source: branches/DEV_r1826_DOC/DOC/TexFiles/Chapters/Chap_DOM.tex @ 1831

% ================================================================
% Chapter 2 Ñ Space and Time Domain (DOM)
% ================================================================
\chapter{Space and Time Domain (DOM) }
\label{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)
%  - daymod: definition of the time domain (nit000, nitend andd the calendar)
%  -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

\gmcomment{STEVEN :maybe a picture of the directory structure in the introduction 
which could be referred to here, would help  ==> to be added}
%%%%


\newpage
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Having defined the continuous equations in Chap.~\ref{PE}, we need to choose a 
discretization on a grid, and numerical algorithms. In the present chapter, we 
provide a general description of the staggered grid used in \NEMO, and other 
information relevant to the main directory routines (time stepping, main program) 
as well as the DOM (DOMain) directory. 

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% ================================================================
% Fundamentals of the Discretisation
% ================================================================
\section{Fundamentals of the Discretisation}
\label{DOM_basics}

% -------------------------------------------------------------------------------------------------------------
%        Arrangement of Variables 
% -------------------------------------------------------------------------------------------------------------
\subsection{Arrangement of Variables}
\label{DOM_cell}

%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
\begin{figure}[!tb] \label{Fig_cell}  \begin{center}
\includegraphics[width=0.90\textwidth]{./TexFiles/Figures/Fig_cell.pdf}
\caption{Arrangement of variables. $t$ indicates scalar points where temperature, 
salinity, density, pressure and horizontal divergence are defined. ($u$,$v$,$w$) 
indicates vector points, and $f$ indicates vorticity points where both relative and 
planetary vorticities are defined}
\end{center}   \end{figure}
%>>>>>>>>>>>>>>>>>>>>>>>>>>>>

The numerical techniques used to solve the Primitive Equations in this model are 
based on the traditional, centred second-order finite difference approximation. 
Special attention has been given to the homogeneity of the solution in the three 
space directions. The arrangement of variables is the same in all directions. 
It consists of cells centred on scalar points ($t$, $S$, $p$, $\rho$) with vector 
points $(u, v, w)$ defined in the centre of each face of the cells (Fig. \ref{Fig_cell}). 
This is the generalisation to three dimensions of the well-known ``C'' grid in 
Arakawa's classification \citep{Mesinger_Arakawa_Bk76}. The relative and 
planetary vorticity, $\zeta$ and $f$, are defined in the centre of each vertical edge 
and the barotropic stream function $\psi$ is defined at horizontal points overlying 
the $\zeta$ and $f$-points.

The ocean mesh ($i.e.$ the position of all the scalar and vector points) is defined 
by the transformation that gives ($\lambda$ ,$\varphi$ ,$z$) as a function of $(i,j,k)$. 
The grid-points are located at integer or integer and a half value of $(i,j,k)$ as 
indicated on Table \ref{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 \eqref{Eq_scale_factors}. As a result, the mesh on which partial 
derivatives $\frac{\partial}{\partial \lambda}, \frac{\partial}{\partial \varphi}$, and 
$\frac{\partial}{\partial z} $ are evaluated is a uniform mesh with a grid size of unity. 
Discrete partial derivatives are formulated by the traditional, centred second order 
finite difference approximation while the scale factors are chosen equal to their 
local analytical value. An important point here is that the partial derivative of the 
scale factors must be evaluated by centred finite difference approximation, not 
from their analytical expression. This preserves the symmetry of the discrete set 
of equations and therefore satisfies many of the continuous properties (see 
Appendix~\ref{Apdx_C}). A similar, related remark can be made about the domain 
size: when needed, an area, volume, or the total ocean depth must be evaluated 
as the sum of the relevant scale factors (see \eqref{DOM_bar}) in the next section). 

\begin{table}[!tb] \label{Tab_cell}
\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{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 \S\ref{DOM_Num_Index})}
\end{center}
\end{table}

% -------------------------------------------------------------------------------------------------------------
%        Vector Invariant Formulation 
% -------------------------------------------------------------------------------------------------------------
\subsection{Discrete Operators}
\label{DOM_operators}

Given the values of a variable $q$ at adjacent points, the differencing and 
averaging operators at the midpoint between them are:
\begin{subequations} \label{Eq_di_mi}
\begin{align}
 \delta _i [q]       &=  \  \    q(i+1/2)  - q(i-1/2)    \\
 \overline q^{\,i} &= \left\{ q(i+1/2) + q(i-1/2) \right\} \; / \; 2
\end{align}
\end{subequations}

Similar operators are defined with respect to $i+1/2$, $j$, $j+1/2$, $k$, and 
$k+1/2$. Following \eqref{Eq_PE_grad} and \eqref{Eq_PE_lap}, the gradient of a 
variable $q$ defined at a $t$-point has its three components defined at $u$-, $v$- 
and $w$-points while its Laplacien is defined at $t$-point. These operators have 
the following discrete forms in the curvilinear $s$-coordinate system:
\begin{equation} \label{Eq_DOM_grad}
\nabla q\equiv    \frac{1}{e_{1u} } \delta _{i+1/2 } [q] \;\,{\rm {\bf i}}
      +  \frac{1}{e_{2v} } \delta _{j+1/2 } [q] \;\,{\rm {\bf j}}
      +  \frac{1}{e_{3w}} \delta _{k+1/2} [q] \;\,{\rm {\bf k}}
\end{equation}
\begin{multline} \label{Eq_DOM_lap}
\Delta q\equiv \frac{1}{e_{1t}\,e_{2t}\,e_{3t} } 
       \;\left(          \delta_i  \left[ \frac{e_{2u}\,e_{3u}} {e_{1u}} \;\delta_{i+1/2} [q] \right]
+                        \delta_j  \left[ \frac{e_{1v}\,e_{3v}}  {e_{2v}} \;\delta_{j+1/2} [q] \right] \;  \right)      \\
+\frac{1}{e_{3t}} \delta_k \left[ \frac{1}{e_{3w} }                     \;\delta_{k+1/2} [q] \right]
\end{multline}

Following \eqref{Eq_PE_curl} and \eqref{Eq_PE_div}, a vector ${\rm {\bf A}}=\left( a_1,a_2,a_3\right)$ 
defined at vector points $(u,v,w)$ has its three curl components defined at $vw$-, $uw$, 
and $f$-points, and its divergence defined at $t$-points:
\begin{eqnarray}  \label{Eq_DOM_curl}
 \nabla \times {\rm {\bf A}}\equiv &
      \frac{1}{e_{2v}\,e_{3vw} } \ \left( \delta_{j +1/2} \left[e_{3w}\,a_3 \right] -\delta_{k+1/2} \left[e_{2v} \,a_2 \right] \right)  &\ \rm{\bf i} \\ 
 +& \frac{1}{e_{2u}\,e_{3uw}} \ \left( \delta_{k+1/2} \left[e_{1u}\,a_1  \right] -\delta_{i +1/2} \left[e_{3w}\,a_3 \right] \right)  &\ \rm{\bf j} \\ 
 +& \frac{1}{e_{1f} \,e_{2f}    } \ \left( \delta_{i +1/2} \left[e_{2v}\,a_2  \right] -\delta_{j +1/2} \left[e_{1u}\,a_1 \right] \right)  &\ \rm{\bf k}
 \end{eqnarray}
\begin{equation} \label{Eq_DOM_div}
\nabla \cdot \rm{\bf A}=\frac{1}{e_{1t}\,e_{2t}\,e_{3t}}\left( \delta_i \left[e_{2u}\,e_{3u}\,a_1 \right]
                                                                                         +\delta_j \left[e_{1v}\,e_{3v}\,a_2 \right] \right)+\frac{1}{e_{3t} }\delta_k \left[a_3 \right]
\end{equation}

