Index: NEMO/branches/2019/dev_r10742_ENHANCE-12_SimonM-Tides/doc/latex/NEMO/subfiles/chap_DOM.tex
===================================================================
--- NEMO/branches/2019/dev_r10742_ENHANCE-12_SimonM-Tides/doc/latex/NEMO/subfiles/chap_DOM.tex	(revision 10744)
+++ NEMO/branches/2019/dev_r10742_ENHANCE-12_SimonM-Tides/doc/latex/NEMO/subfiles/chap_DOM.tex	(revision 12065)
@@ -2,127 +2,174 @@
 
 \begin{document}
-% ================================================================
-% Chapter 2 ——— Space and Time Domain (DOM)
-% ================================================================
+
 \chapter{Space Domain (DOM)}
 \label{chap:DOM}
 
-\minitoc
-
-% Missing things:
-% 	- istate: description of the initial state   ==> this has to be put elsewhere..
-%                  perhaps in MISC ?  By the way the initialisation of T S and dynamics 
-%                  should be put outside of DOM routine (better with TRC staff and off-line
-%                  tracers)
-% 	-geo2ocean:  how to switch from geographic to mesh coordinate
-%     - domclo:  closed sea and lakes.... management of closea sea area : specific to global configuration, both forced and coupled
-
-\newpage
-
-Having defined the continuous equations in \autoref{chap:PE} and chosen a time discretization \autoref{chap:STP},
-we need to choose a discretization on a grid, and numerical algorithms.
+% Missing things
+% -    istate: description of the initial state   ==> this has to be put elsewhere..
+%              perhaps in MISC ?  By the way the initialisation of T S and dynamics
+%              should be put outside of DOM routine (better with TRC staff and off-line
+%              tracers)
+% - geo2ocean: how to switch from geographic to mesh coordinate
+% -    domclo: closed sea and lakes....
+%              management of closea sea area: specific to global cfg, both forced and coupled
+
+\thispagestyle{plain}
+
+\chaptertoc
+
+\paragraph{Changes record} ~\\
+
+{\footnotesize
+  \begin{tabularx}{0.8\textwidth}{l||X|X}
+    Release                                                                                 &
+    Author(s)                                                                               &
+    Modifications                                                                           \\
+    \hline
+    {\em 4.0                                                                              } &
+    {\em Simon M\"{u}ller \& Andrew Coward \newline \newline
+      Simona Flavoni and Tim Graham                                                       } &
+    {\em Compatibility changes: many options moved to external domain configuration tools
+      (see \autoref{apdx:DOMCFG}). \newline
+      Updates                                                                             } \\
+    {\em 3.6                                                                              } &
+    {\em Rachid Benshila, Christian \'{E}th\'{e}, Pierre Mathiot and Gurvan Madec         } &
+    {\em Updates                                                                          } \\
+    {\em $\leq$ 3.4                                                                       } &
+    {\em Gurvan Madec and S\'{e}bastien Masson                                            } &
+    {\em First version                                                                    }
+  \end{tabularx}
+}
+
+\clearpage
+
+Having defined the continuous equations in \autoref{chap:MB} and
+chosen a time discretisation \autoref{chap:TD},
+we need to choose a grid for spatial discretisation and related numerical algorithms.
 In the present chapter, we provide a general description of the staggered grid used in \NEMO,
-and other information relevant to the main directory routines as well as the DOM (DOMain) directory.
-
-% ================================================================
-% Fundamentals of the Discretisation
-% ================================================================
+and other relevant information about the DOM (DOMain) source code modules.
+
+%% =================================================================================================
 \section{Fundamentals of the discretisation}
 \label{sec:DOM_basics}
 
-% -------------------------------------------------------------------------------------------------------------
-%        Arrangement of Variables 
-% -------------------------------------------------------------------------------------------------------------
+%% =================================================================================================
 \subsection{Arrangement of variables}
 \label{subsec:DOM_cell}
 
-%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
-\begin{figure}[!tb]
-  \begin{center}
-    \includegraphics[]{Fig_cell}
-    \caption{
-      \protect\label{fig:cell}
-      Arrangement of variables.
-      $t$ indicates scalar points where temperature, salinity, density, pressure and
-      horizontal divergence are defined.
-      $(u,v,w)$ indicates vector points, and $f$ indicates vorticity points where both relative and
-      planetary vorticities are defined.
-    }
-  \end{center}
+\begin{figure}
+  \centering
+  \includegraphics[width=0.33\textwidth]{DOM_cell}
+  \caption[Arrangement of variables in the unit cell of space domain]{
+    Arrangement of variables in the unit cell of space domain.
+    $t$ indicates scalar points where
+    temperature, salinity, density, pressure and horizontal divergence are defined.
+    $(u,v,w)$ indicates vector points, and $f$ indicates vorticity points where
+    both relative and planetary vorticities are defined.}
+  \label{fig:DOM_cell}
 \end{figure}
-%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
-
-The numerical techniques used to solve the Primitive Equations in this model are based on the traditional,
-centred second-order finite difference approximation.
-Special attention has been given to the homogeneity of the solution in the three space directions.
+
+The numerical techniques used to solve the Primitive Equations in this model are based on
+the traditional, centred second-order finite difference approximation.
+Special attention has been given to the homogeneity of the solution in the three spatial directions.
 The arrangement of variables is the same in all directions.
-It consists of cells centred on scalar points ($t$, $S$, $p$, $\rho$) with vector points $(u, v, w)$ defined in
-the centre of each face of the cells (\autoref{fig:cell}).
-This is the generalisation to three dimensions of the well-known ``C'' grid in Arakawa's classification
-\citep{Mesinger_Arakawa_Bk76}.
-The relative and planetary vorticity, $\zeta$ and $f$, are defined in the centre of each vertical edge and
-the barotropic stream function $\psi$ is defined at horizontal points overlying the $\zeta$ and $f$-points.
-
-The ocean mesh (\ie the position of all the scalar and vector points) is defined by the transformation that
-gives $(\lambda,\varphi,z)$ as a function of $(i,j,k)$.
-The grid-points are located at integer or integer and a half value of $(i,j,k)$ as indicated on \autoref{tab:cell}.
-In all the following, subscripts $u$, $v$, $w$, $f$, $uw$, $vw$ or $fw$ indicate the position of
-the grid-point where the scale factors are defined.
-Each scale factor is defined as the local analytical value provided by \autoref{eq:scale_factors}.
+It consists of cells centred on scalar points ($t$, $S$, $p$, $\rho$) with
+vector points $(u, v, w)$ defined in the centre of each face of the cells (\autoref{fig:DOM_cell}).
+This is the generalisation to three dimensions of the well-known ``C'' grid in
+Arakawa's classification \citep{mesinger.arakawa_bk76}.
+The relative and planetary vorticity, $\zeta$ and $f$, are defined in the centre of each
+vertical edge and the barotropic stream function $\psi$ is defined at horizontal points overlying
+the $\zeta$ and $f$-points.
+
+The ocean mesh (\ie\ the position of all the scalar and vector points) is defined by
+the transformation that gives $(\lambda,\varphi,z)$ as a function of $(i,j,k)$.
+The grid-points are located at integer or integer and a half value of $(i,j,k)$ as indicated on
+\autoref{tab:DOM_cell}.
+In all the following,
+subscripts $u$, $v$, $w$, $f$, $uw$, $vw$ or $fw$ indicate the position of the grid-point where
+the scale factors are defined.
+Each scale factor is defined as the local analytical value provided by \autoref{eq:MB_scale_factors}.
 As a result, the mesh on which partial derivatives $\pd[]{\lambda}$, $\pd[]{\varphi}$ and
-$\pd[]{z}$ are evaluated in a uniform mesh with a grid size of unity.
-Discrete partial derivatives are formulated by the traditional, centred second order finite difference approximation
-while the scale factors are chosen equal to their local analytical value.
+$\pd[]{z}$ are evaluated is a uniform mesh with a grid size of unity.
+Discrete partial derivatives are formulated by
+the traditional, centred second order finite difference approximation while
+the scale factors are chosen equal to their local analytical value.
 An important point here is that the partial derivative of the scale factors must be evaluated by
 centred finite difference approximation, not from their analytical expression.
-This preserves the symmetry of the discrete set of equations and therefore satisfies many of
-the continuous properties (see \autoref{apdx:C}).
+This preserves the symmetry of the discrete set of equations and
+therefore satisfies many of the continuous properties (see \autoref{apdx:INVARIANTS}).
 A similar, related remark can be made about the domain size:
-when needed, an area, volume, or the total ocean depth must be evaluated as the sum of the relevant scale factors
-(see \autoref{eq:DOM_bar} in the next section).
-
-%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
-\begin{table}[!tb]
-  \begin{center}
-    \begin{tabular}{|p{46pt}|p{56pt}|p{56pt}|p{56pt}|}
-      \hline
-      T	& $i 		 $ & $j      $ & $k      $ \\
-      \hline
-      u	& $i + 1/2$ & $j      $ & $k      $ \\
-      \hline
-      v	& $i	 	 $ & $j + 1/2$ & $k      $ \\
-      \hline
-      w	& $i	 	 $ & $j      $ & $k + 1/2$ \\
-      \hline
-      f	& $i + 1/2$ & $j + 1/2$ & $k      $ \\
-      \hline
-      uw	& $i + 1/2$ & $j      $ & $k + 1/2$ \\
-      \hline
-      vw	& $i	 	 $ & $j + 1/2$ & $k + 1/2$ \\
-      \hline
-      fw	& $i + 1/2$ & $j + 1/2$ & $k + 1/2$ \\
-      \hline
-    \end{tabular}
-    \caption{
-      \protect\label{tab:cell}
-      Location of grid-points as a function of integer or integer and a half value of the column, line or level.
-      This indexing is only used for the writing of the semi -discrete equation.
-      In the code, the indexing uses integer values only and has a reverse direction in the vertical
-      (see \autoref{subsec:DOM_Num_Index})
-    }
-  \end{center}
+when needed, an area, volume, or the total ocean depth must be evaluated as
+the product or sum of the relevant scale factors (see \autoref{eq:DOM_bar} in the next section).
+
+\begin{table}
+  \centering
+  \begin{tabular}{|l|l|l|l|}
+    \hline
+    t   & $i      $ & $j      $ & $k      $ \\
+    \hline
+    u   & $i + 1/2$ & $j      $ & $k      $ \\
+    \hline
+    v   & $i      $ & $j + 1/2$ & $k      $ \\
+    \hline
+    w   & $i      $ & $j      $ & $k + 1/2$ \\
+    \hline
+    f   & $i + 1/2$ & $j + 1/2$ & $k      $ \\
+    \hline
+    uw  & $i + 1/2$ & $j      $ & $k + 1/2$ \\
+    \hline
+    vw  & $i      $ & $j + 1/2$ & $k + 1/2$ \\
+    \hline
+    fw  & $i + 1/2$ & $j + 1/2$ & $k + 1/2$ \\
+    \hline
+  \end{tabular}
+  \caption[Location of grid-points]{
+    Location of grid-points as a function of integer or
+    integer and a half value of the column, line or level.
+    This indexing is only used for the writing of the semi-discrete equations.
+    In the code, the indexing uses integer values only and
+    is positive downwards in the vertical with $k=1$ at the surface.
+    (see \autoref{subsec:DOM_Num_Index})}
+  \label{tab:DOM_cell}
 \end{table}
-%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
-
-% -------------------------------------------------------------------------------------------------------------
-%        Vector Invariant Formulation 
-% -------------------------------------------------------------------------------------------------------------
+
+Note that the definition of the scale factors
+(\ie\ as the analytical first derivative of the transformation that
+results in $(\lambda,\varphi,z)$ as a function of $(i,j,k)$)
+is specific to the \NEMO\ model \citep{marti.madec.ea_JGR92}.
+As an example, a scale factor in the $i$ direction is defined locally at a $t$-point,
+whereas many other models on a C grid choose to define such a scale factor as
+the distance between the $u$-points on each side of the $t$-point.
+Relying on an analytical transformation has two advantages:
+firstly, there is no ambiguity in the scale factors appearing in the discrete equations,
+since they are first introduced in the continuous equations;
+secondly, analytical transformations encourage good practice by
+the definition of smoothly varying grids
+(rather than allowing the user to set arbitrary jumps in thickness between adjacent layers)
+\citep{treguier.dukowicz.ea_JGR96}.
+An example of the effect of such a choice is shown in \autoref{fig:DOM_zgr_e3}.
+\begin{figure}
+  \centering
+  \includegraphics[width=0.5\textwidth]{DOM_zgr_e3}
+  \caption[Comparison of grid-point position, vertical grid-size and scale factors]{
+    Comparison of (a) traditional definitions of grid-point position and grid-size in the vertical,
+    and (b) analytically derived grid-point position and scale factors.
+    For both grids here, the same $w$-point depth has been chosen but
+    in (a) the $t$-points are set half way between $w$-points while
+    in (b) they are defined from an analytical function:
+    $z(k) = 5 \, (k - 1/2)^3 - 45 \, (k - 1/2)^2 + 140 \, (k - 1/2) - 150$.
+    Note the resulting difference between the value of the grid-size $\Delta_k$ and
+    those of the scale factor $e_k$.}
+  \label{fig:DOM_zgr_e3}
+\end{figure}
+
+%% =================================================================================================
 \subsection{Discrete operators}
 \label{subsec:DOM_operators}
 
