+(this removes convective processes from the initial Navier-Stokes equations and so
+convective processes must be parameterized instead)
+\textit{(6) Incompressibility hypothesis:} the three dimensional divergence of the velocity vector is assumed to
+be zero.
+Because the gravitational force is so dominant in the equations of large-scale motions,
+it is useful to choose an orthogonal set of unit vectors (\textbf{i},\textbf{j},\textbf{k}) linked to
+the earth such that \textbf{k} is the local upward vector and (\textbf{i},\textbf{j}) are two vectors orthogonal to
+\textbf{k}, $i.e.$ tangent to the geopotential surfaces.
+Let us define the following variables: \textbf{U} the vector velocity, $\textbf{U}=\textbf{U}_h + w\, \textbf{k}$
 (the subscript $h$ denotes the local horizontal vector, $i.e.$ over the (\textbf{i},\textbf{j}) plane),
 $T$ the potential temperature, $S$ the salinity, \textit{$\rho $} the \textit{in situ} density.
 The vector invariant form of the primitive equations in the (\textbf{i},\textbf{j},\textbf{k})
+vector system provides the following six equations (namely the momentum balance, the
+hydrostatic equilibrium, the incompressibility equation, the heat and salt conservation
 equations and an equation of state):
+$T$ the potential temperature, $S$ the salinity, \textit{$\rho $} the \textit{in situ} density.
+The vector invariant form of the primitive equations in the (\textbf{i},\textbf{j},\textbf{k}) vector system
+provides the following six equations
+(namely the momentum balance, the hydrostatic equilibrium, the incompressibility equation,
+the heat and salt conservation equations and an equation of state):
 \begin{subequations} \label{eq:PE}
   \begin{equation}     \label{eq:PE_dyn}
 …
   \end{equation}
 \end{subequations}
+where $\nabla$ is the generalised derivative vector operator in $(\bf i,\bf j, \bf k)$ directions,
+$t$ is the time, $z$ is the vertical coordinate, $\rho $ is the \textit{in situ} density given by
+the equation of state (\autoref{eq:PE_eos}), $\rho_o$ is a reference density, $p$ the pressure,
+$f=2 \bf \Omega \cdot \bf k$ is the Coriolis acceleration (where $\bf \Omega$ is the Earth's
+angular velocity vector), and $g$ is the gravitational acceleration.
+${\rm {\bf D}}^{\rm {\bf U}}$, $D^T$ and $D^S$ are the parameterisations of small-scale
+physics for momentum, temperature and salinity, and ${\rm {\bf F}}^{\rm {\bf U}}$, $F^T$
+and $F^S$ surface forcing terms. Their nature and formulation are discussed in
+\autoref{sec:PE_zdf_ldf} and \autoref{subsec:PE_boundary_condition}.
+.
+where $\nabla$ is the generalised derivative vector operator in $(\bf i,\bf j, \bf k)$ directions, $t$ is the time,
+$z$ is the vertical coordinate, $\rho $ is the \textit{in situ} density given by the equation of state
+(\autoref{eq:PE_eos}), $\rho_o$ is a reference density, $p$ the pressure,
+$f=2 \bf \Omega \cdot \bf k$ is the Coriolis acceleration
+(where $\bf \Omega$ is the Earth's angular velocity vector), and $g$ is the gravitational acceleration.
+${\rm {\bf D}}^{\rm {\bf U}}$, $D^T$ and $D^S$ are the parameterisations of small-scale physics for momentum,
+temperature and salinity, and ${\rm {\bf F}}^{\rm {\bf U}}$, $F^T$ and $F^S$ surface forcing terms.
+Their nature and formulation are discussed in \autoref{sec:PE_zdf_ldf} and \autoref{subsec:PE_boundary_condition}.
 % -------------------------------------------------------------------------------------------------------------
 …
 \label{subsec:PE_boundary_condition}
+An ocean is bounded by complex coastlines, bottom topography at its base and an air-sea
+or ice-sea interface at its top. These boundaries can be defined by two surfaces, $z=-H(i,j)$
+and $z=\eta(i,j,k,t)$, where $H$ is the depth of the ocean bottom and $\eta$ is the height
+of the sea surface. Both $H$ and $\eta$ are usually referenced to a given surface, $z=0$,
+chosen as a mean sea surface (\autoref{fig:ocean_bc}). Through these two boundaries,
+the ocean can exchange fluxes of heat, fresh water, salt, and momentum with the solid earth,
+the continental margins, the sea ice and the atmosphere. However, some of these fluxes are
+so weak that even on climatic time scales of thousands of years they can be neglected.
+In the following, we briefly review the fluxes exchanged at the interfaces between the ocean
+and the other components of the earth system.
+An ocean is bounded by complex coastlines, bottom topography at its base and
+an air-sea or ice-sea interface at its top.
+These boundaries can be defined by two surfaces, $z=-H(i,j)$ and $z=\eta(i,j,k,t)$,
+where $H$ is the depth of the ocean bottom and $\eta$ is the height of the sea surface.
+Both $H$ and $\eta$ are usually referenced to a given surface, $z=0$, chosen as a mean sea surface
+(\autoref{fig:ocean_bc}).
+Through these two boundaries, the ocean can exchange fluxes of heat, fresh water, salt, and momentum with
+the solid earth, the continental margins, the sea ice and the atmosphere.
+However, some of these fluxes are so weak that even on climatic time scales of thousands of years
+they can be neglected.
+In the following, we briefly review the fluxes exchanged at the interfaces between the ocean and
+the other components of the earth system.
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!ht]   \begin{center}
+\includegraphics[width=0.90\textwidth]{Fig_I_ocean_bc}
+\caption{    \protect\label{fig:ocean_bc}
+The ocean is bounded by two surfaces, $z=-H(i,j)$ and $z=\eta(i,j,t)$, where $H$
+is the depth of the sea floor and $\eta$ the height of the sea surface.
+Both $H$ and $\eta$ are referenced to $z=0$.}
+\end{center}   \end{figure}
+\begin{figure}[!ht]
+  \begin{center}
+    \includegraphics[width=0.90\textwidth]{Fig_I_ocean_bc}
+    \caption{   \protect\label{fig:ocean_bc}
+      The ocean is bounded by two surfaces, $z=-H(i,j)$ and $z=\eta(i,j,t)$,
+      where $H$ is the depth of the sea floor and $\eta$ the height of the sea surface.
+      Both $H$ and $\eta$ are referenced to $z=0$.
+    }
+  \end{center}
+\end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 \begin{description}
+\item[Land - ocean interface:] the major flux between continental margins and the ocean is
+a mass exchange of fresh water through river runoff. Such an exchange modifies the sea
+surface salinity especially in the vicinity of major river mouths. It can be neglected for short
+range integrations but has to be taken into account for long term integrations as it influences
+the characteristics of water masses formed (especially at high latitudes). It is required in order
+to close the water cycle of the climate system. It is usually specified as a fresh water flux at
+the air-sea interface in the vicinity of river mouths.
+\item[Solid earth - ocean interface:] heat and salt fluxes through the sea floor are small,
+except in special areas of little extent. They are usually neglected in the model
+\footnote{In fact, it has been shown that the heat flux associated with the solid Earth cooling
+($i.e.$the geothermal heating) is not negligible for the thermohaline circulation of the world
+ocean (see \autoref{subsec:TRA_bbc}).}.
+The boundary condition is thus set to no flux of heat and salt across solid boundaries.
+For momentum, the situation is different. There is no flow across solid boundaries,
+$i.e.$ the velocity normal to the ocean bottom and coastlines is zero (in other words,
+the bottom velocity is parallel to solid boundaries). This kinematic boundary condition
+can be expressed as:
+\begin{equation} \label{eq:PE_w_bbc}
+w = -{\rm {\bf U}}_h \cdot  \nabla _h \left( H \right)
+\end{equation}
+In addition, the ocean exchanges momentum with the earth through frictional processes.
+Such momentum transfer occurs at small scales in a boundary layer. It must be parameterized
+in terms of turbulent fluxes using bottom and/or lateral boundary conditions. Its specification
+depends on the nature of the physical parameterisation used for ${\rm {\bf D}}^{\rm {\bf U}}$
+in \autoref{eq:PE_dyn}. It is discussed in \autoref{eq:PE_zdf}.% and Chap. III.6 to 9.
+\item[Atmosphere - ocean interface:] the kinematic surface condition plus the mass flux
+of fresh water PE  (the precipitation minus evaporation budget) leads to:
+\begin{equation} \label{eq:PE_w_sbc}
+w = \frac{\partial \eta }{\partial t}
+    + \left. {{\rm {\bf U}}_h } \right|_{z=\eta } \cdot  \nabla _h \left( \eta \right)
+\item[Land - ocean interface:]
+  the major flux between continental margins and the ocean is a mass exchange of fresh water through river runoff.
+  Such an exchange modifies the sea surface salinity especially in the vicinity of major river mouths.
+  It can be neglected for short range integrations but has to be taken into account for long term integrations as
+  it influences the characteristics of water masses formed (especially at high latitudes).
+  It is required in order to close the water cycle of the climate system.
+  It is usually specified as a fresh water flux at the air-sea interface in the vicinity of river mouths.
+\item[Solid earth - ocean interface:]
+  heat and salt fluxes through the sea floor are small, except in special areas of little extent.
+  They are usually neglected in the model \footnote{
+    In fact, it has been shown that the heat flux associated with the solid Earth cooling
+    ($i.e.$the geothermal heating) is not negligible for the thermohaline circulation of the world ocean
+    (see \autoref{subsec:TRA_bbc}).
+  }.
+  The boundary condition is thus set to no flux of heat and salt across solid boundaries.
+  For momentum, the situation is different. There is no flow across solid boundaries,
+  $i.e.$ the velocity normal to the ocean bottom and coastlines is zero (in other words,
+  the bottom velocity is parallel to solid boundaries). This kinematic boundary condition
+  can be expressed as:
+  \begin{equation} \label{eq:PE_w_bbc}
+    w = -{\rm {\bf U}}_h \cdot  \nabla _h \left( H \right)
+  \end{equation}
+  In addition, the ocean exchanges momentum with the earth through frictional processes.
+  Such momentum transfer occurs at small scales in a boundary layer.
+  It must be parameterized in terms of turbulent fluxes using bottom and/or lateral boundary conditions.
+  Its specification depends on the nature of the physical parameterisation used for
+  ${\rm {\bf D}}^{\rm {\bf U}}$ in \autoref{eq:PE_dyn}.
+  It is discussed in \autoref{eq:PE_zdf}.% and Chap. III.6 to 9.
+\item[Atmosphere - ocean interface:]
+  the kinematic surface condition plus the mass flux of fresh water PE (the precipitation minus evaporation budget)
+  leads to:
+  \begin{equation} \label{eq:PE_w_sbc}
+    w = \frac{\partial \eta }{\partial t}
+    + \left. {{\rm {\bf U}}_h } \right|_{z=\eta } \cdot  \nabla _h \left( \eta \right)
     + P-E
 \end{equation}
+The dynamic boundary condition, neglecting the surface tension (which removes capillary
+waves from the system) leads to the continuity of pressure across the interface $z=\eta$.
 The atmosphere and ocean also exchange horizontal momentum (wind stress), and heat.
 \item[Sea ice - ocean interface:] the ocean and sea ice exchange heat, salt, fresh water
+and momentum. The sea surface temperature is constrained to be at the freezing point
+at the interface. Sea ice salinity is very low ($\sim4-6 \,psu$) compared to those of the
+ocean ($\sim34 \,psu$). The cycle of freezing/melting is associated with fresh water and
 salt fluxes that cannot be neglected.
+  \end{equation}
+  The dynamic boundary condition, neglecting the surface tension (which removes capillary waves from the system)
+  leads to the continuity of pressure across the interface $z=\eta$.
+  The atmosphere and ocean also exchange horizontal momentum (wind stress), and heat.
+\item[Sea ice - ocean interface:]
+  the ocean and sea ice exchange heat, salt, fresh water and momentum.
+  The sea surface temperature is constrained to be at the freezing point at the interface.
+  Sea ice salinity is very low ($\sim4-6 \,psu$) compared to those of the ocean ($\sim34 \,psu$).
+  The cycle of freezing/melting is associated with fresh water and salt fluxes that cannot be neglected.
 \end{description}
 …
 \label{subsec:PE_p_formulation}
+The total pressure at a given depth $z$ is composed of a surface pressure $p_s$ at a
+reference geopotential surface ($z=0$) and a hydrostatic pressure $p_h$ such that:
+$p(i,j,k,t)=p_s(i,j,t)+p_h(i,j,k,t)$. The latter is computed by integrating (\autoref{eq:PE_hydrostatic}),
+assuming that pressure in decibars can be approximated by depth in meters in (\autoref{eq:PE_eos}).
+The total pressure at a given depth $z$ is composed of a surface pressure $p_s$ at
+a reference geopotential surface ($z=0$) and a hydrostatic pressure $p_h$ such that:
+$p(i,j,k,t)=p_s(i,j,t)+p_h(i,j,k,t)$.
+The latter is computed by integrating (\autoref{eq:PE_hydrostatic}),
+assuming that pressure in decibars can be approximated by depth in meters in (\autoref{eq:PE_eos}).
 The hydrostatic pressure is then given by:
 \begin{equation} \label{eq:PE_pressure}
 …
  = \int_{\varsigma =z}^{\varsigma =0} {g\;\rho \left( {T,S,\varsigma} \right)\;d\varsigma }
 \end{equation}
+ Two strategies can be considered for the surface pressure term: $(a)$ introduce of a
+ new variable $\eta$, the free-surface elevation, for which a prognostic equation can be
+ established and solved; $(b)$ assume that the ocean surface is a rigid lid, on which the
+ pressure (or its horizontal gradient) can be diagnosed. When the former strategy is used,
+ one solution of the free-surface elevation consists of the excitation of external gravity waves.
+ The flow is barotropic and the surface moves up and down with gravity as the restoring force.
+ The phase speed of such waves is high (some hundreds of metres per second) so that
+ the time step would have to be very short if they were present in the model. The latter
+ strategy filters out these waves since the rigid lid approximation implies $\eta=0$, $i.e.$
+ the sea surface is the surface $z=0$. This well known approximation increases the surface
+ wave speed to infinity and modifies certain other longwave dynamics ($e.g.$ barotropic
+ Rossby or planetary waves). The rigid-lid hypothesis is an obsolescent feature in modern
+ OGCMs. It has been available until the release 3.1 of  \NEMO, and it has been removed
+ in release 3.2 and followings. Only the free surface formulation is now described in the
+ this document (see the next sub-section).
+Two strategies can be considered for the surface pressure term:
+$(a)$ introduce of a  new variable $\eta$, the free-surface elevation,
+for which a prognostic equation can be established and solved;
+$(b)$ assume that the ocean surface is a rigid lid,
+on which the pressure (or its horizontal gradient) can be diagnosed.
+When the former strategy is used, one solution of the free-surface elevation consists of
+the excitation of external gravity waves.
+The flow is barotropic and the surface moves up and down with gravity as the restoring force.
+The phase speed of such waves is high (some hundreds of metres per second) so that
+the time step would have to be very short if they were present in the model.
+The latter strategy filters out these waves since the rigid lid approximation implies $\eta=0$,
+$i.e.$ the sea surface is the surface $z=0$.
+This well known approximation increases the surface wave speed to infinity and
+modifies certain other longwave dynamics ($e.g.$ barotropic Rossby or planetary waves).
+The rigid-lid hypothesis is an obsolescent feature in modern OGCMs.
+It has been available until the release 3.1 of  \NEMO, and it has been removed in release 3.2 and followings.
+Only the free surface formulation is now described in the this document (see the next sub-section).
 % -------------------------------------------------------------------------------------------------------------
 …
 \label{subsec:PE_free_surface}
 In the free surface formulation, a variable $\eta$, the sea-surface height, is introduced
+which describes the shape of the air-sea interface. This variable is solution of a
+prognostic equation which is established by forming the vertical average of the kinematic
 surface condition (\autoref{eq:PE_w_bbc}):
+In the free surface formulation, a variable $\eta$, the sea-surface height,
+is introduced which describes the shape of the air-sea interface.
+This variable is solution of a prognostic equation which is established by forming the vertical average of
+the kinematic surface condition (\autoref{eq:PE_w_bbc}):
 \begin{equation} \label{eq:PE_ssh}
 \frac{\partial \eta }{\partial t}=-D+P-E
 …
 and using (\autoref{eq:PE_hydrostatic}) the surface pressure is given by: $p_s = \rho \, g \, \eta$.
 Allowing the air-sea interface to move introduces the external gravity waves (EGWs)
 as a class of solution of the primitive equations. These waves are barotropic because
+of hydrostatic assumption, and their phase speed is quite high. Their time scale is
 short with respect to the other processes described by the primitive equations.
 Two choices can be made regarding the implementation of the free surface in the model,
+Allowing the air-sea interface to move introduces the external gravity waves (EGWs) as
+a class of solution of the primitive equations.
+These waves are barotropic because of hydrostatic assumption, and their phase speed is quite high.
+Their time scale is short with respect to the other processes described by the primitive equations.
+Two choices can be made regarding the implementation of the free surface in the model,
 depending on the physical processes of interest.
+$\bullet$ If one is interested in EGWs, in particular the tides and their interaction
+with the baroclinic structure of the ocean (internal waves) possibly in shallow seas,
+then a non linear free surface is the most appropriate. This means that no
+approximation is made in (\autoref{eq:PE_ssh}) and that the variation of the ocean
+volume is fully taken into account. Note that in order to study the fast time scales
+associated with EGWs it is necessary to minimize time filtering effects (use an
+explicit time scheme with very small time step, or a split-explicit scheme with
+reasonably small time step, see \autoref{subsec:DYN_spg_exp} or \autoref{subsec:DYN_spg_ts}.
+$\bullet$ If one is not interested in EGW but rather sees them as high frequency
+noise, it is possible to apply an explicit filter to slow down the fastest waves while
+not altering the slow barotropic Rossby waves. If further, an approximative conservation
+of heat and salt contents is sufficient for the problem solved, then it is
+sufficient to solve a linearized version of (\autoref{eq:PE_ssh}), which still allows
+to take into account freshwater fluxes applied at the ocean surface \citep{Roullet_Madec_JGR00}.
+$\bullet$ If one is interested in EGWs, in particular the tides and their interaction with
+the baroclinic structure of the ocean (internal waves) possibly in shallow seas,
+then a non linear free surface is the most appropriate.
+This means that no approximation is made in (\autoref{eq:PE_ssh}) and that
+the variation of the ocean volume is fully taken into account.
+Note that in order to study the fast time scales associated with EGWs it is necessary to
+minimize time filtering effects
+(use an explicit time scheme with very small time step, or a split-explicit scheme with reasonably small time step,
+see \autoref{subsec:DYN_spg_exp} or \autoref{subsec:DYN_spg_ts}).
+$\bullet$ If one is not interested in EGW but rather sees them as high frequency noise,
+it is possible to apply an explicit filter to slow down the fastest waves while
+not altering the slow barotropic Rossby waves.
+If further, an approximative conservation of heat and salt contents is sufficient for the problem solved,
+then it is sufficient to solve a linearized version of (\autoref{eq:PE_ssh}),
+which still allows to take into account freshwater fluxes applied at the ocean surface \citep{Roullet_Madec_JGR00}.
 Nevertheless, with the linearization, an exact conservation of heat and salt contents is lost.
+The filtering of EGWs in models with a free surface is usually a matter of discretisation
+of the temporal derivatives, using a split-explicit method \citep{Killworth_al_JPO91, Zhang_Endoh_JGR92}
+or the implicit scheme \citep{Dukowicz1994} or the addition of a filtering force in the momentum equation
+\citep{Roullet_Madec_JGR00}. With the present release, \NEMO offers the choice between
+an explicit free surface (see \autoref{subsec:DYN_spg_exp}) or a split-explicit scheme strongly
+inspired the one proposed by \citet{Shchepetkin_McWilliams_OM05} (see \autoref{subsec:DYN_spg_ts}).
+The filtering of EGWs in models with a free surface is usually a matter of discretisation of
+the temporal derivatives,
+using a split-explicit method \citep{Killworth_al_JPO91, Zhang_Endoh_JGR92} or
+the implicit scheme \citep{Dukowicz1994} or
+the addition of a filtering force in the momentum equation \citep{Roullet_Madec_JGR00}.
+With the present release, \NEMO offers the choice between
+an explicit free surface (see \autoref{subsec:DYN_spg_exp}) or
+a split-explicit scheme strongly inspired the one proposed by \citet{Shchepetkin_McWilliams_OM05}
+(see \autoref{subsec:DYN_spg_ts}).
 %\newpage
 …
 \label{subsec:PE_tensorial}
+In many ocean circulation problems, the flow field has regions of enhanced dynamics
+($i.e.$ surface layers, western boundary currents, equatorial currents, or ocean fronts).
+The representation of such dynamical processes can be improved by specifically increasing
+the model resolution in these regions. As well, it may be convenient to use a lateral
+boundary-following coordinate system to better represent coastal dynamics. Moreover,
+the common geographical coordinate system has a singular point at the North Pole that
+cannot be easily treated in a global model without filtering. A solution consists of introducing
+an appropriate coordinate transformation that shifts the singular point onto land
+\citep{Madec_Imbard_CD96, Murray_JCP96}. As a consequence, it is important to solve the primitive
+equations in various curvilinear coordinate systems. An efficient way of introducing an
+appropriate coordinate transform can be found when using a tensorial formalism.
+This formalism is suited to any multidimensional curvilinear coordinate system. Ocean
+modellers mainly use three-dimensional orthogonal grids on the sphere (spherical earth
+approximation), with preservation of the local vertical. Here we give the simplified equations
+for this particular case. The general case is detailed by \citet{Eiseman1980} in their survey
+of the conservation laws of fluid dynamics.
+Let (\textit{i},\textit{j},\textit{k}) be a set of orthogonal curvilinear coordinates on the sphere
+associated with the positively oriented orthogonal set of unit vectors (\textbf{i},\textbf{j},\textbf{k})
