source: trunk/DOC/TexFiles/Chapters/Chap_DOM.tex @ 5120

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#1473. Re-organisation and optimisation of ice shelf cavity option. This commit merges changes from the dev_r5094_UKMO_ISFCLEAN branch onto the trunk. Results will change, even with ln_isfcav=F, due to a return to original definitions of the vertical metrics. All changes have been reviewed and SETTE tested.

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1% ================================================================
2% Chapter 2 � Space and Time Domain (DOM)
3% ================================================================
4\chapter{Space Domain (DOM) }
5\label{DOM}
6\minitoc
7
8% Missing things:
9%  - istate: description of the initial state   ==> this has to be put elsewhere..
10%                  perhaps in MISC ?  By the way the initialisation of T S and dynamics
11%                  should be put outside of DOM routine (better with TRC staff and off-line
12%                  tracers)
13%  -geo2ocean:  how to switch from geographic to mesh coordinate
14%     - domclo:  closed sea and lakes.... management of closea sea area : specific to global configuration, both forced and coupled
15
16
17\newpage
18$\ $\newline    % force a new ligne
19
20Having defined the continuous equations in Chap.~\ref{PE} and chosen a time
21discretization Chap.~\ref{STP}, we need to choose a discretization on a grid,
22and numerical algorithms. In the present chapter, we provide a general description
23of the staggered grid used in \NEMO, and other information relevant to the main
24directory routines as well as the DOM (DOMain) directory.
25
26$\ $\newline    % force a new lign
27
28% ================================================================
29% Fundamentals of the Discretisation
30% ================================================================
31\section{Fundamentals of the Discretisation}
32\label{DOM_basics}
33
34% -------------------------------------------------------------------------------------------------------------
35%        Arrangement of Variables
36% -------------------------------------------------------------------------------------------------------------
37\subsection{Arrangement of Variables}
38\label{DOM_cell}
39
40%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
41\begin{figure}[!tb]    \begin{center}
42\includegraphics[width=0.90\textwidth]{./TexFiles/Figures/Fig_cell.pdf}
43\caption{ \label{Fig_cell}   
44Arrangement of variables. $t$ indicates scalar points where temperature,
45salinity, density, pressure and horizontal divergence are defined. ($u$,$v$,$w$)
46indicates vector points, and $f$ indicates vorticity points where both relative and
47planetary vorticities are defined}
48\end{center}   \end{figure}
49%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
50
51The numerical techniques used to solve the Primitive Equations in this model are
52based on the traditional, centred second-order finite difference approximation.
53Special attention has been given to the homogeneity of the solution in the three
54space directions. The arrangement of variables is the same in all directions.
55It consists of cells centred on scalar points ($t$, $S$, $p$, $\rho$) with vector
56points $(u, v, w)$ defined in the centre of each face of the cells (Fig. \ref{Fig_cell}).
57This is the generalisation to three dimensions of the well-known ``C'' grid in
58Arakawa's classification \citep{Mesinger_Arakawa_Bk76}. The relative and
59planetary vorticity, $\zeta$ and $f$, are defined in the centre of each vertical edge
60and the barotropic stream function $\psi$ is defined at horizontal points overlying
61the $\zeta$ and $f$-points.
62
63The ocean mesh ($i.e.$ the position of all the scalar and vector points) is defined
64by the transformation that gives ($\lambda$ ,$\varphi$ ,$z$) as a function of $(i,j,k)$.
65The grid-points are located at integer or integer and a half value of $(i,j,k)$ as
66indicated on Table \ref{Tab_cell}. In all the following, subscripts $u$, $v$, $w$,
67$f$, $uw$, $vw$ or $fw$ indicate the position of the grid-point where the scale
68factors are defined. Each scale factor is defined as the local analytical value
69provided by \eqref{Eq_scale_factors}. As a result, the mesh on which partial
70derivatives $\frac{\partial}{\partial \lambda}, \frac{\partial}{\partial \varphi}$, and
71$\frac{\partial}{\partial z} $ are evaluated is a uniform mesh with a grid size of unity.
72Discrete partial derivatives are formulated by the traditional, centred second order
73finite difference approximation while the scale factors are chosen equal to their
74local analytical value. An important point here is that the partial derivative of the
75scale factors must be evaluated by centred finite difference approximation, not
76from their analytical expression. This preserves the symmetry of the discrete set
77of equations and therefore satisfies many of the continuous properties (see
78Appendix~\ref{Apdx_C}). A similar, related remark can be made about the domain
79size: when needed, an area, volume, or the total ocean depth must be evaluated
80as the sum of the relevant scale factors (see \eqref{DOM_bar}) in the next section).