-Given the values of a variable $q$ at adjacent points, the differencing and averaging operators at
-the midpoint between them are:
+Given the values of a variable $q$ at adjacent points,
+the differencing and averaging operators at the midpoint between them are:
 \begin{alignat*}{2}
-  % \label{eq:di_mi}
+  % \label{eq:DOM_di_mi}
   \delta_i [q]      &= &       &q (i + 1/2) - q (i - 1/2) \\
   \overline q^{\, i} &= &\big\{ &q (i + 1/2) + q (i - 1/2) \big\} / 2
@@ -130,27 +177,27 @@
 
 Similar operators are defined with respect to $i + 1/2$, $j$, $j + 1/2$, $k$, and $k + 1/2$.
-Following \autoref{eq:PE_grad} and \autoref{eq:PE_lap}, the gradient of a variable $q$ defined at
-a $t$-point has its three components defined at $u$-, $v$- and $w$-points while
-its Laplacian is defined at $t$-point.
+Following \autoref{eq:MB_grad} and \autoref{eq:MB_lap},
+the gradient of a variable $q$ defined at a $t$-point has
+its three components defined at $u$-, $v$- and $w$-points while
+its Laplacian is defined at the $t$-point.
 These operators have the following discrete forms in the curvilinear $s$-coordinates system:
-\[
+\begin{gather*}
   % \label{eq:DOM_grad}
   \nabla q \equiv   \frac{1}{e_{1u}} \delta_{i + 1/2} [q] \; \, \vect i
                   + \frac{1}{e_{2v}} \delta_{j + 1/2} [q] \; \, \vect j
-                  + \frac{1}{e_{3w}} \delta_{k + 1/2} [q] \; \, \vect k
-\]
-\begin{multline*}
+                  + \frac{1}{e_{3w}} \delta_{k + 1/2} [q] \; \, \vect k \\
   % \label{eq:DOM_lap}
   \Delta q \equiv   \frac{1}{e_{1t} \, e_{2t} \, e_{3t}}
                     \; \lt[   \delta_i \lt( \frac{e_{2u} \, e_{3u}}{e_{1u}} \; \delta_{i + 1/2} [q] \rt)
-                            + \delta_j \lt( \frac{e_{1v} \, e_{3v}}{e_{2v}} \; \delta_{j + 1/2} [q] \rt) \; \rt] \\
+                            + \delta_j \lt( \frac{e_{1v} \, e_{3v}}{e_{2v}} \; \delta_{j + 1/2} [q] \rt) \; \rt]
                   + \frac{1}{e_{3t}}
                               \delta_k \lt[ \frac{1              }{e_{3w}} \; \delta_{k + 1/2} [q] \rt]
-\end{multline*}
-
-Following \autoref{eq:PE_curl} and \autoref{eq:PE_div}, a vector $\vect A = (a_1,a_2,a_3)$ defined at
-vector points $(u,v,w)$ has its three curl components defined at $vw$-, $uw$, and $f$-points, and
+\end{gather*}
+
+Following \autoref{eq:MB_curl} and \autoref{eq:MB_div},
+a vector $\vect A = (a_1,a_2,a_3)$ defined at vector points $(u,v,w)$ has
+its three curl components defined at $vw$-, $uw$, and $f$-points, and
 its divergence defined at $t$-points:
-\begin{multline}
+\begin{multline*}
 % \label{eq:DOM_curl}
   \nabla \times \vect A \equiv   \frac{1}{e_{2v} \, e_{3vw}}
@@ -163,14 +210,14 @@
                                  \Big[   \delta_{i + 1/2} (e_{2v} \, a_2)
                                        - \delta_{j + 1/2} (e_{1u} \, a_1) \Big] \vect k
-\end{multline}
-\begin{equation}
+\end{multline*}
+\[
 % \label{eq:DOM_div}
   \nabla \cdot \vect A \equiv   \frac{1}{e_{1t} \, e_{2t} \, e_{3t}}
                                 \Big[ \delta_i (e_{2u} \, e_{3u} \, a_1) + \delta_j (e_{1v} \, e_{3v} \, a_2) \Big]
                               + \frac{1}{e_{3t}} \delta_k (a_3)
-\end{equation}
-
-The vertical average over the whole water column denoted by an overbar becomes for a quantity $q$ which
-is a masked field (i.e. equal to zero inside solid area):
+\]
+
+The vertical average over the whole water column is denoted by an overbar and
+is for a masked field $q$ (\ie\ a quantity that is equal to zero inside solid areas):
 \begin{equation}
   \label{eq:DOM_bar}
@@ -178,6 +225,7 @@
 \end{equation}
 where $H_q$  is the ocean depth, which is the masked sum of the vertical scale factors at $q$ points,
-$k^b$ and $k^o$ are the bottom and surface $k$-indices, and the symbol $k^o$ refers to a summation over
-all grid points of the same type in the direction indicated by the subscript (here $k$).
+$k^b$ and $k^o$ are the bottom and surface $k$-indices,
+and the symbol $\sum \limits_k$ refers to a summation over all grid points of the same type in
+the direction indicated by the subscript (here $k$).
 
 In continuous form, the following properties are satisfied:
@@ -189,13 +237,14 @@
 \end{gather}
 
-It is straightforward to demonstrate that these properties are verified locally in discrete form as soon as
-the scalar $q$ is taken at $t$-points and the vector $\vect A$ has its components defined at
+It is straightforward to demonstrate that these properties are verified locally in discrete form as
+soon as the scalar $q$ is taken at $t$-points and the vector $\vect A$ has its components defined at
 vector points $(u,v,w)$.
 
-Let $a$ and $b$ be two fields defined on the mesh, with value zero inside continental area.
-Using integration by parts it can be shown that the differencing operators ($\delta_i$, $\delta_j$ and $\delta_k$)
-are skew-symmetric linear operators, and further that the averaging operators $\overline{\cdots}^{\, i}$,
-$\overline{\cdots}^{\, j}$ and $\overline{\cdots}^{\, k}$) are symmetric linear operators, \ie
-\begin{alignat}{4}
+Let $a$ and $b$ be two fields defined on the mesh, with a value of zero inside continental areas.
+It can be shown that the differencing operators ($\delta_i$, $\delta_j$ and
+$\delta_k$) are skew-symmetric linear operators,
+and further that the averaging operators ($\overline{\cdots}^{\, i}$, $\overline{\cdots}^{\, j}$ and
+$\overline{\cdots}^{\, k}$) are symmetric linear operators, \ie
+\begin{alignat}{5}
   \label{eq:DOM_di_adj}
   &\sum \limits_i a_i \; \delta_i [b]      &\equiv &- &&\sum \limits_i \delta      _{   i + 1/2} [a] &b_{i + 1/2} \\
@@ -204,840 +253,447 @@
 \end{alignat}
 
-In other words, the adjoint of the differencing and averaging operators are $\delta_i^* = \delta_{i + 1/2}$ and 
+In other words,
+the adjoint of the differencing and averaging operators are $\delta_i^* = \delta_{i + 1/2}$ and
 $(\overline{\cdots}^{\, i})^* = \overline{\cdots}^{\, i + 1/2}$, respectively.
-These two properties will be used extensively in the \autoref{apdx:C} to
+These two properties will be used extensively in the \autoref{apdx:INVARIANTS} to
 demonstrate integral conservative properties of the discrete formulation chosen.
 
-% -------------------------------------------------------------------------------------------------------------
-%        Numerical Indexing 
-% -------------------------------------------------------------------------------------------------------------
+%% =================================================================================================
 \subsection{Numerical indexing}
 \label{subsec:DOM_Num_Index}
 
-%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
-\begin{figure}[!tb]
-  \begin{center}
-    \includegraphics[]{Fig_index_hor}
-    \caption{
-      \protect\label{fig:index_hor}
-      Horizontal integer indexing used in the \fortran code.
-      The dashed area indicates the cell in which variables contained in arrays have the same $i$- and $j$-indices
-    }
-  \end{center}
+\begin{figure}
+  \centering
+  \includegraphics[width=0.33\textwidth]{DOM_index_hor}
+  \caption[Horizontal integer indexing]{
+    Horizontal integer indexing used in the \fortran\ code.
+    The dashed area indicates the cell in which
+    variables contained in arrays have the same $i$- and $j$-indices}
+  \label{fig:DOM_index_hor}
 \end{figure}
-%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
-
-The array representation used in the \fortran code requires an integer indexing while
-the analytical definition of the mesh (see \autoref{subsec:DOM_cell}) is associated with the use of
-integer values for $t$-points and both integer and integer and a half values for all the other points.
-Therefore a specific integer indexing must be defined for points other than $t$-points
-(\ie velocity and vorticity grid-points).
-Furthermore, the direction of the vertical indexing has been changed so that the surface level is at $k = 1$.
-
-% -----------------------------------
-%        Horizontal Indexing 
-% -----------------------------------
+
+The array representation used in the \fortran\ code requires an integer indexing.
+However, the analytical definition of the mesh (see \autoref{subsec:DOM_cell}) is associated with
+the use of integer values for $t$-points only while
+all the other points involve integer and a half values.
+Therefore, a specific integer indexing has been defined for points other than $t$-points
+(\ie\ velocity and vorticity grid-points).
+Furthermore, the direction of the vertical indexing has been reversed and
+the surface level set at $k = 1$.
+
+%% =================================================================================================
 \subsubsection{Horizontal indexing}
 \label{subsec:DOM_Num_Index_hor}
 