+linked to the earth such that \textbf{k} is the local upward vector and (\textbf{i},\textbf{j}) are
+two vectors orthogonal to \textbf{k}, $i.e.$ along geopotential surfaces (\autoref{fig:referential}).
+Let $(\lambda,\varphi,z)$ be the geographical coordinate system in which a position is defined
+by the latitude $\varphi(i,j)$, the longitude $\lambda(i,j)$ and the distance from the centre of
+the earth $a+z(k)$ where $a$ is the earth's radius and $z$ the altitude above a reference sea
+level (\autoref{fig:referential}). The local deformation of the curvilinear coordinate system is
+given by $e_1$, $e_2$ and $e_3$, the three scale factors:
+In many ocean circulation problems, the flow field has regions of enhanced dynamics
+($i.e.$ surface layers, western boundary currents, equatorial currents, or ocean fronts).
+The representation of such dynamical processes can be improved by
+specifically increasing the model resolution in these regions.
+As well, it may be convenient to use a lateral boundary-following coordinate system to
+better represent coastal dynamics.
+Moreover, the common geographical coordinate system has a singular point at the North Pole that
+cannot be easily treated in a global model without filtering.
+A solution consists of introducing an appropriate coordinate transformation that
+shifts the singular point onto land \citep{Madec_Imbard_CD96, Murray_JCP96}.
+As a consequence, it is important to solve the primitive equations in various curvilinear coordinate systems.
+An efficient way of introducing an appropriate coordinate transform can be found when using a tensorial formalism.
+This formalism is suited to any multidimensional curvilinear coordinate system.
+Ocean modellers mainly use three-dimensional orthogonal grids on the sphere (spherical earth approximation),
+with preservation of the local vertical. Here we give the simplified equations for this particular case.
+The general case is detailed by \citet{Eiseman1980} in their survey of the conservation laws of fluid dynamics.
+Let (\textit{i},\textit{j},\textit{k}) be a set of orthogonal curvilinear coordinates on
+the sphere associated with the positively oriented orthogonal set of unit vectors
+(\textbf{i},\textbf{j},\textbf{k}) linked to the earth such that
+\textbf{k} is the local upward vector and (\textbf{i},\textbf{j}) are two vectors orthogonal to \textbf{k},
+$i.e.$ along geopotential surfaces (\autoref{fig:referential}).
+Let $(\lambda,\varphi,z)$ be the geographical coordinate system in which a position is defined by
+the latitude $\varphi(i,j)$, the longitude $\lambda(i,j)$ and
+the distance from the centre of the earth $a+z(k)$ where $a$ is the earth's radius and
+$z$ the altitude above a reference sea level (\autoref{fig:referential}).
+The local deformation of the curvilinear coordinate system is given by $e_1$, $e_2$ and $e_3$,
+the three scale factors:
 \begin{equation} \label{eq:scale_factors}
+\begin{aligned}
+ e_1 &=\left( {a+z} \right)\;\left[ {\left( {\frac{\partial \lambda
+}{\partial i}\cos \varphi } \right)^2+\left( {\frac{\partial \varphi
+}{\partial i}} \right)^2} \right]^{1/2} \\
+ e_2 &=\left( {a+z} \right)\;\left[ {\left( {\frac{\partial \lambda
+}{\partial j}\cos \varphi } \right)^2+\left( {\frac{\partial \varphi
+}{\partial j}} \right)^2} \right]^{1/2} \\
+ e_3 &=\left( {\frac{\partial z}{\partial k}} \right) \\
+ \end{aligned}
+ \end{equation}
+  \begin{aligned}
+    e_1 &=\left( {a+z} \right)\;\left[ {\left( {\frac{\partial \lambda}{\partial i}\cos \varphi } \right)^2
+        +\left( {\frac{\partial \varphi }{\partial i}} \right)^2} \right]^{1/2} \\
+    e_2 &=\left( {a+z} \right)\;\left[ {\left( {\frac{\partial \lambda }{\partial j}\cos \varphi } \right)^2+
+        \left( {\frac{\partial \varphi }{\partial j}} \right)^2} \right]^{1/2} \\
+    e_3 &=\left( {\frac{\partial z}{\partial k}} \right) \\
+  \end{aligned}
+\end{equation}
+% >>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!tb]
+  \begin{center}
+    \includegraphics[width=0.60\textwidth]{Fig_I_earth_referential}
+    \caption{  \protect\label{fig:referential}
+      the geographical coordinate system $(\lambda,\varphi,z)$ and the curvilinear
+      coordinate system (\textbf{i},\textbf{j},\textbf{k}). }
+  \end{center}
+\end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!tb]   \begin{center}
+\includegraphics[width=0.60\textwidth]{Fig_I_earth_referential}
+\caption{   \protect\label{fig:referential}
+the geographical coordinate system $(\lambda,\varphi,z)$ and the curvilinear
+coordinate system (\textbf{i},\textbf{j},\textbf{k}). }
+\end{center}   \end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+Since the ocean depth is far smaller than the earth's radius, $a+z$, can be replaced by
+$a$ in (\autoref{eq:scale_factors}) (thin-shell approximation). The resulting horizontal scale
+factors $e_1$, $e_2$  are independent of $k$ while the vertical scale factor is a single
+function of $k$ as \textbf{k} is parallel to \textbf{z}. The scalar and vector operators that
+appear in the primitive equations (\autoref{eq:PE_dyn} to \autoref{eq:PE_eos}) can
+be written in the tensorial form, invariant in any orthogonal horizontal curvilinear coordinate
+system transformation:
+Since the ocean depth is far smaller than the earth's radius, $a+z$, can be replaced by $a$ in
+(\autoref{eq:scale_factors}) (thin-shell approximation).
+The resulting horizontal scale factors $e_1$, $e_2$  are independent of $k$ while
+the vertical scale factor is a single function of $k$ as \textbf{k} is parallel to \textbf{z}.
+The scalar and vector operators that appear in the primitive equations
+(\autoref{eq:PE_dyn} to \autoref{eq:PE_eos}) can be written in the tensorial form,
+invariant in any orthogonal horizontal curvilinear coordinate system transformation:
 \begin{subequations} \label{eq:PE_discrete_operators}
 \begin{equation} \label{eq:PE_grad}
 …
 \label{subsec:PE_zco_Eq}
+In order to express the Primitive Equations in tensorial formalism, it is necessary to compute
+the horizontal component of the non-linear and viscous terms of the equation using
+\autoref{eq:PE_grad}) to \autoref{eq:PE_lap_vector}.
+Let us set $\vect U=(u,v,w)={\vect{U}}_h +w\;\vect{k}$, the velocity in the $(i,j,k)$ coordinate
+system and define the relative vorticity $\zeta$ and the divergence of the horizontal velocity
+field $\chi$, by:
+In order to express the Primitive Equations in tensorial formalism,
+it is necessary to compute the horizontal component of the non-linear and viscous terms of the equation using
+\autoref{eq:PE_grad}) to \autoref{eq:PE_lap_vector}.
+Let us set $\vect U=(u,v,w)={\vect{U}}_h +w\;\vect{k}$, the velocity in the $(i,j,k)$ coordinate system and
+define the relative vorticity $\zeta$ and the divergence of the horizontal velocity field $\chi$, by:
 \begin{equation} \label{eq:PE_curl_Uh}
 \zeta =\frac{1}{e_1 e_2 }\left[ {\frac{\partial \left( {e_2 \,v}
 …
 \end{equation}
 Using the fact that the horizontal scale factors $e_1$ and $e_2$ are independent of $k$
+and that $e_3$  is a function of the single variable $k$, the nonlinear term of
 \autoref{eq:PE_dyn} can be transformed as follows:
+Using the fact that the horizontal scale factors $e_1$ and $e_2$ are independent of $k$ and that
+$e_3$  is a function of the single variable $k$,
+the nonlinear term of \autoref{eq:PE_dyn} can be transformed as follows:
 \begin{flalign*}
 &\left[ {\left( { \nabla \times {\rm {\bf U}}    } \right) \times {\rm {\bf U}}
 …
 \end{flalign*}
 The last term of the right hand side is obviously zero, and thus the nonlinear term of
+The last term of the right hand side is obviously zero, and thus the nonlinear term of
 \autoref{eq:PE_dyn} is written in the $(i,j,k)$ coordinate system:
 \begin{equation} \label{eq:PE_vector_form}
 …
 \end{equation}
 This is the so-called \textit{vector invariant form} of the momentum advection term.
 For some purposes, it can be advantageous to write this term in the so-called flux form,
 $i.e.$ to write it as the divergence of fluxes. For example, the first component of
 \autoref{eq:PE_vector_form} (the $i$-component) is transformed as follows:
+This is the so-called \textit{vector invariant form} of the momentum advection term.
+For some purposes, it can be advantageous to write this term in the so-called flux form,
+$i.e.$ to write it as the divergence of fluxes.
+For example, the first component of \autoref{eq:PE_vector_form} (the $i$-component) is transformed as follows:
 \begin{flalign*}
 &{ \begin{array}{*{20}l}
 …
 \end{multline}
 The flux form has two terms, the first one is expressed as the divergence of momentum
+fluxes (hence the flux form name given to this formulation) and the second one is due to
+the curvilinear nature of the coordinate system used. The latter is called the \emph{metric}
 term and can be viewed as a modification of the Coriolis parameter:
+The flux form has two terms,
+the first one is expressed as the divergence of momentum fluxes (hence the flux form name given to this formulation)
+and the second one is due to the curvilinear nature of the coordinate system used.
+The latter is called the \emph{metric} term and can be viewed as a modification of the Coriolis parameter:
 \begin{equation} \label{eq:PE_cor+metric}
 f \to f + \frac{1}{e_1\;e_2}  \left(  v \frac{\partial e_2}{\partial i}
 …
 \end{equation}
 Note that in the case of geographical coordinate, $i.e.$ when $(i,j) \to (\lambda ,\varphi )$
+and $(e_1 ,e_2) \to (a \,\cos \varphi ,a)$, we recover the commonly used modification of
 the Coriolis parameter $f \to f+(u/a) \tan \varphi$.
+Note that in the case of geographical coordinate,
+$i.e.$ when $(i,j) \to (\lambda ,\varphi )$ and $(e_1 ,e_2) \to (a \,\cos \varphi ,a)$,
+we recover the commonly used modification of the Coriolis parameter $f \to f+(u/a) \tan \varphi$.
 $\ $\newline    % force a new ligne
 To sum up, the curvilinear $z$-coordinate equations solved by the ocean model can be
 written in the following tensorial formalism:
+To sum up, the curvilinear $z$-coordinate equations solved by the ocean model can be written in
+the following tensorial formalism:
 \vspace{+10pt}
 …
 \end{multline}
 \end{subequations}
 where $\zeta$, the relative vorticity, is given by \autoref{eq:PE_curl_Uh} and $p_s $,
 the surface pressure, is given by:
+where $\zeta$, the relative vorticity, is given by \autoref{eq:PE_curl_Uh} and
+$p_s $, the surface pressure, is given by:
 \begin{equation} \label{eq:PE_spg}
 p_s =  \rho \,g \,\eta
 \end{equation}
 with $\eta$ is solution of \autoref{eq:PE_ssh}
+with $\eta$ is solution of \autoref{eq:PE_ssh}.
 The vertical velocity and the hydrostatic pressure are diagnosed from the following equations:
 …
 \end{equation}
 The expression of \textbf{D}$^{U}$, $D^{S}$ and $D^{T}$ depends on the subgrid scale
+parameterisation used. It will be defined in \autoref{eq:PE_zdf}. The nature and formulation of
+${\rm {\bf F}}^{\rm {\bf U}}$, $F^T$ and $F^S$, the surface forcing terms, are discussed
 in \autoref{chap:SBC}.
+The expression of \textbf{D}$^{U}$, $D^{S}$ and $D^{T}$ depends on the subgrid scale parameterisation used.
+It will be defined in \autoref{eq:PE_zdf}.
+The nature and formulation of ${\rm {\bf F}}^{\rm {\bf U}}$, $F^T$ and $F^S$, the surface forcing terms,
+are discussed in \autoref{chap:SBC}.
 …
 \label{sec:PE_gco}
+The ocean domain presents a huge diversity of situation in the vertical. First the ocean surface is a time dependent surface (moving surface). Second the ocean floor depends on the geographical position, varying from more than 6,000 meters in abyssal trenches to zero at the coast. Last but not least, the ocean stratification exerts a strong barrier to vertical motions and mixing.
+Therefore, in order to represent the ocean with respect to the first point a space and time dependent vertical coordinate that follows the variation of the sea surface height $e.g.$ an $z$*-coordinate; for the second point, a space variation to fit the change of bottom topography $e.g.$ a terrain-following or $\sigma$-coordinate; and for the third point, one will be tempted to use a space and time dependent coordinate that follows the isopycnal surfaces, $e.g.$ an isopycnic coordinate.
+In order to satisfy two or more constrains one can even be tempted to mixed these coordinate systems, as in HYCOM (mixture of $z$-coordinate at the surface, isopycnic coordinate in the ocean interior and $\sigma$ at the ocean bottom) \citep{Chassignet_al_JPO03}  or OPA (mixture of $z$-coordinate in vicinity the surface and steep topography areas and $\sigma$-coordinate elsewhere) \citep{Madec_al_JPO96} among others.
+In fact one is totally free to choose any space and time vertical coordinate by introducing an arbitrary vertical coordinate :
+The ocean domain presents a huge diversity of situation in the vertical.
+First the ocean surface is a time dependent surface (moving surface).
+Second the ocean floor depends on the geographical position,
+varying from more than 6,000 meters in abyssal trenches to zero at the coast.
+Last but not least, the ocean stratification exerts a strong barrier to vertical motions and mixing.
+Therefore, in order to represent the ocean with respect to
+the first point a space and time dependent vertical coordinate that follows the variation of the sea surface height
+$e.g.$ an $z$*-coordinate;
+for the second point, a space variation to fit the change of bottom topography
+$e.g.$ a terrain-following or $\sigma$-coordinate;
+and for the third point, one will be tempted to use a space and time dependent coordinate that
+follows the isopycnal surfaces, $e.g.$ an isopycnic coordinate.
+In order to satisfy two or more constrains one can even be tempted to mixed these coordinate systems, as in
+HYCOM (mixture of $z$-coordinate at the surface, isopycnic coordinate in the ocean interior and $\sigma$ at
+the ocean bottom) \citep{Chassignet_al_JPO03} or
+OPA (mixture of $z$-coordinate in vicinity the surface and steep topography areas and $\sigma$-coordinate elsewhere)
+\citep{Madec_al_JPO96} among others.
+In fact one is totally free to choose any space and time vertical coordinate by
+introducing an arbitrary vertical coordinate :
 \begin{equation} \label{eq:PE_s}
 s=s(i,j,k,t)
 \end{equation}
+with the restriction that the above equation gives a single-valued monotonic relationship between $s$ and $k$, when $i$, $j$ and $t$ are held fixed. \autoref{eq:PE_s} is a transformation from the $(i,j,k,t)$ coordinate system with independent variables into the $(i,j,s,t)$ generalised coordinate system with $s$ depending on the other three variables through \autoref{eq:PE_s}.
+This so-called \textit{generalised vertical coordinate} \citep{Kasahara_MWR74} is in fact an Arbitrary Lagrangian--Eulerian (ALE) coordinate. Indeed, choosing an expression for $s$ is an arbitrary choice that determines which part of the vertical velocity (defined from a fixed referential) will cross the levels (Eulerian part) and which part will be used to move them (Lagrangian part).
+The coordinate is also sometime referenced as an adaptive coordinate \citep{Hofmeister_al_OM09}, since the coordinate system is adapted in the course of the simulation. Its most often used implementation is via an ALE algorithm, in which a pure lagrangian step is followed by regridding and remapping steps, the later step implicitly embedding the vertical advection \citep{Hirt_al_JCP74, Chassignet_al_JPO03, White_al_JCP09}. Here we follow the \citep{Kasahara_MWR74} strategy : a regridding step (an update of the vertical coordinate) followed by an eulerian step with an explicit computation of vertical advection relative to the moving s-surfaces.
+with the restriction that the above equation gives a single-valued monotonic relationship between $s$ and $k$,
+when $i$, $j$ and $t$ are held fixed.
+\autoref{eq:PE_s} is a transformation from the $(i,j,k,t)$ coordinate system with independent variables into
+the $(i,j,s,t)$ generalised coordinate system with $s$ depending on the other three variables through
+\autoref{eq:PE_s}.
+This so-called \textit{generalised vertical coordinate} \citep{Kasahara_MWR74} is in fact
+an Arbitrary Lagrangian--Eulerian (ALE) coordinate.
+Indeed, choosing an expression for $s$ is an arbitrary choice that determines
+which part of the vertical velocity (defined from a fixed referential) will cross the levels (Eulerian part) and
+which part will be used to move them (Lagrangian part).
+The coordinate is also sometime referenced as an adaptive coordinate \citep{Hofmeister_al_OM09},
+since the coordinate system is adapted in the course of the simulation.
+Its most often used implementation is via an ALE algorithm,
+in which a pure lagrangian step is followed by regridding and remapping steps,
+the later step implicitly embedding the vertical advection
+\citep{Hirt_al_JCP74, Chassignet_al_JPO03, White_al_JCP09}.
+Here we follow the \citep{Kasahara_MWR74} strategy:
+a regridding step (an update of the vertical coordinate) followed by an eulerian step with
+an explicit computation of vertical advection relative to the moving s-surfaces.
 %\gmcomment{
 …
 %A key point here is that the $s$-coordinate depends on $(i,j)$ ==> horizontal pressure gradient...
+the generalized vertical coordinates used in ocean modelling are not orthogonal,
+which contrasts with many other applications in mathematical physics.
+Hence, it is useful to keep in mind the following properties that may seem
+odd on initial encounter.
+The horizontal velocity in ocean models measures motions in the horizontal plane,
+perpendicular to the local gravitational field. That is, horizontal velocity is mathematically
+the same regardless the vertical coordinate, be it geopotential, isopycnal, pressure,
+or terrain following. The key motivation for maintaining the same horizontal velocity
+component is that the hydrostatic and geostrophic balances are dominant in the large-scale ocean.
+Use of an alternative quasi-horizontal velocity, for example one oriented parallel
+to the generalized surface, would lead to unacceptable numerical errors.
+Correspondingly, the vertical direction is anti-parallel to the gravitational force in all
+of the coordinate systems. We do not choose the alternative of a quasi-vertical
+direction oriented normal to the surface of a constant generalized vertical coordinate.
+It is the method used to measure transport across the generalized vertical coordinate
+surfaces which differs between the vertical coordinate choices. That is, computation
+of the dia-surface velocity component represents the fundamental distinction between
+the various coordinates. In some models, such as geopotential, pressure, and
+terrain following, this transport is typically diagnosed from volume or mass conservation.
+In other models, such as isopycnal layered models, this transport is prescribed based
+on assumptions about the physical processes producing a flux across the layer interfaces.
+In this section we first establish the PE in the generalised vertical $s$-coordinate,
+the generalized vertical coordinates used in ocean modelling are not orthogonal,
+which contrasts with many other applications in mathematical physics.
+Hence, it is useful to keep in mind the following properties that may seem odd on initial encounter.
+The horizontal velocity in ocean models measures motions in the horizontal plane,
+perpendicular to the local gravitational field.
+That is, horizontal velocity is mathematically the same regardless the vertical coordinate, be it geopotential,
+isopycnal, pressure, or terrain following.
+The key motivation for maintaining the same horizontal velocity component is that
+the hydrostatic and geostrophic balances are dominant in the large-scale ocean.
+Use of an alternative quasi-horizontal velocity, for example one oriented parallel to the generalized surface,
+would lead to unacceptable numerical errors.
+Correspondingly, the vertical direction is anti-parallel to the gravitational force in
+all of the coordinate systems.
+We do not choose the alternative of a quasi-vertical direction oriented normal to
+the surface of a constant generalized vertical coordinate.
+It is the method used to measure transport across the generalized vertical coordinate surfaces which differs between
+the vertical coordinate choices.
+That is, computation of the dia-surface velocity component represents the fundamental distinction between
+the various coordinates.
+In some models, such as geopotential, pressure, and terrain following, this transport is typically diagnosed from
+volume or mass conservation.
+In other models, such as isopycnal layered models, this transport is prescribed based on assumptions about
+the physical processes producing a flux across the layer interfaces.
+In this section we first establish the PE in the generalised vertical $s$-coordinate,
 then we discuss the particular cases available in \NEMO, namely $z$, $z$*, $s$, and $\tilde z$.
 %}
 …
 \subsection{\textit{S-}coordinate formulation}
 Starting from the set of equations established in \autoref{sec:PE_zco} for the special case $k=z$
+and thus $e_3=1$, we introduce an arbitrary vertical coordinate $s=s(i,j,k,t)$, which includes
+$z$-, \textit{z*}- and $\sigma-$coordinates as special cases ($s=z$, $s=\textit{z*}$, and
+$s=\sigma=z/H$ or $=z/\left(H+\eta \right)$, resp.). A formal derivation of the transformed
+equations is given in \autoref{apdx:A}. Let us define the vertical scale factor by
+$e_3=\partial_s z$  ($e_3$ is now a function of $(i,j,k,t)$ ), and the slopes in the
 (\textbf{i},\textbf{j}) directions between $s-$ and $z-$surfaces by :
+Starting from the set of equations established in \autoref{sec:PE_zco} for the special case $k=z$ and thus $e_3=1$,
+we introduce an arbitrary vertical coordinate $s=s(i,j,k,t)$,
+which includes $z$-, \textit{z*}- and $\sigma-$coordinates as special cases
+($s=z$, $s=\textit{z*}$, and $s=\sigma=z/H$ or $=z/\left(H+\eta \right)$, resp.).
+A formal derivation of the transformed equations is given in \autoref{apdx:A}.
+Let us define the vertical scale factor by $e_3=\partial_s z$  ($e_3$ is now a function of $(i,j,k,t)$ ),
+and the slopes in the (\textbf{i},\textbf{j}) directions between $s-$ and $z-$surfaces by:
 \begin{equation} \label{eq:PE_sco_slope}
 \sigma _1 =\frac{1}{e_1 }\;\left. {\frac{\partial z}{\partial i}} \right|_s
 …
 \end{equation}
+The equations solved by the ocean model \autoref{eq:PE} in $s-$coordinate can be written as follows (see \autoref{sec:A_momentum}):
+The equations solved by the ocean model \autoref{eq:PE} in $s-$coordinate can be written as follows
+(see \autoref{sec:A_momentum}):
  \vspace{0.5cm}
 …
 \end{multline}
 where the relative vorticity, \textit{$\zeta $}, the surface pressure gradient, and the hydrostatic
+pressure have the same expressions as in $z$-coordinates although they do not represent
+exactly the same quantities. $\omega$ is provided by the continuity equation