In the special case of a pure $z$-coordinate system, \eqref{Eq_DOM_lap} and 
\eqref{Eq_DOM_div} can be simplified. In this case, the vertical scale factor 
becomes a function of the single variable $k$ and thus does not depend on the 
horizontal location of a grid point. For example \eqref{Eq_DOM_div} reduces to: 
\begin{equation*}
\nabla \cdot \rm{\bf A}=\frac{1}{e_{1t}\,e_{2t}} \left( \delta_i \left[e_{2u}\,a_1 \right] 
                                                                              +\delta_j \left[e_{1v}\, a_2 \right]  \right)
                                                     +\frac{1}{e_{3t}} \delta_k \left[             a_3 \right]
\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):
\begin{equation} \label{DOM_bar}
\bar q   =         \frac{1}{H}    \int_{k^b}^{k^o} {q\;e_{3q} \,dk} 
      \equiv \frac{1}{H_q }\sum\limits_k {q\;e_{3q} }
\end{equation}
where $H_q$  is the ocean depth, which is the masked sum of the vertical scale 
factors at $q$ points, $k^b$ and $k^o$ are the bottom and surface $k$-indices, 
and the symbol $k^o$ refers to a summation over all grid points of the same type 
in the direction indicated by the subscript (here $k$). 

In continuous form, the following properties are satisfied:
\begin{equation} \label{Eq_DOM_curl_grad}
\nabla \times \nabla q ={\rm {\bf {0}}}
\end{equation}
\begin{equation} \label{Eq_DOM_div_curl}
\nabla \cdot \left( {\nabla \times {\rm {\bf A}}} \right)=0
\end{equation}

It is straightforward to demonstrate that these properties are verified locally in 
discrete form as soon as the scalar $q$ is taken at $t$-points and the vector 
\textbf{A} has its components defined at vector points $(u,v,w)$.

Let $a$ and $b$ be two fields defined on the mesh, with value zero inside 
continental area. Using integration by parts it can be shown that the differencing 
operators ($\delta_i$, $\delta_j$ and $\delta_k$) are anti-symmetric linear 
operators, and further that the averaging operators $\overline{\,\cdot\,}^{\,i}$, 
$\overline{\,\cdot\,}^{\,k}$ and $\overline{\,\cdot\,}^{\,k}$) are symmetric linear 
operators, $i.e.$
\begin{align} 
\label{DOM_di_adj}
\sum\limits_i { a_i \;\delta _i \left[ b \right]} 
   &\equiv -\sum\limits_i {\delta _{i+1/2} \left[ a \right]\;b_{i+1/2} }      \\
\label{DOM_mi_adj}
\sum\limits_i { a_i \;\overline b^{\,i}} 
   & \equiv \quad \sum\limits_i {\overline a ^{\,i+1/2}\;b_{i+1/2} } 
\end{align}

In other words, the adjoint of the differencing and averaging operators are 
$\delta_i^*=\delta_{i+1/2}$ and 
${(\overline{\,\cdot \,}^{\,i})}^*= \overline{\,\cdot\,}^{\,i+1/2}$, respectively. 
These two properties will be used extensively in the Appendix~\ref{Apdx_C} to 
demonstrate integral conservative properties of the discrete formulation chosen.

% -------------------------------------------------------------------------------------------------------------
%        Numerical Indexing 
% -------------------------------------------------------------------------------------------------------------
\subsection{Numerical Indexing}
\label{DOM_Num_Index}

%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
\begin{figure}[!tb] \label{Fig_index_hor}  \begin{center}
\includegraphics[width=0.90\textwidth]{./TexFiles/Figures/Fig_index_hor.pdf}
\caption{Horizontal integer indexing used in the \textsc{Fortran} code. The dashed 
area indicates the cell in which variables contained in arrays have the same 
$i$- and $j$-indices}
\end{center}   \end{figure}
%>>>>>>>>>>>>>>>>>>>>>>>>>>>>

The array representation used in the \textsc{Fortran} code requires an integer 
indexing while the analytical definition of the mesh (see \S\ref{DOM_cell}) is 
associated with the use of integer values for $t$-points and both integer and 
integer and a half values for all the other points. Therefore a specific integer 
indexing must be defined for points other than $t$-points ($i.e.$ velocity and 
vorticity grid-points). Furthermore, the direction of the vertical indexing has 
been changed so that the surface level is at $k=1$.

% -----------------------------------
%        Horizontal Indexing 
% -----------------------------------
\subsubsection{Horizontal Indexing}
\label{DOM_Num_Index_hor}

The indexing in the horizontal plane has been chosen as shown in Fig.\ref{Fig_index_hor}. 
For an increasing $i$ index ($j$ index), the $t$-point and the eastward $u$-point 
(northward $v$-point) have the same index (see the dashed area in Fig.\ref{Fig_index_hor}). 
A $t$-point and its nearest northeast $f$-point have the same $i$-and $j$-indices.

% -----------------------------------
%        Vertical indexing 
% -----------------------------------
\subsubsection{Vertical Indexing}
\label{DOM_Num_Index_vertical}

In the vertical, the chosen indexing requires special attention since the 
$k$-axis is re-orientated downward in the \textsc{Fortran} code compared 
to the indexing used in the semi-discrete equations and given in \S\ref{DOM_cell}. 
The sea surface corresponds to the $w$-level $k=1$ which is the same index 
as $t$-level just below (Fig.\ref{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 (Fig.\ref{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 
Fig.\ref{Fig_index_hor} and \ref{Fig_index_vert}). Since the scale factors are 
chosen to be strictly positive, a \emph{minus sign} appears in the \textsc{Fortran} 
code \emph{before all the vertical derivatives} of the discrete equations given in 
this documentation.