-The indexing in the horizontal plane has been chosen as shown in \autoref{fig:index_hor}.
+The indexing in the horizontal plane has been chosen as shown in \autoref{fig:DOM_index_hor}.
 For an increasing $i$ index ($j$ index),
 the $t$-point and the eastward $u$-point (northward $v$-point) have the same index
-(see the dashed area in \autoref{fig:index_hor}).
-A $t$-point and its nearest northeast $f$-point have the same $i$-and $j$-indices.
-
-% -----------------------------------
-%        Vertical indexing 
-% -----------------------------------
+(see the dashed area in \autoref{fig:DOM_index_hor}).
+A $t$-point and its nearest north-east $f$-point have the same $i$-and $j$-indices.
+
+%% =================================================================================================
 \subsubsection{Vertical indexing}
 \label{subsec:DOM_Num_Index_vertical}
 
-In the vertical, the chosen indexing requires special attention since the $k$-axis is re-orientated downward in
-the \fortran code compared to the indexing used in the semi -discrete equations and
-given in \autoref{subsec:DOM_cell}.
-The sea surface corresponds to the $w$-level $k = 1$ which is the same index as $t$-level just below
-(\autoref{fig:index_vert}).
-The last $w$-level ($k = jpk$) either corresponds to the ocean floor or is inside the bathymetry while
-the last $t$-level is always inside the bathymetry (\autoref{fig:index_vert}).
-Note that for an increasing $k$ index, a $w$-point and the $t$-point just below have the same $k$ index,
-in opposition to what is done in the horizontal plane where
-it is the $t$-point and the nearest velocity points in the direction of the horizontal axis that
-have the same $i$ or $j$ index
-(compare the dashed area in \autoref{fig:index_hor} and \autoref{fig:index_vert}).
+In the vertical, the chosen indexing requires special attention since
+the direction of the $k$-axis in the \fortran\ code is the reverse of
+that used in the semi-discrete equations and given in \autoref{subsec:DOM_cell}.
+The sea surface corresponds to the $w$-level $k = 1$,
+which is the same index as the $t$-level just below (\autoref{fig:DOM_index_vert}).
+The last $w$-level ($k = jpk$) either corresponds to or is below the ocean floor while
+the last $t$-level is always outside the ocean domain (\autoref{fig:DOM_index_vert}).
+Note that a $w$-point and the directly underlaying $t$-point have a common $k$ index
+(\ie\ $t$-points and their nearest $w$-point neighbour in negative index direction),
+in contrast to the indexing on the horizontal plane where
+the $t$-point has the same index as the nearest velocity points in
+the positive direction of the respective horizontal axis index
+(compare the dashed area in \autoref{fig:DOM_index_hor} and \autoref{fig:DOM_index_vert}).
 Since the scale factors are chosen to be strictly positive,
-a \textit{minus sign} appears in the \fortran code \textit{before all the vertical derivatives} of
-the discrete equations given in this documentation.
-
-%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
-\begin{figure}[!pt]
-  \begin{center}
-    \includegraphics[]{Fig_index_vert}
-    \caption{
-      \protect\label{fig:index_vert}
-      Vertical integer indexing used in the \fortran code.
-      Note that the $k$-axis is orientated downward.
-      The dashed area indicates the cell in which variables contained in arrays have the same $k$-index.
-    }
-  \end{center}
+a \textit{minus sign} is included in the \fortran\ implementations of
+\textit{all the vertical derivatives} of the discrete equations given in this manual in order to
+accommodate the opposing vertical index directions in implementation and documentation.
+
+\begin{figure}
+  \centering
+  \includegraphics[width=0.33\textwidth]{DOM_index_vert}
+  \caption[Vertical integer indexing]{
+    Vertical integer indexing used in the \fortran\ code.
+    Note that the $k$-axis is oriented downward.
+    The dashed area indicates the cell in which
+    variables contained in arrays have a common $k$-index.}
+  \label{fig:DOM_index_vert}
 \end{figure}
-%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
-
-% -----------------------------------
-%        Domain Size
-% -----------------------------------
-\subsubsection{Domain size}
+
+%% =================================================================================================
+\section{Spatial domain configuration}
+\label{subsec:DOM_config}
+
+Two typical methods are available to specify the spatial domain configuration;
+they can be selected using parameter \np{ln_read_cfg}{ln\_read\_cfg} parameter in
+namelist \nam{cfg}{cfg}.
+
+If \np{ln_read_cfg}{ln\_read\_cfg} is set to \forcode{.true.},
+the domain-specific parameters and fields are read from a NetCDF input file,
+whose name (without its .nc suffix) can be specified as
+the value of the \np{cn_domcfg}{cn\_domcfg} parameter in namelist \nam{cfg}{cfg}.
+
+If \np{ln_read_cfg}{ln\_read\_cfg} is set to \forcode{.false.},
+the domain-specific parameters and fields can be provided (\eg\ analytically computed) by
+subroutines \mdl{usrdef\_hgr} and \mdl{usrdef\_zgr}.
+These subroutines can be supplied in the \path{MY_SRC} directory of the configuration,
+and default versions that configure the spatial domain for the GYRE reference configuration are
+present in the \path{./src/OCE/USR} directory.
+
+In version 4.0 there are no longer any options for reading complex bathymetries and
+performing a vertical discretisation at run-time.
+Whilst it is occasionally convenient to have a common bathymetry file and, for example,
+to run similar models with and without partial bottom boxes and/or sigma-coordinates,
+supporting such choices leads to overly complex code.
+Worse still is the difficulty of ensuring the model configurations intended to be identical are
+indeed so when the model domain itself can be altered by runtime selections.
+The code previously used to perform vertical discretisation has been incorporated into
+an external tool (\path{./tools/DOMAINcfg}) which is briefly described in \autoref{apdx:DOMCFG}.
+
+The next subsections summarise the parameter and fields related to
+the configuration of the whole model domain.
+These represent the minimum information that must be provided either via
+the \np{cn_domcfg}{cn\_domcfg} file or
+set by code inserted into user-supplied versions of the \texttt{usrdef\_*} subroutines.
+The requirements are presented in three sections:
+the domain size (\autoref{subsec:DOM_size}), the horizontal mesh (\autoref{subsec:DOM_hgr}),
+and the vertical grid (\autoref{subsec:DOM_zgr}).
+
+%% =================================================================================================
+\subsection{Domain size}
 \label{subsec:DOM_size}
 