 (see \autoref{apdx:A}):
+where the relative vorticity, \textit{$\zeta $}, the surface pressure gradient,
+and the hydrostatic pressure have the same expressions as in $z$-coordinates although
+they do not represent exactly the same quantities.
+$\omega$ is provided by the continuity equation (see \autoref{apdx:A}):
 \begin{equation} \label{eq:PE_sco_continuity}
 \frac{\partial e_3}{\partial t} + e_3 \; \chi + \frac{\partial \omega }{\partial s} = 0
 …
 \end{multline}
 The equation of state has the same expression as in $z$-coordinate, and similar expressions
 are used for mixing and forcing terms.
+The equation of state has the same expression as in $z$-coordinate,
+and similar expressions are used for mixing and forcing terms.
 \gmcomment{
 …
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!b]    \begin{center}
+\includegraphics[width=1.0\textwidth]{Fig_z_zstar}
+\caption{   \protect\label{fig:z_zstar}
+(a) $z$-coordinate in linear free-surface case ;
+(b) $z-$coordinate in non-linear free surface case ;
+(c) re-scaled height coordinate (become popular as the \textit{z*-}coordinate
+\citep{Adcroft_Campin_OM04} ).}
+\end{center}   \end{figure}
+\begin{figure}[!b]
+  \begin{center}
+    \includegraphics[width=1.0\textwidth]{Fig_z_zstar}
+    \caption{  \protect\label{fig:z_zstar}
+      (a) $z$-coordinate in linear free-surface case ;
+      (b) $z-$coordinate in non-linear free surface case ;
+      (c) re-scaled height coordinate (become popular as the \textit{z*-}coordinate
+      \citep{Adcroft_Campin_OM04} ).
+    }
+  \end{center}
+\end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+In that case, the free surface equation is nonlinear, and the variations of volume are fully
+taken into account. These coordinates systems is presented in a report \citep{Levier2007}
+available on the \NEMO web site.
+In that case, the free surface equation is nonlinear, and the variations of volume are fully taken into account.
+These coordinates systems is presented in a report \citep{Levier2007} available on the \NEMO web site.
 %\gmcomment{
+The \textit{z*} coordinate approach is an unapproximated, non-linear free surface implementation
+which allows one to deal with large amplitude free-surface
+variations relative to the vertical resolution \citep{Adcroft_Campin_OM04}. In
+the  \textit{z*} formulation, the variation of the column thickness due to sea-surface
+undulations is not concentrated in the surface level, as in the $z$-coordinate formulation,
+but is equally distributed over the full water column. Thus vertical
+levels naturally follow sea-surface variations, with a linear attenuation with
+depth, as illustrated by figure fig.1c . Note that with a flat bottom, such as in
+fig.1c, the bottom-following  $z$ coordinate and  \textit{z*} are equivalent.
+The definition and modified oceanic equations for the rescaled vertical coordinate
+ \textit{z*}, including the treatment of fresh-water flux at the surface, are
+detailed in Adcroft and Campin (2004). The major points are summarized
+here. The position ( \textit{z*}) and vertical discretization (\textit{z*}) are expressed as:
+The \textit{z*} coordinate approach is an unapproximated, non-linear free surface implementation which allows one to
+deal with large amplitude free-surface variations relative to the vertical resolution \citep{Adcroft_Campin_OM04}.
+In the \textit{z*} formulation,
+the variation of the column thickness due to sea-surface undulations is not concentrated in the surface level,
+as in the $z$-coordinate formulation, but is equally distributed over the full water column.
+Thus vertical levels naturally follow sea-surface variations, with a linear attenuation with depth,
+as illustrated by figure fig.1c.
+Note that with a flat bottom, such as in fig.1c, the bottom-following $z$ coordinate and \textit{z*} are equivalent.
+The definition and modified oceanic equations for the rescaled vertical coordinate  \textit{z*},
+including the treatment of fresh-water flux at the surface, are detailed in Adcroft and Campin (2004).
+The major points are summarized here.
+The position ( \textit{z*}) and vertical discretization (\textit{z*}) are expressed as:
 \begin{equation} \label{eq:z-star}
 H +  \textit{z*} = (H + z) / r \quad \text{and} \ \delta \textit{z*} = \delta z / r \quad \text{with} \ r = \frac{H+\eta} {H}
 \end{equation}
 Since the vertical displacement of the free surface is incorporated in the vertical
+coordinate  \textit{z*}, the upper and lower boundaries are at fixed  \textit{z*} position,
 $\textit{z*} = 0$ and  $\textit{z*} = -H$ respectively. Also the divergence of the flow field
 is no longer zero as shown by the continuity equation:
+Since the vertical displacement of the free surface is incorporated in the vertical coordinate \textit{z*},
+the upper and lower boundaries are at fixed  \textit{z*} position,
+$\textit{z*} = 0$ and  $\textit{z*} = -H$ respectively.
+Also the divergence of the flow field is no longer zero as shown by the continuity equation:
 \begin{equation*}
 \frac{\partial r}{\partial t} = \nabla_{\textit{z*}} \cdot \left( r \; \rm{\bf U}_h \right)
 …
 % from MOM4p1 documentation
+To overcome problems with vanishing surface and/or bottom cells, we consider the
+zstar coordinate
+To overcome problems with vanishing surface and/or bottom cells, we consider the zstar coordinate
 \begin{equation} \label{eq:PE_}
    z^\star = H \left( \frac{z-\eta}{H+\eta} \right)
 \end{equation}
+This coordinate is closely related to the "eta" coordinate used in many atmospheric
+models (see Black (1994) for a review of eta coordinate atmospheric models). It
+was originally used in ocean models by Stacey et al. (1995) for studies of tides
+next to shelves, and it has been recently promoted by Adcroft and Campin (2004)
+for global climate modelling.
+The surfaces of constant $z^\star$ are quasi-horizontal. Indeed, the $z^\star$ coordinate reduces to $z$ when $\eta$ is zero. In general, when noting the large differences between
+undulations of the bottom topography versus undulations in the surface height, it
+is clear that surfaces constant $z^\star$ are very similar to the depth surfaces. These properties greatly reduce difficulties of computing the horizontal pressure gradient relative to terrain following sigma models discussed in \autoref{subsec:PE_sco}.
+Additionally, since $z^\star$ when $\eta = 0$, no flow is spontaneously generated in an
+unforced ocean starting from rest, regardless the bottom topography. This behaviour is in contrast to the case with "s"-models, where pressure gradient errors in
+the presence of nontrivial topographic variations can generate nontrivial spontaneous flow from a resting state, depending on the sophistication of the pressure
+gradient solver. The quasi-horizontal nature of the coordinate surfaces also facilitates the implementation of neutral physics parameterizations in $z^\star$ models using
+the same techniques as in $z$-models (see Chapters 13-16 of \cite{Griffies_Bk04}) for a
+discussion of neutral physics in $z$-models, as well as \autoref{sec:LDF_slp}
+in this document for treatment in \NEMO).
+The range over which $z^\star$ varies is time independent $-H \leq z^\star \leq 0$. Hence, all
+cells remain nonvanishing, so long as the surface height maintains $\eta > ?H$. This
+is a minor constraint relative to that encountered on the surface height when using
+$s = z$ or $s = z - \eta$.
+Because $z^\star$ has a time independent range, all grid cells have static increments
+ds, and the sum of the ver tical increments yields the time independent ocean
+depth %·k ds = H.
+The $z^\star$ coordinate is therefore invisible to undulations of the
+free surface, since it moves along with the free surface. This proper ty means that
+no spurious ver tical transpor t is induced across surfaces of constant $z^\star$ by the
+motion of external gravity waves. Such spurious transpor t can be a problem in
+z-models, especially those with tidal forcing. Quite generally, the time independent
+range for the $z^\star$ coordinate is a very convenient proper ty that allows for a nearly
+arbitrary ver tical resolution even in the presence of large amplitude fluctuations of
+the surface height, again so long as $\eta > -H$.
+This coordinate is closely related to the "eta" coordinate used in many atmospheric models
+(see Black (1994) for a review of eta coordinate atmospheric models).
+It was originally used in ocean models by Stacey et al. (1995) for studies of tides next to shelves,
+and it has been recently promoted by Adcroft and Campin (2004) for global climate modelling.
+The surfaces of constant $z^\star$ are quasi-horizontal.
+Indeed, the $z^\star$ coordinate reduces to $z$ when $\eta$ is zero.
+In general, when noting the large differences between
+undulations of the bottom topography versus undulations in the surface height,
+it is clear that surfaces constant $z^\star$ are very similar to the depth surfaces.
+These properties greatly reduce difficulties of computing the horizontal pressure gradient relative to
+terrain following sigma models discussed in \autoref{subsec:PE_sco}.
+Additionally, since $z^\star$ when $\eta = 0$,
+no flow is spontaneously generated in an unforced ocean starting from rest, regardless the bottom topography.
+This behaviour is in contrast to the case with "s"-models, where pressure gradient errors in the presence of
+nontrivial topographic variations can generate nontrivial spontaneous flow from a resting state,
+depending on the sophistication of the pressure gradient solver.
+The quasi-horizontal nature of the coordinate surfaces also facilitates the implementation of
+neutral physics parameterizations in $z^\star$ models using the same techniques as in $z$-models
+(see Chapters 13-16 of \cite{Griffies_Bk04}) for a discussion of neutral physics in $z$-models,
+as well as \autoref{sec:LDF_slp} in this document for treatment in \NEMO).
+The range over which $z^\star$ varies is time independent $-H \leq z^\star \leq 0$.
+Hence, all cells remain nonvanishing, so long as the surface height maintains $\eta > ?H$.
+This is a minor constraint relative to that encountered on the surface height when using $s = z$ or $s = z - \eta$.
+Because $z^\star$ has a time independent range, all grid cells have static increments ds,
+and the sum of the ver tical increments yields the time independent ocean depth. %·k ds = H.
+The $z^\star$ coordinate is therefore invisible to undulations of the free surface,
+since it moves along with the free surface.
+This proper ty means that no spurious vertical transport is induced across surfaces of constant $z^\star$ by
+the motion of external gravity waves.
+Such spurious transpor t can be a problem in z-models, especially those with tidal forcing.
+Quite generally, the time independent range for the $z^\star$ coordinate is a very convenient property that
+allows for a nearly arbitrary ver tical resolution even in the presence of large amplitude fluctuations of
+the surface height, again so long as $\eta > -H$.
 %end MOM doc %%%
 …
 \subsubsection{Introduction}
+Several important aspects of the ocean circulation are influenced by bottom topography.
+Of course, the most important is that bottom topography determines deep ocean sub-basins,
+barriers, sills and channels that strongly constrain the path of water masses, but more subtle
+effects exist. For example, the topographic $\beta$-effect is usually larger than the planetary
+one along continental slopes. Topographic Rossby waves can be excited and can interact
+with the mean current. In the $z-$coordinate system presented in the previous section
+(\autoref{sec:PE_zco}), $z-$surfaces are geopotential surfaces. The bottom topography is
+discretised by steps. This often leads to a misrepresentation of a gradually sloping bottom
+and to large localized depth gradients associated with large localized vertical velocities.
+The response to such a velocity field often leads to numerical dispersion effects.
+One solution to strongly reduce this error is to use a partial step representation of bottom
+topography instead of a full step one \cite{Pacanowski_Gnanadesikan_MWR98}.
+Another solution is to introduce a terrain-following coordinate system (hereafter $s-$coordinate)
+The $s$-coordinate avoids the discretisation error in the depth field since the layers of
+computation are gradually adjusted with depth to the ocean bottom. Relatively small
+topographic features as well as  gentle, large-scale slopes of the sea floor in the deep
+ocean, which would be ignored in typical $z$-model applications with the largest grid
+spacing at greatest depths, can easily be represented (with relatively low vertical resolution).
+A terrain-following model (hereafter $s-$model) also facilitates the modelling of the
+boundary layer flows over a large depth range, which in the framework of the $z$-model
+would require high vertical resolution over the whole depth range. Moreover, with a
+$s$-coordinate it is possible, at least in principle, to have the bottom and the sea surface
+as the only boundaries of the domain (nomore lateral boundary condition to specify).
+Nevertheless, a $s$-coordinate also has its drawbacks. Perfectly adapted to a
+homogeneous ocean, it has strong limitations as soon as stratification is introduced.
+The main two problems come from the truncation error in the horizontal pressure
+gradient and a possibly increased diapycnal diffusion. The horizontal pressure force
+in $s$-coordinate consists of two terms (see \autoref{apdx:A}),
+Several important aspects of the ocean circulation are influenced by bottom topography.
+Of course, the most important is that bottom topography determines deep ocean sub-basins, barriers, sills and
+channels that strongly constrain the path of water masses, but more subtle effects exist.
+For example, the topographic $\beta$-effect is usually larger than the planetary one along continental slopes.
+Topographic Rossby waves can be excited and can interact with the mean current.
+In the $z-$coordinate system presented in the previous section (\autoref{sec:PE_zco}),
+$z-$surfaces are geopotential surfaces.
+The bottom topography is discretised by steps.
+This often leads to a misrepresentation of a gradually sloping bottom and to
+large localized depth gradients associated with large localized vertical velocities.
+The response to such a velocity field often leads to numerical dispersion effects.
+One solution to strongly reduce this error is to use a partial step representation of bottom topography instead of
+a full step one \cite{Pacanowski_Gnanadesikan_MWR98}.
+Another solution is to introduce a terrain-following coordinate system (hereafter $s-$coordinate).
+The $s$-coordinate avoids the discretisation error in the depth field since the layers of
+computation are gradually adjusted with depth to the ocean bottom.
+Relatively small topographic features as well as  gentle, large-scale slopes of the sea floor in the deep ocean,
+which would be ignored in typical $z$-model applications with the largest grid spacing at greatest depths,
+can easily be represented (with relatively low vertical resolution).
+A terrain-following model (hereafter $s-$model) also facilitates the modelling of the boundary layer flows over
+a large depth range, which in the framework of the $z$-model would require high vertical resolution over
+the whole depth range.
+Moreover, with a $s$-coordinate it is possible, at least in principle, to have the bottom and the sea surface as
+the only boundaries of the domain (no more lateral boundary condition to specify).
+Nevertheless, a $s$-coordinate also has its drawbacks. Perfectly adapted to a homogeneous ocean,
+it has strong limitations as soon as stratification is introduced.
+The main two problems come from the truncation error in the horizontal pressure gradient and
+a possibly increased diapycnal diffusion.
+The horizontal pressure force in $s$-coordinate consists of two terms (see \autoref{apdx:A}),
 \begin{equation} \label{eq:PE_p_sco}
 …
 \end{equation}
+The second term in \autoref{eq:PE_p_sco} depends on the tilt of the coordinate surface
+and introduces a truncation error that is not present in a $z$-model. In the special case
+of a $\sigma-$coordinate (i.e. a depth-normalised coordinate system $\sigma = z/H$),
+\citet{Haney1991} and \citet{Beckmann1993} have given estimates of the magnitude
+of this truncation error. It depends on topographic slope, stratification, horizontal and
+vertical resolution, the equation of state, and the finite difference scheme. This error
+limits the possible topographic slopes that a model can handle at a given horizontal
+and vertical resolution. This is a severe restriction for large-scale applications using
+realistic bottom topography. The large-scale slopes require high horizontal resolution,
+and the computational cost becomes prohibitive. This problem can be at least partially
+overcome by mixing $s$-coordinate and step-like representation of bottom topography \citep{Gerdes1993a,Gerdes1993b,Madec_al_JPO96}. However, the definition of the model
+domain vertical coordinate becomes then a non-trivial thing for a realistic bottom
+topography: a envelope topography is defined in $s$-coordinate on which a full or
+partial step bottom topography is then applied in order to adjust the model depth to
+the observed one (see \autoref{sec:DOM_zgr}.
+For numerical reasons a minimum of diffusion is required along the coordinate surfaces
+of any finite difference model. It causes spurious diapycnal mixing when coordinate
+surfaces do not coincide with isoneutral surfaces. This is the case for a $z$-model as
+well as for a $s$-model. However, density varies more strongly on $s-$surfaces than
+on horizontal surfaces in regions of large topographic slopes, implying larger diapycnal
+diffusion in a $s$-model than in a $z$-model. Whereas such a diapycnal diffusion in a
+$z$-model tends to weaken horizontal density (pressure) gradients and thus the horizontal
+circulation, it usually reinforces these gradients in a $s$-model, creating spurious circulation.
+For example, imagine an isolated bump of topography in an ocean at rest with a horizontally
+uniform stratification. Spurious diffusion along $s$-surfaces will induce a bump of isoneutral
+surfaces over the topography, and thus will generate there a baroclinic eddy. In contrast,
+the ocean will stay at rest in a $z$-model. As for the truncation error, the problem can be reduced by introducing the terrain-following coordinate below the strongly stratified portion of the water column
+($i.e.$ the main thermocline) \citep{Madec_al_JPO96}. An alternate solution consists of rotating
+the lateral diffusive tensor to geopotential or to isoneutral surfaces (see \autoref{subsec:PE_ldf}.
+Unfortunately, the slope of isoneutral surfaces relative to the $s$-surfaces can very large,
+The second term in \autoref{eq:PE_p_sco} depends on the tilt of the coordinate surface and
+introduces a truncation error that is not present in a $z$-model.
+In the special case of a $\sigma-$coordinate (i.e. a depth-normalised coordinate system $\sigma = z/H$),
+\citet{Haney1991} and \citet{Beckmann1993} have given estimates of the magnitude of this truncation error.
+It depends on topographic slope, stratification, horizontal and vertical resolution, the equation of state,
+and the finite difference scheme.
+This error limits the possible topographic slopes that a model can handle at
+a given horizontal and vertical resolution.
+This is a severe restriction for large-scale applications using realistic bottom topography.
+The large-scale slopes require high horizontal resolution, and the computational cost becomes prohibitive.
+This problem can be at least partially overcome by mixing $s$-coordinate and
+step-like representation of bottom topography \citep{Gerdes1993a,Gerdes1993b,Madec_al_JPO96}.
+However, the definition of the model domain vertical coordinate becomes then a non-trivial thing for
+a realistic bottom topography:
+a envelope topography is defined in $s$-coordinate on which a full or
+partial step bottom topography is then applied in order to adjust the model depth to the observed one
+(see \autoref{sec:DOM_zgr}.
+For numerical reasons a minimum of diffusion is required along the coordinate surfaces of any finite difference model.
+It causes spurious diapycnal mixing when coordinate surfaces do not coincide with isoneutral surfaces.
+This is the case for a $z$-model as well as for a $s$-model.
+However, density varies more strongly on $s-$surfaces than on horizontal surfaces in regions of
+large topographic slopes, implying larger diapycnal diffusion in a $s$-model than in a $z$-model.
+Whereas such a diapycnal diffusion in a $z$-model tends to weaken horizontal density (pressure) gradients and thus
+the horizontal circulation, it usually reinforces these gradients in a $s$-model, creating spurious circulation.
+For example, imagine an isolated bump of topography in an ocean at rest with a horizontally uniform stratification.
+Spurious diffusion along $s$-surfaces will induce a bump of isoneutral surfaces over the topography,
+and thus will generate there a baroclinic eddy.
+In contrast, the ocean will stay at rest in a $z$-model.
+As for the truncation error, the problem can be reduced by introducing the terrain-following coordinate below
+the strongly stratified portion of the water column ($i.e.$ the main thermocline) \citep{Madec_al_JPO96}.
+An alternate solution consists of rotating the lateral diffusive tensor to geopotential or to isoneutral surfaces
+(see \autoref{subsec:PE_ldf}).
+Unfortunately, the slope of isoneutral surfaces relative to the $s$-surfaces can very large,
 strongly exceeding the stability limit of such a operator when it is discretized (see \autoref{chap:LDF}).
 The $s-$coordinates introduced here \citep{Lott_al_OM90,Madec_al_JPO96} differ mainly in two
+aspects from similar models:  it allows  a representation of bottom topography with mixed
+full or partial step-like/terrain following topography ; It also offers a completely general
 transformation, $s=s(i,j,z)$ for the vertical coordinate.
+The $s-$coordinates introduced here \citep{Lott_al_OM90,Madec_al_JPO96} differ mainly in two aspects from
+similar models:
+it allows a representation of bottom topography with mixed full or partial step-like/terrain following topography;
+It also offers a completely general transformation, $s=s(i,j,z)$ for the vertical coordinate.
 …
 The $\tilde{z}$-coordinate has been developed by \citet{Leclair_Madec_OM11}.
+It is available in \NEMO since the version 3.4. Nevertheless, it is currently not robust enough
+to be used in all possible configurations. Its use is therefore not recommended.
+It is available in \NEMO since the version 3.4.
+Nevertheless, it is currently not robust enough to be used in all possible configurations.
