source: trunk/NEMO/DOC/BETA/Chapters/Chap_DOM.tex @ 705

% ================================================================
% 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
%  - 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





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 OPA, and other information relevant to the main directory routines (time stepping, main program) as well as the DOM (DOMain) directory. 

% ================================================================
% 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]{./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 in 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. 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 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 allows satisfying many of the continuous properties (see { \colorbox{yellow}{Annexe 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 indexation is only used for the writing of semi-discrete equation. In the code, the indexation use integer value 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 derivation 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 $T$-point has its three components defined at $(u,v,w)$ 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} \left[ q \right]\;\,{\rm {\bf i}}
         +  \frac{1}{e_{2v} }\delta _{j+1/2} \left[ q \right]\;\,{\rm {\bf j}}
         +  \frac{1}{e_{3w} }\delta _{k+1/2} \left[ q \right]\;\,{\rm {\bf k}}
\end{equation}
\begin{multline} \label{Eq_DOM_lap}
\Delta q\equiv \frac{1}{e_{1T} {\kern 1pt}e_{2T} {\kern 1pt}e_{3T} }\;\left( 
{\delta _i \left[ {\frac{e_{2u} e_{3u} }{e_{1u} }\;\delta _{i+1/2} 
\left[ q \right]} \right]
+\delta _j \left[ {\frac{e_{1v} e_{3v} }{e_{2v} 
}\;\delta _{j+1/2} \left[ q \right]} \right]\;} \right)     \\
+\frac{1}{e_{3T} }\delta _k \left[ {\frac{1}{e_{3w} }\;\delta _{k+1/2} 
\left[ q \right]} \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 $(vw,uw,f)$ and its divergence defined at $T$-points:
\begin{equation} \label{Eq_DOM_curl}
\begin{split}
 \nabla \times {\rm {\bf A}}\equiv \frac{1}{e_{2v} {\kern 1pt}e_{3vw} 
}{\kern 1pt}\,\;\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} {\kern 1pt}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{e_{3f} }{e_{1f} {\kern 1pt}e_{2f} }\,{\kern 1pt}\;\left( {\delta 
_{i+1/2} \left[ {e_{2v} a_2 } \right]-\delta _{j+12} \left[ {e_{1u} a_1 } \right]} 
\right)  &\;\;{\rm {\bf k}}
 \end{split}
\end{equation}
\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 pure $z$-coordinates 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. It can be simplified from outside and inside the $\delta _i$ and $\delta_j$ operators. 

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 $H_q$  the ocean depth, is the masked sum of the vertical scale factors at q points, $k^b$ and $k^o$ are the bottom and surface $k$-index, and the symbol $k^o$ referring to a summation over all grid points of the same species in the direction indicated by the subscript (here $k$). 

In continuous, 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 straight forward 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 ocean mesh, extended to zero inside continental area. By integration by part it can be shown that the derivation operators ($\delta_i$, $\delta_j$ and $\delta_k$) are anti-symmetric linear operators, and further that the averaging operators $\overline{\cdot}^i$, $\overline{\cdot}^j$ and $\overline{\cdot}^k$) are symmetric linear operators, i.e.,
\begin{equation} \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} }
\end{equation}
\begin{equation} \label{DOM_mi_adj}
\sum\limits_i {a_i \;\overline b ^i} \equiv \sum\limits_i {\overline a ^{i+1/2}\;b_{i+1/2} } 
\end{equation}

In other words, the adjoint of the derivation 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 \colorbox{yellow} {Appendix C} to 
demonstrate integral conservative properties of the discrete formulation chosen.

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

%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
\begin{figure}[!tb] \label{Fig_index_hor}  \begin{center}
\includegraphics[width=0.90\textwidth]{./Figures/Fig_index_hor.pdf}
\caption{Horizontal integer indexation 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 indexation while the analytical definition of the mesh (see \S\ref{DOM_cell}) is associated with the use of integer values of $(i,j,k)$ for $T$-points whereas all the other points use both integer and integer and a half values of $(i,j,k)$. Therefore a specific integer indexation must be defined for the latter grid-points 
(i.e. velocity and vorticity grid-points). Furthermore, it has been chosen to change the direction of the vertical indexation so that the surface level is at $k=1$.