-The total size of the computational domain is set by the parameters \np{jpiglo},
-\np{jpjglo} and \np{jpkglo} in the $i$, $j$ and $k$ directions respectively.
-Parameters $jpi$ and $jpj$ refer to the size of each processor subdomain when
-the code is run in parallel using domain decomposition (\key{mpp\_mpi} defined,
-see \autoref{sec:LBC_mpp}).
-
-% ================================================================
-% Domain: List of fields needed
-% ================================================================
-\section{Needed fields}
-\label{sec:DOM_fields}
-The ocean mesh (\ie the position of all the scalar and vector points) is defined by the transformation that
-gives $(\lambda,\varphi,z)$ as a function of $(i,j,k)$.
-The grid-points are located at integer or integer and a half values of as indicated in \autoref{tab:cell}.
-The associated scale factors are defined using the analytical first derivative of the transformation
-\autoref{eq:scale_factors}.
-Necessary fields for configuration definition are:
+The total size of the computational domain is set by the parameters \jp{jpiglo}, \jp{jpjglo} and
+\jp{jpkglo} for the $i$, $j$ and $k$ directions, respectively.
+Note, that the variables \texttt{jpi} and \texttt{jpj} refer to
+the size of each processor subdomain when the code is run in parallel using domain decomposition
+(\key{mpp\_mpi} defined, see \autoref{sec:LBC_mpp}).
+
+The name of the configuration is set through parameter \np{cn_cfg}{cn\_cfg},
+and the nominal resolution through parameter \np{nn_cfg}{nn\_cfg}
+(unless in the input file both of variables \texttt{ORCA} and \texttt{ORCA\_index} are present,
+in which case \np{cn_cfg}{cn\_cfg} and \np{nn_cfg}{nn\_cfg} are set from these values accordingly).
+
+The global lateral boundary condition type is selected from 8 options using parameter \jp{jperio}.
+See \autoref{sec:LBC_jperio} for details on the available options and
+the corresponding values for \jp{jperio}.
+
+%% =================================================================================================
+\subsection[Horizontal grid mesh (\textit{domhgr.F90}]{Horizontal grid mesh (\protect\mdl{domhgr})}
+\label{subsec:DOM_hgr}
+
+%% =================================================================================================
+\subsubsection{Required fields}
+\label{sec:DOM_hgr_fields}
+
+The explicit specification of a range of mesh-related fields are required for
+the definition of a configuration.
+These include:
+
+\begin{clines}
+int    jpiglo, jpjglo, jpkglo     /* global domain sizes                                    */
+int    jperio                     /* lateral global domain b.c.                             */
+double glamt, glamu, glamv, glamf /* geographic longitude (t,u,v and f points respectively) */
+double gphit, gphiu, gphiv, gphif /* geographic latitude                                    */
+double e1t, e1u, e1v, e1f         /* horizontal scale factors                               */
+double e2t, e2u, e2v, e2f         /* horizontal scale factors                               */
+\end{clines}
+
+The values of the geographic longitude and latitude arrays at indices $i,j$ correspond to
+the analytical expressions of the longitude $\lambda$ and latitude $\varphi$ as a function of $(i,j)$,
+evaluated at the values as specified in \autoref{tab:DOM_cell} for the respective grid-point position.
+The calculation of the values of the horizontal scale factor arrays in general additionally involves
+partial derivatives of $\lambda$ and $\varphi$ with respect to $i$ and $j$,
+evaluated for the same arguments as $\lambda$ and $\varphi$.
+
+%% =================================================================================================
+\subsubsection{Optional fields}
+
+\begin{clines}
+                        /* Optional:                                                 */
+int    ORCA, ORCA_index /* configuration name, configuration resolution              */
+double e1e2u, e1e2v     /* U and V surfaces (if grid size reduction in some straits) */
+double ff_f, ff_t       /* Coriolis parameter (if not on the sphere)                 */
+\end{clines}
+
+\NEMO\ can support the local reduction of key strait widths by
+altering individual values of e2u or e1v at the appropriate locations.
+This is particularly useful for locations such as Gibraltar or Indonesian Throughflow pinch-points
+(see \autoref{sec:MISC_strait} for illustrated examples).
+The key is to reduce the faces of $T$-cell
+(\ie\ change the value of the horizontal scale factors at $u$- or $v$-point) but
+not the volume of the cells.
+Doing otherwise can lead to numerical instability issues.
+In normal operation the surface areas are computed from $e1u * e2u$ and $e1v * e2v$ but
+in cases where a gridsize reduction is required,
+the unaltered surface areas at $u$ and $v$ grid points
+(\texttt{e1e2u} and \texttt{e1e2v}, respectively) must be read or pre-computed in \mdl{usrdef\_hgr}.
+If these arrays are present in the \np{cn_domcfg}{cn\_domcfg} file they are read and
+the internal computation is suppressed.
+Versions of \mdl{usrdef\_hgr} which set their own values of \texttt{e1e2u} and \texttt{e1e2v} should
+set the surface-area computation flag:
+\texttt{ie1e2u\_v} to a non-zero value to suppress their re-computation.
+
+\smallskip
+Similar logic applies to the other optional fields:
+\texttt{ff\_f} and \texttt{ff\_t} which can be used to
+provide the Coriolis parameter at F- and T-points respectively if the mesh is not on a sphere.
+If present these fields will be read and used and
+the normal calculation ($2 * \Omega * \sin(\varphi)$) suppressed.
+Versions of \mdl{usrdef\_hgr} which set their own values of \texttt{ff\_f} and \texttt{ff\_t} should
+set the Coriolis computation flag:
+\texttt{iff} to a non-zero value to suppress their re-computation.
+
+Note that longitudes, latitudes, and scale factors at $w$ points are exactly equal to
+those of $t$ points, thus no specific arrays are defined at $w$ points.
+
+%% =================================================================================================
+\subsection[Vertical grid (\textit{domzgr.F90})]{Vertical grid (\protect\mdl{domzgr})}
+\label{subsec:DOM_zgr}
+
+\begin{listing}
+  \nlst{namdom}
+  \caption{\forcode{&namdom}}
+  \label{lst:namdom}
+\end{listing}
+
+In the vertical, the model mesh is determined by four things:
+\begin{enumerate}
+\item the bathymetry given in meters;
+\item the number of levels of the model (\jp{jpk});
+\item the analytical transformation $z(i,j,k)$ and the vertical scale factors
+  (derivatives of the transformation); and
+\item the masking system,
+  \ie\ the number of wet model levels at each $(i,j)$ location of the horizontal grid.
+\end{enumerate}
+
+\begin{figure}
+  \centering
+  \includegraphics[width=0.5\textwidth]{DOM_z_zps_s_sps}
+  \caption[Ocean bottom regarding coordinate systems ($z$, $s$ and hybrid $s-z$)]{
+    The ocean bottom as seen by the model:
+    \begin{enumerate*}[label=(\textit{\alph*})]
+    \item $z$-coordinate with full step,
+    \item $z$-coordinate with partial step,
+    \item $s$-coordinate: terrain following representation,
+    \item hybrid $s-z$ coordinate,
+    \item hybrid $s-z$ coordinate with partial step, and
+    \item same as (e) but in the non-linear free surface
+      (\protect\np[=.false.]{ln_linssh}{ln\_linssh}).
+  \end{enumerate*}
+  Note that the non-linear free surface can be used with any of the 5 coordinates (a) to (e).}
+  \label{fig:DOM_z_zps_s_sps}
+\end{figure}
+
+The choice of a vertical coordinate is made when setting up the configuration;
+it is not intended to be an option which can be changed in the middle of an experiment.
+The one exception to this statement being the choice of linear or non-linear free surface.
+In v4.0 the linear free surface option is implemented as
+a special case of the non-linear free surface.
+This is computationally wasteful since it uses the structures for time-varying 3D metrics
+for fields that (in the linear free surface case) are fixed.
+However, the linear free-surface is rarely used and
+implementing it this way means a single configuration file can support both options.
+
+By default a non-linear free surface is used
+(\np{ln_linssh}{ln\_linssh} set to \forcode{=.false.} in \nam{dom}{dom}):
+the coordinate follow the time-variation of the free surface so that
+the transformation is time dependent: $z(i,j,k,t)$ (\eg\ \autoref{fig:DOM_z_zps_s_sps}f).
+When a linear free surface is assumed
+(\np{ln_linssh}{ln\_linssh} set to \forcode{=.true.} in \nam{dom}{dom}),
+the vertical coordinates are fixed in time, but
+the seawater can move up and down across the $z_0$ surface
+(in other words, the top of the ocean in not a rigid lid).
+
+Note that settings:
+\np{ln_zco}{ln\_zco}, \np{ln_zps}{ln\_zps}, \np{ln_sco}{ln\_sco} and \np{ln_isfcav}{ln\_isfcav}
+mentioned in the following sections appear to be namelist options but
+they are no longer truly namelist options for \NEMO.
+Their value is written to and read from the domain configuration file and
+they should be treated as fixed parameters for a particular configuration.
+They are namelist options for the \texttt{DOMAINcfg} tool that can be used to
+build the configuration file and serve both to provide a record of the choices made whilst
+building the configuration and to trigger appropriate code blocks within \NEMO.
+These values should not be altered in the \np{cn_domcfg}{cn\_domcfg} file.
+
+\medskip
+The decision on these choices must be made when the \np{cn_domcfg}{cn\_domcfg} file is constructed.
+Three main choices are offered (\autoref{fig:DOM_z_zps_s_sps}a-c):
 
