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chap_TRA.tex

Timestamp:

2018-11-21T17:59:55+01:00 (5 years ago)

Author:

nicolasmartin

Message:

Vast edition of LaTeX subfiles to improve the readability by cutting sentences in a more suitable way
Every sentence begins in a new line and if necessary is splitted around 110 characters lenght for side-by-side visualisation,
this setting may not be adequate for everyone but something has to be set.
The punctuation was the primer trigger for the cutting process, otherwise subordinators and coordinators, in order to mostly keep a meaning for each line

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: 1 edited

NEMO/trunk/doc/latex/NEMO/subfiles/chap_TRA.tex (modified) (50 diffs)

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NEMO/trunk/doc/latex/NEMO/subfiles/chap_TRA.tex

-                      r10146
+                      r10354
 %$\ $\newline    % force a new ligne
+Using the representation described in \autoref{chap:DOM}, several semi-discrete
+space forms of the tracer equations are available depending on the vertical
+coordinate used and on the physics used. In all the equations presented
+here, the masking has been omitted for simplicity. One must be aware that
+all the quantities are masked fields and that each time a mean or difference
+operator is used, the resulting field is multiplied by a mask.
+The two active tracers are potential temperature and salinity. Their prognostic
+equations can be summarized as follows:
+Using the representation described in \autoref{chap:DOM},
+several semi-discrete space forms of the tracer equations are available depending on
+the vertical coordinate used and on the physics used.
+In all the equations presented here, the masking has been omitted for simplicity.
+One must be aware that all the quantities are masked fields and
+that each time a mean or difference operator is used,
+the resulting field is multiplied by a mask.
+The two active tracers are potential temperature and salinity.
+Their prognostic equations can be summarized as follows:
 \begin{equation*}
 \text{NXT} = \text{ADV}+\text{LDF}+\text{ZDF}+\text{SBC}
 …
 \end{equation*}
+NXT stands for next, referring to the time-stepping. From left to right, the terms
+on the rhs of the tracer equations are the advection (ADV), the lateral diffusion
+(LDF), the vertical diffusion (ZDF), the contributions from the external forcings
+(SBC: Surface Boundary Condition, QSR: penetrative Solar Radiation, and BBC:
+Bottom Boundary Condition), the contribution from the bottom boundary Layer
+(BBL) parametrisation, and an internal damping (DMP) term. The terms QSR,
+BBC, BBL and DMP are optional. The external forcings and parameterisations
+require complex inputs and complex calculations ($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}

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