%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
\begin{figure}[!pt] \label{Fig_index_vert}  \begin{center}
\includegraphics[width=.90\textwidth]{./TexFiles/Figures/Fig_index_vert.pdf}
\caption{Vertical integer indexing used in the \textsc{Fortran } code. Note that 
the $k$-axis is orientated downward. The dashed area indicates the cell in 
which variables contained in arrays have the same $k$-index.}
\end{center}   \end{figure}
%>>>>>>>>>>>>>>>>>>>>>>>>>>>>

% -----------------------------------
%        Domain Size
% -----------------------------------
\subsubsection{Domain Size}
\label{DOM_size}

The total size of the computational domain is set by the parameters \jp{jpiglo}, 
\jp{jpjglo} and \jp{jpk} in the $i$, $j$ and $k$ directions respectively. They are 
given as parameters in the \mdl{par\_oce} module\footnote{When a specific 
configuration is used (ORCA2 global ocean, etc...) the parameter are actually 
defined in additional files introduced by \mdl{par\_oce} module via CPP 
\textit{include} command. For example, ORCA2 parameters are set in 
\textit{par\_ORCA\_R2.h90} file}. The use of parameters rather than variables 
(together with dynamic allocation of arrays) was chosen because it ensured that 
the compiler would optimize the executable code efficiently, especially on vector 
machines (optimization may be less efficient when the problem size is unknown 
at the time of compilation). Nevertheless, it is possible to set up the code with full 
dynamical allocation by using the AGRIF packaged \citep{Debreu_al_CG2008}. 
%
\gmcomment{  add the following ref 
\colorbox{yellow}{(ref part of the doc)} } 
%
Note that are other parameters in \mdl{par\_oce} that refer to the domain size. 
The two parameters $jpidta$ and $jpjdta$ may be larger than $jpiglo$, $jpjglo$ 
when the user wants to use only a sub-region of a given configuration. This is 
the "zoom" capability described in \S\ref{MISC_zoom}. In most applications of 
the model, $jpidta=jpiglo$, $jpjdta=jpjglo$, and $jpizoom=jpjzoom=1$. 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 
\S\ref{LBC_mpp}).


$\ $\newline    % force a new ligne

% ================================================================
% Domain: Horizontal Grid (mesh) 
% ================================================================
\section  [Domain: Horizontal Grid (mesh) (\textit{domhgr})]               
      {Domain: Horizontal Grid (mesh) \small{(\mdl{domhgr} module)} }
\label{DOM_hgr}

% -------------------------------------------------------------------------------------------------------------
%        Coordinates and scale factors 
% -------------------------------------------------------------------------------------------------------------
\subsection{Coordinates and scale factors}
\label{DOM_hgr_coord_e}

The ocean mesh ($i.e.$ the position of all the scalar and vector points) is defined 
by the transformation that gives $(\lambda,\varphi,z)$ as a function of $(i,j,k)$. 
The grid-points are located at integer or integer and a half values of as indicated 
in Table~\ref{Tab_cell}. The associated scale factors are defined using the 
analytical first derivative of the transformation \eqref{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 
\eqref{Eq_scale_factors}. For example, when the longitude and latitude are 
function of a single value ($i$ and $j$, respectively) (geographical configuration 
of the mesh), the horizontal mesh definition reduces to define the wanted 
$\lambda(i)$, $\varphi(j)$, and their derivatives $\lambda'(i)$ $\varphi'(j)$ in the 
\mdl{domhgr} module. The model computes the grid-point positions and scale 
factors in the horizontal plane as follows:
\begin{flalign*}
\lambda_t &\equiv \text{glamt}= \lambda(i)     & \varphi_t &\equiv \text{gphit} = \varphi(j)\\
\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) 
\end{flalign*}
\begin{flalign*}
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)|
\end{flalign*}
where the last letter of each computational name indicates the grid point 
considered and $r_a$ is the earth radius (defined in \mdl{phycst} along with 
all universal constants). Note that the horizontal position of and scale factors 
at $w$-points are exactly equal to those of $t$-points, thus no specific arrays 
are defined at $w$-points. 

Note that the definition of the scale factors ($i.e.$ as the analytical first derivative 
of the transformation that gives $(\lambda,\varphi,z)$ as a function of $(i,j,k)$) is 
specific to the \NEMO model \citep{Marti_al_JGR92}. As an example, $e_{1t}$ is defined 
locally at a $t$-point, whereas many other models on a C grid choose to define 
such a scale factor as the distance between the $U$-points on each side of the 
$t$-point. Relying on an analytical transformation has two advantages: firstly, there 
is no ambiguity in the scale factors appearing in the discrete equations, since they 
are first introduced in the continuous equations; secondly, analytical transformations 
encourage good practice by the definition of smoothly varying grids (rather than 
allowing the user to set arbitrary jumps in thickness between adjacent layers) 
\citep{Treguier1996}. An example of the effect of such a choice is shown in 
Fig.~\ref{Fig_zgr_e3}.
%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
\begin{figure}[!t] \label{Fig_zgr_e3}  \begin{center}
\includegraphics[width=0.90\textwidth]{./TexFiles/Figures/Fig_zgr_e3.pdf}
\caption{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: $z(k)=5\,(i-1/2)^3 - 45\,(i-1/2)^2 + 140\,(i-1/2) - 150$. 
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{DOM_hgr_msh_choice}

The user has three options available in defining a horizontal grid, which involve 
the parameter $jphgr\_mesh$ of the \mdl{par\_oce} module. 
\begin{description}
\item[\jp{jphgr\_mesh}=0]  The most general curvilinear orthogonal grids.
The coordinates and their first derivatives with respect to $i$ and $j$ are 
provided in a file, read in \rou{hgr\_read} subroutine of the domhgr module.
\item[\jp{jphgr\_mesh}=1 to 5] A few simple analytical grids are provided (see below). 
For other analytical grids, the \mdl{domhgr} module must be modified by the user. 
\end{description}

There are two simple cases of geographical grids on the sphere. With 
\jp{jphgr\_mesh}=1, the grid (expressed in degrees) is regular in space, 
with grid sizes specified by parameters \pp{ppe1\_deg} and \pp{ppe2\_deg}, 
respectively. Such a geographical grid can be very anisotropic at high latitudes 
because of the convergence of meridians (the zonal scale factors $e_1$ 
become much smaller than the meridional scale factors $e_2$). The Mercator 
grid (\jp{jphgr\_mesh}=4) avoids this anisotropy by refining the meridional scale 
factors in the same way as the zonal ones. In this case, meridional scale factors 
and latitudes are calculated analytically using the formulae appropriate for 
a Mercator projection, based on \pp{ppe1\_deg} which is a reference grid spacing 
at the equator (this applies even when the geographical equator is situated outside 
the model domain). 
%%%
\gmcomment{ give here the analytical expression of the Mercator mesh}
%%%
In these two cases (\jp{jphgr\_mesh}=1 or 4), the grid position is defined by the 
longitude and latitude of the south-westernmost point (\pp{ppglamt0} 
and \pp{ppgphi0}). Note that for the Mercator grid the user need only provide 
an approximate starting latitude: the real latitude will be recalculated analytically, 
in order to ensure that the equator corresponds to line passing through $t$- 
and $u$-points.  

Rectangular grids ignoring the spherical geometry are defined with 
\jp{jphgr\_mesh} = 2, 3, 5. The domain is either an $f$-plane (\jp{jphgr\_mesh} = 2, 
Coriolis factor is constant) or a beta-plane (\jp{jphgr\_mesh} = 3, the Coriolis factor 
is linear in the $j$-direction). The grid size is uniform in meter in each direction, 
and given by the parameters \pp{ppe1\_m} and \pp{ppe2\_m} respectively. 
The zonal grid coordinate (\textit{glam} arrays) is in kilometers, starting at zero 
with the first $t$-point. The meridional coordinate (gphi. arrays) is in kilometers, 
and the second $t$-point corresponds to coordinate $gphit=0$. The input 
parameter \pp{ppglam0} is ignored. \pp{ppgphi0} is used to set the reference 
latitude for computation of the Coriolis parameter. In the case of the beta plane, 
\pp{ppgphi0} corresponds to the center of the domain. Finally, the special case 
\jp{jphgr\_mesh}=5 corresponds to a beta plane in a rotated domain for the 
GYRE configuration, representing a classical mid-latitude double gyre system. 
The rotation allows us to maximize the jet length relative to the gyre areas 
(and the number of grid points). 