 \begin{itemize}
-\item
-  Geographic position:
-  longitude with \texttt{glamt}, \texttt{glamu}, \texttt{glamv}, \texttt{glamf} and
-  latitude  with \texttt{gphit}, \texttt{gphiu}, \texttt{gphiv}, \texttt{gphif}
-  (all respectively at T, U, V and F point)
-\item
-  Coriolis parameter (if domain not on the sphere): \texttt{ff\_f} and \texttt{ff\_t}
-  (at T and F point)
-\item
-  Scale factors:
-  \texttt{e1t}, \texttt{e1u}, \texttt{e1v} and \texttt{e1f} (on i direction),
-  \texttt{e2t}, \texttt{e2u}, \texttt{e2v} and \texttt{e2f} (on j direction) and
-  \texttt{ie1e2u\_v}, \texttt{e1e2u}, \texttt{e1e2v}. \\
-  \texttt{e1e2u}, \texttt{e1e2v} are u and v surfaces (if gridsize reduction in some straits), 
-  \texttt{ie1e2u\_v} is to flag set u and v surfaces are neither read nor computed.
+\item $z$-coordinate with full step bathymetry (\np[=.true.]{ln_zco}{ln\_zco}),
+\item $z$-coordinate with partial step ($zps$) bathymetry (\np[=.true.]{ln_zps}{ln\_zps}),
+\item Generalized, $s$-coordinate (\np[=.true.]{ln_sco}{ln\_sco}).
 \end{itemize}
- 
-These fields can be read in an domain input file which name is setted in \np{cn\_domcfg} parameter specified in
-\ngn{namcfg}.
-
-\nlst{namcfg}
-
-Or they can be defined in an analytical way in \path{MY_SRC} directory of the configuration.
-For Reference Configurations of NEMO input domain files are supplied by NEMO System Team.
-For analytical definition of input fields two routines are supplied: \mdl{usrdef\_hgr} and \mdl{usrdef\_zgr}.
-They are an example of GYRE configuration parameters, and they are available in \path{src/OCE/USR} directory,
-they provide the horizontal and vertical mesh.
-% -------------------------------------------------------------------------------------------------------------
-%        Needed fields 
-% -------------------------------------------------------------------------------------------------------------
-%\subsection{List of needed fields to build DOMAIN}
-%\label{subsec:DOM_fields_list}
-
-
-% ================================================================
-% Domain: Horizontal Grid (mesh) 
-% ================================================================
-\section{Horizontal grid mesh (\protect\mdl{domhgr})}
-\label{sec:DOM_hgr}
-
-% -------------------------------------------------------------------------------------------------------------
-%        Coordinates and scale factors 
-% -------------------------------------------------------------------------------------------------------------
-\subsection{Coordinates and scale factors}
-\label{subsec:DOM_hgr_coord_e}
-
-The ocean mesh (\ie the position of all the scalar and vector points) is defined by
-the transformation that gives $(\lambda,\varphi,z)$ as a function of $(i,j,k)$.
-The grid-points are located at integer or integer and a half values of as indicated in \autoref{tab:cell}.
-The associated scale factors are defined using the analytical first derivative of the transformation
-\autoref{eq:scale_factors}.
-These definitions are done in two modules, \mdl{domhgr} and \mdl{domzgr},
-which provide the horizontal and vertical meshes, respectively.
-This section deals with the horizontal mesh parameters.
-
-In a horizontal plane, the location of all the model grid points is defined from
-the analytical expressions of the longitude $\lambda$ and latitude $\varphi$ as a function of $(i,j)$.
-The horizontal scale factors are calculated using \autoref{eq:scale_factors}.
-For example, when the longitude and latitude are function of a single value
-($i$ and $j$, respectively) (geographical configuration of the mesh),
-the horizontal mesh definition reduces to define the wanted $\lambda(i)$, $\varphi(j)$,
-and their derivatives $\lambda'(i) \ \varphi'(j)$ in the \mdl{domhgr} module.
-The model computes the grid-point positions and scale factors in the horizontal plane as follows:
+
+Additionally, hybrid combinations of the three main coordinates are available:
+$s-z$ or $s-zps$ coordinate (\autoref{fig:DOM_z_zps_s_sps}d and \autoref{fig:DOM_z_zps_s_sps}e).
+
+A further choice related to vertical coordinate concerns
+the presence (or not) of ocean cavities beneath ice shelves within the model domain.
+A setting of \np{ln_isfcav}{ln\_isfcav} as \forcode{.true.} indicates that
+the domain contains ocean cavities,
+otherwise the top, wet layer of the ocean will always be at the ocean surface.
+This option is currently only available for $z$- or $zps$-coordinates.
+In the latter case, partial steps are also applied at the ocean/ice shelf interface.
+
+Within the model,
+the arrays describing the grid point depths and vertical scale factors are
+three set of three dimensional arrays $(i,j,k)$ defined at
+\textit{before}, \textit{now} and \textit{after} time step.
+The time at which they are defined is indicated by a suffix: $\_b$, $\_n$, or $\_a$, respectively.
+They are updated at each model time step.
+The initial fixed reference coordinate system is held in variable names with a $\_0$ suffix.
+When the linear free surface option is used (\np[=.true.]{ln_linssh}{ln\_linssh}),
+\textit{before}, \textit{now} and \textit{after} arrays are initially set to
+their reference counterpart and remain fixed.
+
+%% =================================================================================================
+\subsubsection{Required fields}
+\label{sec:DOM_zgr_fields}
+
+The explicit specification of a range of fields related to the vertical grid are required for
+the definition of a configuration.
+These include:
+
+\begin{clines}
+int    ln_zco, ln_zps, ln_sco            /* flags for z-coord, z-coord with partial steps and s-coord    */
+int    ln_isfcav                         /* flag  for ice shelf cavities                                 */
+double e3t_1d, e3w_1d                    /* reference vertical scale factors at T and W points           */
+double e3t_0, e3u_0, e3v_0, e3f_0, e3w_0 /* vertical scale factors 3D coordinate at T,U,V,F and W points */
+double e3uw_0, e3vw_0                    /* vertical scale factors 3D coordinate at UW and VW points     */
+int    bottom_level, top_level           /* last wet T-points, 1st wet T-points (for ice shelf cavities) */
+                                         /* For reference:                                               */
+float  bathy_metry                       /* bathymetry used in setting top and bottom levels             */
+\end{clines}
+
+This set of vertical metrics is sufficient to describe the initial depth and thickness of
+every gridcell in the model regardless of the choice of vertical coordinate.
+With constant z-levels, e3 metrics will be uniform across each horizontal level.
+In the partial step case each e3 at the \jp{bottom\_level}
+(and, possibly, \jp{top\_level} if ice cavities are present)
+may vary from its horizontal neighbours.
+And, in s-coordinates, variations can occur throughout the water column.
+With the non-linear free-surface, all the coordinates behave more like the s-coordinate in that
+variations occur throughout the water column with displacements related to the sea surface height.
+These variations are typically much smaller than those arising from bottom fitted coordinates.
+The values for vertical metrics supplied in the domain configuration file can be considered as
+those arising from a flat sea surface with zero elevation.
+
+The \jp{bottom\_level} and \jp{top\_level} 2D arrays define
+the \jp{bottom\_level} and top wet levels in each grid column.
+Without ice cavities, \jp{top\_level} is essentially a land mask (0 on land; 1 everywhere else).
+With ice cavities, \jp{top\_level} determines the first wet point below the overlying ice shelf.
+
+%% =================================================================================================
+\subsubsection{Level bathymetry and mask}
+\label{subsec:DOM_msk}
+
+From \jp{top\_level} and \jp{bottom\_level} fields, the mask fields are defined as follows:
 \begin{align*}
-   \lambda_t &\equiv \text{glamt} =      \lambda (i      )
-  &\varphi_t &\equiv \text{gphit} =      \varphi (j      ) \\
-   \lambda_u &\equiv \text{glamu} =      \lambda (i + 1/2)
-  &\varphi_u &\equiv \text{gphiu} =      \varphi (j      ) \\
-   \lambda_v &\equiv \text{glamv} =      \lambda (i      )
-  &\varphi_v &\equiv \text{gphiv} =      \varphi (j + 1/2) \\
-   \lambda_f &\equiv \text{glamf} =      \lambda (i + 1/2)
-  &\varphi_f &\equiv \text{gphif} =      \varphi (j + 1/2) \\
-   e_{1t}    &\equiv \text{e1t}   = r_a |\lambda'(i      ) \; \cos\varphi(j      ) |
-  &e_{2t}    &\equiv \text{e2t}   = r_a |\varphi'(j      )                         | \\
-   e_{1u}    &\equiv \text{e1t}   = r_a |\lambda'(i + 1/2) \; \cos\varphi(j      ) |
-  &e_{2u}    &\equiv \text{e2t}   = r_a |\varphi'(j      )                         | \\
-   e_{1v}    &\equiv \text{e1t}   = r_a |\lambda'(i      ) \; \cos\varphi(j + 1/2) |
-  &e_{2v}    &\equiv \text{e2t}   = r_a |\varphi'(j + 1/2)                         | \\
-   e_{1f}    &\equiv \text{e1t}   = r_a |\lambda'(i + 1/2) \; \cos\varphi(j + 1/2) |
-  &e_{2f}    &\equiv \text{e2t}   = r_a |\varphi'(j + 1/2)                         |
+  tmask(i,j,k) &=
+  \begin{cases}
+    0 &\text{if $                             k <    top\_level(i,j)$} \\
+    1 &\text{if $     bottom\_level(i,j) \leq k \leq top\_level(i,j)$} \\
+    0 &\text{if $k >  bottom\_level(i,j)                            $}
+  \end{cases} \\
+  umask(i,j,k) &= tmask(i,j,k) * tmask(i + 1,j,    k) \\
+  vmask(i,j,k) &= tmask(i,j,k) * tmask(i    ,j + 1,k) \\
+  fmask(i,j,k) &= tmask(i,j,k) * tmask(i + 1,j,    k) * tmask(i,j,k) * tmask(i + 1,j,    k) \\
+  wmask(i,j,k) &= tmask(i,j,k) * tmask(i    ,j,k - 1) \\
+  \text{with~} wmask(i,j,1) &= tmask(i,j,1)
 \end{align*}
-where the last letter of each computational name indicates the grid point considered and
-$r_a$ is the earth radius (defined in \mdl{phycst} along with all universal constants).
-Note that the horizontal position of and scale factors at $w$-points are exactly equal to those of $t$-points,
-thus no specific arrays are defined at $w$-points.
-
-Note that the definition of the scale factors
-(\ie as the analytical first derivative of the transformation that
-gives $(\lambda,\varphi,z)$ as a function of $(i,j,k)$)
-is specific to the \NEMO model \citep{Marti_al_JGR92}.
-As an example, $e_{1t}$ is defined locally at a $t$-point,
-whereas many other models on a C grid choose to define such a scale factor as
-the distance between the $U$-points on each side of the $t$-point.
-Relying on an analytical transformation has two advantages:
-firstly, there is no ambiguity in the scale factors appearing in the discrete equations,
-since they are first introduced in the continuous equations;
-secondly, analytical transformations encourage good practice by the definition of smoothly varying grids
-(rather than allowing the user to set arbitrary jumps in thickness between adjacent layers) \citep{Treguier1996}.
-An example of the effect of such a choice is shown in \autoref{fig:zgr_e3}.
-%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
-\begin{figure}[!t]
-  \begin{center}
-    \includegraphics[]{Fig_zgr_e3}
-    \caption{
-      \protect\label{fig:zgr_e3}
-      Comparison of (a) traditional definitions of grid-point position and grid-size in the vertical,
-      and (b) analytically derived grid-point position and scale factors.
-      For both grids here, the same $w$-point depth has been chosen but
-      in (a) the $t$-points are set half way between $w$-points while
-      in (b) they are defined from an analytical function:
-      $z(k) = 5 \, (k - 1/2)^3 - 45 \, (k - 1/2)^2 + 140 \, (k - 1/2) - 150$.
-      Note the resulting difference between the value of the grid-size $\Delta_k$ and
-      those of the scale factor $e_k$.
-    }
-  \end{center}
-\end{figure}
-%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
-
-% -------------------------------------------------------------------------------------------------------------
-%        Choice of horizontal grid
-% -------------------------------------------------------------------------------------------------------------
-\subsection{Choice of horizontal grid}
-\label{subsec:DOM_hgr_msh_choice}
-
-% -------------------------------------------------------------------------------------------------------------
-%        Grid files
-% -------------------------------------------------------------------------------------------------------------
-\subsection{Output grid files}
-\label{subsec:DOM_hgr_files}
-