+Its use is therefore not recommended.
 …
 \label{sec:PE_zdf_ldf}
+The primitive equations describe the behaviour of a geophysical fluid at
+space and time scales larger than a few kilometres in the horizontal, a few
+meters in the vertical and a few minutes. They are usually solved at larger
+scales: the specified grid spacing and time step of the numerical model. The
+effects of smaller scale motions (coming from the advective terms in the
+Navier-Stokes equations) must be represented entirely in terms of
+large-scale patterns to close the equations. These effects appear in the
+equations as the divergence of turbulent fluxes ($i.e.$ fluxes associated with
+the mean correlation of small scale perturbations). Assuming a turbulent
+closure hypothesis is equivalent to choose a formulation for these fluxes.
+It is usually called the subgrid scale physics. It must be emphasized that
+this is the weakest part of the primitive equations, but also one of the
+most important for long-term simulations as small scale processes \textit{in fine}
+balance the surface input of kinetic energy and heat.
+The control exerted by gravity on the flow induces a strong anisotropy
+between the lateral and vertical motions. Therefore subgrid-scale physics
+\textbf{D}$^{\vect{U}}$, $D^{S}$ and $D^{T}$  in \autoref{eq:PE_dyn},
+\autoref{eq:PE_tra_T} and \autoref{eq:PE_tra_S} are divided into a lateral part
+\textbf{D}$^{l \vect{U}}$, $D^{lS}$ and $D^{lT}$ and a vertical part
+\textbf{D}$^{vU}$, $D^{vS}$ and $D^{vT}$. The formulation of these terms
+and their underlying physics are briefly discussed in the next two subsections.
+The primitive equations describe the behaviour of a geophysical fluid at space and time scales larger than
+a few kilometres in the horizontal, a few meters in the vertical and a few minutes.
+They are usually solved at larger scales: the specified grid spacing and time step of the numerical model.
+The effects of smaller scale motions (coming from the advective terms in the Navier-Stokes equations) must be represented entirely in terms of large-scale patterns to close the equations.
+These effects appear in the equations as the divergence of turbulent fluxes
+($i.e.$ fluxes associated with the mean correlation of small scale perturbations).
+Assuming a turbulent closure hypothesis is equivalent to choose a formulation for these fluxes.
+It is usually called the subgrid scale physics.
+It must be emphasized that this is the weakest part of the primitive equations,
+but also one of the most important for long-term simulations as
+small scale processes \textit{in fine} balance the surface input of kinetic energy and heat.
+The control exerted by gravity on the flow induces a strong anisotropy between the lateral and vertical motions.
+Therefore subgrid-scale physics \textbf{D}$^{\vect{U}}$, $D^{S}$ and $D^{T}$  in
+\autoref{eq:PE_dyn}, \autoref{eq:PE_tra_T} and \autoref{eq:PE_tra_S} are divided into
+a lateral part  \textbf{D}$^{l \vect{U}}$, $D^{lS}$ and $D^{lT}$ and
+a vertical part  \textbf{D}$^{vU}$, $D^{vS}$ and $D^{vT}$.
+The formulation of these terms and their underlying physics are briefly discussed in the next two subsections.
 % -------------------------------------------------------------------------------------------------------------
 …
 \label{subsec:PE_zdf}
+The model resolution is always larger than the scale at which the major
+sources of vertical turbulence occur (shear instability, internal wave
+breaking...). Turbulent motions are thus never explicitly solved, even
+partially, but always parameterized. The vertical turbulent fluxes are
+assumed to depend linearly on the gradients of large-scale quantities (for
+example, the turbulent heat flux is given by $\overline{T'w'}=-A^{vT} \partial_z \overline T$,
+where $A^{vT}$ is an eddy coefficient). This formulation is
+analogous to that of molecular diffusion and dissipation. This is quite
+clearly a necessary compromise: considering only the molecular viscosity
+acting on large scale severely underestimates the role of turbulent
+diffusion and dissipation, while an accurate consideration of the details of
+turbulent motions is simply impractical. The resulting vertical momentum and
+tracer diffusive operators are of second order:
+The model resolution is always larger than the scale at which the major sources of vertical turbulence occur
+(shear instability, internal wave breaking...).
+Turbulent motions are thus never explicitly solved, even partially, but always parameterized.
+The vertical turbulent fluxes are assumed to depend linearly on the gradients of large-scale quantities
+(for example, the turbulent heat flux is given by $\overline{T'w'}=-A^{vT} \partial_z \overline T$,
+where $A^{vT}$ is an eddy coefficient).
+This formulation is analogous to that of molecular diffusion and dissipation.
+This is quite clearly a necessary compromise: considering only the molecular viscosity acting on
+large scale severely underestimates the role of turbulent diffusion and dissipation,
+while an accurate consideration of the details of turbulent motions is simply impractical.
+The resulting vertical momentum and tracer diffusive operators are of second order:
 \begin{equation} \label{eq:PE_zdf}
    \begin{split}
 …
    \end{split}
 \end{equation}
+where $A^{vm}$ and $A^{vT}$ are the vertical eddy viscosity and diffusivity coefficients,
+respectively. At the sea surface and at the bottom, turbulent fluxes of momentum, heat
+and salt must be specified (see \autoref{chap:SBC} and \autoref{chap:ZDF} and \autoref{sec:TRA_bbl}).
+All the vertical physics is embedded in the specification of the eddy coefficients.
+They can be assumed to be either constant, or function of the local fluid properties
+($e.g.$ Richardson number, Brunt-Vais\"{a}l\"{a} frequency...), or computed from a
+turbulent closure model. The choices available in \NEMO are discussed in \autoref{chap:ZDF}).
+where $A^{vm}$ and $A^{vT}$ are the vertical eddy viscosity and diffusivity coefficients, respectively.
+At the sea surface and at the bottom, turbulent fluxes of momentum, heat and salt must be specified
+(see \autoref{chap:SBC} and \autoref{chap:ZDF} and \autoref{sec:TRA_bbl}).
+All the vertical physics is embedded in the specification of the eddy coefficients.
+They can be assumed to be either constant, or function of the local fluid properties
+($e.g.$ Richardson number, Brunt-Vais\"{a}l\"{a} frequency...),
+or computed from a turbulent closure model.
+The choices available in \NEMO are discussed in \autoref{chap:ZDF}).
 % -------------------------------------------------------------------------------------------------------------
 …
 \label{subsec:PE_ldf}
+Lateral turbulence can be roughly divided into a mesoscale turbulence
+associated with eddies (which can be solved explicitly if the resolution is
+sufficient since their underlying physics are included in the primitive
+equations), and a sub mesoscale turbulence which is never explicitly solved
+even partially, but always parameterized. The formulation of lateral eddy
+fluxes depends on whether the mesoscale is below or above the grid-spacing
+Lateral turbulence can be roughly divided into a mesoscale turbulence associated with eddies
+(which can be solved explicitly if the resolution is sufficient since
+their underlying physics are included in the primitive equations),
+and a sub mesoscale turbulence which is never explicitly solved even partially, but always parameterized.
+The formulation of lateral eddy fluxes depends on whether the mesoscale is below or above the grid-spacing
 ($i.e.$ the model is eddy-resolving or not).
+In non-eddy-resolving configurations, the closure is similar to that used
+for the vertical physics. The lateral turbulent fluxes are assumed to depend
+linearly on the lateral gradients of large-scale quantities. The resulting
+lateral diffusive and dissipative operators are of second order.
+Observations show that lateral mixing induced by mesoscale turbulence tends
+to be along isopycnal surfaces (or more precisely neutral surfaces \cite{McDougall1987})
+rather than across them.
+As the slope of neutral surfaces is small in the ocean, a common
+approximation is to assume that the `lateral' direction is the horizontal,
+$i.e.$ the lateral mixing is performed along geopotential surfaces. This leads
+to a geopotential second order operator for lateral subgrid scale physics.
+This assumption can be relaxed: the eddy-induced turbulent fluxes can be
+better approached by assuming that they depend linearly on the gradients of
+large-scale quantities computed along neutral surfaces. In such a case,
+the diffusive operator is an isoneutral second order operator and it has
+components in the three space directions. However, both horizontal and
+isoneutral operators have no effect on mean ($i.e.$ large scale) potential
+energy whereas potential energy is a main source of turbulence (through
+baroclinic instabilities). \citet{Gent1990} have proposed a
+parameterisation of mesoscale eddy-induced turbulence which associates an
+eddy-induced velocity to the isoneutral diffusion. Its mean effect is to
+reduce the mean potential energy of the ocean. This leads to a formulation
+of lateral subgrid-scale physics made up of an isoneutral second order
+operator and an eddy induced advective part. In all these lateral diffusive
+formulations, the specification of the lateral eddy coefficients remains the
+problematic point as there is no really satisfactory formulation of these
+coefficients as a function of large-scale features.
+In eddy-resolving configurations, a second order operator can be used, but
+usually the more scale selective biharmonic operator is preferred as the
+grid-spacing is usually not small enough compared to the scale of the
+eddies. The role devoted to the subgrid-scale physics is to dissipate the
+energy that cascades toward the grid scale and thus to ensure the stability of
+the model while not interfering with the resolved mesoscale activity. Another approach
+is becoming more and more popular: instead of specifying explicitly a sub-grid scale
+term in the momentum and tracer time evolution equations, one uses a advective
+scheme which is diffusive enough to maintain the model stability. It must be emphasised
+that then, all the sub-grid scale physics is included in the formulation of the
+advection scheme.
+All these parameterisations of subgrid scale physics have advantages and
+drawbacks. There are not all available in \NEMO. For active tracers (temperature and
+salinity) the main ones are: Laplacian and bilaplacian operators acting along
+geopotential or iso-neutral surfaces, \citet{Gent1990} parameterisation,
+and various slightly diffusive advection schemes.
+For momentum, the main ones are: Laplacian and bilaplacian operators acting along
+geopotential surfaces, and UBS advection schemes when flux form is chosen for the momentum advection.
+In non-eddy-resolving configurations, the closure is similar to that used for the vertical physics.
+The lateral turbulent fluxes are assumed to depend linearly on the lateral gradients of large-scale quantities.
+The resulting lateral diffusive and dissipative operators are of second order.
+Observations show that lateral mixing induced by mesoscale turbulence tends to be along isopycnal surfaces
+(or more precisely neutral surfaces \cite{McDougall1987}) rather than across them.
+As the slope of neutral surfaces is small in the ocean, a common approximation is to assume that
+the `lateral' direction is the horizontal, $i.e.$ the lateral mixing is performed along geopotential surfaces.
+This leads to a geopotential second order operator for lateral subgrid scale physics.
+This assumption can be relaxed: the eddy-induced turbulent fluxes can be better approached by assuming that
+they depend linearly on the gradients of large-scale quantities computed along neutral surfaces.
+In such a case, the diffusive operator is an isoneutral second order operator and
+it has components in the three space directions.
+However,
+both horizontal and isoneutral operators have no effect on mean ($i.e.$ large scale) potential energy whereas
+potential energy is a main source of turbulence (through baroclinic instabilities).
+\citet{Gent1990} have proposed a parameterisation of mesoscale eddy-induced turbulence which
+associates an eddy-induced velocity to the isoneutral diffusion.
+Its mean effect is to reduce the mean potential energy of the ocean.
+This leads to a formulation of lateral subgrid-scale physics made up of an isoneutral second order operator and
+an eddy induced advective part.
+In all these lateral diffusive formulations,
+the specification of the lateral eddy coefficients remains the problematic point as
+there is no really satisfactory formulation of these coefficients as a function of large-scale features.
+In eddy-resolving configurations, a second order operator can be used,
+but usually the more scale selective biharmonic operator is preferred as
+the grid-spacing is usually not small enough compared to the scale of the eddies.
+The role devoted to the subgrid-scale physics is to dissipate the energy that
+cascades toward the grid scale and thus to ensure the stability of the model while
+not interfering with the resolved mesoscale activity.
+Another approach is becoming more and more popular:
+instead of specifying explicitly a sub-grid scale term in the momentum and tracer time evolution equations,
+one uses a advective scheme which is diffusive enough to maintain the model stability.
+It must be emphasised that then, all the sub-grid scale physics is included in the formulation of
+the advection scheme.
+All these parameterisations of subgrid scale physics have advantages and drawbacks.
+There are not all available in \NEMO. For active tracers (temperature and salinity) the main ones are:
+Laplacian and bilaplacian operators acting along geopotential or iso-neutral surfaces,
+\citet{Gent1990} parameterisation, and various slightly diffusive advection schemes.
+For momentum, the main ones are: Laplacian and bilaplacian operators acting along geopotential surfaces,
+and UBS advection schemes when flux form is chosen for the momentum advection.
 \subsubsection{Lateral laplacian tracer diffusive operator}
 …
 \end{array} }} \right)
 \end{equation}
+where $r_1 \;\mbox{and}\;r_2 $ are the slopes between the surface along
+which the diffusive operator acts and the model level ($e. g.$ $z$- or
+$s$-surfaces). Note that the formulation \autoref{eq:PE_iso_tensor} is exact for the
+rotation between geopotential and $s$-surfaces, while it is only an approximation
+for the rotation between isoneutral and $z$- or $s$-surfaces. Indeed, in the latter
+case, two assumptions are made to simplify \autoref{eq:PE_iso_tensor} \citep{Cox1987}.
+First, the horizontal contribution of the dianeutral mixing is neglected since the ratio
+between iso and dia-neutral diffusive coefficients is known to be several orders of
+magnitude smaller than unity. Second, the two isoneutral directions of diffusion are
+assumed to be independent since the slopes are generally less than $10^{-2}$ in the
+ocean (see \autoref{apdx:B}).
+For \textit{iso-level} diffusion, $r_1$ and $r_2 $ are zero. $\Re $ reduces to the identity
+in the horizontal direction, no rotation is applied.
+For \textit{geopotential} diffusion, $r_1$ and $r_2 $ are the slopes between the
+geopotential and computational surfaces: they are equal to $\sigma _1$ and $\sigma _2$,
+respectively (see \autoref{eq:PE_sco_slope}).
+For \textit{isoneutral} diffusion $r_1$ and $r_2$ are the slopes between the isoneutral
+and computational surfaces. Therefore, they are different quantities,
+but have similar expressions in $z$- and $s$-coordinates. In $z$-coordinates:
+where $r_1 \;\mbox{and}\;r_2 $ are the slopes between the surface along which the diffusive operator acts and
+the model level ($e. g.$ $z$- or $s$-surfaces).
+Note that the formulation \autoref{eq:PE_iso_tensor} is exact for
+the rotation between geopotential and $s$-surfaces,
+while it is only an approximation for the rotation between isoneutral and $z$- or $s$-surfaces.
+Indeed, in the latter case, two assumptions are made to simplify \autoref{eq:PE_iso_tensor} \citep{Cox1987}.
+First, the horizontal contribution of the dianeutral mixing is neglected since the ratio between iso and
+dia-neutral diffusive coefficients is known to be several orders of magnitude smaller than unity.
+Second, the two isoneutral directions of diffusion are assumed to be independent since
+the slopes are generally less than $10^{-2}$ in the ocean (see \autoref{apdx:B}).
+For \textit{iso-level} diffusion, $r_1$ and $r_2 $ are zero.
+$\Re $ reduces to the identity in the horizontal direction, no rotation is applied.
+For \textit{geopotential} diffusion,
+$r_1$ and $r_2 $ are the slopes between the geopotential and computational surfaces:
+they are equal to $\sigma _1$ and $\sigma _2$, respectively (see \autoref{eq:PE_sco_slope}).
+For \textit{isoneutral} diffusion $r_1$ and $r_2$ are the slopes between the isoneutral and computational surfaces.
+Therefore, they are different quantities, but have similar expressions in $z$- and $s$-coordinates.
+In $z$-coordinates:
 \begin{equation} \label{eq:PE_iso_slopes}
 r_1 =\frac{e_3 }{e_1 }  \left( \pd[\rho]{i} \right) \left( \pd[\rho]{k} \right)^{-1} \, \quad
 …
 \subsubsection{Eddy induced velocity}
+ When the \textit{eddy induced velocity} parametrisation (eiv) \citep{Gent1990} is used,
+When the \textit{eddy induced velocity} parametrisation (eiv) \citep{Gent1990} is used,
 an additional tracer advection is introduced in combination with the isoneutral diffusion of tracers:
 \begin{equation} \label{eq:PE_iso+eiv}
 …
            +\nabla \cdot \left( {{\vect{U}}^\ast \,T} \right)
 \end{equation}
 where ${\vect{U}}^\ast =\left( {u^\ast ,v^\ast ,w^\ast } \right)$ is a non-divergent,
+where ${\vect{U}}^\ast =\left( {u^\ast ,v^\ast ,w^\ast } \right)$ is a non-divergent,
 eddy-induced transport velocity. This velocity field is defined by:
 \begin{equation} \label{eq:PE_eiv}
 …
    \end{split}
 \end{equation}
 where $A^{eiv}$ is the eddy induced velocity coefficient (or equivalently the isoneutral
+thickness diffusivity coefficient), and $\tilde{r}_1$ and $\tilde{r}_2$ are the slopes
+between isoneutral and \emph{geopotential} surfaces. Their values are
 thus independent of the vertical coordinate, but their expression depends on the coordinate:
+where $A^{eiv}$ is the eddy induced velocity coefficient
+(or equivalently the isoneutral thickness diffusivity coefficient),
+and $\tilde{r}_1$ and $\tilde{r}_2$ are the slopes between isoneutral and \emph{geopotential} surfaces.
+Their values are thus independent of the vertical coordinate, but their expression depends on the coordinate:
 \begin{align} \label{eq:PE_slopes_eiv}
 \tilde{r}_n = \begin{cases}
 …
 \end{align}
+The normal component of the eddy induced velocity is zero at all the boundaries.
+This can be achieved in a model by tapering either the eddy coefficient or the slopes
+to zero in the vicinity of the boundaries. The latter strategy is used in \NEMO (cf. \autoref{chap:LDF}).
+The normal component of the eddy induced velocity is zero at all the boundaries.
+This can be achieved in a model by tapering either the eddy coefficient or
+the slopes to zero in the vicinity of the boundaries.
+The latter strategy is used in \NEMO (cf. \autoref{chap:LDF}).
 \subsubsection{Lateral bilaplacian tracer diffusive operator}
 …
 D^{lT}= - \Delta \left( \;\Delta T \right)
 \qquad \text{where} \;\; \Delta \bullet = \nabla \left( {\sqrt{B^{lT}\,}\;\Re \;\nabla \bullet} \right)
  \end{equation}
 It is the Laplacian operator given by \autoref{eq:PE_iso_tensor} applied twice with
+\end{equation}
+It is the Laplacian operator given by \autoref{eq:PE_iso_tensor} applied twice with
 the harmonic eddy diffusion coefficient set to the square root of the biharmonic one.
 …
 \subsubsection{Lateral Laplacian momentum diffusive operator}
 The Laplacian momentum diffusive operator along $z$- or $s$-surfaces is found by
+The Laplacian momentum diffusive operator along $z$- or $s$-surfaces is found by
 applying \autoref{eq:PE_lap_vector} to the horizontal velocity vector (see \autoref{apdx:B}):
 \begin{equation} \label{eq:PE_lapU}
 …
 \end{equation}
 Such a formulation ensures a complete separation between the vorticity and
+horizontal divergence fields (see \autoref{apdx:C}).
 Unfortunately, it is only available in \textit{iso-level} direction.
 When a rotation is required ($i.e.$ geopotential diffusion in $s-$coordinates
+or isoneutral diffusion in both $z$- and $s$-coordinates), the $u$ and $v-$fields
 are considered as independent scalar fields, so that the diffusive operator is given by:
+Such a formulation ensures a complete separation between the vorticity and horizontal divergence fields
+(see \autoref{apdx:C}).
+Unfortunately, it is only available in \textit{iso-level} direction.
+When a rotation is required
+($i.e.$ geopotential diffusion in $s-$coordinates or isoneutral diffusion in both $z$- and $s$-coordinates),
+the $u$ and $v-$fields are considered as independent scalar fields, so that the diffusive operator is given by:
 \begin{equation} \label{eq:PE_lapU_iso}
 \begin{split}
  D_u^{l{\rm {\bf U}}} &= \nabla .\left( {A^{lm} \;\Re \;\nabla u} \right) \\
  D_v^{l{\rm {\bf U}}} &= \nabla .\left( {A^{lm} \;\Re \;\nabla v} \right)
+ \end{split}
+ \end{equation}
+where $\Re$ is given by \autoref{eq:PE_iso_tensor}. It is the same expression as
+those used for diffusive operator on tracers. It must be emphasised that such a
+formulation is only exact in a Cartesian coordinate system, $i.e.$ on a $f-$ or
+$\beta-$plane, not on the sphere. It is also a very good approximation in vicinity
+of the Equator in a geographical coordinate system \citep{Lengaigne_al_JGR03}.
+\end{split}
+\end{equation}
+where $\Re$ is given by \autoref{eq:PE_iso_tensor}.
+It is the same expression as those used for diffusive operator on tracers.
+It must be emphasised that such a formulation is only exact in a Cartesian coordinate system,
+$i.e.$ on a $f-$ or $\beta-$plane, not on the sphere.
+It is also a very good approximation in vicinity of the Equator in
+a geographical coordinate system \citep{Lengaigne_al_JGR03}.
 \subsubsection{lateral bilaplacian momentum diffusive operator}
+As for tracers, the bilaplacian order momentum diffusive operator is a
+re-entering Laplacian operator with the harmonic eddy diffusion coefficient
+set to the square root of the biharmonic one. Nevertheless it is currently
+not available in the iso-neutral case.
+As for tracers, the bilaplacian order momentum diffusive operator is a re-entering Laplacian operator with
+the harmonic eddy diffusion coefficient set to the square root of the biharmonic one.
+Nevertheless it is currently not available in the iso-neutral case.
 \end{document}

NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_model_basics_zstar.tex

-                      r10146
+                      r10368
 \colorbox{yellow}{ to be updated }
+In that case, the free surface equation is nonlinear, and the variations of
+volume are fully taken into account. These coordinates systems is presented in
+a report \citep{Levier2007} available on the \NEMO web site.
+In that case, the free surface equation is nonlinear, and the variations of volume are fully taken into account.
+These coordinates systems is presented in a report \citep{Levier2007} available on the \NEMO web site.
 \colorbox{yellow}{  end of to be updated}
 …
 % from MOM4p1 documentation
+To overcome problems with vanishing surface and/or bottom cells, we consider the
+zstar coordinate
+To overcome problems with vanishing surface and/or bottom cells, we consider the zstar coordinate
 \begin{equation} \label{eq:PE_}
    z^\star = H \left( \frac{z-\eta}{H+\eta} \right)
 \end{equation}
+This coordinate is closely related to the "eta" coordinate used in many atmospheric
+models (see Black (1994) for a review of eta coordinate atmospheric models). It
+was originally used in ocean models by Stacey et al. (1995) for studies of tides
+next to shelves, and it has been recently promoted by Adcroft and Campin (2004)
+for global climate modelling.
+The surfaces of constant $z^\star$ are quasi-horizontal. Indeed, the $z^\star$ coordinate reduces to $z$ when $\eta$ is zero. In general, when noting the large differences between
+undulations of the bottom topography versus undulations in the surface height, it
+is clear that surfaces constant $z^\star$ are very similar to the depth surfaces. These properties greatly reduce difficulties of computing the horizontal pressure gradient relative to terrain following sigma models discussed in \autoref{subsec:PE_sco}.
+Additionally, since $z^\star$ when $\eta = 0$, no flow is spontaneously generated in an
+unforced ocean starting from rest, regardless the bottom topography. This behaviour is in contrast to the case with "s"-models, where pressure gradient errors in
+the presence of nontrivial topographic variations can generate nontrivial spontaneous flow from a resting state, depending on the sophistication of the pressure
+gradient solver. The quasi-horizontal nature of the coordinate surfaces also facilitates the implementation of neutral physics parameterizations in $z^\star$ models using
+the same techniques as in $z$-models (see Chapters 13-16 of Griffies (2004) for a
+discussion of neutral physics in $z$-models, as well as  \autoref{sec:LDF_slp}
+in this document for treatment in \NEMO).
+The range over which $z^\star$ varies is time independent $-H \leq z^\star \leq 0$. Hence, all
+cells remain nonvanishing, so long as the surface height maintains $\eta > ?H$. This
+is a minor constraint relative to that encountered on the surface height when using
+$s = z$ or $s = z - \eta$.
+Because $z^\star$ has a time independent range, all grid cells have static increments
+ds, and the sum of the ver tical increments yields the time independent ocean
+depth %�k ds = H.
+The $z^\star$ coordinate is therefore invisible to undulations of the
+free surface, since it moves along with the free surface. This proper ty means that
+no spurious ver tical transpor t is induced across surfaces of constant $z^\star$ by the
+motion of external gravity waves. Such spurious transpor t can be a problem in
+z-models, especially those with tidal forcing. Quite generally, the time independent
+range for the $z^\star$ coordinate is a very convenient proper ty that allows for a nearly
+arbitrary ver tical resolution even in the presence of large amplitude fluctuations of
+This coordinate is closely related to the "eta" coordinate used in many atmospheric models
+(see Black (1994) for a review of eta coordinate atmospheric models).
+It was originally used in ocean models by Stacey et al. (1995) for studies of tides next to shelves,
+and it has been recently promoted by Adcroft and Campin (2004) for global climate modelling.
+The surfaces of constant $z^\star$ are quasi-horizontal.
+Indeed, the $z^\star$ coordinate reduces to $z$ when $\eta$ is zero.
+In general, when noting the large differences between undulations of the bottom topography versus undulations in
+the surface height, it is clear that surfaces constant $z^\star$ are very similar to the depth surfaces.
+These properties greatly reduce difficulties of computing the horizontal pressure gradient relative to
+terrain following sigma models discussed in \autoref{subsec:PE_sco}.
+Additionally, since $z^\star$ when $\eta = 0$, no flow is spontaneously generated in
+an unforced ocean starting from rest, regardless the bottom topography.
+This behaviour is in contrast to the case with "s"-models, where pressure gradient errors in the presence of
+nontrivial topographic variations can generate nontrivial spontaneous flow from a resting state,
+depending on the sophistication of the pressure gradient solver.
+The quasi-horizontal nature of the coordinate surfaces also facilitates the implementation of
+neutral physics parameterizations in $z^\star$ models using the same techniques as in $z$-models
+(see Chapters 13-16 of Griffies (2004) for a discussion of neutral physics in $z$-models,
+as well as  \autoref{sec:LDF_slp} in this document for treatment in \NEMO).
+The range over which $z^\star$ varies is time independent $-H \leq z^\star \leq 0$.
+Hence, all cells remain nonvanishing, so long as the surface height maintains $\eta > ?H$.
+This is a minor constraint relative to that encountered on the surface height when using $s = z$ or $s = z - \eta$.
+Because $z^\star$ has a time independent range, all grid cells have static increments ds,
+and the sum of the ver tical increments yields the time independent ocean depth %�k ds = H.
+The $z^\star$ coordinate is therefore invisible to undulations of the free surface,
+since it moves along with the free surface.
+This proper ty means that no spurious ver tical transpor t is induced across surfaces of
+constant $z^\star$ by the motion of external gravity waves.
+Such spurious transpor t can be a problem in z-models, especially those with tidal forcing.
+Quite generally, the time independent range for the $z^\star$ coordinate is a very convenient property that
+allows for a nearly arbitrary vertical resolution even in the presence of large amplitude fluctuations of
 the surface height, again so long as $\eta > -H$.
 …
 \nlst{nam_dynspg}
 %------------------------------------------------------------------------------------------------------------
+Options are defined through the  \ngn{nam\_dynspg} namelist variables.
+The surface pressure gradient term is related to the representation of the free surface (\autoref{sec:PE_hor_pg}). The main distinction is between the fixed volume case (linear free surface or rigid lid) and the variable volume case (nonlinear free surface, \key{vvl} is active). In the linear free surface case (\autoref{subsec:PE_free_surface}) and rigid lid (\autoref{PE_rigid_lid}), the vertical scale factors $e_{3}$ are fixed in time, while in the nonlinear case (\autoref{subsec:PE_free_surface}) they are time-dependent. With both linear and nonlinear free surface, external gravity waves are allowed in the equations, which imposes a very small time step when an explicit time stepping is used. Two methods are proposed to allow a longer time step for the three-dimensional equations: the filtered free surface, which is a modification of the continuous equations (see \autoref{eq:PE_flt}), and the split-explicit free surface described below. The extra term introduced in the filtered method is calculated implicitly, so that the update of the next velocities is done in module \mdl{dynspg\_flt} and not in \mdl{dynnxt}.
+Options are defined through the \ngn{nam\_dynspg} namelist variables.
+The surface pressure gradient term is related to the representation of the free surface (\autoref{sec:PE_hor_pg}).
+The main distinction is between the fixed volume case (linear free surface or rigid lid) and
+the variable volume case (nonlinear free surface, \key{vvl} is active).
+In the linear free surface case (\autoref{subsec:PE_free_surface}) and rigid lid (\autoref{PE_rigid_lid}),
+the vertical scale factors $e_{3}$ are fixed in time,
+while in the nonlinear case (\autoref{subsec:PE_free_surface}) they are time-dependent.
+With both linear and nonlinear free surface, external gravity waves are allowed in the equations,
+which imposes a very small time step when an explicit time stepping is used.
+Two methods are proposed to allow a longer time step for the three-dimensional equations:
+the filtered free surface, which is a modification of the continuous equations (see \autoref{eq:PE_flt}),
+and the split-explicit free surface described below.
+The extra term introduced in the filtered method is calculated implicitly,
+so that the update of the next velocities is done in module \mdl{dynspg\_flt} and not in \mdl{dynnxt}.
 %-------------------------------------------------------------
 …
 \label{subsec:DYN_spg_exp}
+In the explicit free surface formulation, the model time step is chosen small enough to describe the external gravity waves (typically a few ten seconds). The sea surface height is given by :
+In the explicit free surface formulation, the model time step is chosen small enough to
+describe the external gravity waves (typically a few ten seconds).
+The sea surface height is given by:
 \begin{equation} \label{eq:dynspg_ssh}
 \frac{\partial \eta }{\partial t}\equiv \frac{\text{EMP}}{\rho _w }+\frac{1}{e_{1T}
 …
 \end{equation}
+where EMP is the surface freshwater budget (evaporation minus precipitation, and minus river runoffs (if the later are introduced as a surface freshwater flux, see \autoref{chap:SBC}) expressed in $Kg.m^{-2}.s^{-1}$, and $\rho _w =1,000\,Kg.m^{-3}$ is the volumic mass of pure water. The sea-surface height is evaluated using a leapfrog scheme in combination with an Asselin time filter, i.e. the velocity appearing in (\autoref{eq:dynspg_ssh}) is centred in time (\textit{now} velocity).
+The surface pressure gradient, also evaluated using a leap-frog scheme, is then simply given by :
+where EMP is the surface freshwater budget (evaporation minus precipitation, and minus river runoffs
+(if the later are introduced as a surface freshwater flux, see \autoref{chap:SBC}) expressed in $Kg.m^{-2}.s^{-1}$,
+and $\rho _w =1,000\,Kg.m^{-3}$ is the volumic mass of pure water.
+The sea-surface height is evaluated using a leapfrog scheme in combination with an Asselin time filter,
+i.e. the velocity appearing in (\autoref{eq:dynspg_ssh}) is centred in time (\textit{now} velocity).
+The surface pressure gradient, also evaluated using a leap-frog scheme, is then simply given by:
 \begin{equation} \label{eq:dynspg_exp}
 \left\{ \begin{aligned}
 …
 \end{equation}
+Consistent with the linearization, a $\left. \rho \right|_{k=1} / \rho _o$ factor is omitted in (\autoref{eq:dynspg_exp}).
+Consistent with the linearization, a $\left. \rho \right|_{k=1} / \rho _o$ factor is omitted in
+(\autoref{eq:dynspg_exp}).
 %-------------------------------------------------------------
 …
 \nlst{namdom}
 %--------------------------------------------------------------------------------------------------------------
+The split-explicit free surface formulation used in OPA follows the one proposed by \citet{Griffies2004}. The general idea is to solve the free surface equation with a small time step, while the three dimensional prognostic variables are solved with a longer time step that is a multiple of \np{rdtbt}
+in the  \ngn{namdom} namelist.
+(Figure III.3).
+The split-explicit free surface formulation used in OPA follows the one proposed by \citet{Griffies2004}.
+The general idea is to solve the free surface equation with a small time step,
+while the three dimensional prognostic variables are solved with a longer time step that
+is a multiple of \np{rdtbt} in the \ngn{namdom} namelist (Figure III.3).
 %>   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >
 …
 \includegraphics[width=0.90\textwidth]{Fig_DYN_dynspg_ts}
 \caption{    \protect\label{fig:DYN_dynspg_ts}
+Schematic of the split-explicit time stepping scheme for the barotropic and baroclinic modes,
+after \citet{Griffies2004}. Time increases to the right. Baroclinic time steps are denoted by
+$t-\Delta t$, $t, t+\Delta t$, and $t+2\Delta t$. The curved line represents a leap-frog time step,
+and the smaller barotropic time steps $N \Delta t=2\Delta t$ are denoted by the zig-zag line.
+The vertically integrated forcing \textbf{M}(t) computed at baroclinic time step t represents
+the interaction between the barotropic and baroclinic motions. While keeping the total depth,
+tracer, and freshwater forcing fields fixed, a leap-frog integration carries the surface height
+and vertically integrated velocity from t to $t+2 \Delta t$ using N barotropic time steps of length
+$\Delta t$. Time averaging the barotropic fields over the N+1 time steps (endpoints included)
+centers the vertically integrated velocity at the baroclinic timestep $t+\Delta t$.
+A baroclinic leap-frog time step carries the surface height to $t+\Delta t$ using the convergence
+of the time averaged vertically integrated velocity taken from baroclinic time step t. }
+  Schematic of the split-explicit time stepping scheme for the barotropic and baroclinic modes,
+  after \citet{Griffies2004}.
+  Time increases to the right.
+  Baroclinic time steps are denoted by $t-\Delta t$, $t, t+\Delta t$, and $t+2\Delta t$.
+  The curved line represents a leap-frog time step,
+  and the smaller barotropic time steps $N \Delta t=2\Delta t$ are denoted by the zig-zag line.
+  The vertically integrated forcing \textbf{M}(t) computed at
+  baroclinic time step t represents the interaction between the barotropic and baroclinic motions.
+  While keeping the total depth, tracer, and freshwater forcing fields fixed,
+  a leap-frog integration carries the surface height and vertically integrated velocity from
+  t to $t+2 \Delta t$ using N barotropic time steps of length $\Delta t$.
+  Time averaging the barotropic fields over the N+1 time steps (endpoints included)
+  centers the vertically integrated velocity at the baroclinic timestep $t+\Delta t$.
+  A baroclinic leap-frog time step carries the surface height to $t+\Delta t$ using the convergence of
+  the time averaged vertically integrated velocity taken from baroclinic time step t. }
 \end{center}
 \end{figure}
 %>   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >
+The split-explicit formulation has a damping effect on external gravity waves, which is weaker than the filtered free surface but still significant as shown by \citet{Levier2007} in the case of an analytical barotropic Kelvin wave.
+The split-explicit formulation has a damping effect on external gravity waves,
+which is weaker than the filtered free surface but still significant as shown by \citet{Levier2007} in
+the case of an analytical barotropic Kelvin wave.
 %from griffies book: .....   copy past !
 …
 \textbf{title: Time stepping the barotropic system }
 Assume knowledge of the full velocity and tracer fields at baroclinic time $\tau$. Hence,
+we can update the surface height and vertically integrated velocity with a leap-frog
+scheme using the small barotropic time step $\Delta t$. We have
+Assume knowledge of the full velocity and tracer fields at baroclinic time $\tau$.
+Hence, we can update the surface height and vertically integrated velocity with a leap-frog scheme using
+the small barotropic time step $\Delta t$.
+We have
 \begin{equation} \label{eq:DYN_spg_ts_eta}
 \eta^{(b)}(\tau,t_{n+1}) - \eta^{(b)}(\tau,t_{n+1}) (\tau,t_{n-1})
 …
+\
+In these equations, araised (b) denotes values of surface height and vertically integrated velocity updated with the barotropic time steps. The $\tau$ time label on $\eta^{(b)}$
+and $U^{(b)}$ denotes the baroclinic time at which the vertically integrated forcing $\textbf{M}(\tau)$ (note that this forcing includes the surface freshwater forcing), the tracer fields, the freshwater flux $\text{EMP}_w(\tau)$, and total depth of the ocean $H(\tau)$ are held for the duration of the barotropic time stepping over a single cycle. This is also the time
+that sets the barotropic time steps via
+In these equations, araised (b) denotes values of surface height and
+vertically integrated velocity updated with the barotropic time steps.
+The $\tau$ time label on $\eta^{(b)}$ and $U^{(b)}$ denotes the baroclinic time at which
+the vertically integrated forcing $\textbf{M}(\tau)$
+(note that this forcing includes the surface freshwater forcing), the tracer fields,
+the freshwater flux $\text{EMP}_w(\tau)$, and total depth of the ocean $H(\tau)$ are held for
+the duration of the barotropic time stepping over a single cycle.
+This is also the time that sets the barotropic time steps via
 \begin{equation} \label{eq:DYN_spg_ts_t}
 t_n=\tau+n\Delta t
 \end{equation}
+with $n$ an integer. The density scaled surface pressure is evaluated via
+with $n$ an integer.
+The density scaled surface pressure is evaluated via
 \begin{equation} \label{eq:DYN_spg_ts_ps}
 p_s^{(b)}(\tau,t_{n}) = \begin{cases}
 …
  \overline{\eta^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N \eta^{(b)}(\tau-\Delta t,t_{n})
 \end{equation}
+the time averaged surface height taken from the previous barotropic cycle. Likewise,
+the time averaged surface height taken from the previous barotropic cycle.
+Likewise,
 \begin{equation} \label{eq:DYN_spg_ts_u}
 \textbf{U}^{(b)}(\tau,t_{n=0}) = \overline{\textbf{U}^{(b)}(\tau)}   \\
 …
    = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau-\Delta t,t_{n})
 \end{equation}
+the time averaged vertically integrated transport. Notably, there is no Robert-Asselin time filter used in the barotropic portion of the integration.
+Upon reaching $t_{n=N} = \tau + 2\Delta \tau$ , the vertically integrated velocity is time averaged to produce the updated vertically integrated velocity at baroclinic time $\tau + \Delta \tau$
+the time averaged vertically integrated transport.
+Notably, there is no Robert-Asselin time filter used in the barotropic portion of the integration.
+Upon reaching $t_{n=N} = \tau + 2\Delta \tau$ , the vertically integrated velocity is time averaged to
+produce the updated vertically integrated velocity at baroclinic time $\tau + \Delta \tau$
 \begin{equation} \label{eq:DYN_spg_ts_u}
 \textbf{U}(\tau+\Delta t) = \overline{\textbf{U}^{(b)}(\tau+\Delta t)}
    = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau,t_{n})
 \end{equation}
 The surface height on the new baroclinic time step is then determined via a baroclinic leap-frog using the following form
+The surface height on the new baroclinic time step is then determined via
+a baroclinic leap-frog using the following form
 \begin{equation} \label{eq:DYN_spg_ts_ssh}
 \eta(\tau+\Delta) - \eta^{F}(\tau-\Delta) = 2\Delta t \ \left[ - \nabla \cdot \textbf{U}(\tau) + \text{EMP}_w \right]
 \end{equation}
+ The use of this "big-leap-frog" scheme for the surface height ensures compatibility between the mass/volume budgets and the tracer budgets. More discussion of this point is provided in Chapter 10 (see in particular Section 10.2).
+The use of this "big-leap-frog" scheme for the surface height ensures compatibility between
+the mass/volume budgets and the tracer budgets.
+More discussion of this point is provided in Chapter 10 (see in particular Section 10.2).
 In general, some form of time filter is needed to maintain integrity of the surface
+height field due to the leap-frog splitting mode in equation \autoref{eq:DYN_spg_ts_ssh}. We
+have tried various forms of such filtering, with the following method discussed in
+Griffies et al. (2001) chosen due to its stability and reasonably good maintenance of
 tracer conservation properties (see ??)
+In general, some form of time filter is needed to maintain integrity of the surface height field due to
+the leap-frog splitting mode in equation \autoref{eq:DYN_spg_ts_ssh}.
+We have tried various forms of such filtering,
+with the following method discussed in Griffies et al. (2001) chosen due to its stability and
+reasonably good maintenance of tracer conservation properties (see ??)
 \begin{equation} \label{eq:DYN_spg_ts_sshf}
 …
 \end{equation}
+which is useful since it isolates all the time filtering aspects into the term multiplied
+by $\alpha$. This isolation allows for an easy check that tracer conservation is exact when
+eliminating tracer and surface height time filtering (see ?? for more complete discussion). However, in the general case with a non-zero $\alpha$, the filter \autoref{eq:DYN_spg_ts_sshf} was found to be more conservative, and so is recommended.
+which is useful since it isolates all the time filtering aspects into the term multiplied by $\alpha$.
+This isolation allows for an easy check that tracer conservation is exact when eliminating tracer and
+surface height time filtering (see ?? for more complete discussion).
+However, in the general case with a non-zero $\alpha$, the filter \autoref{eq:DYN_spg_ts_sshf} was found to
+be more conservative, and so is recommended.
 …
 \label{subsec:DYN_spg_flt}
+The filtered formulation follows the \citet{Roullet2000} implementation. The extra term introduced in the equations (see {\S}I.2.2) is solved implicitly. The elliptic solvers available in the code are
+documented in \autoref{chap:MISC}. The amplitude of the extra term is given by the namelist variable \np{rnu}. The default value is 1, as recommended by \citet{Roullet2000}
+The filtered formulation follows the \citet{Roullet2000} implementation.
+The extra term introduced in the equations (see {\S}I.2.2) is solved implicitly.
+The elliptic solvers available in the code are documented in \autoref{chap:MISC}.
+The amplitude of the extra term is given by the namelist variable \np{rnu}.
+The default value is 1, as recommended by \citet{Roullet2000}
 \colorbox{red}{\np{rnu}\forcode{ = 1} to be suppressed from namelist !}
 …
 \label{subsec:DYN_spg_vvl}
+In the non-linear free surface formulation, the variations of volume are fully taken into account. This option is presented in a report \citep{Levier2007} available on the NEMO web site. The three time-stepping methods (explicit, split-explicit and filtered) are the same as in \autoref{DYN_spg_linear} except that the ocean depth is now time-dependent. In particular, this means that in filtered case, the matrix to be inverted has to be recomputed at each time-step.
+In the non-linear free surface formulation, the variations of volume are fully taken into account.
+This option is presented in a report \citep{Levier2007} available on the NEMO web site.
+The three time-stepping methods (explicit, split-explicit and filtered) are the same as in
+\autoref{DYN_spg_linear} except that the ocean depth is now time-dependent.
+In particular, this means that in filtered case, the matrix to be inverted has to be recomputed at each time-step.

NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_time_domain.tex

-                      r10146
+                      r10368
 \gmcomment{STEVEN :maybe a picture of the directory structure in the introduction
 which could be referred to here, would help  ==> to be added}
+\gmcomment{STEVEN :maybe a picture of the directory structure in the introduction which could be referred to here,
+  would help  ==> to be added}
 %%%%
 …
+Having defined the continuous equations in \autoref{chap:PE}, we need now to choose
+a time discretization, a key feature of an ocean model as it exerts a strong influence
+on the structure of the computer code ($i.e.$ on its flowchart).
+In the present chapter, we provide a general description of the \NEMO
+time stepping strategy and the consequences for the order in which the equations are
+solved.
+Having defined the continuous equations in \autoref{chap:PE}, we need now to choose a time discretization,
+a key feature of an ocean model as it exerts a strong influence on the structure of the computer code
+($i.e.$ on its flowchart).
+In the present chapter, we provide a general description of the \NEMO time stepping strategy and
+the consequences for the order in which the equations are solved.
 $\ $\newline    % force a new ligne
 …
 \label{sec:STP_environment}
+The time stepping used in \NEMO is a three level scheme that can be
+represented as follows:
+The time stepping used in \NEMO is a three level scheme that can be represented as follows:
 \begin{equation} \label{eq:STP}
    x^{t+\rdt} = x^{t-\rdt} + 2 \, \rdt \  \text{RHS}_x^{t-\rdt,\,t,\,t+\rdt}
 \end{equation}
 where $x$ stands for $u$, $v$, $T$ or $S$; RHS is the Right-Hand-Side of the
+corresponding time evolution equation; $\rdt$ is the time step; and the
+superscripts indicate the time at which a quantity is evaluated. Each term of the
+RHS is evaluated at a specific time step depending on the physics with which
+it is associated.
 The choice of the time step used for this evaluation is discussed below as
+well as the implications for starting or restarting a model
+simulation. Note that the time stepping calculation is generally performed in a single
+operation. With such a complex and nonlinear system of equations it would be
 dangerous to let a prognostic variable evolve in time for each term separately.
 The three level scheme requires three arrays for each prognostic variable.
 For each variable $x$ there is $x_b$ (before), $x_n$ (now) and $x_a$. The third array,
+although referred to as $x_a$ (after) in the code, is usually not the variable at
+the after time step; but rather it is used to store the time derivative (RHS in
+\autoref{eq:STP}) prior to time-stepping the equation. Generally, the time
+stepping is performed once at each time step in the \mdl{tranxt} and \mdl{dynnxt}
+modules, except when using implicit vertical diffusion or calculating sea surface height
 in which case time-splitting options are used.
+where $x$ stands for $u$, $v$, $T$ or $S$;
+RHS is the Right-Hand-Side of the corresponding time evolution equation;
+$\rdt$ is the time step;
+and the superscripts indicate the time at which a quantity is evaluated.
+Each term of the RHS is evaluated at a specific time step depending on the physics with which it is associated.
+The choice of the time step used for this evaluation is discussed below as well as
+the implications for starting or restarting a model simulation.
+Note that the time stepping calculation is generally performed in a single operation.
+With such a complex and nonlinear system of equations it would be dangerous to let a prognostic variable evolve in
+time for each term separately.
+The three level scheme requires three arrays for each prognostic variable.
+For each variable $x$ there is $x_b$ (before), $x_n$ (now) and $x_a$.
+The third array, although referred to as $x_a$ (after) in the code,
+is usually not the variable at the after time step;
+but rather it is used to store the time derivative (RHS in \autoref{eq:STP}) prior to time-stepping the equation.
+Generally, the time stepping is performed once at each time step in the \mdl{tranxt} and \mdl{dynnxt} modules,
+except when using implicit vertical diffusion or calculating sea surface height in which
+case time-splitting options are used.
 % -------------------------------------------------------------------------------------------------------------
 …
 \label{sec:STP_leap_frog}
 The time stepping used for processes other than diffusion is the well-known leapfrog
+scheme \citep{Mesinger_Arakawa_Bk76}.  This scheme is widely used for advection
+processes in low-viscosity fluids. It is a time centred scheme, $i.e.$
+the RHS in \autoref{eq:STP} is evaluated at time step $t$, the now time step.
 It may be used for momentum and tracer advection,
 pressure gradient, and Coriolis terms, but not for diffusion terms.
 It is an efficient method that achieves
+second-order accuracy with just one right hand side evaluation per time step.
 Moreover, it does not artificially damp linear oscillatory motion nor does it produce
+instability by amplifying the oscillations. These advantages are somewhat diminished
+by the large phase-speed error of the leapfrog scheme, and the unsuitability
+of leapfrog differencing for the representation of diffusion and Rayleigh
+damping processes. However, the scheme allows the coexistence of a numerical
+and a physical mode due to its leading third order dispersive error. In other words a
+divergence of odd and even time steps may occur. To prevent it, the leapfrog scheme
+is often used in association with a Robert-Asselin time filter (hereafter the LF-RA scheme).
+This filter, first designed by \citet{Robert_JMSJ66} and more comprehensively studied
+by \citet{Asselin_MWR72}, is a kind of laplacian diffusion in time that mixes odd and
 even time steps:
+The time stepping used for processes other than diffusion is the well-known leapfrog scheme
+\citep{Mesinger_Arakawa_Bk76}.
+This scheme is widely used for advection processes in low-viscosity fluids.
+It is a time centred scheme, $i.e.$ the RHS in \autoref{eq:STP} is evaluated at time step $t$, the now time step.
+It may be used for momentum and tracer advection, pressure gradient, and Coriolis terms,
+but not for diffusion terms.
+It is an efficient method that achieves second-order accuracy with
+just one right hand side evaluation per time step.
+Moreover, it does not artificially damp linear oscillatory motion nor does it produce instability by
+amplifying the oscillations.
+These advantages are somewhat diminished by the large phase-speed error of the leapfrog scheme,
+and the unsuitability of leapfrog differencing for the representation of diffusion and Rayleigh damping processes.
+However, the scheme allows the coexistence of a numerical and a physical mode due to
+its leading third order dispersive error.
+In other words a divergence of odd and even time steps may occur.
+To prevent it, the leapfrog scheme is often used in association with a Robert-Asselin time filter
+(hereafter the LF-RA scheme).
+This filter, first designed by \citet{Robert_JMSJ66} and more comprehensively studied by \citet{Asselin_MWR72},
+is a kind of laplacian diffusion in time that mixes odd and even time steps:
 \begin{equation} \label{eq:STP_asselin}
 x_F^t  = x^t + \gamma \, \left[ x_F^{t-\rdt} - 2 x^t + x^{t+\rdt} \right]
 \end{equation}
+where the subscript $F$ denotes filtered values and $\gamma$ is the Asselin
+coefficient. $\gamma$ is initialized as \np{rn\_atfp} (namelist parameter).
+Its default value is \np{rn\_atfp}\forcode{ = 10.e-3} (see \autoref{sec:STP_mLF}),
+causing only a weak dissipation of high frequency motions (\citep{Farge1987}).
+The addition of a time filter degrades the accuracy of the
+calculation from second to first order. However, the second order truncation
+error is proportional to $\gamma$, which is small compared to 1. Therefore,
+the LF-RA is a quasi second order accurate scheme. The LF-RA scheme
+is preferred to other time differencing schemes such as
+predictor corrector or trapezoidal schemes, because the user has an explicit
+and simple control of the magnitude of the time diffusion of the scheme.
+When used with the 2nd order space centred discretisation of the
+advection terms in the momentum and tracer equations, LF-RA avoids implicit
+numerical diffusion: diffusion is set explicitly by the user through the Robert-Asselin
+where the subscript $F$ denotes filtered values and $\gamma$ is the Asselin coefficient.
+$\gamma$ is initialized as \np{rn\_atfp} (namelist parameter).
+Its default value is \np{rn\_atfp}\forcode{ = 10.e-3} (see \autoref{sec:STP_mLF}),
+causing only a weak dissipation of high frequency motions (\citep{Farge1987}).
+The addition of a time filter degrades the accuracy of the calculation from second to first order.
+However, the second order truncation error is proportional to $\gamma$, which is small compared to 1.
+Therefore, the LF-RA is a quasi second order accurate scheme.
+The LF-RA scheme is preferred to other time differencing schemes such as predictor corrector or trapezoidal schemes,
+because the user has an explicit and simple control of the magnitude of the time diffusion of the scheme.
+When used with the 2nd order space centred discretisation of the advection terms in
+the momentum and tracer equations, LF-RA avoids implicit numerical diffusion:
+diffusion is set explicitly by the user through the Robert-Asselin
 filter parameter and the viscosity and diffusion coefficients.
 …
 \label{sec:STP_forward_imp}
+The leapfrog differencing scheme is unsuitable for the representation of
+diffusion and damping processes. For a tendancy $D_x$, representing a
+diffusion term or a restoring term to a tracer climatology
+(when present, see \autoref{sec:TRA_dmp}), a forward time differencing scheme
+ is used :
+The leapfrog differencing scheme is unsuitable for the representation of diffusion and damping processes.
+For a tendancy $D_x$, representing a diffusion term or a restoring term to a tracer climatology
+(when present, see \autoref{sec:TRA_dmp}), a forward time differencing scheme is used :
 \begin{equation} \label{eq:STP_euler}
    x^{t+\rdt} = x^{t-\rdt} + 2 \, \rdt \ {D_x}^{t-\rdt}
 \end{equation}
 This is diffusive in time and conditionally stable. The
 conditions for stability of second and fourth order horizontal diffusion schemes are \citep{Griffies_Bk04}:
+This is diffusive in time and conditionally stable.
+The conditions for stability of second and fourth order horizontal diffusion schemes are \citep{Griffies_Bk04}:
 \begin{equation} \label{eq:STP_euler_stability}
 A^h < \left\{
 …
 \right.
 \end{equation}
+where $e$ is the smallest grid size in the two horizontal directions and $A^h$ is
+the mixing coefficient. The linear constraint \autoref{eq:STP_euler_stability}
+is a necessary condition, but not sufficient. If it is not satisfied, even mildly,
+then the model soon becomes wildly unstable. The instability can be removed
+by either reducing the length of the time steps or reducing the mixing coefficient.
+For the vertical diffusion terms, a forward time differencing scheme can be
+used, but usually the numerical stability condition imposes a strong
+constraint on the time step. Two solutions are available in \NEMO to overcome
+the stability constraint: $(a)$ a forward time differencing scheme using a
+time splitting technique (\np{ln\_zdfexp}\forcode{ = .true.}) or $(b)$ a backward (or implicit)
+time differencing scheme (\np{ln\_zdfexp}\forcode{ = .false.}). In $(a)$, the master
+time step $\Delta $t is cut into $N$ fractional time steps so that the
+stability criterion is reduced by a factor of $N$. The computation is performed as
+follows:
+where $e$ is the smallest grid size in the two horizontal directions and $A^h$ is the mixing coefficient.
+The linear constraint \autoref{eq:STP_euler_stability} is a necessary condition, but not sufficient.
+If it is not satisfied, even mildly, then the model soon becomes wildly unstable.
+The instability can be removed by either reducing the length of the time steps or reducing the mixing coefficient.
+For the vertical diffusion terms, a forward time differencing scheme can be used,
+but usually the numerical stability condition imposes a strong constraint on the time step.
+Two solutions are available in \NEMO to overcome the stability constraint:
+$(a)$ a forward time differencing scheme using a time splitting technique (\np{ln\_zdfexp}\forcode{ = .true.}) or
+$(b)$ a backward (or implicit) time differencing scheme                   (\np{ln\_zdfexp}\forcode{ = .false.}).
+In $(a)$, the master time step $\Delta $t is cut into $N$ fractional time steps so that
+the stability criterion is reduced by a factor of $N$.
+The computation is performed as follows:
 \begin{equation} \label{eq:STP_ts}
 \begin{split}
 …
 \end{split}
 \end{equation}
 with DF a vertical diffusion term. The number of fractional time steps, $N$, is given
+by setting \np{nn\_zdfexp}, (namelist parameter). The scheme $(b)$ is unconditionally
 stable but diffusive. It can be written as follows:
+with DF a vertical diffusion term.
+The number of fractional time steps, $N$, is given by setting \np{nn\_zdfexp}, (namelist parameter).
+The scheme $(b)$ is unconditionally stable but diffusive. It can be written as follows:
 \begin{equation} \label{eq:STP_imp}
    x^{t+\rdt} = x^{t-\rdt} + 2 \, \rdt \  \text{RHS}_x^{t+\rdt}
 …
 %%gm
+This scheme is rather time consuming since it requires a matrix inversion,
+but it becomes attractive since a value of 3 or more is needed for N in
+the forward time differencing scheme. For example, the finite difference
+approximation of the temperature equation is:
+This scheme is rather time consuming since it requires a matrix inversion,
+but it becomes attractive since a value of 3 or more is needed for N in the forward time differencing scheme.
+For example, the finite difference approximation of the temperature equation is:
 \begin{equation} \label{eq:STP_imp_zdf}
 \frac{T(k)^{t+1}-T(k)^{t-1}}{2\;\rdt}\equiv \text{RHS}+\frac{1}{e_{3t} }\delta
 …
 \right]
 \end{equation}
 where RHS is the right hand side of the equation except for the vertical diffusion term.
+where RHS is the right hand side of the equation except for the vertical diffusion term.
 We rewrite \autoref{eq:STP_imp} as:
 \begin{equation} \label{eq:STP_imp_mat}
 …
 \end{align*}
 \autoref{eq:STP_imp_mat} is a linear system of equations with an associated
+matrix which is tridiagonal. Moreover, $c(k)$ and $d(k)$ are positive and the diagonal
+term is greater than the sum of the two extra-diagonal terms, therefore a special
+adaptation of the Gauss elimination procedure is used to find the solution
+\autoref{eq:STP_imp_mat} is a linear system of equations with an associated matrix which is tridiagonal.
+Moreover,
+$c(k)$ and $d(k)$ are positive and the diagonal term is greater than the sum of the two extra-diagonal terms,
+therefore a special adaptation of the Gauss elimination procedure is used to find the solution
 (see for example \citet{Richtmyer1967}).
 …
 %\gmcomment{
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!t]     \begin{center}
+\includegraphics[width=0.7\textwidth]{Fig_TimeStepping_flowchart}
+\caption{   \protect\label{fig:TimeStep_flowchart}
+Sketch of the leapfrog time stepping sequence in \NEMO from \citet{Leclair_Madec_OM09}.
+The use of a semi-implicit computation of the hydrostatic pressure gradient requires
+the tracer equation to be stepped forward prior to the momentum equation.
+The need for knowledge of the vertical scale factor (here denoted as $h$)
+requires the sea surface height and the continuity equation to be stepped forward
+prior to the computation of the tracer equation.
+Note that the method for the evaluation of the surface pressure gradient $\nabla p_s$ is not presented here
+(see \autoref{sec:DYN_spg}). }
+\begin{figure}[!t]
+  \begin{center}
+    \includegraphics[width=0.7\textwidth]{Fig_TimeStepping_flowchart}
+    \caption{  \protect\label{fig:TimeStep_flowchart}
+      Sketch of the leapfrog time stepping sequence in \NEMO from \citet{Leclair_Madec_OM09}.
+      The use of a semi-implicit computation of the hydrostatic pressure gradient requires the tracer equation to
+      be stepped forward prior to the momentum equation.
+      The need for knowledge of the vertical scale factor (here denoted as $h$) requires the sea surface height and
+      the continuity equation to be stepped forward prior to the computation of the tracer equation.
+      Note that the method for the evaluation of the surface pressure gradient $\nabla p_s$ is not presented here
+      (see \autoref{sec:DYN_spg}).
+    }
 \end{center}   \end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 …
 \label{sec:STP_mLF}
+Significant changes have been introduced by \cite{Leclair_Madec_OM09} in the
+LF-RA scheme in order to ensure tracer conservation and to allow the use of
+a much smaller value of the Asselin filter parameter. The modifications affect
+both the forcing and filtering treatments in the LF-RA scheme.
+In a classical LF-RA environment, the forcing term is centred in time, $i.e.$
+it is time-stepped over a $2\rdt$ period:  $x^t  = x^t + 2\rdt Q^t $ where $Q$
+is the forcing applied to $x$, and the time filter is given by \autoref{eq:STP_asselin}
+so that $Q$ is redistributed over several time step.
+Significant changes have been introduced by \cite{Leclair_Madec_OM09} in the LF-RA scheme in order to ensure tracer conservation and to allow the use of a much smaller value of the Asselin filter parameter.
+The modifications affect both the forcing and filtering treatments in the LF-RA scheme.
+In a classical LF-RA environment, the forcing term is centred in time,
+$i.e.$ it is time-stepped over a $2\rdt$ period:
+$x^t  = x^t + 2\rdt Q^t $ where $Q$ is the forcing applied to $x$,
+and the time filter is given by \autoref{eq:STP_asselin} so that $Q$ is redistributed over several time step.
 In the modified LF-RA environment, these two formulations have been replaced by:
 \begin{align}
 …
            - \gamma\,\rdt \, \left[ Q^{t+\rdt/2} -  Q^{t-\rdt/2} \right]                          \label{eq:STP_RA}
 \end{align}
 The change in the forcing formulation given by \autoref{eq:STP_forcing}
+(see \autoref{fig:MLF_forcing}) has a significant effect: the forcing term no
 longer excites the divergence of odd and even time steps \citep{Leclair_Madec_OM09}.
+The change in the forcing formulation given by \autoref{eq:STP_forcing} (see \autoref{fig:MLF_forcing})
+has a significant effect:
+the forcing term no longer excites the divergence of odd and even time steps \citep{Leclair_Madec_OM09}.
 % forcing seen by the model....
 This property improves the LF-RA scheme in two respects.
+This property improves the LF-RA scheme in two respects.
 First, the LF-RA can now ensure the local and global conservation of tracers.
+Indeed, time filtering is no longer required on the forcing part. The influence of
+the Asselin filter on the forcing is be removed by adding a new term in the filter
+(last term in \autoref{eq:STP_RA} compared to \autoref{eq:STP_asselin}). Since
+the filtering of the forcing was the source of non-conservation in the classical
+LF-RA scheme, the modified formulation becomes conservative  \citep{Leclair_Madec_OM09}.
+Second, the LF-RA becomes a truly quasi-second order scheme. Indeed,
+\autoref{eq:STP_forcing} used in combination with a careful treatment of static
+instability (\autoref{subsec:ZDF_evd}) and of the TKE physics (\autoref{subsec:ZDF_tke_ene}),
+the two other main sources of time step divergence, allows a reduction by
+two orders of magnitude of the Asselin filter parameter.
+Note that the forcing is now provided at the middle of a time step: $Q^{t+\rdt/2}$
+is the forcing applied over the $[t,t+\rdt]$ time interval. This and the change
+in the time filter, \autoref{eq:STP_RA}, allows an exact evaluation of the
+contribution due to the forcing term between any two time steps,
+even if separated by only $\rdt$ since the time filter is no longer applied to the
+forcing term.
+Indeed, time filtering is no longer required on the forcing part.
+The influence of the Asselin filter on the forcing is be removed by adding a new term in the filter
+(last term in \autoref{eq:STP_RA} compared to \autoref{eq:STP_asselin}).
+Since the filtering of the forcing was the source of non-conservation in the classical LF-RA scheme,
+the modified formulation becomes conservative \citep{Leclair_Madec_OM09}.
+Second, the LF-RA becomes a truly quasi-second order scheme.
+Indeed, \autoref{eq:STP_forcing} used in combination with a careful treatment of static instability
+(\autoref{subsec:ZDF_evd}) and of the TKE physics (\autoref{subsec:ZDF_tke_ene}),
+the two other main sources of time step divergence,
+allows a reduction by two orders of magnitude of the Asselin filter parameter.
+Note that the forcing is now provided at the middle of a time step:
+$Q^{t+\rdt/2}$ is the forcing applied over the $[t,t+\rdt]$ time interval.
+This and the change in the time filter, \autoref{eq:STP_RA},
+allows an exact evaluation of the contribution due to the forcing term between any two time steps,
+even if separated by only $\rdt$ since the time filter is no longer applied to the forcing term.
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!t]     \begin{center}
+\includegraphics[width=0.90\textwidth]{Fig_MLF_forcing}
+\caption{   \protect\label{fig:MLF_forcing}
+Illustration of forcing integration methods.
+(top) ''Traditional'' formulation : the forcing is defined at the same time as the variable
+to which it is applied (integer value of the time step index) and it is applied over a $2\rdt$ period.
+(bottom)  modified formulation : the forcing is defined in the middle of the time (integer and a half
+value of the time step index) and the mean of two successive forcing values ($n-1/2$, $n+1/2$).
+is applied over a $2\rdt$ period.}
+\end{center}   \end{figure}
+\begin{figure}[!t]
+  \begin{center}
+    \includegraphics[width=0.90\textwidth]{Fig_MLF_forcing}
+    \caption{  \protect\label{fig:MLF_forcing}
+      Illustration of forcing integration methods.
+      (top) ''Traditional'' formulation:
+      the forcing is defined at the same time as the variable to which it is applied
+      (integer value of the time step index) and it is applied over a $2\rdt$ period.
+      (bottom)  modified formulation:
+      the forcing is defined in the middle of the time (integer and a half value of the time step index) and
+      the mean of two successive forcing values ($n-1/2$, $n+1/2$) is applied over a $2\rdt$ period.
+    }
+  \end{center}
+\end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 …
 %--------------------------------------------------------------------------------------------------------------
 The first time step of this three level scheme when starting from initial conditions
+is a forward step (Euler time integration):
+The first time step of this three level scheme when starting from initial conditions is a forward step
+(Euler time integration):
 \begin{equation} \label{eq:DOM_euler}
    x^1 = x^0 + \rdt \ \text{RHS}^0
 \end{equation}
+This is done simply by keeping the leapfrog environment ($i.e.$ the \autoref{eq:STP}
+three level time stepping) but setting all $x^0$ (\textit{before}) and $x^{1}$ (\textit{now}) fields
+equal at the first time step and using half the value of $\rdt$.
+It is also possible to restart from a previous computation, by using a
+restart file. The restart strategy is designed to ensure perfect
+restartability of the code: the user should obtain the same results to
+machine precision either by running the model for $2N$ time steps in one go,
+or by performing two consecutive experiments of $N$ steps with a restart.
+This requires saving two time levels and many auxiliary data in the restart
+files in machine precision.
+Note that when a semi-implicit scheme is used to evaluate the hydrostatic pressure
+gradient (see \autoref{subsec:DYN_hpg_imp}), an extra three-dimensional field has to be
+added to the restart file to ensure an exact restartability. This is done optionally
+via the  \np{nn\_dynhpg\_rst} namelist parameter, so that the size of the
+restart file can be reduced when restartability is not a key issue (operational
+oceanography or in ensemble simulations for seasonal forecasting).
+Note the size of the time step used, $\rdt$, is also saved in the restart file.
+When restarting, if the the time step has been changed, a restart using an Euler time
+stepping scheme is imposed.
+This is done simply by keeping the leapfrog environment ($i.e.$ the \autoref{eq:STP} three level time stepping) but
+setting all $x^0$ (\textit{before}) and $x^{1}$ (\textit{now}) fields equal at the first time step and
+using half the value of $\rdt$.
+It is also possible to restart from a previous computation, by using a restart file.
+The restart strategy is designed to ensure perfect restartability of the code:
+the user should obtain the same results to machine precision either by
+running the model for $2N$ time steps in one go,
+or by performing two consecutive experiments of $N$ steps with a restart.
+This requires saving two time levels and many auxiliary data in the restart files in machine precision.
+Note that when a semi-implicit scheme is used to evaluate the hydrostatic pressure gradient
+(see \autoref{subsec:DYN_hpg_imp}), an extra three-dimensional field has to
+be added to the restart file to ensure an exact restartability.
+This is done optionally via the  \np{nn\_dynhpg\_rst} namelist parameter,
+so that the size of the restart file can be reduced when restartability is not a key issue
+(operational oceanography or in ensemble simulations for seasonal forecasting).
+Note the size of the time step used, $\rdt$, is also saved in the restart file.
+When restarting, if the the time step has been changed, a restart using an Euler time stepping scheme is imposed.
 Options are defined through the  \ngn{namrun} namelist variables.
 %%%

NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/introduction.tex

-                      r9407
+                      r10368
 \chapter{Introduction}
+The Nucleus for European Modelling of the Ocean (\NEMO) is a framework of ocean
+related engines, namely OPA\footnote{OPA = Oc\'{e}an PArall\'{e}lis\'{e}} for the
+ocean dynamics and thermodynamics, LIM\footnote{LIM = Louvain la-neuve Ice
+Model} for the sea-ice dynamics and thermodynamics, TOP\footnote{TOP = Tracer
+in the Ocean Paradigm} for the biogeochemistry (both transport (TRP) and sources
+minus sinks (LOBSTER\footnote{LOBSTER = Lodyc Ocean Biogeochemical SysTem for
+Ecosystem and Resources}, PISCES\footnote{PISCES = Pelagic Interactions Scheme for
+Carbon and Ecosystem Studies}). It is intended to be a flexible tool for studying
+the ocean and its interactions with the other components of the earth climate system
+(atmosphere, sea-ice, biogeochemical tracers, ...) over a wide range of space and time scales.
+This documentation provides information about the physics represented by the ocean
+component of \NEMO and the rationale for the choice of numerical schemes and
+the model design. More specific information about running the model on different
+computers, or how to set up a configuration, are found on the \NEMO web site
+(www.nemo-ocean.eu).
+The ocean component of \NEMO has been developed from the OPA model,
+release 8.2, described in \citet{Madec1998}. This model has been used for a wide
+range of applications, both regional or global, as a forced ocean model and as a
+model coupled with the sea-ice and/or the atmosphere.
+This manual is organised in as follows. \autoref{chap:PE} presents the model basics,
+$i.e.$ the equations and their assumptions, the vertical coordinates used, and the
+subgrid scale physics. This part deals with the continuous equations of the model
+(primitive equations, with temperature, salinity and an equation of seawater).
+The equations are written in a curvilinear coordinate system, with a choice of vertical
+coordinates ($z$, $s$, \textit{z*}, \textit{s*}, $\tilde{z}$, $\tilde{s}$, and a mixture of them).
+Momentum equations are formulated in vector invariant or flux form.
+Dimensional units in the meter, kilogram, second (MKS) international system
+are used throughout.
+The following chapters deal with the discrete equations. \autoref{chap:STP} presents the
+time domain. The model time stepping environment is a three level scheme in which
+the tendency terms of the equations are evaluated either centered  in time, or forward,
+or backward depending of the nature of the term.
+\autoref{chap:DOM} presents the space domain. The model is discretised on a staggered
+grid (Arakawa C grid) with masking of land areas. Vertical discretisation used depends
+on both how the bottom topography is represented and whether the free surface is linear or not.
+Full step or partial step $z$-coordinate or $s$- (terrain-following) coordinate is used
+with linear free surface (level position are then fixed in time). In non-linear free surface,
+the corresponding rescaled height coordinate formulation (\textit{z*} or \textit{s*}) is used
+(the level position then vary in time as a function of the sea surface heigh).
+The following two chapters (\autoref{chap:TRA} and \autoref{chap:DYN}) describe the discretisation of the
+prognostic equations for the active tracers and the momentum. Explicit, split-explicit
+and filtered free surface formulations are implemented.
+A number of numerical schemes are available for momentum advection, for the computation
+of the pressure gradients, as well as for the advection of tracers (second or higher
+order advection schemes, including positive ones).
+Surface boundary conditions (\autoref{chap:SBC}) can be implemented as prescribed
+fluxes, or bulk formulations for the surface fluxes (wind stress, heat, freshwater). The
+model allows penetration of solar radiation  There is an optional geothermal heating at
+the ocean bottom. Within the \NEMO system the ocean model is interactively coupled
+with a sea ice model (LIM) and with biogeochemistry models (PISCES, LOBSTER).
+Interactive coupling to Atmospheric models is possible via the OASIS coupler
+\citep{OASIS2006}. Two-way nesting is also available through an interface to the
+AGRIF package (Adaptative Grid Refinement in \textsc{Fortran}) \citep{Debreu_al_CG2008}.
+The interface code for coupling to an alternative sea ice model (CICE, \citet{Hunke2008})
+has now been upgraded so that it works for both global and regional domains, although AGRIF
+is still not available.
+Other model characteristics are the lateral boundary conditions (\autoref{chap:LBC}).
+Global configurations of the model make use of the ORCA tripolar grid, with special north
+fold boundary condition. Free-slip or no-slip boundary conditions are allowed at land
+boundaries. Closed basin geometries as well as periodic domains and open boundary
+conditions are possible.
+Physical parameterisations are described in \autoref{chap:LDF} and \autoref{chap:ZDF}. The
+model includes an implicit treatment of vertical viscosity and diffusivity. The lateral
+Laplacian and biharmonic viscosity and diffusion can be rotated following a geopotential
+or neutral direction. There is an optional eddy induced velocity \citep{Gent1990} with a
+space and time variable coefficient \citet{Treguier1997}. The model has vertical harmonic
+viscosity and diffusion with a space and time variable coefficient, with options to compute
+the coefficients with \citet{Blanke1993}, \citet{Pacanowski_Philander_JPO81},
+The Nucleus for European Modelling of the Ocean (\NEMO) is a framework of ocean related engines,
+namely OPA\footnote{OPA = Oc\'{e}an PArall\'{e}lis\'{e}} for the ocean dynamics and thermodynamics,
+LIM\footnote{LIM = Louvain la-neuve Ice Model} for the sea-ice dynamics and thermodynamics,
+TOP\footnote{TOP = Tracer in the Ocean Paradigm} for the biogeochemistry (both transport (TRP) and sources
+minus sinks (LOBSTER\footnote{LOBSTER = Lodyc Ocean Biogeochemical SysTem for Ecosystem and Resources},
+PISCES\footnote{PISCES = Pelagic Interactions Scheme for Carbon and Ecosystem Studies})).
+It is intended to be a flexible tool for studying the ocean and its interactions with the other components of
+the earth climate system (atmosphere, sea-ice, biogeochemical tracers, ...) over
+a wide range of space and time scales.
+This documentation provides information about the physics represented by the ocean component of \NEMO and
+the rationale for the choice of numerical schemes and the model design.
+More specific information about running the model on different computers, or how to set up a configuration,
+are found on the \NEMO web site (www.nemo-ocean.eu).
+The ocean component of \NEMO has been developed from the OPA model, release 8.2, described in \citet{Madec1998}.
+This model has been used for a wide range of applications, both regional or global,
+as a forced ocean model and as a model coupled with the sea-ice and/or the atmosphere.
+This manual is organised in as follows.
+\autoref{chap:PE} presents the model basics, $i.e.$ the equations and their assumptions,
+the vertical coordinates used, and the subgrid scale physics.
+This part deals with the continuous equations of the model
+(primitive equations, with temperature, salinity and an equation of seawater).
+The equations are written in a curvilinear coordinate system, with a choice of vertical coordinates
+($z$, $s$, \textit{z*}, \textit{s*}, $\tilde{z}$, $\tilde{s}$, and a mixture of them).
+Momentum equations are formulated in vector invariant or flux form.
+Dimensional units in the meter, kilogram, second (MKS) international system are used throughout.
+The following chapters deal with the discrete equations.
+\autoref{chap:STP} presents the time domain.
+The model time stepping environment is a three level scheme in which the tendency terms of
+the equations are evaluated either centered in time, or forward, or backward depending of the nature of the term.
+\autoref{chap:DOM} presents the space domain.
+The model is discretised on a staggered grid (Arakawa C grid) with masking of land areas.
+Vertical discretisation used depends on both how the bottom topography is represented and
+whether the free surface is linear or not.
+Full step or partial step $z$-coordinate or $s$- (terrain-following) coordinate is used with
+linear free surface (level position are then fixed in time).
+In non-linear free surface,
+the corresponding rescaled height coordinate formulation (\textit{z*} or \textit{s*}) is used
+(the level position then vary in time as a function of the sea surface heigh).
+The following two chapters (\autoref{chap:TRA} and \autoref{chap:DYN}) describe the discretisation of
+the prognostic equations for the active tracers and the momentum.
+Explicit, split-explicit and filtered free surface formulations are implemented.
+A number of numerical schemes are available for momentum advection, for the computation of the pressure gradients,
+as well as for the advection of tracers (second or higher order advection schemes, including positive ones).
+Surface boundary conditions (\autoref{chap:SBC}) can be implemented as prescribed fluxes,
+or bulk formulations for the surface fluxes (wind stress, heat, freshwater).
+The model allows penetration of solar radiation.
+There is an optional geothermal heating at the ocean bottom.
+Within the \NEMO system the ocean model is interactively coupled with a sea ice model (LIM) and
+with biogeochemistry models (PISCES, LOBSTER).
+Interactive coupling to Atmospheric models is possible via the OASIS coupler \citep{OASIS2006}.
+Two-way nesting is also available through an interface to the AGRIF package
+(Adaptative Grid Refinement in \textsc{Fortran}) \citep{Debreu_al_CG2008}.
+The interface code for coupling to an alternative sea ice model (CICE, \citet{Hunke2008}) has now been upgraded so
+that it works for both global and regional domains, although AGRIF is still not available.
+Other model characteristics are the lateral boundary conditions (\autoref{chap:LBC}).
+Global configurations of the model make use of the ORCA tripolar grid, with special north fold boundary condition.
+Free-slip or no-slip boundary conditions are allowed at land boundaries.
+Closed basin geometries as well as periodic domains and open boundary conditions are possible.
+Physical parameterisations are described in \autoref{chap:LDF} and \autoref{chap:ZDF}.
+The model includes an implicit treatment of vertical viscosity and diffusivity.
+The lateral Laplacian and biharmonic viscosity and diffusion can be rotated following
+a geopotential or neutral direction.
+There is an optional eddy induced velocity \citep{Gent1990} with a space and time variable coefficient
+\citet{Treguier1997}.
+The model has vertical harmonic viscosity and diffusion with a space and time variable coefficient,
+with options to compute the coefficients with \citet{Blanke1993}, \citet{Pacanowski_Philander_JPO81},
 or \citet{Umlauf_Burchard_JMS03} mixing schemes.
  \vspace{1cm}
 …
 \noindent \index{CPP keys} CPP keys \newline
+Some CPP keys are implemented in the FORTRAN code to allow code selection at compiling step. This selection of code at compilation time reduces the reliability of the whole platform since it changes the code from one set of CPP keys to the other. It is used only when the addition/suppression of the part of code highly changes the amount of memory at run time.
+Some CPP keys are implemented in the FORTRAN code to allow code selection at compiling step.
+This selection of code at compilation time reduces the reliability of the whole platform since
+it changes the code from one set of CPP keys to the other.
+It is used only when the addition/suppression of the part of code highly changes the amount of memory at run time.
 Usual coding looks like :
 \begin{forlines}
 …
 \noindent \index{Namelist} Namelists
+The namelist allows to input variables (character, logical, real and integer) into the code. There is one namelist file for each component of NEMO (dynamics, sea-ice, biogeochemistry...) containing all the FOTRAN namelists needed. The implementation in NEMO uses a two step process. For each FORTRAN namelist, two files are read:
+\begin{enumerate}
+\item A reference namelist (in \path{CONFIG/SHARED/namelist_ref}) is read first. This file contains all the namelist variables which are initialised to default values
+\item A configuration namelist (in \path{CONFIG/CFG_NAME/EXP00/namelist_cfg}) is read aferwards. This file contains only the namelist variables which are changed from default values, and overwrites those.
+The namelist allows to input variables (character, logical, real and integer) into the code.
+There is one namelist file for each component of NEMO (dynamics, sea-ice, biogeochemistry...)
+containing all the FOTRAN namelists needed.
+The implementation in NEMO uses a two step process. For each FORTRAN namelist, two files are read:
+\begin{enumerate}
+\item
+  A reference namelist (in \path{CONFIG/SHARED/namelist_ref}) is read first.
+  This file contains all the namelist variables which are initialised to default values
+\item
+  A configuration namelist (in \path{CONFIG/CFG_NAME/EXP00/namelist_cfg}) is read aferwards.
+  This file contains only the namelist variables which are changed from default values, and overwrites those.
 \end{enumerate}
 A template can be found in \path{NEMO/OPA_SRC/module.example}.
+The effective namelist, taken in account during the run, is stored at execution time in an output\_namelist\_dyn (or \_ice or \_top) file.
+ \vspace{1cm}
+The effective namelist, taken in account during the run, is stored at execution time in
+an output\_namelist\_dyn (or \_ice or \_top) file.
+\vspace{1cm}
 %%gm  end
 Model outputs management and specific online diagnostics are described in \autoref{chap:DIA}.
+The diagnostics includes the output of all the tendencies of the momentum and tracers equations,
+the output of tracers tendencies averaged over the time evolving mixed layer, the output of
+the tendencies of the barotropic vorticity equation, the computation of on-line floats trajectories...
+\autoref{chap:OBS} describes a tool which reads in observation files (profile temperature
+and salinity, sea surface temperature, sea level anomaly and sea ice concentration)
+and calculates an interpolated model equivalent value at the observation location
+and nearest model timestep. Originally developed of data assimilation, it is a fantastic
+tool for model and data comparison. \autoref{chap:ASM} describes how increments
+produced by data assimilation may be applied to the model equations.
+Finally, \autoref{chap:CFG} provides a brief introduction to the pre-defined model
+configurations (water column model, ORCA and GYRE families of configurations).
+The model is implemented in \textsc{Fortran 90}, with preprocessing (C-pre-processor).
+It runs under UNIX. It is optimized for vector computers and parallelised by domain
+decomposition with MPI. All input and output is done in NetCDF (Network Common Data
+Format) with a optional direct access format for output. To ensure the clarity and
+readability of the code it is necessary to follow coding rules. The coding rules for OPA
+include conventions for naming variables, with different starting letters for different types
+of variables (real, integer, parameter\ldots). Those rules are briefly presented in
+\autoref{apdx:D} and a more complete document is available on the \NEMO web site.
+The model is organized with a high internal modularity based on physics. For example,
+each trend ($i.e.$, a term in the RHS of the prognostic equation) for momentum and
+tracers is computed in a dedicated module.  To make it easier for the user to find his way
+around the code, the module names follow a three-letter rule. For example, \mdl{traldf}
+is a module related to the TRAcers equation, computing the Lateral DiFfussion.
+The diagnostics includes the output of all the tendencies of the momentum and tracers equations,
+the output of tracers tendencies averaged over the time evolving mixed layer,
+the output of the tendencies of the barotropic vorticity equation,
+the computation of on-line floats trajectories...
+\autoref{chap:OBS} describes a tool which reads in observation files
+(profile temperature and salinity, sea surface temperature, sea level anomaly and sea ice concentration)
+and calculates an interpolated model equivalent value at the observation location and nearest model timestep.
+Originally developed of data assimilation, it is a fantastic tool for model and data comparison.
+\autoref{chap:ASM} describes how increments produced by data assimilation may be applied to the model equations.