The choice of the grid must be consistent with the boundary conditions specified 
by the parameter \jp{jperio} (see {\S\ref{LBC}).

% -------------------------------------------------------------------------------------------------------------
%        Grid files
% -------------------------------------------------------------------------------------------------------------
\subsection{Grid files}
\label{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 $\np{nmsh} \not= 0$ 
(namelist parameter). 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 $\np{nmsh} \not=0$ is no more equal to the input grid.

$\ $\newline    % force a new ligne

% ================================================================
% Domain: Vertical Grid (domzgr)
% ================================================================
\section  [Domain: Vertical Grid (\textit{domzgr})]
      {Domain: Vertical Grid \small{(\mdl{domzgr} module)} }
\label{DOM_zgr}
%-----------------------------------------nam_zgr & namdom-------------------------------------------
\namdisplay{namzgr} 
\namdisplay{namdom} 
%-------------------------------------------------------------------------------------------------------------

In the vertical, the model mesh is determined by four things: 
(1) the bathymetry given in meters ; 
(2) the number of levels of the model (\jp{jpk}) ; 
(3) the analytical transformation $z(i,j,k)$ and the vertical scale factors 
(derivatives of the transformation) ; 
and (4) the masking system, $i.e.$ the number of wet model levels at each 
$(i,j)$ column of points.

%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
\begin{figure}[!tb] \label{Fig_z_zps_s_sps}  \begin{center}
\includegraphics[width=1.0\textwidth]{./TexFiles/Figures/Fig_z_zps_s_sps.pdf}
\caption{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 with variable volume associated with the non-linear free surface. 
Note that the variable volume option (\key{vvl}) 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 a namelist parameter, 
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 (Fig.~\ref{Fig_z_zps_s_sps}a to c): $z$-coordinate with full step 
bathymetry (\np{ln\_zco}=true), $z$-coordinate with partial step bathymetry 
(\np{ln\_zps}=true), or generalized, $s$-coordinate (\np{ln\_sco}=true). 
Hybridation of the three main coordinates are available: $s-z$ or $s-zps$ coordinate 
(Fig.~\ref{Fig_z_zps_s_sps}d and \ref{Fig_z_zps_s_sps}e). When using the variable 
volume option \key{vvl}) ($i.e.$ non-linear free surface), the coordinate follow the 
time-variation of the free surface so that the transformation is time dependent: 
$z(i,j,k,t)$ (Fig.~\ref{Fig_z_zps_s_sps}f). This option can be used with full step 
bathymetry or $s$-coordinate (hybride and partial step coordinates have not 
yet been tested in NEMO v2.3). 

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)\footnote{N.B. in full step $z$-coordinate, a \textit{bathy\_level} file can 
replace the \textit{bathy\_meter} file, so that the computation of the number of 
wet ocean point in each water column is by-passed}. After reading the bathymetry, 
the algorithm for vertical grid definition differs between the different options:
\begin{description}
\item[\textit{zco}] set a reference coordinate transformation $z_0 (k)$, and set $z(i,j,k,t)=z_0 (k)$.
\item[\textit{zps}] set a reference coordinate transformation $z_0 (k)$, and 
calculate the thickness of the deepest level at each $(i,j)$ point using the 
bathymetry, to obtain the final three-dimensional depth and scale factor arrays.
\item[\textit{sco}] smooth the bathymetry to fulfil the hydrostatic consistency 
criteria and set the three-dimensional transformation.
\item[\textit{s-z} and \textit{s-zps}] smooth the bathymetry to fulfil the hydrostatic 
consistency criteria and set the three-dimensional transformation $z(i,j,k)$, and 
possibly introduce masking of extra land points to better fit the original bathymetry file
\end{description}
%%%
\gmcomment{   add the description of the smoothing:  envelop topography...}
%%%

Generally, the arrays describing the grid point depths and vertical scale factors 
are three dimensional arrays $(i,j,k)$. In the special case of $z$-coordinates with 
full step bottom topography, it is possible to define those arrays as one-dimensional, 
in order to save memory. This is performed by defining the \key{zco} 
C-Pre-Processor (CPP) key. To improve the code readability while providing this 
flexibility, the vertical coordinate and scale factors are defined as functions of 
$(i,j,k)$ with "fs" as prefix (examples: \textit{fsdeptht, fse3t,} etc) that can be either 
three-dimensional arrays, or a one dimensional array when \key{zco} is defined. 
These functions are defined in the file \hf{domzgr\_substitute} of the DOM directory. 
They are used throughout the code, and replaced by the corresponding arrays at 
the time of pre-processing (CPP capability).

% -------------------------------------------------------------------------------------------------------------
%        Meter Bathymetry
% -------------------------------------------------------------------------------------------------------------
\subsection{Meter Bathymetry}
\label{DOM_bathy}

Three options are possible for defining the bathymetry, according to the 
namelist variable \np{ntopo}: 
\begin{description}
\item[\np{ntopo} = 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 \jp{jperio}. 
\item[\np{ntopo} = -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{ntopo} = 1] read a bathymetry. The bathymetry file (Netcdf format) 
provides the ocean depth (positive, in meters) at each grid point of the model grid. 
The bathymetry is usually built by interpolating a standard bathymetry product 
($e.g.$ ETOPO2) onto the horizontal ocean mesh. Defining the bathymetry also 
defines the coastline: where the bathymetry is zero, no model levels are defined 
(all levels are masked).
\end{description}

When using the rigid lid approximation (\key{dynspg\_rl} is defined) isolated land 
masses (islands) must be identified by negative integers in the input bathymetry file 
(see \S\ref{MISC_solisl}).

When a global ocean is coupled to an atmospheric model it is better to represent 
all large water bodies (e.g, great lakes, Caspian sea...) even if the model 
resolution does not allow their communication with the rest of the ocean. 
This is unnecessary when the ocean is forced by fixed atmospheric conditions, 
so these seas can be removed from the ocean domain. The user has the option 
to set the bathymetry in closed seas to zero (see \S\ref{MISC_closea}), but the 
code has to be adapted to the user's configuration. 

% -------------------------------------------------------------------------------------------------------------
%        z-coordinate  and reference coordinate transformation
% -------------------------------------------------------------------------------------------------------------
\subsection[$z$-coordinate (\np{ln\_zco} or \key{zco})]
        {$z$-coordinate (\np{ln\_zco}=.true. or \key{zco}) and reference coordinate}
\label{DOM_zco}

%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
\begin{figure}[!tb] \label{Fig_zgr}  \begin{center}
\includegraphics[width=0.90\textwidth]{./TexFiles/Figures/Fig_zgr.pdf}
\caption{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 \eqref{DOM_zgr_ana} using \eqref{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 
Fig.\ref{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 \textit{par\_oce.h90} file. 