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

The indexation 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 nearby northeast $f$-point have the same $i$-and $j$-indices.

% -----------------------------------
%        Vertical Indexation 
% -----------------------------------
\subsubsection{Vertical Indexation}
\label{DOM_Num_Index_vertical}

In the vertical plane, the chosen indexation requires special attention since the $k$-axis is re-oriented downward in the \textsc{Fortran} code compared to the indexation used for the semi-discrete equations and given in \S\ref{DOM_cell}. The sea surface corresponds to the $w$-level $k=1$ like the $T-$level just below (Fig.\ref{Fig_index_vert}). The last $w$-level ($k=jpk$) is either the ocean bottom or inside the ocean floor while the last $T-$level is always inside the floor (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 nearby 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}). As the scale factors are chosen to be strictly positive, \emph{a minus sign appears in the \textsc{Fortran} code 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]{./Figures/Fig_index_vert.pdf}
\caption{Vertical integer indexation used in the \textsc{Fortran } code. Note that the $k$-axis is oriented downward. The dashed area indicates the cell in which variables contained in arrays have the same $k$-index.}
\end{center}   \end{figure}
%>>>>>>>>>>>>>>>>>>>>>>>>>>>>

% -----------------------------------
%        Vertical Indexation 
% -----------------------------------
\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 (or additional files included in this module such as \textit{par\_ORCA\_R2.h90}, specific to a given configuration). The use of parameters rather than variables (together with dynamic allocation of arrays) was made 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 \colorbox{yellow}{(ref agrif!+ 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}).

% ================================================================
% Domain: Horizontal Grid (mesh) 
% ================================================================
\section{Domain: Horizontal Grid (mesh) (\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 latitude $\varphi$ and the longitude $\lambda$ as a function of  $(i,j)$. The horizontal scale factors are calculated using \eqref{Eq_scale_factors}. For example, when the latitude and longitude are function of a single value ($j$ and $i$, respectively) (geographical configuration of the mesh), the horizontal mesh definition reduces to define the wanted $\varphi(j)$, $\varphi'(j)$, $\lambda(i)$, and $\lambda'(i)$ 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 and scale factors of $w$-points are exactly equal to those of $T-$points, thus no specific arrays are defined at those grid-points. 

Note that the definition of the scale factors --- as the analytical first derivative of the transformation that gives $(\lambda,\varphi,z)$ as a function of $(i,j,k)$ --- is specific to the OPA model \citep{Marti1992}. 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 smooth grids \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]{./Figures/Fig_zgr_e3.pdf}
\caption{(a) Traditional definition of grid-point position and grid-size in the vertical versus (b) analytically derived grid-point position and scale factors. For both grid here,a same $w$-point depth has been chosen but in (a) the $T$-points are set at the middle of $w$-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 grids 
% -------------------------------------------------------------------------------------------------------------
\subsection{Choice of horizontal grids}
\label{DOM_hgr_msh_choice}

The user has three options to define a horizontal grid, involving the parameter $jphgr\_mesh$ of the \mdl{par\_oce} module. 
\begin{enumerate}
\item For 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: \jp{jphgr\_mesh}=0.
\item A few simple analytical grids are provided as examples, that can be selected by setting \jp{jphgr\_mesh}=1 to 5 (see below) 
\item For other analytical grids, the \mdl{domhgr} module must be modified by the user. 
\end{enumerate}

There are two simple cases of geographical grids on the sphere. With \jp{jphgr\_mesh}=1, the grid is regular in space, with grid sizes specified by parameters \pp{ppe1\_deg} and \pp{ppe2\_deg}, respectively. 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 that 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). In those two cases (\jp{jphgr\_mesh}=1 or 4), the grid position is defined by the longitude and latitude of the south-westhernmost 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, so as to ensure that the equator corresponds to a $T$- and$ U$-point.  

Rectangular grids ignoring the spherical geometry are defined with \jp{jphgr\_mesh} = 2, 3, 5. The domain is either a $f$-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 each direction, and given in meters by the parameters \pp{ppe1\_m} and \pp{ppe2\_m} respectively. The zonal grid coordinate (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 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 related 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 the insufficient model resolution). On example is Lombok Strait in the ORCA2 configuration. When such modifications are done, the output grid written when $\np{nmsh} \not=0$ is not exactly equal to the input grid.