-All the arrays relating to a particular ocean model configuration (grid-point position, scale factors, masks)
-can be saved in files if \np{nn\_msh} $\not = 0$ (namelist variable in \ngn{namdom}).
-This can be particularly useful for plots and off-line diagnostics.
-In some cases, the user may choose to make a local modification of a scale factor in the code.
-This is the case in global configurations when restricting the width of a specific strait
-(usually a one-grid-point strait that happens to be too wide due to insufficient model resolution).
-An example is Gibraltar Strait in the ORCA2 configuration.
-When such modifications are done,
-the output grid written when \np{nn\_msh} $\not = 0$ is no more equal to the input grid.
-
-% ================================================================
-% Domain: Vertical Grid (domzgr)
-% ================================================================
-\section{Vertical grid (\protect\mdl{domzgr})}
-\label{sec:DOM_zgr}
-%-----------------------------------------nam_zgr & namdom-------------------------------------------
-%
-%\nlst{namzgr} 
-
-\nlst{namdom} 
-%-------------------------------------------------------------------------------------------------------------
-
-Variables are defined through the \ngn{namzgr} and \ngn{namdom} namelists.
-In the vertical, the model mesh is determined by four things: 
-(1) the bathymetry given in meters; 
-(2) the number of levels of the model (\jp{jpk}); 
-(3) the analytical transformation $z(i,j,k)$ and the vertical scale factors (derivatives of the transformation); and
-(4) the masking system, \ie the number of wet model levels at each 
-$(i,j)$ column of points.
-
-%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
-\begin{figure}[!tb]
-  \begin{center}
-    \includegraphics[]{Fig_z_zps_s_sps}
-    \caption{
-      \protect\label{fig:z_zps_s_sps}
-      The ocean bottom as seen by the model:
-      (a) $z$-coordinate with full step,
-      (b) $z$-coordinate with partial step,
-      (c) $s$-coordinate: terrain following representation,
-      (d) hybrid $s-z$ coordinate,
-      (e) hybrid $s-z$ coordinate with partial step, and
-      (f) same as (e) but in the non-linear free surface (\protect\np{ln\_linssh}~\forcode{= .false.}).
-      Note that the non-linear free surface can be used with any of the 5 coordinates (a) to (e).
-    }
-  \end{center}
-\end{figure}
-%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
-
-The choice of a vertical coordinate, even if it is made through \ngn{namzgr} namelist parameters, 
-must be done once of all at the beginning of an experiment.
-It is not intended as an option which can be enabled or disabled in the middle of an experiment.
-Three main choices are offered (\autoref{fig:z_zps_s_sps}):
-$z$-coordinate with full step bathymetry (\np{ln\_zco}~\forcode{= .true.}),
-$z$-coordinate with partial step bathymetry (\np{ln\_zps}~\forcode{= .true.}),
-or generalized, $s$-coordinate (\np{ln\_sco}~\forcode{= .true.}).
-Hybridation of the three main coordinates are available:
-$s-z$ or $s-zps$ coordinate (\autoref{fig:z_zps_s_sps} and \autoref{fig:z_zps_s_sps}).
-By default a non-linear free surface is used: the coordinate follow the time-variation of the free surface so that
-the transformation is time dependent: $z(i,j,k,t)$ (\autoref{fig:z_zps_s_sps}).
-When a linear free surface is assumed (\np{ln\_linssh}~\forcode{= .true.}),
-the vertical coordinate are fixed in time, but the seawater can move up and down across the $z_0$ surface
-(in other words, the top of the ocean in not a rigid-lid).
-The last choice in terms of vertical coordinate concerns the presence (or not) in
-the model domain of ocean cavities beneath ice shelves.
-Setting \np{ln\_isfcav} to true allows to manage ocean cavities, otherwise they are filled in.
-This option is currently only available in $z$- or $zps$-coordinate,
-and partial step are also applied at the ocean/ice shelf interface.
-
-Contrary to the horizontal grid, the vertical grid is computed in the code and no provision is made for
-reading it from a file.
-The only input file is the bathymetry (in meters) (\ifile{bathy\_meter})
-\footnote{
-  N.B. in full step $z$-coordinate, a \ifile{bathy\_level} file can replace the \ifile{bathy\_meter} file,
-  so that the computation of the number of wet ocean point in each water column is by-passed}.
-If \np{ln\_isfcav}~\forcode{= .true.}, an extra file input file (\ifile{isf\_draft\_meter}) describing
-the ice shelf draft (in meters) is needed.
-
-After reading the bathymetry, the algorithm for vertical grid definition differs between the different options:
-\begin{description}
-\item[\textit{zco}]
-  set a reference coordinate transformation $z_0(k)$, and set $z(i,j,k,t) = z_0(k)$.
-\item[\textit{zps}]
-  set a reference coordinate transformation $z_0(k)$, and calculate the thickness of the deepest level at
-  each $(i,j)$ point using the bathymetry, to obtain the final three-dimensional depth and scale factor arrays.
-\item[\textit{sco}]
-  smooth the bathymetry to fulfill the hydrostatic consistency criteria and
-  set the three-dimensional transformation.
-\item[\textit{s-z} and \textit{s-zps}]
-  smooth the bathymetry to fulfill the hydrostatic consistency criteria and
-  set the three-dimensional transformation $z(i,j,k)$,
-  and possibly introduce masking of extra land points to better fit the original bathymetry file.
-\end{description}
-%%%
-\gmcomment{   add the description of the smoothing:  envelop topography...}
-%%%
-
-Unless a linear free surface is used (\np{ln\_linssh}~\forcode{= .false.}),
-the arrays describing the grid point depths and vertical scale factors are three set of
-three dimensional arrays $(i,j,k)$ defined at \textit{before}, \textit{now} and \textit{after} time step.
-The time at which they are defined is indicated by a suffix: $\_b$, $\_n$, or $\_a$, respectively.
-They are updated at each model time step using a fixed reference coordinate system which
-computer names have a $\_0$ suffix.
-When the linear free surface option is used (\np{ln\_linssh}~\forcode{= .true.}), \textit{before},
-\textit{now} and \textit{after} arrays are simply set one for all to their reference counterpart.
-
-% -------------------------------------------------------------------------------------------------------------
-%        Meter Bathymetry
-% -------------------------------------------------------------------------------------------------------------
-\subsection{Meter bathymetry}
-\label{subsec:DOM_bathy}
-
-Three options are possible for defining the bathymetry, according to the namelist variable \np{nn\_bathy}
-(found in \ngn{namdom} namelist): 
-\begin{description}
-\item[\np{nn\_bathy}~\forcode{= 0}]:
-  a flat-bottom domain is defined.
-  The total depth $z_w (jpk)$ is given by the coordinate transformation.
-  The domain can either be a closed basin or a periodic channel depending on the parameter \np{jperio}.
-\item[\np{nn\_bathy}~\forcode{= -1}]:
-  a domain with a bump of topography one third of the domain width at the central latitude.
-  This is meant for the "EEL-R5" configuration, a periodic or open boundary channel with a seamount.
-\item[\np{nn\_bathy}~\forcode{= 1}]:
-  read a bathymetry and ice shelf draft (if needed).
-  The \ifile{bathy\_meter} file (Netcdf format) provides the ocean depth (positive, in meters) at
-  each grid point of the model grid.
-  The bathymetry is usually built by interpolating a standard bathymetry product (\eg ETOPO2) onto
-  the horizontal ocean mesh.
-  Defining the bathymetry also defines the coastline: where the bathymetry is zero,
-  no model levels are defined (all levels are masked).
-
-  The \ifile{isfdraft\_meter} file (Netcdf format) provides the ice shelf draft (positive, in meters) at
-  each grid point of the model grid.
-  This file is only needed if \np{ln\_isfcav}~\forcode{= .true.}.
-  Defining the ice shelf draft will also define the ice shelf edge and the grounding line position.
-\end{description}
-
-When a global ocean is coupled to an atmospheric model it is better to represent all large water bodies
-(\eg great lakes, Caspian sea...) even if the model resolution does not allow their communication with
-the rest of the ocean.
+
+Note that, without ice shelves cavities,
+masks at $t-$ and $w-$points are identical with the numerical indexing used
+(\autoref{subsec:DOM_Num_Index}).
+Nevertheless,
+$wmask$ are required with ocean cavities to deal with the top boundary (ice shelf/ocean interface)
+exactly in the same way as for the bottom boundary.
+
+%% The specification of closed lateral boundaries requires that at least
+%% the first and last rows and columns of the \textit{mbathy} array are set to zero.
+%% In the particular case of an east-west cyclical boundary condition, \textit{mbathy} has its last column equal to
+%% the second one and its first column equal to the last but one (and so too the mask arrays)
+%% (see \autoref{fig:LBC_jperio}).
+
+%        Closed seas
+%% =================================================================================================
+\subsection{Closed seas}
+\label{subsec:DOM_closea}
+
+When a global ocean is coupled to an atmospheric model it is better to
+represent all large water bodies (\eg\ Great Lakes, Caspian sea, \dots) even if
+the model resolution does not allow their communication with the rest of the ocean.
 This is unnecessary when the ocean is forced by fixed atmospheric conditions,
 so these seas can be removed from the ocean domain.
-The user has the option to set the bathymetry in closed seas to zero (see \autoref{sec:MISC_closea}),
-but the code has to be adapted to the user's configuration.
-
-% -------------------------------------------------------------------------------------------------------------
-%        z-coordinate  and reference coordinate transformation
-% -------------------------------------------------------------------------------------------------------------
-\subsection[$Z$-coordinate (\protect\np{ln\_zco}~\forcode{= .true.}) and ref. coordinate]
-				{$Z$-coordinate (\protect\np{ln\_zco}~\forcode{= .true.}) and reference coordinate}
-\label{subsec:DOM_zco}
-
-%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
-\begin{figure}[!tb]
-  \begin{center}
-    \includegraphics[]{Fig_zgr}
-    \caption{
-      \protect\label{fig:zgr}
-      Default vertical mesh for ORCA2: 30 ocean levels (L30).
-      Vertical level functions for (a) T-point depth and (b) the associated scale factor as computed from
-      \autoref{eq:DOM_zgr_ana_1} using \autoref{eq:DOM_zgr_coef} in $z$-coordinate.
-    }
-  \end{center}
-\end{figure}
-%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
-
-The reference coordinate transformation $z_0(k)$ defines the arrays $gdept_0$ and $gdepw_0$ for $t$- and $w$-points,
-respectively.
-As indicated on \autoref{fig:index_vert} \jp{jpk} is the number of $w$-levels.
-$gdepw_0(1)$ is the ocean surface.
-There are at most \jp{jpk}-1 $t$-points inside the ocean,
-the additional $t$-point at $jk = jpk$ is below the sea floor and is not used.
-The vertical location of $w$- and $t$-levels is defined from the analytic expression of the depth $z_0(k)$ whose
-analytical derivative with respect to $k$ provides the vertical scale factors.
-The user must provide the analytical expression of both $z_0$ and its first derivative with respect to $k$.
-This is done in routine \mdl{domzgr} through statement functions,
-using parameters provided in the \ngn{namcfg} namelist.
-
-It is possible to define a simple regular vertical grid by giving zero stretching (\np{ppacr}~\forcode{= 0}).
-In that case, the parameters \jp{jpk} (number of $w$-levels) and
-\np{pphmax} (total ocean depth in meters) fully define the grid.
-
-For climate-related studies it is often desirable to concentrate the vertical resolution near the ocean surface.
-The following function is proposed as a standard for a $z$-coordinate (with either full or partial steps): 
-\begin{gather}
-  \label{eq:DOM_zgr_ana_1}
-    z_0  (k) = h_{sur} - h_0 \; k - \; h_1 \; \log  \big[ \cosh ((k - h_{th}) / h_{cr}) \big] \\
-    e_3^0(k) = \lt|    - h_0      -    h_1 \; \tanh \big[        (k - h_{th}) / h_{cr}  \big] \rt|
-\end{gather}
-where $k = 1$ to \jp{jpk} for $w$-levels and $k = 1$ to $k = 1$ for $T-$levels.
-Such an expression allows us to define a nearly uniform vertical location of levels at the ocean top and bottom with
-a smooth hyperbolic tangent transition in between (\autoref{fig:zgr}).
-
-If the ice shelf cavities are opened (\np{ln\_isfcav}~\forcode{= .true.}), the definition of $z_0$ is the same.