It is possible to define a simple regular vertical grid by giving zero stretching (\pp{ppacr=0}). 
In that case, the parameters \jp{jpk} (number of $w$-levels) and \pp{pphmax} 
(total ocean depth in meters) fully define the grid. 

For climate-related studies it is often desirable to concentrate the vertical resolution 
near the ocean surface. The following function is proposed as a standard for a 
$z$-coordinate (with either full or partial steps): 
\begin{equation} \label{DOM_zgr_ana}
\begin{split}
 z_0 (k)    &= h_{sur} -h_0 \;k-\;h_1 \;\log \left[ {\,\cosh \left( {{(k-h_{th} )} / {h_{cr} }} \right)\,} \right] \\ 
 e_3^0 (k)  &= \left| -h_0 -h_1 \;\tanh \left( {{(k-h_{th} )} / {h_{cr} }} \right) \right| 
\end{split}
\end{equation}
where $k=1$ to \jp{jpk} for $w$-levels and $k=1$ to $k=1$ for $T-$levels. Such an 
expression allows us to define a nearly uniform vertical location of levels at the 
ocean top and bottom with a smooth hyperbolic tangent transition in between 
(Fig.~\ref{Fig_zgr}).

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{DOM_zgr_coef}
\begin{split}
 e_3 (1+1/2)      &=10. \\ 
 e_3 (jpk-1/2) &=500. \\ 
 z(1)       &=0. \\ 
 z(jpk)        &=-5000. \\ 
\end{split}
\end{equation}

With the choice of the stretching $h_{cr} =3$ and the number of levels 
\jp{jpk}=$31$, the four coefficients $h_{sur}$, $h_{0}$, $h_{1}$, and $h_{th}$ in 
\eqref{DOM_zgr_ana} have been determined such that \eqref{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 Fig.~\ref{Fig_zgr} and 
given in Table \ref{Tab_orca_zgr}. Those values correspond to the parameters 
\pp{ppsur}, \pp{ppa0}, \pp{ppa1}, \pp{ppkth} in the parameter file \mdl{par\_oce}. 

Rather than entering parameters $h_{sur}$, $h_{0}$, and $h_{1}$ directly, it is 
possible to recalculate them. In that case the user sets 
\pp{ppsur}=\pp{ppa0}=\pp{ppa1}=\pp{pp\_to\_be\_computed}, in \mdl{par\_oce}, 
and specifies instead the four following parameters:
\begin{itemize}
\item    \pp{ppacr}=$h_{cr} $: stretching factor (nondimensional). The larger 
\pp{ppacr}, the smaller the stretching. Values from $3$ to $10$ are usual.
\item    \pp{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    \pp{ppdzmin}: minimum thickness for the top layer (in meters)
\item    \pp{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, \pp{ppacr}=9, \pp{ppkth}=23.563, \pp{ppdzmin}=6m, 
\pp{pphmax}=5750m.

%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
\begin{table} \label{Tab_orca_zgr}
\begin{center} \begin{tabular}{c||r|r|r|r}
\hline
\textbf{LEVEL}& \textbf{gdept}& \textbf{gdepw}& \textbf{e3t }& \textbf{e3w  } \\ \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{Default vertical mesh in $z$-coordinate for 30 layers ORCA2 configuration 
as computed from \eqref{DOM_zgr_ana} using the coefficients given in 
\eqref{DOM_zgr_coef}}
\end{table}
%>>>>>>>>>>>>>>>>>>>>>>>>>>>>

% -------------------------------------------------------------------------------------------------------------
%        z-coordinate with partial step
% -------------------------------------------------------------------------------------------------------------
\subsection   [$z$-coordinate with partial step (\np{ln\_zps})]
         {$z$-coordinate with partial step (\np{ln\_zps}=.true.)}
\label{DOM_zps}
%--------------------------------------------namdom-------------------------------------------------------
\namdisplay{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, \emph{except} in the bottom layer. The thickness of the bottom layer is 
allowed to vary as a function of geographical location $(\lambda,\varphi)$ to allow a 
better representation of the bathymetry, especially in the case of small 
slopes (where the bathymetry varies by less than one level thickness from 
one grid point to the next). The reference layer thicknesses $e_{3t}^0$ have been 
defined in the absence of bathymetry. With partial steps, layers from 1 to 
\jp{jpk}-2 can have a thickness smaller than $e_{3t}(jk)$. The model deepest layer (\jp{jpk}-1) 
is allowed to have either a smaller or larger thickness than $e_{3t}(jpk)$: the 
maximum thickness allowed is $2*e_{3t}(jpk-1)$. This has to be kept in mind when 
specifying the maximum depth \pp{pphmax} in partial steps: for example, with 
\pp{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 $e_{3t}(jk)$).

 \colorbox{yellow}{Add a figure here of pstep especially at last ocean level }

% -------------------------------------------------------------------------------------------------------------
%        s-coordinate
% -------------------------------------------------------------------------------------------------------------
\subsection   [$s$-coordinate (\np{ln\_sco})]
           {$s$-coordinate (\np{ln\_sco}=true)}
\label{DOM_sco}
%------------------------------------------nam_zgr_sco---------------------------------------------------
\namdisplay{namzgr_sco} 
%--------------------------------------------------------------------------------------------------------------
In $s$-coordinate (\key{sco} is defined), the depth and thickness of the model 
levels are defined from the product of a depth field and either a stretching 
function or its derivative, respectively:
\begin{equation} \label{DOM_sco_ana}
\begin{split}
 z(k)       &= h(i,j) \; z_0(k)  \\
 e_3(k)  &= h(i,j) \; z_0'(k)
\end{split}
\end{equation}
where $h$ is the depth of the last $w$-level ($z_0(k)$) defined at the $t$-point 
location in the horizontal and $z_0(k)$ is a function which varies from $0$ at the sea 
surface to $1$ at the ocean bottom. The depth field $h$ is not necessary the ocean 
depth, since a mixed step-like and bottom-following representation of the 
topography can be used (Fig.~\ref{Fig_z_zps_s_sps}d-e). In the example provided 
(\hf{zgr\_s} file) $h$ is a smooth envelope bathymetry and steps are used to represent 
sharp bathymetric gradients.

A new flexible stretching function, modified from \citet{Song_Haidvogel_JCP94} is provided as an example:
\begin{equation} \label{DOM_sco_function}
\begin{split}
z  &= h_c +( h-h_c)\;c s)  \\
c(s)  &=  \frac{ \left[   \tanh{ \left( \theta \, (s+b) \right)} 
               - \tanh{ \left(  \theta \, b      \right)}  \right]}
            {2\;\sinh \left( \theta \right)}
\end{split}
\end{equation}
where $h_c$ is the thermocline depth and $\theta$ and $b$ 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 (Fig.~\ref{Fig_sco_function}).