+Finally, \autoref{chap:CFG} provides a brief introduction to the pre-defined model configurations
+(water column model, ORCA and GYRE families of configurations).
+The model is implemented in \textsc{Fortran 90}, with preprocessing (C-pre-processor).
+It runs under UNIX.
+It is optimized for vector computers and parallelised by domain decomposition with MPI.
+All input and output is done in NetCDF (Network Common Data Format) with a optional direct access format for output.
+To ensure the clarity and readability of the code it is necessary to follow coding rules.
+The coding rules for OPA include conventions for naming variables,
+with different starting letters for different types of variables (real, integer, parameter\ldots).
+Those rules are briefly presented in \autoref{apdx:D} and a more complete document is available on
+the \NEMO web site.
+The model is organized with a high internal modularity based on physics.
+For example, each trend ($i.e.$, a term in the RHS of the prognostic equation) for momentum and tracers
+is computed in a dedicated module.
+To make it easier for the user to find his way around the code, the module names follow a three-letter rule.
+For example, \mdl{traldf} is a module related to the TRAcers equation, computing the Lateral DiFfussion.
 %The complete list of module names is presented in \autoref{apdx:D}.      %====>>>> to be done !
 Furthermore, modules are organized in a few directories that correspond to their category,
+Furthermore, modules are organized in a few directories that correspond to their category,
 as indicated by the first three letters of their name (\autoref{tab:chap}).
+The manual mirrors the organization of the model.
+After the presentation of the continuous equations (\autoref{chap:PE}), the following chapters
+refer to specific terms of the equations each associated with a group of modules (\autoref{tab:chap}).
+The manual mirrors the organization of the model.
+After the presentation of the continuous equations (\autoref{chap:PE}),
+the following chapters refer to specific terms of the equations each associated with
+a group of modules (\autoref{tab:chap}).
 %--------------------------------------------------TABLE--------------------------------------------------
 \begin{table}[!t]
+%\begin{center} \begin{tabular}{|p{143pt}|l|l|} \hline
+\caption{ \protect\label{tab:chap}   Organization of Chapters mimicking the one of the model directories. }
+\begin{center}    \begin{tabular}{|l|l|l|}   \hline
+\autoref{chap:STP}   & -                 & model time STePping environment \\    \hline
+\autoref{chap:DOM}   & DOM    & model DOMain \\    \hline
+\autoref{chap:TRA}   & TRA    & TRAcer equations (potential temperature and salinity) \\   \hline
+\autoref{chap:DYN}   & DYN    & DYNamic equations (momentum) \\      \hline
+\autoref{chap:SBC}   & SBC    & Surface Boundary Conditions \\       \hline
+\autoref{chap:LBC}   & LBC    & Lateral Boundary Conditions (also OBC and BDY)  \\     \hline
+\autoref{chap:LDF}   & LDF    & Lateral DiFfusion (parameterisations) \\   \hline
+\autoref{chap:ZDF}   & ZDF    & vertical (Z) DiFfusion (parameterisations)  \\      \hline
+\autoref{chap:DIA}   & DIA    & I/O and DIAgnostics (also IOM, FLO and TRD) \\      \hline
+\autoref{chap:OBS}   & OBS    & OBServation and model comparison  \\    \hline
+\autoref{chap:ASM}   & ASM    & ASsiMilation increment  \\     \hline
+\autoref{chap:MISC}  & SOL    & Miscellaneous  topics (including solvers)  \\       \hline
+\autoref{chap:CFG}   &  -        & predefined configurations (including C1D) \\     \hline
+\end{tabular}
+\end{center}   \end{table}
+  % \begin{center} \begin{tabular}{|p{143pt}|l|l|} \hline
+  \caption{ \protect\label{tab:chap}   Organization of Chapters mimicking the one of the model directories. }
+  \begin{center}
+    \begin{tabular}{|l|l|l|}  \hline
+      \autoref{chap:STP}   & -                 & model time STePping environment \\    \hline
+      \autoref{chap:DOM}   & DOM    & model DOMain \\    \hline
+      \autoref{chap:TRA}   & TRA    & TRAcer equations (potential temperature and salinity) \\   \hline
+      \autoref{chap:DYN}   & DYN    & DYNamic equations (momentum) \\      \hline
+      \autoref{chap:SBC}   & SBC    & Surface Boundary Conditions \\       \hline
+      \autoref{chap:LBC}   & LBC    & Lateral Boundary Conditions (also OBC and BDY)  \\     \hline
+      \autoref{chap:LDF}   & LDF    & Lateral DiFfusion (parameterisations) \\   \hline
+      \autoref{chap:ZDF}   & ZDF    & vertical (Z) DiFfusion (parameterisations)  \\      \hline
+      \autoref{chap:DIA}   & DIA    & I/O and DIAgnostics (also IOM, FLO and TRD) \\      \hline
+      \autoref{chap:OBS}   & OBS    & OBServation and model comparison  \\    \hline
+      \autoref{chap:ASM}   & ASM    & ASsiMilation increment  \\     \hline
+      \autoref{chap:MISC}  & SOL    & Miscellaneous  topics (including solvers)  \\       \hline
+      \autoref{chap:CFG}   &  -        & predefined configurations (including C1D) \\     \hline
+    \end{tabular}
+  \end{center}
+\end{table}
 %--------------------------------------------------------------------------------------------------------------
 \subsubsection{Changes between releases}
 NEMO/OPA, like all research tools, is in perpetual evolution. The present document describes
+the OPA version include in the release 3.4 of NEMO.  This release differs significantly
 from version 8, documented in \citet{Madec1998}.\\
+NEMO/OPA, like all research tools, is in perpetual evolution.
+The present document describes the OPA version include in the release 3.4 of NEMO.
+This release differs significantly from version 8, documented in \citet{Madec1998}.\\
 $\bullet$ The main modifications from OPA v8 and NEMO/OPA v3.2 are :\\
 \begin{enumerate}
+\item transition to full native \textsc{Fortran} 90, deep code restructuring and drastic
+reduction of CPP keys;
+\item introduction of partial step representation of bottom topography \citep{Barnier_al_OD06, Le_Sommer_al_OM09, Penduff_al_OS07};
+\item partial reactivation of a terrain-following vertical coordinate ($s$- and hybrid $s$-$z$)
+with the addition of several options for pressure gradient computation \footnote{Partial
+support of $s$-coordinate: there is presently no support for neutral physics in $s$-
+coordinate and for the new options for horizontal pressure gradient computation with
+a non-linear equation of state.};
+\item more choices for the treatment of the free surface: full explicit, split-explicit or filtered
+schemes, and suppression of the rigid-lid option;
+\item non linear free surface associated with the rescaled height coordinate
+\textit{z*} or  \textit{s};
+\item additional schemes for vector and flux forms of the momentum  advection;
+\item additional advection schemes for tracers;
+\item implementation of the AGRIF package (Adaptative Grid Refinement in \textsc{Fortran}) \citep{Debreu_al_CG2008};
+\item online diagnostics : tracers trend in the mixed layer and vorticity balance;
+\item rewriting of the I/O management with the use of an I/O server;
+\item generalized ocean-ice-atmosphere-CO2 coupling interface, interfaced with OASIS 3 coupler ;
+\item surface module (SBC) that simplify the way the ocean is forced and include two
+bulk formulea (CLIO and CORE) and which includes an on-the-fly interpolation of input forcing fields ;
+\item RGB light penetration and optional use of ocean color
+\item major changes in the TKE schemes: it now includes a Langmuir cell parameterization  \citep{Axell_JGR02},
+the \citet{Mellor_Blumberg_JPO04} surface wave breaking parameterization, and has a time discretization
+which is energetically consistent with the ocean model equations \citep{Burchard_OM02, Marsaleix_al_OM08};
+\item tidal mixing parametrisation (bottom intensification) + Indonesian specific tidal mixing \citep{Koch-Larrouy_al_GRL07};
+\item introduction of LIM-3, the new Louvain-la-Neuve sea-ice model (C-grid rheology and
+new thermodynamics including bulk ice salinity) \citep{Vancoppenolle_al_OM09a, Vancoppenolle_al_OM09b}
+\item
+  transition to full native \textsc{Fortran} 90, deep code restructuring and drastic reduction of CPP keys;
+\item
+  introduction of partial step representation of bottom topography
+  \citep{Barnier_al_OD06, Le_Sommer_al_OM09, Penduff_al_OS07};
+\item
+  partial reactivation of a terrain-following vertical coordinate ($s$- and hybrid $s$-$z$) with
+  the addition of several options for pressure gradient computation
+  \footnote{Partial support of $s$-coordinate: there is presently no support for neutral physics in
+    $s$-coordinate and for the new options for horizontal pressure gradient computation with
+    a non-linear equation of state.
+  };
+\item
+  more choices for the treatment of the free surface: full explicit, split-explicit or filtered schemes,
+  and suppression of the rigid-lid option;
+\item
+  non linear free surface associated with the rescaled height coordinate \textit{z*} or \textit{s};
+\item
+  additional schemes for vector and flux forms of the momentum advection;
+\item
+  additional advection schemes for tracers;
+\item
+  implementation of the AGRIF package (Adaptative Grid Refinement in \textsc{Fortran}) \citep{Debreu_al_CG2008};
+\item
+  online diagnostics : tracers trend in the mixed layer and vorticity balance;
+\item
+  rewriting of the I/O management with the use of an I/O server;
+\item
+  generalized ocean-ice-atmosphere-CO2 coupling interface, interfaced with OASIS 3 coupler;
+\item
+  surface module (SBC) that simplify the way the ocean is forced and include two bulk formulea (CLIO and CORE) and
+  which includes an on-the-fly interpolation of input forcing fields;
+\item
+  RGB light penetration and optional use of ocean color
+\item
+  major changes in the TKE schemes: it now includes a Langmuir cell parameterization \citep{Axell_JGR02},
+  the \citet{Mellor_Blumberg_JPO04} surface wave breaking parameterization, and has a time discretization which
+  is energetically consistent with the ocean model equations \citep{Burchard_OM02, Marsaleix_al_OM08};
+\item
+  tidal mixing parametrisation (bottom intensification) + Indonesian specific tidal mixing
+  \citep{Koch-Larrouy_al_GRL07};
+\item
+  introduction of LIM-3, the new Louvain-la-Neuve sea-ice model
+  (C-grid rheology and new thermodynamics including bulk ice salinity)
+  \citep{Vancoppenolle_al_OM09a, Vancoppenolle_al_OM09b}
 \end{enumerate}
  \vspace{1cm}
+$\bullet$ The main modifications from NEMO/OPA v3.2 and  v3.3 are :\\
+\begin{enumerate}
+\item introduction of a modified leapfrog-Asselin filter time stepping scheme \citep{Leclair_Madec_OM09};
+\item additional scheme for iso-neutral mixing \citep{Griffies_al_JPO98}, although it is still a "work in progress";
+\item a rewriting of the bottom boundary layer scheme, following \citet{Campin_Goosse_Tel99};
+\item addition of a Generic Length Scale vertical mixing scheme, following \citet{Umlauf_Burchard_JMS03};
+\item addition of the atmospheric pressure as an external forcing on both ocean and sea-ice dynamics;
+\item addition of a diurnal cycle on solar radiation \citep{Bernie_al_CD07};
+\item river runoffs added through a non-zero depth, and having its own temperature and salinity;
+\item CORE II normal year forcing set as the default forcing of ORCA2-LIM configuration ;
+\item generalisation of the use of \mdl{fldread} for all input fields (ocean climatology, sea-ice damping...) ;
+\item addition of an on-line observation and model comparison (thanks to NEMOVAR project);
+\item optional application of an assimilation increment (thanks to NEMOVAR project);
+\item coupling interface adjusted for WRF atmospheric model;
+\item C-grid ice rheology now available fro both LIM-2 and LIM-3 \citep{Bouillon_al_OM09};
+\item LIM-3 ice-ocean momentum coupling applied to LIM-2 ;
+\item a deep re-writting and simplification of the off-line tracer component (OFF\_SRC) ;
+\item the merge of passive and active advection and diffusion modules ;
+\item  Use of the Flexible Configuration Manager (FCM) to build configurations, generate the Makefile and produce the executable ;
+\item Linear-tangent and Adjoint component (TAM) added, phased with v3.0
+\end{enumerate}
+ \vspace{1cm}
+In addition, several minor modifications in the coding have been introduced with the constant
+concern of improving the model performance.
+ \vspace{1cm}
+$\bullet$ The main modifications from NEMO/OPA v3.2 and v3.3 are :\\
+\begin{enumerate}
+\item
+  introduction of a modified leapfrog-Asselin filter time stepping scheme
+  \citep{Leclair_Madec_OM09};
+\item
+  additional scheme for iso-neutral mixing \citep{Griffies_al_JPO98}, although it is still a "work in progress";
+\item
+  a rewriting of the bottom boundary layer scheme, following \citet{Campin_Goosse_Tel99};
+\item
+  addition of a Generic Length Scale vertical mixing scheme, following \citet{Umlauf_Burchard_JMS03};
+\item
+  addition of the atmospheric pressure as an external forcing on both ocean and sea-ice dynamics;
+\item
+  addition of a diurnal cycle on solar radiation \citep{Bernie_al_CD07};
+\item
+  river runoffs added through a non-zero depth, and having its own temperature and salinity;
+\item
+  CORE II normal year forcing set as the default forcing of ORCA2-LIM configuration;
+\item
+  generalisation of the use of \mdl{fldread} for all input fields (ocean climatology, sea-ice damping...);
+\item
+  addition of an on-line observation and model comparison (thanks to NEMOVAR project);
+\item
+  optional application of an assimilation increment (thanks to NEMOVAR project);
+\item
+  coupling interface adjusted for WRF atmospheric model;
+\item
+  C-grid ice rheology now available fro both LIM-2 and LIM-3 \citep{Bouillon_al_OM09};
+\item
+  LIM-3 ice-ocean momentum coupling applied to LIM-2;
+\item
+  a deep re-writting and simplification of the off-line tracer component (OFF\_SRC);
+\item
+  the merge of passive and active advection and diffusion modules;
+\item
+  Use of the Flexible Configuration Manager (FCM) to build configurations,
+  generate the Makefile and produce the executable;
+\item
+  Linear-tangent and Adjoint component (TAM) added, phased with v3.0
+\end{enumerate}
+\vspace{1cm}
+In addition, several minor modifications in the coding have been introduced with the constant concern of
+improving the model performance.
+\vspace{1cm}
 $\bullet$ The main modifications from NEMO/OPA v3.3 and  v3.4 are :\\
 \begin{enumerate}
 \item finalisation of above iso-neutral mixing \citep{Griffies_al_JPO98}";
+\item finalisation of above iso-neutral mixing \citep{Griffies_al_JPO98}";
 \item "Neptune effect" parametrisation;
 \item horizontal pressure gradient suitable for s-coordinate;
+\item horizontal pressure gradient suitable for s-coordinate;
 \item semi-implicit bottom friction;
 \item finalisation of the merge of passive and active tracers advection-diffusion modules;
+\item finalisation of the merge of passive and active tracers advection-diffusion modules;
 \item a new bulk formulae (so-called MFS);
 \item use fldread for the off-line tracer component (OFF\_SRC) ;
+\item use fldread for the off-line tracer component (OFF\_SRC);
 \item use MPI point to point communications  for north fold;
 \item diagnostic of transport ;
+\item diagnostic of transport;
 \end{enumerate}

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NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_ASM.tex

NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_CONFIG.tex

NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_DIA.tex

NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_DIU.tex

NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_DOM.tex

NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_DYN.tex

NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_LBC.tex

NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_LDF.tex

NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_OBS.tex

NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_SBC.tex

NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_STO.tex

NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_TRA.tex

NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_ZDF.tex

NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_conservation.tex

NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_misc.tex

NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_model_basics.tex

NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_model_basics_zstar.tex

NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_time_domain.tex

NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/introduction.tex

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