% ================================================================
% Domain: Vertical Grid (domzgr)
% ================================================================
\section{Domain: Vertical Grid (\mdl{domzgr} module)}
\label{DOM_zgr}
%-----------------------------------------nam_zgr & namdom-------------------------------------------
\namdisplay{nam_zgr} 
\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 $(i,j)$.

%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
\begin{figure}[!tb] \label{Fig_z_zps_s_sps}  \begin{center}
\includegraphics[width=1.0\textwidth]{./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 level associated with the non-linear free surface. Note that the variable volume level (\key{vvl}) could be used with any of the 5 coordinates (a) to (e).}
\end{center}   \end{figure}
%>>>>>>>>>>>>>>>>>>>>>>>>>>>>

The choice of a vertical coordinate among all those offered in NEMO, 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: hybrid $s-z$ 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}), 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 $s$-coordinates (hybride and partial step coordinates not yet implemented 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). 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 height at the deepest levels using the bathymetry, to obtain the final three-dimensional depth and scale factor arrays.
\item[\textit{sco}] Smooth the bathymetry to fullfill the hydrostatic consistency criteria and set the three-dimensional transformation.
\item[\textit{s-z} and \textit{s-zps}] Smooth the bathymetry to fullfill 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}

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 coordinates and scale factors are defined as functions of $(i,j,k)$ with "fs" as prefix (examples: \textit{fsdeptht, fse3t,} etc) that can be equal to three-dimensional arrays, or a one dimensional array when \key{zco} is defined. These functions are defined in the file 
\textit{domzgr\_substitute.h90} of the DOM directory. They are used through 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 a closed basin or a periodic channel according to the parameter \jp{jperio}. 
\item[\np{ntopo} = -1] a domain with a bump of topography at the central latitude and 1/3 of the domain width. 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. The bathymetry file 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} defined) isolated land masses (islands) must be identified by negative integers in the input bathymetry file (see \S\ref{MISC_solisl}).

When the 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 to represent their communication with the rest of 
the ocean. This is unnecessary when the ocean is forced by fixed atmospheric 
conditions. A possibility is offered to the user to set to zero the 
bathymetry in rectangular regions covering those closed seas (see \S\ref{MISC_closea})

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

%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
\begin{figure}[!tb] \label{Fig_zgr}  \begin{center}
\includegraphics[width=0.90\textwidth]{./Figures/Fig_zgr.pdf}
\caption{Default vertical mesh for ORCA2-L30. Vertical level functions for (a) T-point depth and (b) the associated scale factor as computed from (III.2.1) in z-coordinates.}
\end{center}   \end{figure}
%>>>>>>>>>>>>>>>>>>>>>>>>>>>>

The reference coordinate transformation $z_0 (k)$ defines the arrays \textit{gdept0} and \textit{gdepw0} for $T$- and $w$-points, respectively. As indicated on Fig.\ref{Fig_index_vert} \jp{jpk} is the number of $w$-levels. $gdepw(1)$ being the ocean surface. There are at most \jp{jpk}-1 $T$-points in 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 analytic 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 $z$-coordinates and 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 first grid defined for ORCA2 had $10~m$ ($500~m)$ 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 vertial 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 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}=T)}
\label{DOM_zps}
%--------------------------------------------namdom-----------------------------------------------------
\namdisplay{namdom} 
%--------------------------------------------------------------------------------------------------------------

In that case, the depths of the model levels are still defined by the 
reference analytical function $z_0 (k)$ as described in the previous 
section, excepted 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 layers 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{e3zpsmin} 
(thickness in meters, usually $20~m$) or $e_{3t}(jk)*\np{e3zps\_rat}$ (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}=T)}
\label{DOM_sco}
%--------------------------------------------nam_zgr_sco---------------------------------------------------
\namdisplay{nam_zgr_sco} 
%--------------------------------------------------------------------------------------------------------------
In s-coordinate (\key{sco} defined), the depths of the model 
levels are defined from the product of a depth field and a stretching 
function and 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 $T-$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 as 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{Song1994} 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 
$0\leqslant b\leqslant 1$. Examples of the stretching function applied to a seamount are given in Fig.~\ref{Fig_sco_function}.