%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
\begin{figure}[!tb] \label{Fig_sco_function}  \begin{center}
\includegraphics[width=1.0\textwidth]{./TexFiles/Figures/Fig_sco_function.pdf}
\caption{Examples of the stretching function applied to a sea mont; from left to right: 
surface, surface and bottom, and bottom intensified resolutions}
\end{center}   \end{figure}
%>>>>>>>>>>>>>>>>>>>>>>>>>>>>

% -------------------------------------------------------------------------------------------------------------
%        z*- or s*-coordinate
% -------------------------------------------------------------------------------------------------------------
\subsection{$z^*$- or $s^*$-coordinate (add \key{vvl}) }
\label{DOM_zgr_vvl}

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{DOM_msk}

Whatever the vertical coordinate used, the model offers the possibility of 
representing the bottom topography with steps that follow the face of the 
model cells (step like topography) \citep{Madec_al_JPO96}. The distribution of 
the steps in the horizontal is defined in a 2D integer array, mbathy, which 
gives the number of ocean levels ($i.e.$ those that are not masked) at each 
$t$-point. mbathy is computed from the meter bathymetry using the definiton of 
gdept as the number of $t$-points which gdept $\leq$ bathy. Note that in version 
NEMO v2.3, the user still has to provide the "level" bathymetry in a NetCDF
file when using the full step option (\np{ln\_zco}), rather than the bathymetry 
in meters: both will be allowed in future versions.

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.

In the case of the rigid-lid approximation when islands occur in the computational 
domain (\np{ln\_dynspg\_rl}=.true. and \key{island} is defined), the \textit{mbathy} 
array must be provided and takes values from $-N$ to \jp{jpk}-1. It provides the 
following information: $mbathy(i,j) = -n, \ n \in \left] 0,N \right]$, $t$-points are 
land points on the $n^{th}$ island ; $mbathy(i,j) =0$, $t$-points are land points 
on the main land (continent) ; $mbathy(i,j) =k$, the first $k$ $t$- and $w$-points 
are ocean points, the others are points below the ocean floor. 

This is used to compute the island barotropic stream function used in the rigid lid 
computation (see \S\ref{MISC_solisl}).

From the \textit{mbathy} array, the mask fields are defined as follows:
\begin{align*}
tmask(i,j,k) &= \begin{cases}   \; 1&   \text{ if $k\leq mbathy(i,j)$  }    \\
                                                \; 0&   \text{ if $k\leq mbathy(i,j)$  }    \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)
\end{align*}

\gmcomment{ STEVEN: are the dots multiplications?}      

Note that \textit{wmask} is not defined as it is exactly equal to \textit{tmask} with 
the numerical indexing used (\S~\ref{DOM_Num_Index}). Moreover, 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 \S~\ref{LBC_jperio}).

%%%
\gmcomment{   \colorbox{yellow}{Add one word on tricky trick !} mbathy in further modified in zdfbfr{\ldots}.  }
%%%

$\ $\newline    % force a new ligne

% ================================================================
% Time Discretisation
% ================================================================
\section{Time Discretisation}
\label{DOM_nxt}

The time stepping used in \NEMO is a three level scheme that can be 
represented as follows:
\begin{equation} \label{Eq_DOM_nxt}
   x^{t+\Delta t} = x^{t-\Delta t} + 2 \, \Delta t \  \text{RHS}_x^{t-\Delta t,t,t+\Delta t}
\end{equation} 
where $x$ stands for $u$, $v$, $t$ or $S$; RHS is the Right-Hand-Side of the 
corresponding time evolution equation; $\Delta t$ is the time step; and the 
superscripts indicate the time at which a quantity is evaluated. Each term of the 
RHS is evaluated at a specific time step(s) depending on the physics with which 
it is associated. 

The choice of the time step used for this evaluation is discussed below as 
well as the implications in terms of starting or restarting a model 
simulation. Note that the time stepping is generally performed in a one step 
operation. With such a complex and nonlinear system of equations it would be 
dangerous to let a prognostic variable evolve in time for each term separately.

The three level scheme requires three arrays for each prognostic variables. 
For each variable $x$ there is $x_b$ (before) and $x_n$ (now). The third array, 
although referred to as $x_a$ (after) in the code, is usually not the variable at 
the next time step; but rather it is used to store the time derivative (RHS in 
\eqref{Eq_DOM_nxt}) prior to time-stepping the equation. Generally, the time 
stepping is performed once at each time step in \mdl{tranxt} and \mdl{dynnxt} 
modules, except for implicit vertical diffusion or sea surface height when 
time-splitting options are used.

% -------------------------------------------------------------------------------------------------------------
%        Non-Diffusive Part---Leapfrog Scheme
% -------------------------------------------------------------------------------------------------------------
\subsection{Non-Diffusive Part --- Leapfrog Scheme}
\label{DOM_nxt_leap_frog}

The time stepping used for non-diffusive processes is the well-known 
leapfrog scheme. It is a time centred scheme, i.e. the RHS is evaluated at 
time step $t$, the now time step. It is only used for non-diffusive terms, 
that is momentum and tracer advection, pressure gradient, and Coriolis 
terms. This scheme is widely used for advective processes in low-viscosity 
fluids. It is an efficient method that achieves second-order accuracy with 
just one right hand side evaluation per time step. Moreover, it does not 
artificially damp linear oscillatory motion nor does it produce instability 
by amplifying the oscillations. These advantages are somewhat diminished by 
the large phase-speed error of the leapfrog scheme, and the unsuitability of 
leapfrog differencing for the representation of diffusive and Rayleigh 
damping processes. However, the most serious problem associated with the 
leapfrog scheme is a high-frequency computational noise called 
"time-splitting" \citep{Haltiner1980} that develops when the method 
is used to model non linear fluid dynamics: the even and odd time steps tend 
to diverge into a physical and a computational mode. Time splitting can 
be controlled through the use of an Asselin time filter (first designed by 
\citep{Robert1966} and more comprehensively studied by \citet{Asselin1972}), 
or by periodically reinitialising the leapfrog solution through a single 
integration step with a two-level scheme. In \NEMO we follow the first 
strategy:
\begin{equation} \label{Eq_DOM_nxt_asselin}
x_F^t  = x^t + \gamma \, \left[ x_f^{t-\Delta t} - 2 x^t + x^{t+\Delta t} \right]
\end{equation} 
where the subscript $f$ denotes filtered values and $\gamma$ is the Asselin 
coefficient. $\gamma$ is initialized as \np{atfp} (namelist parameter). 
Its default value is \np{atfp}=0.1.  This default value causes a significant dissipation 
of high frequency motions. Recommended values in idealized studies of shallow 
water turbulence are two orders of magnitude smaller (\citep{Farge1987}). 
Both strategies do, nevertheless, degrade the accuracy of the calculation from 
second to first order. The leapfrog scheme combined with a Robert-Asselin 
time filter has been preferred to other time differencing schemes such as 
predictor corrector or trapezoidal schemes, because the user has an explicit 
and simple control of the magnitude of the time diffusion of the scheme. 
In association with the 2nd order centred space discretisation of the 
advective terms in the momentum and tracer equations, it avoids implicit 
numerical diffusion in both the time and space discretisations of the 
advective term: they are both set explicitly by the user through the Robert-Asselin 
filter parameter and the viscous and diffusive coefficients.