-However, definition of $e_3^0$ at $t$- and $w$-points is respectively changed to:
-\begin{equation}
-  \label{eq:DOM_zgr_ana_2}
-  \begin{split}
-    e_3^T(k) &= z_W (k + 1) - z_W (k    ) \\
-    e_3^W(k) &= z_T (k    ) - z_T (k - 1)
-  \end{split}
-\end{equation}
-This formulation decrease the self-generated circulation into the ice shelf cavity 
-(which can, in extreme case, leads to blow up).\\
- 
-The most used vertical grid for ORCA2 has $10~m$ ($500~m$) resolution in the surface (bottom) layers and
-a depth which varies from 0 at the sea surface to a minimum of $-5000~m$.
-This leads to the following conditions:
-\begin{equation}
-  \label{eq:DOM_zgr_coef}
-  \begin{array}{ll}
-    e_3 (1   + 1/2) =  10. & z(1  ) =     0. \\
-    e_3 (jpk - 1/2) = 500. & z(jpk) = -5000.
-  \end{array}
-\end{equation}
-
-With the choice of the stretching $h_{cr} = 3$ and the number of levels \jp{jpk}~$= 31$,
-the four coefficients $h_{sur}$, $h_0$, $h_1$, and $h_{th}$ in
-\autoref{eq:DOM_zgr_ana_2} have been determined such that
-\autoref{eq:DOM_zgr_coef} is satisfied, through an optimisation procedure using a bisection method.
-For the first standard ORCA2 vertical grid this led to the following values:
-$h_{sur} = 4762.96$, $h_0 = 255.58, h_1 = 245.5813$, and $h_{th} = 21.43336$.
-The resulting depths and scale factors as a function of the model levels are shown in
-\autoref{fig:zgr} and given in \autoref{tab:orca_zgr}.
-Those values correspond to the parameters \np{ppsur}, \np{ppa0}, \np{ppa1}, \np{ppkth} in \ngn{namcfg} namelist.
-
-Rather than entering parameters $h_{sur}$, $h_0$, and $h_1$ directly, it is possible to recalculate them.
-In that case the user sets \np{ppsur}~$=$~\np{ppa0}~$=$~\np{ppa1}~$= 999999$.,
-in \ngn{namcfg} namelist, and specifies instead the four following parameters:
-\begin{itemize}
-\item
-  \np{ppacr}~$= h_{cr}$: stretching factor (nondimensional).
-  The larger \np{ppacr}, the smaller the stretching.
-  Values from $3$ to $10$ are usual.
-\item
-  \np{ppkth}~$= h_{th}$: is approximately the model level at which maximum stretching occurs
-  (nondimensional, usually of order 1/2 or 2/3 of \jp{jpk})
-\item
-  \np{ppdzmin}: minimum thickness for the top layer (in meters).
-\item
-  \np{pphmax}: total depth of the ocean (meters).
-\end{itemize}
-As an example, for the $45$ layers used in the DRAKKAR configuration those parameters are:
-\jp{jpk}~$= 46$, \np{ppacr}~$= 9$, \np{ppkth}~$= 23.563$, \np{ppdzmin}~$= 6~m$, \np{pphmax}~$= 5750~m$.
-
-%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
-\begin{table}
-  \begin{center}
-    \begin{tabular}{c||r|r|r|r}
-      \hline
-      \textbf{LEVEL} & \textbf{gdept\_1d} & \textbf{gdepw\_1d} & \textbf{e3t\_1d } & \textbf{e3w\_1d} \\
-      \hline
-      1              & \textbf{     5.00} &               0.00 & \textbf{   10.00} &            10.00 \\
-      \hline
-      2              & \textbf{    15.00} &	            10.00 & \textbf{   10.00} &            10.00 \\
-      \hline
-      3              & \textbf{    25.00} &              20.00 & \textbf{   10.00} &            10.00 \\
-      \hline
-      4              & \textbf{    35.01} &	            30.00 & \textbf{   10.01} &            10.00 \\
-      \hline
-      5              & \textbf{    45.01} &              40.01 & \textbf{   10.01} &            10.01 \\
-      \hline
-      6              & \textbf{    55.03} &              50.02 & \textbf{   10.02} &            10.02 \\
-      \hline
-      7              & \textbf{    65.06} &              60.04 & \textbf{   10.04} &            10.03 \\
-      \hline
-      8              & \textbf{    75.13} &              70.09 & \textbf{   10.09} &            10.06 \\
-      \hline
-      9              & \textbf{    85.25} &              80.18 & \textbf{   10.17} &            10.12 \\
-      \hline
-      10             & \textbf{    95.49} &              90.35 & \textbf{   10.33} &            10.24 \\
-      \hline
-      11             & \textbf{   105.97} &             100.69 & \textbf{   10.65} &            10.47 \\
-      \hline
-      12             & \textbf{   116.90} &             111.36 & \textbf{   11.27} &            10.91 \\
-      \hline
-      13             & \textbf{   128.70} &             122.65 & \textbf{   12.47} &            11.77 \\
-      \hline
-      14             & \textbf{   142.20} &             135.16 & \textbf{   14.78} &            13.43 \\
-      \hline
-      15             & \textbf{   158.96} &             150.03 & \textbf{   19.23} &            16.65 \\
-      \hline
-      16             & \textbf{   181.96} &             169.42 & \textbf{   27.66} &            22.78 \\
-      \hline
-      17             & \textbf{   216.65} &             197.37 & \textbf{   43.26} &            34.30 \\
-      \hline
-      18             & \textbf{   272.48} &             241.13 & \textbf{   70.88} &            55.21 \\
-      \hline
-      19             & \textbf{   364.30} &             312.74 & \textbf{  116.11} &            90.99 \\
-      \hline
-      20             & \textbf{   511.53} &             429.72 & \textbf{  181.55} &           146.43 \\
-      \hline
-      21             & \textbf{   732.20} &             611.89 & \textbf{  261.03} &           220.35 \\
-      \hline
-      22             & \textbf{  1033.22} &             872.87 & \textbf{  339.39} &           301.42 \\
-      \hline
-      23             & \textbf{  1405.70} &            1211.59 & \textbf{  402.26} &           373.31 \\
-      \hline
-      24             & \textbf{  1830.89} &            1612.98 & \textbf{  444.87} &           426.00 \\
-      \hline
-      25             & \textbf{  2289.77} &            2057.13 & \textbf{  470.55} &           459.47 \\
-      \hline
-      26             & \textbf{  2768.24} &            2527.22 & \textbf{  484.95} &           478.83 \\
-      \hline
-      27             & \textbf{  3257.48} &            3011.90 & \textbf{  492.70} &           489.44 \\
-      \hline
-      28             & \textbf{  3752.44} &            3504.46 & \textbf{  496.78} &           495.07 \\
-      \hline
-      29             & \textbf{  4250.40} &            4001.16 & \textbf{  498.90} &           498.02 \\
-      \hline
-      30             & \textbf{  4749.91} &            4500.02 & \textbf{  500.00} &           499.54 \\
-      \hline
-      31             & \textbf{  5250.23} &            5000.00 & \textbf{  500.56} &           500.33 \\
-      \hline
-    \end{tabular}
-  \end{center}
-  \caption{
-    \protect\label{tab:orca_zgr}
-    Default vertical mesh in $z$-coordinate for 30 layers ORCA2 configuration as computed from
-    \autoref{eq:DOM_zgr_ana_2} using the coefficients given in \autoref{eq:DOM_zgr_coef}
-  }
-\end{table}
-%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
-
-% -------------------------------------------------------------------------------------------------------------
-%        z-coordinate with partial step
-% -------------------------------------------------------------------------------------------------------------
-\subsection{$Z$-coordinate with partial step (\protect\np{ln\_zps}~\forcode{= .true.})}
-\label{subsec:DOM_zps}
-%--------------------------------------------namdom-------------------------------------------------------
-
-\nlst{namdom} 
-%--------------------------------------------------------------------------------------------------------------
-
-In $z$-coordinate partial step,
-the depths of the model levels are defined by the reference analytical function $z_0(k)$ as described in
-the previous section, \textit{except} in the bottom layer.
-The thickness of the bottom layer is allowed to vary as a function of geographical location $(\lambda,\varphi)$ to
-allow a better representation of the bathymetry, especially in the case of small slopes
-(where the bathymetry varies by less than one level thickness from one grid point to the next).
-The reference layer thicknesses $e_{3t}^0$ have been defined in the absence of bathymetry.
-With partial steps, layers from 1 to \jp{jpk}-2 can have a thickness smaller than $e_{3t}(jk)$.
-The model deepest layer (\jp{jpk}-1) is allowed to have either a smaller or larger thickness than $e_{3t}(jpk)$:
-the maximum thickness allowed is $2*e_{3t}(jpk - 1)$.
-This has to be kept in mind when specifying values in \ngn{namdom} namelist,
-as the maximum depth \np{pphmax} in partial steps:
-for example, with \np{pphmax}~$= 5750~m$ for the DRAKKAR 45 layer grid,
-the maximum ocean depth allowed is actually $6000~m$ (the default thickness $e_{3t}(jpk - 1)$ being $250~m$).
-Two variables in the namdom namelist are used to define the partial step vertical grid.
-The mimimum water thickness (in meters) allowed for a cell partially filled with bathymetry at level jk is
-the minimum of \np{rn\_e3zps\_min} (thickness in meters, usually $20~m$) or $e_{3t}(jk)*$\np{rn\_e3zps\_rat}
-(a fraction, usually 10\%, of the default thickness $e_{3t}(jk)$).
-
-\gmcomment{ \colorbox{yellow}{Add a figure here of pstep especially at last ocean level }  }
-
-% -------------------------------------------------------------------------------------------------------------
-%        s-coordinate
-% -------------------------------------------------------------------------------------------------------------
-\subsection{$S$-coordinate (\protect\np{ln\_sco}~\forcode{= .true.})}
-\label{subsec:DOM_sco}
-%------------------------------------------nam_zgr_sco---------------------------------------------------
-%
-%\nlst{namzgr_sco} 
-%--------------------------------------------------------------------------------------------------------------
-Options are defined in \ngn{namzgr\_sco}.
-In $s$-coordinate (\np{ln\_sco}~\forcode{= .true.}), the depth and thickness of the model levels are defined from
-the product of a depth field and either a stretching function or its derivative, respectively:
-
-\begin{align*}
-  % \label{eq:DOM_sco_ana}
-  z(k)   &= h(i,j) \; z_0 (k) \\
-  e_3(k) &= h(i,j) \; z_0'(k)
-\end{align*}
-
-where $h$ is the depth of the last $w$-level ($z_0(k)$) defined at the $t$-point location in the horizontal and
-$z_0(k)$ is a function which varies from $0$ at the sea surface to $1$ at the ocean bottom.
-The depth field $h$ is not necessary the ocean depth,
-since a mixed step-like and bottom-following representation of the topography can be used
-(\autoref{fig:z_zps_s_sps}) or an envelop bathymetry can be defined (\autoref{fig:z_zps_s_sps}).
-The namelist parameter \np{rn\_rmax} determines the slope at which
-the terrain-following coordinate intersects the sea bed and becomes a pseudo z-coordinate.
-The coordinate can also be hybridised by specifying \np{rn\_sbot\_min} and \np{rn\_sbot\_max} as
-the minimum and maximum depths at which the terrain-following vertical coordinate is calculated.
-
-Options for stretching the coordinate are provided as examples,
-but care must be taken to ensure that the vertical stretch used is appropriate for the application.
-
-The original default NEMO s-coordinate stretching is available if neither of the other options are specified as true
-(\np{ln\_s\_SH94}~\forcode{= .false.} and \np{ln\_s\_SF12}~\forcode{= .false.}).
-This uses a depth independent $\tanh$ function for the stretching \citep{Madec_al_JPO96}:
-
-\[
-  z = s_{min} + C (s) (H - s_{min})
-  % \label{eq:SH94_1}
-\]
-
-where $s_{min}$ is the depth at which the $s$-coordinate stretching starts and
-allows a $z$-coordinate to placed on top of the stretched coordinate,
-and $z$ is the depth (negative down from the asea surface).
-\begin{gather*}
-  s = - \frac{k}{n - 1} \quad \text{and} \quad 0 \leq k \leq n - 1
-  % \label{eq:DOM_s}
- \\
-  % \label{eq:DOM_sco_function}
-  C(s) = \frac{[\tanh(\theta \, (s + b)) - \tanh(\theta \, b)]}{2 \; \sinh(\theta)}
-\end{gather*}
-
-A stretching function,
-modified from the commonly used \citet{Song_Haidvogel_JCP94} stretching (\np{ln\_s\_SH94}~\forcode{= .true.}),
-is also available and is more commonly used for shelf seas modelling:
-
-\[
-  C(s) =   (1 - b) \frac{\sinh(\theta s)}{\sinh(\theta)}
-         + b       \frac{\tanh \lt[ \theta \lt(s + \frac{1}{2} \rt) \rt] -   \tanh \lt( \frac{\theta}{2} \rt)}
-                        {                                                  2 \tanh \lt( \frac{\theta}{2} \rt)}
-  % \label{eq:SH94_2}
-\]
-
-%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
-\begin{figure}[!ht]
-  \begin{center}
-    \includegraphics[]{Fig_sco_function}
-    \caption{
-      \protect\label{fig:sco_function}
-      Examples of the stretching function applied to a seamount;