%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
\begin{figure}[!tb] \label{Fig_sco_function}  \begin{center}
\includegraphics[width=1.0\textwidth]{./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 resolution}
\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{Madec1996}. 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 case of rigid-lid approximation and islands in the computational domain (\np{ln\_dynspg\_rl}=true and \key{island} 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 of the $n^{th}$ island ; $mbathy(i,j) =0$, $T-$points are land points of the main land (continent) ; $mbathy(i,j) =k$, the first $k$ $T$- and $w$-points are ocean points, the others land points. This is used to compute the island barotropic stream function used in rigid lid computation (see \S\ref{LBC_isl}).

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)  \\
umask(i,j,k) &= \; tmask(i,j,k) \;.\; tmask(i,j+1,k)  \\
umask(i,j,k) &= \; tmask(i,j,k) \;.\; tmask(i+1,j,k)  \\
                  & \quad . \; tmask(i,j,k) \;.\; tmask(i+1,j,k)
\end{align*}

Note that \textit{wmask} is not defined as it is exactly equal to \textit{tmask} with the numerical 
indexation 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 
\textit{mbathy} array are set to zero. In the particular case of an east-west cyclic 
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 the mask arrays) (see \S~\ref{LBC_jperio}).

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

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

The time stepping used in OPA is a three level scheme that can be presented 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 $x$ stand 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 overscripts indicate 
the time at which a quantity is evaluated. Each term of the RHS is evaluated at 
specific time step(s) depending on the physics to which it is associated. 
The choice of the time step used for this evaluation is discussed below as 
well as the implication in term of starting or restarting a model 
simulation. Note that the time stepping is generally performed in a one step 
operation: it would be dangerous to let a prognostic variable evolve in time for each term successively.

The three level scheme requires three arrays for the 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 $x_a$ at the next time step; 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, excepted 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 are 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 between 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 OPA 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. Recommanded values in idealized studies of shallow water turbulence are two order of magnitude lower (\citep{Farge1987}). Both strategies do, nevertheless, degrade the accuracy of the calculation from second to first order. The leapfrog scheme associated to a Robert-Asselin 
time filter has been preferred to other time differencing schemes such as 
predictor corrector or trapezoidal schemes because the user can better 
control the magnitude and the spatial structure of the time diffusion of the 
scheme. In association with the centred space discretisation of the 
advective terms in the momentum and tracer equations, it avoids implicit 
numerical diffusion in both the time and space discretisation of the 
advective term: they are both set explicitly by the user through the Robert-Asselin filter parameter and the viscous and diffusive coefficients.

%gm - reflexion about leapfrog: ongoing work with Matthieu Leclair
% to be updated latter with addition of new time stepping strategy
\amtcomment{
\colorbox{yellow}{Note}: 
1- There is no reason why one should apply a same value of $\gamma$ on both momentum and tracer equations. In climate applications, one could found useful to use a lower value on tracer (quantity that one wants to conserve) than on the dynamics. We never explore this possibility.
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 have to be taken into account in all budget of the equations (mass, heat content, salt content, kinetic energy...). Nevertheless, we stress here that it is small and does not systematic biases. Indeed let evaluates how it contributes to 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 is unsuitable for the representation of diffusive 
and damping processes. For $D$, a horizontal diffusive terms and/or the 
restoring terms to a tracer climatology (when they are 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 \  \text{RHS}_x^{t-\Delta t}
\end{equation} 

This is diffusive in time and conditionally stable. For example, the 
condition of stability for a second and fourth order horizontal diffusions are \citep{Griffies2004}:
\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 direction and $A^h$ 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 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 OPA to overcome 
the stability constraint: $(a)$ a forward time differencing scheme using a 
time splitting technique (\np{ln\_zdfexp}=T) or $(b)$ a backward (or implicit) 
time differencing scheme by \np{ln\_zdfexp}=F). 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 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. All the elements of the corresponding matrix vanish except those on the diagonals. 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): $x^1 = x^0 + \Delta t \ \text{RHS}^0$.

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 double precision. 


%-------------------------------------------------------------------------------------------------------------
%        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}