\gmcomment{
%gm - reflexion about leapfrog: ongoing work with Matthieu Leclair
% to be updated latter with addition of new time stepping strategy
\colorbox{yellow}{Note}: 
The Robert-Asselin time filter slightly departs from a simple second order time 
diffusive operator computed with a forward time stepping due to the presence of 
$x_f^{t-\Delta t}$ in the right hand side of  \ref{Eq_DOM_nxt_asselin}. The original 
willing of Robert1966 and Asselin1972 was to design a time filter that allow much 
larger parameter than 0.5.   is due to computer saving consideration. In the original 
asselin filter, $x^{t-\Delta t}$ is used instead:
 \begin{equation} \label{Eq_DOM_nxt_asselin_true}
x_f^t  = x^t + \gamma \, \left[ x^{t-\Delta t} - 2 x^t + x^{t+\Delta t} \right]
\end{equation} 
Applying a "true" Asselin time filter is nothing more than adding a harmonic 
diffusive operator in time. Indeed, equations \ref{Eq_DOM_nxt} and 
\ref{Eq_DOM_nxt_asselin_true} can be rewritten together as:
\begin{equation} \label{Eq_DOM_nxt2}
\begin{split}
  \frac{ x^{t+\Delta t} - x^{t-\Delta t} } { 2 \,\Delta t } 
  &=  \text{RHS}_x^{t-\Delta t,t,t+\Delta t} + \frac{ x_f^t  - x^t }{2 \,\Delta t} \\
  &=  \text{RHS}_x^{t-\Delta t,t,t+\Delta t} 
    + \gamma\ \frac{  \, \left[ x^{t-\Delta t} - 2 x^t + x^{t+\Delta t} \right] }{2 \,\Delta t}  \\
  &=  \text{RHS}_x^{t-\Delta t,t,t+\Delta t} 
  + 2 \Delta t \ \gamma \ \frac{1}{{2 \Delta t}^2}  
   \,\delta_{t-1}\,\left[ \delta_{t+1/2}\left[ x^t \right] \right] 
  \end{split}
\end{equation} 
expressing this again in a continuous form, one finds that the Asselin filter leads to :
\begin{equation} \label{Eq_DOM_nxt3}
  \frac{ \partial x} { \partial t } =  \text{RHS} + 2 \,\Delta t \ \gamma \ \frac{ {\partial}^2 x}{ \partial t ^2 }
\end{equation} 

Equations  \ref{Eq_DOM_nxt2} and \ref{Eq_DOM_nxt3} suggest several remarks. 
First the Asselin filter is definitively a second order time diffusive operator which is 
evaluated at centered time step. The magnitude of this diffusion is proportional to 
the time step (with $\gamma$ usually taken between $10^{-1}$ to $10^{-3}$). 
Second, this term has to be taken into account in all budgets of the equations 
(mass, heat content, salt content, kinetic energy...). Nevertheless,we stress here 
that it is small and does not create systematic biases. Indeed let us evaluate how 
it contributes to the time evolution of $x$ between $t_o$ and $t_1$:
\begin{equation} \label{Eq_DOM_nxt4}
\begin{split}
 t_1-t_o &= \int_{t_o}^{t_1} \frac{ \partial x} { \partial t } dt \\
      &= \int_{t_o}^{t_1} \text{RHS} dt + 2 \,\Delta t \ \gamma \left(
        \left. \frac{ \partial x}{ \partial t } \right|_1 
      - \left. \frac{ \partial x}{ \partial t } \right|_o  \right)
 \end{split}
\end{equation} 
}

Alternative time stepping schemes are currently under investigation.

% -------------------------------------------------------------------------------------------------------------
%        Diffusive Part---Forward or Backward Scheme
% -------------------------------------------------------------------------------------------------------------
\subsection{Diffusive Part --- Forward or Backward Scheme}
\label{DOM_nxt_forward_imp}

The leapfrog differencing scheme is unsuitable for the representation of 
diffusive and damping processes. For a tendancy $D_x$, representing a 
diffusive term or a restoring term to a tracer climatology 
(when present, see \S~\ref{TRA_dmp}), a forward time differencing scheme
 is used :
\begin{equation} \label{Eq_DOM_nxt_euler}
   x^{t+\Delta t} = x^{t-\Delta t} + 2 \, \Delta t \ {D_x}^{t-\Delta t}
\end{equation} 

This is diffusive in time and conditionally stable. For example, the 
conditions for stability of second and fourth order horizontal diffusion schemes are \citep{Griffies_Bk04}:
\begin{equation} \label{Eq_DOM_nxt_euler_stability}
A^h < \left\{
\begin{aligned}
                    &\frac{e^2}{  8 \; \Delta t }  &&\quad \text{laplacian diffusion}  \\
                    &\frac{e^4}{64 \; \Delta t }   &&\quad \text{bilaplacian diffusion} 
            \end{aligned}
\right.
\end{equation}
where $e$ is the smallest grid size in the two horizontal directions and $A^h$ is 
the mixing coefficient. The linear constraint \eqref{Eq_DOM_nxt_euler_stability} 
is a necessary condition, but not sufficient. If it is not satisfied, even mildly, 
then the model soon becomes wildly unstable. The instability can be removed 
by either reducing the length of the time steps or reducing the mixing coefficient.

For the vertical diffusion terms, a forward time differencing scheme can be 
used, but usually the numerical stability condition implies a strong 
constraint on the time step. Two solutions are available in \NEMO to overcome 
the stability constraint: $(a)$ a forward time differencing scheme using a 
time splitting technique (\np{ln\_zdfexp}=.true.) or $(b)$ a backward (or implicit) 
time differencing scheme by \np{ln\_zdfexp}=.false.). In $(a)$, the master 
time step $\Delta $t is cut into $N$ fractional time steps so that the 
stability criterion is reduced by a factor of $N$. The computation is done as 
follows:
\begin{equation} \label{Eq_DOM_nxt_ts}
\begin{split}
& u_\ast ^{t-\Delta t} = u^{t-\Delta t}   \\
& u_\ast ^{t-\Delta t+L\frac{2\Delta t}{N}}=u_\ast ^{t-\Delta t+\left( {L-1} 
\right)\frac{2\Delta t}{N}}+\frac{2\Delta t}{N}\;\text{DF}^{t-\Delta t+\left( {L-1} \right)\frac{2\Delta t}{N}}
        \quad \text{for $L=1$ to $N$}      \\
& u^{t+\Delta t} = u_\ast^{t+\Delta t}
\end{split}
\end{equation}
with DF a vertical diffusion term. The number of fractional time steps, $N$, is given 
by setting \np{n\_zdfexp}, (namelist parameter). The scheme $(b)$ is unconditionally 
stable but diffusive. It can be written as follows:
\begin{equation} \label{Eq_DOM_nxt_imp}
   x^{t+\Delta t} = x^{t-\Delta t} + 2 \, \Delta t \  \text{RHS}_x^{t+\Delta t}
\end{equation} 