-      from left to right: surface, surface and bottom, and bottom intensified resolutions
-    }
-  \end{center}
-\end{figure}
-%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
-
-where $H_c$ is the critical depth (\np{rn\_hc}) at which the coordinate transitions from pure $\sigma$ to
-the stretched coordinate, and $\theta$ (\np{rn\_theta}) and $b$ (\np{rn\_bb}) are the surface and
-bottom control parameters such that $0 \leqslant \theta \leqslant 20$, and $0 \leqslant b \leqslant 1$.
-$b$ has been designed to allow surface and/or bottom increase of the vertical resolution
-(\autoref{fig:sco_function}).
-
-Another example has been provided at version 3.5 (\np{ln\_s\_SF12}) that allows a fixed surface resolution in
-an analytical terrain-following stretching \citet{Siddorn_Furner_OM12}.
-In this case the a stretching function $\gamma$ is defined such that:
-
-\begin{equation}
-  z = - \gamma h \quad \text{with} \quad 0 \leq \gamma \leq 1
-  % \label{eq:z}
-\end{equation}
-
-The function is defined with respect to $\sigma$, the unstretched terrain-following coordinate:
-
-\begin{gather*}
-  % \label{eq:DOM_gamma_deriv}
-  \gamma =   A \lt( \sigma   - \frac{1}{2} (\sigma^2     + f (\sigma)) \rt)
-           + B \lt( \sigma^3 - f           (\sigma) \rt) + f (\sigma)       \\
-  \intertext{Where:}
-  % \label{eq:DOM_gamma}
-  f(\sigma) = (\alpha + 2) \sigma^{\alpha + 1} - (\alpha + 1) \sigma^{\alpha + 2}
-  \quad \text{and} \quad \sigma = \frac{k}{n - 1}
-\end{gather*}
-
-This gives an analytical stretching of $\sigma$ that is solvable in $A$ and $B$ as a function of
-the user prescribed stretching parameter $\alpha$ (\np{rn\_alpha}) that stretches towards
-the surface ($\alpha > 1.0$) or the bottom ($\alpha < 1.0$) and
-user prescribed surface (\np{rn\_zs}) and bottom depths.
-The bottom cell depth in this example is given as a function of water depth:
-
-\[
-  % \label{eq:DOM_zb}
-  Z_b = h a + b
-\]
-
-where the namelist parameters \np{rn\_zb\_a} and \np{rn\_zb\_b} are $a$ and $b$ respectively.
-
-%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
-\begin{figure}[!ht]
-  \includegraphics[]{Fig_DOM_compare_coordinates_surface}
-  \caption{
-    A comparison of the \citet{Song_Haidvogel_JCP94} $S$-coordinate (solid lines),
-    a 50 level $Z$-coordinate (contoured surfaces) and
-    the \citet{Siddorn_Furner_OM12} $S$-coordinate (dashed lines) in the surface $100~m$ for
-    a idealised bathymetry that goes from $50~m$ to $5500~m$ depth.
-    For clarity every third coordinate surface is shown.
-  }
-  \label{fig:fig_compare_coordinates_surface}
-\end{figure}
- % >>>>>>>>>>>>>>>>>>>>>>>>>>>>
-
-This gives a smooth analytical stretching in computational space that is constrained to
-given specified surface and bottom grid cell thicknesses in real space.
-This is not to be confused with the hybrid schemes that
-superimpose geopotential coordinates on terrain following coordinates thus
-creating a non-analytical vertical coordinate that
-therefore may suffer from large gradients in the vertical resolutions.
-This stretching is less straightforward to implement than the \citet{Song_Haidvogel_JCP94} stretching,
-but has the advantage of resolving diurnal processes in deep water and has generally flatter slopes.
-
-As with the \citet{Song_Haidvogel_JCP94} stretching the stretch is only applied at depths greater than
-the critical depth $h_c$.
-In this example two options are available in depths shallower than $h_c$,
-with pure sigma being applied if the \np{ln\_sigcrit} is true and pure z-coordinates if it is false
-(the z-coordinate being equal to the depths of the stretched coordinate at $h_c$).
-
-Minimising the horizontal slope of the vertical coordinate is important in terrain-following systems as
-large slopes lead to hydrostatic consistency.
-A hydrostatic consistency parameter diagnostic following \citet{Haney1991} has been implemented,
-and is output as part of the model mesh file at the start of the run.
-
-% -------------------------------------------------------------------------------------------------------------
-%        z*- or s*-coordinate
-% -------------------------------------------------------------------------------------------------------------
-\subsection{\zstar- or \sstar-coordinate (\protect\np{ln\_linssh}~\forcode{= .false.})}
-\label{subsec:DOM_zgr_star}
-
-This option is described in the Report by Levier \textit{et al.} (2007), available on the \NEMO web site.
-
-%gm% key advantage: minimise the diffusion/dispertion associated with advection in response to high frequency surface disturbances
-
-% -------------------------------------------------------------------------------------------------------------
-%        level bathymetry and mask 
-% -------------------------------------------------------------------------------------------------------------
-\subsection{Level bathymetry and mask}
-\label{subsec:DOM_msk}
-
-Whatever the vertical coordinate used, the model offers the possibility of representing the bottom topography with
-steps that follow the face of the model cells (step like topography) \citep{Madec_al_JPO96}.
-The distribution of the steps in the horizontal is defined in a 2D integer array, mbathy, which
-gives the number of ocean levels (\ie those that are not masked) at each $t$-point.
-mbathy is computed from the meter bathymetry using the definiton of gdept as the number of $t$-points which
-gdept $\leq$ bathy.
-
-Modifications of the model bathymetry are performed in the \textit{bat\_ctl} routine (see \mdl{domzgr} module) after
-mbathy is computed.
-Isolated grid points that do not communicate with another ocean point at the same level are eliminated.
-
-As for the representation of bathymetry, a 2D integer array, misfdep, is created.
-misfdep defines the level of the first wet $t$-point.
-All the cells between $k = 1$ and $misfdep(i,j) - 1$ are masked.
-By default, $misfdep(:,:) = 1$ and no cells are masked.
-
-In case of ice shelf cavities, modifications of the model bathymetry and ice shelf draft into 
-the cavities are performed in the \textit{zgr\_isf} routine.
-The compatibility between ice shelf draft and bathymetry is checked.
-All the locations where the isf cavity is thinnest than \np{rn\_isfhmin} meters are grounded (\ie masked).
-If only one cell on the water column is opened at $t$-, $u$- or $v$-points,
-the bathymetry or the ice shelf draft is dug to fit this constrain.
-If the incompatibility is too strong (need to dig more than 1 cell), the cell is masked.
-
-From the \textit{mbathy} and \textit{misfdep} array, the mask fields are defined as follows:
-\begin{alignat*}{2}
-  tmask(i,j,k) &= &  &
-    \begin{cases}
-                  0 &\text{if $                  k  <    misfdep(i,j)$} \\
-                  1 &\text{if $misfdep(i,j) \leq k \leq   mbathy(i,j)$} \\
-                  0 &\text{if $                  k  >     mbathy(i,j)$}
-    \end{cases}
-  \\
-  umask(i,j,k) &= &  &tmask(i,j,k) * tmask(i + 1,j,    k) \\
-  vmask(i,j,k) &= &  &tmask(i,j,k) * tmask(i    ,j + 1,k) \\
-  fmask(i,j,k) &= &  &tmask(i,j,k) * tmask(i + 1,j,    k) \\
-               &  &* &tmask(i,j,k) * tmask(i + 1,j,    k) \\
-  wmask(i,j,k) &= &  &tmask(i,j,k) * tmask(i    ,j,k - 1) \\
-  \text{with~} wmask(i,j,1) &= & &tmask(i,j,1)
-\end{alignat*}
-
-Note that, without ice shelves cavities,
-masks at $t-$ and $w-$points are identical with the numerical indexing used (\autoref{subsec:DOM_Num_Index}).
-Nevertheless, $wmask$ are required with ocean cavities to deal with the top boundary (ice shelf/ocean interface) 
-exactly in the same way as for the bottom boundary.
-
-The specification of closed lateral boundaries requires that at least
-the first and last rows and columns of the \textit{mbathy} array are set to zero.
-In the particular case of an east-west cyclical boundary condition, \textit{mbathy} has its last column equal to
-the second one and its first column equal to the last but one (and so too the mask arrays)
-(see \autoref{fig:LBC_jperio}).
-
-% ================================================================
-% Domain: Initial State (dtatsd & istate)
-% ================================================================
-\section{Initial state (\protect\mdl{istate} and \protect\mdl{dtatsd})}
-\label{sec:DTA_tsd}
-%-----------------------------------------namtsd-------------------------------------------
-
-\nlst{namtsd} 
-%------------------------------------------------------------------------------------------
-
-Options are defined in \ngn{namtsd}.
-By default, the ocean start from rest (the velocity field is set to zero) and the initialization of temperature and
-salinity fields is controlled through the \np{ln\_tsd\_ini} namelist parameter.
+The user has the option to
+set the bathymetry in closed seas to zero (see \autoref{sec:MISC_closea}) and to
+optionally decide on the fate of any freshwater imbalance over the area.
+The options are explained in \autoref{sec:MISC_closea} but
+it should be noted here that a successful use of these options requires
+appropriate mask fields to be present in the domain configuration file.
+Among the possibilities are:
+
+\begin{clines}
+int closea_mask     /* non-zero values in closed sea areas for optional masking                */
+int closea_mask_rnf /* non-zero values in closed sea areas with runoff locations (precip only) */
+int closea_mask_emp /* non-zero values in closed sea areas with runoff locations (total emp)   */
+\end{clines}
+
+%% =================================================================================================
+\subsection{Output grid files}
+\label{subsec:DOM_meshmask}
+
+Most of the arrays relating to a particular ocean model configuration discussed in this chapter
+(grid-point position, scale factors) can be saved in a file if
+namelist parameter \np{ln_write_cfg}{ln\_write\_cfg} (namelist \nam{cfg}{cfg}) is set to
+\forcode{.true.};
+the output filename is set through parameter \np{cn_domcfg_out}{cn\_domcfg\_out}.
+This is only really useful if
+the fields are computed in subroutines \mdl{usrdef\_hgr} or \mdl{usrdef\_zgr} and
+checking or confirmation is required.
+
+Alternatively, all the arrays relating to a particular ocean model configuration
+(grid-point position, scale factors, depths and masks) can be saved in
+a file called \texttt{mesh\_mask} if
+namelist parameter \np{ln_meshmask}{ln\_meshmask} (namelist \nam{dom}{dom}) is set to
+\forcode{.true.}.
+This file contains additional fields that can be useful for post-processing applications.
+
+%% =================================================================================================
+\section[Initial state (\textit{istate.F90} and \textit{dtatsd.F90})]{Initial state (\protect\mdl{istate} and \protect\mdl{dtatsd})}
+\label{sec:DOM_DTA_tsd}
+
+\begin{listing}
+  \nlst{namtsd}
+  \caption{\forcode{&namtsd}}
+  \label{lst:namtsd}
+\end{listing}
+
+Basic initial state options are defined in \nam{tsd}{tsd}.
+By default, the ocean starts from rest (the velocity field is set to zero) and
+the initialization of temperature and salinity fields is controlled through the \np{ln_tsd_init}{ln\_tsd\_init} namelist parameter.
+
 \begin{description}
-\item[\np{ln\_tsd\_init}~\forcode{= .true.}]
-  use a T and S input files that can be given on the model grid itself or on their native input data grid.
+\item [{\np[=.true.]{ln_tsd_init}{ln\_tsd\_init}}] Use T and S input files that can be given on
+  the model grid itself or on their native input data grids.
   In the latter case,
   the data will be interpolated on-the-fly both in the horizontal and the vertical to the model grid
   (see \autoref{subsec:SBC_iof}).
-  The information relative to the input files are given in the \np{sn\_tem} and \np{sn\_sal} structures.
+  The information relating to the input files are specified in
+  the \np{sn_tem}{sn\_tem} and \np{sn_sal}{sn\_sal} structures.
   The computation is done in the \mdl{dtatsd} module.
-\item[\np{ln\_tsd\_init}~\forcode{= .false.}]
-  use constant salinity value of $35.5~psu$ and an analytical profile of temperature
-  (typical of the tropical ocean), see \rou{istate\_t\_s} subroutine called from \mdl{istate} module.
+\item [{\np[=.false.]{ln_tsd_init}{ln\_tsd\_init}}] Initial values for T and S are set via
+  a user supplied \rou{usr\_def\_istate} routine contained in \mdl{userdef\_istate}.
+  The default version sets horizontally uniform T and profiles as used in the GYRE configuration
+  (see \autoref{sec:CFGS_gyre}).
 \end{description}
 
-\biblio
-
-\pindex
+\subinc{\input{../../global/epilogue}}
 
 \end{document}