This scheme is rather time consuming since it requires a matrix inversion, 
but it becomes attractive since a splitting factor of 3 or more is needed 
for the forward time differencing scheme. For example, the finite difference 
approximation of the temperature equation is:
\begin{equation} \label{Eq_DOM_nxt_imp_zdf}
\frac{T(k)^{t+1}-T(k)^{t-1}}{2\;\Delta t}\equiv \text{RHS}+\frac{1}{e_{3t} }\delta 
_k \left[ {\frac{A_w^{vT} }{e_{3w} }\delta _{k+1/2} \left[ {T^{t+1}} \right]} 
\right]
\end{equation}
where RHS is the right hand side of the equation except for the vertical diffusion term. 
We rewrite \eqref{Eq_DOM_nxt_imp} as:
\begin{equation} \label{Eq_DOM_nxt_imp_mat}
-c(k+1)\;u^{t+1}(k+1)+d(k)\;u^{t+1}(k)-\;c(k)\;u^{t+1}(k-1) \equiv b(k)
\end{equation}
where 
\begin{align*} 
 c(k) &= A_w^{vm} (k) \, / \, e_{3uw} (k)     \\
 d(k) &= e_{3u} (k)       \, / \, (2\Delta t) + c_k + c_{k+1}    \\
 b(k) &= e_{3u} (k) \; \left( u^{t-1}(k) \, / \, (2\Delta t) + \text{RHS} \right)    
\end{align*}

\eqref{Eq_DOM_nxt_imp_mat} is a linear system of equations which associated 
matrix is tridiagonal. Moreover, $c(k)$ and $d(k)$ are positive and the diagonal 
term is greater than the sum of the two extra-diagonal terms, therefore a special 
adaptation of the Gauss elimination procedure is used to find the solution 
(see for example \citet{Richtmyer1967}).

% -------------------------------------------------------------------------------------------------------------
%        Start/Restart strategy
% -------------------------------------------------------------------------------------------------------------
\subsection{Start/Restart strategy}
\label{DOM_nxt_rst}
%--------------------------------------------namrun-------------------------------------------
\namdisplay{namrun}         
%--------------------------------------------------------------------------------------------------------------

The first time step of this three level scheme when starting from initial conditions 
is a forward step (Euler time integration): 
\begin{equation} \label{Eq_DOM_euler}
   x^1 = x^0 + \Delta t \ \text{RHS}^0
\end{equation}

It is also possible to restart from a previous computation, by using a 
restart file. The restart strategy is designed to ensure perfect 
restartability of the code: the user should obtain the same results to 
machine precision either by running the model for $2N$ time steps in one go, 
or by performing two consecutive experiments of $N$ steps with a restart. 
This requires saving two time levels and many auxiliary data in the restart 
files in machine precision. 

Note that when a semi-implicit scheme is used to evaluate the hydrostatic pressure 
gradient (see \S\ref{DYN_hpg_imp}), an extra three-dimensional field has to be 
added in the restart file to ensure an exact restartability. This is done only optionally 
via the namelist parameter \np{nn\_dynhpg\_rst}, so that a reduction of the size of 
restart file can be obtained when the restartability is not a key issue (operational 
oceanography or ensemble simulation for seasonal forcast).
%%%
\gmcomment{add here how to force the restart to contain only one time step for operational purposes}
%%%

\gmcomment{       % add a subsection here  

%-------------------------------------------------------------------------------------------------------------
%        Time Domain
% -------------------------------------------------------------------------------------------------------------
\subsection{Time domain}
\label{DOM_nxt_time}

 \colorbox{yellow}{add here a few word on nit000 and nitend}

 \colorbox{yellow}{Write documentation on the calendar and the key variable adatrj}

}        %% end add





Implicit time stepping in case of variable volume thickness.

Tracer case (NB for momentum in vector invariant form take care!)

\begin{flalign*}
&\frac{\left( e_{3t}\,T \right)_k^{t+1}-\left( e_{3t}\,T \right)_k^{t-1}}{2\Delta t}
\equiv \text{RHS}+ \delta _k \left[ {\frac{A_w^{vt} }{e_{3w}^{t+1} }\delta _{k+1/2} \left[ {T^{t+1}} \right]} 
\right]      \\
&\left( e_{3t}\,T \right)_k^{t+1}-\left( e_{3t}\,T \right)_k^{t-1}
\equiv {2\Delta t} \ \text{RHS}+ {2\Delta t} \ \delta _k \left[ {\frac{A_w^{vt} }{e_{3w}^{t+1} }\delta _{k+1/2} \left[ {T^{t+1}} \right]} 
\right]      \\
&\left( e_{3t}\,T \right)_k^{t+1}-\left( e_{3t}\,T \right)_k^{t-1}
\equiv 2\Delta t \ \text{RHS}
+ 2\Delta t \ \left\{ \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k+1/2} [ T_{k+1}^{t+1} - T_k      ^{t+1} ]
                          - \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k-1/2} [ T_k       ^{t+1} - T_{k-1}^{t+1} ]  \right\}     \\
&\\
&\left( e_{3t}\,T \right)_k^{t+1}
-  {2\Delta t} \           \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k+1/2}                  T_{k+1}^{t+1} 
+ {2\Delta t} \ \left\{  \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k+1/2} 
                            +  \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k-1/2}     \right\}   T_{k    }^{t+1}
-  {2\Delta t} \           \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k-1/2}                  T_{k-1}^{t+1}      \\
&\equiv \left( e_{3t}\,T \right)_k^{t-1} + {2\Delta t} \ \text{RHS}    \\
%
\end{flalign*}

\begin{flalign*}
\allowdisplaybreaks
\intertext{ Tracer case }
%
&  \qquad \qquad  \quad   -  {2\Delta t}                  \ \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k+1/2}   
                                                                                                      \qquad \qquad \qquad  \qquad  T_{k+1}^{t+1}   \\
&+ {2\Delta t} \ \biggl\{  (e_{3t})_{k   }^{t+1}  \bigg. +    \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k+1/2}  
                                                                               +   \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k-1/2} \bigg. \biggr\}  \ \ \ T_{k   }^{t+1}  &&\\
& \qquad \qquad  \qquad \qquad \qquad \quad \ \ -  {2\Delta t} \                          \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k-1/2}                          \quad \ \ T_{k-1}^{t+1}     
\ \equiv \ \left( e_{3t}\,T \right)_k^{t-1} + {2\Delta t} \ \text{RHS}  \\
%
\end{flalign*}
\begin{flalign*}
\allowdisplaybreaks
\intertext{ Tracer content case }
%
& -  {2\Delta t} \              & \frac{(A_w^{vt})_{k+1/2}} {(e_{3w})_{k+1/2}^{t+1}\;(e_{3t})_{k+1}^{t+1}}  && \  \left( e_{3t}\,T \right)_{k+1}^{t+1}   &\\
& + {2\Delta t} \ \left[ 1  \right.+ & \frac{(A_w^{vt})_{k+1/2}} {(e_{3w})_{k+1/2}^{t+1}\;(e_{3t})_k^{t+1}} 
                                    + & \frac{(A_w^{vt})_{k -1/2}} {(e_{3w})_{k -1/2}^{t+1}\;(e_{3t})_k^{t+1}}  \left.  \right]  & \left( e_{3t}\,T \right)_{k   }^{t+1}  &\\
& -  {2\Delta t} \               & \frac{(A_w^{vt})_{k -1/2}} {(e_{3w})_{k -1/2}^{t+1}\;(e_{3t})_{k-1}^{t+1}}     &\  \left( e_{3t}\,T \right)_{k-1}^{t+1}   
\equiv \left( e_{3t}\,T \right)_k^{t-1} + {2\Delta t} \ \text{RHS}  &
\end{flalign*}