81
82%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
83\begin{table}[!tb]
84\begin{center} \begin{tabular}{|p{46pt}|p{56pt}|p{56pt}|p{56pt}|}
85\hline
86&$i$     & $j$    & $k$     \\ \hline
87& $i+1/2$   & $j$    & $k$    \\ \hline
88& $i$    & $j+1/2$   & $k$    \\ \hline
89& $i$    & $j$    & $k+1/2$   \\ \hline
90& $i+1/2$   & $j+1/2$   & $k$    \\ \hline
91uw & $i+1/2$   & $j$    & $k+1/2$   \\ \hline
92vw & $i$    & $j+1/2$   & $k+1/2$   \\ \hline
93fw & $i+1/2$   & $j+1/2$   & $k+1/2$   \\ \hline
94\end{tabular}
95\caption{ \label{Tab_cell}
96Location of grid-points as a function of integer or integer and a half value of the column,
97line or level. This indexing is only used for the writing of the semi-discrete equation.
98In the code, the indexing uses integer values only and has a reverse direction
99in the vertical (see \S\ref{DOM_Num_Index})}
100\end{center}
101\end{table}
102%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
103
104% -------------------------------------------------------------------------------------------------------------
105%        Vector Invariant Formulation
106% -------------------------------------------------------------------------------------------------------------
107\subsection{Discrete Operators}
108\label{DOM_operators}
109
110Given the values of a variable $q$ at adjacent points, the differencing and
111averaging operators at the midpoint between them are:
112\begin{subequations} \label{Eq_di_mi}
113\begin{align}
114 \delta _i [q]       &\  \    q(i+1/2)  - q(i-1/2)    \\
115 \overline q^{\,i} &= \left\{ q(i+1/2) + q(i-1/2) \right\} \; / \; 2
116\end{align}
117\end{subequations}
118
119Similar operators are defined with respect to $i+1/2$, $j$, $j+1/2$, $k$, and
120$k+1/2$. Following \eqref{Eq_PE_grad} and \eqref{Eq_PE_lap}, the gradient of a
121variable $q$ defined at a $t$-point has its three components defined at $u$-, $v$-
122and $w$-points while its Laplacien is defined at $t$-point. These operators have
123the following discrete forms in the curvilinear $s$-coordinate system:
124\begin{equation} \label{Eq_DOM_grad}
125\nabla q\equiv    \frac{1}{e_{1u} } \delta _{i+1/2 } [q] \;\,\mathbf{i}
126      +  \frac{1}{e_{2v} } \delta _{j+1/2 } [q] \;\,\mathbf{j}
127      +  \frac{1}{e_{3w}} \delta _{k+1/2} [q] \;\,\mathbf{k}
128\end{equation}
129\begin{multline} \label{Eq_DOM_lap}
130\Delta q\equiv \frac{1}{e_{1t}\,e_{2t}\,e_{3t} }
131       \;\left(          \delta_\left[ \frac{e_{2u}\,e_{3u}} {e_{1u}} \;\delta_{i+1/2} [q] \right]
132+                        \delta_\left[ \frac{e_{1v}\,e_{3v}}  {e_{2v}} \;\delta_{j+1/2} [q] \right] \;  \right)      \\
133+\frac{1}{e_{3t}} \delta_k \left[ \frac{1}{e_{3w} }                     \;\delta_{k+1/2} [q] \right]
134\end{multline}
135
136Following \eqref{Eq_PE_curl} and \eqref{Eq_PE_div}, a vector ${\rm {\bf A}}=\left( a_1,a_2,a_3\right)$ 
137defined at vector points $(u,v,w)$ has its three curl components defined at $vw$-, $uw$,
138and $f$-points, and its divergence defined at $t$-points:
139\begin{eqnarray}  \label{Eq_DOM_curl}
140 \nabla \times {\rm {\bf A}}\equiv &
141      \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&\ \mathbf{i} \\ 
142 +& \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&\ \mathbf{j} \\
143 +& \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&\ \mathbf{k}
144 \end{eqnarray}
145\begin{equation} \label{Eq_DOM_div}
146\nabla \cdot \rm{\bf A}=\frac{1}{e_{1t}\,e_{2t}\,e_{3t}}\left( \delta_i \left[e_{2u}\,e_{3u}\,a_1 \right]
147                                                                                         +\delta_j \left[e_{1v}\,e_{3v}\,a_2 \right] \right)+\frac{1}{e_{3t} }\delta_k \left[a_3 \right]
148\end{equation}
149
150In the special case of a pure $z$-coordinate system, \eqref{Eq_DOM_lap} and
151\eqref{Eq_DOM_div} can be simplified. In this case, the vertical scale factor
152becomes a function of the single variable $k$ and thus does not depend on the
153horizontal location of a grid point. For example \eqref{Eq_DOM_div} reduces to:
154\begin{equation*}
155\nabla \cdot \rm{\bf A}=\frac{1}{e_{1t}\,e_{2t}} \left( \delta_i \left[e_{2u}\,a_1 \right] 
156                                                                              +\delta_j \left[e_{1v}\, a_2 \right]  \right)
157                                                     +\frac{1}{e_{3t}} \delta_k \left[             a_3 \right]
158\end{equation*}
159
160The vertical average over the whole water column denoted by an overbar becomes
161for a quantity $q$ which is a masked field (i.e. equal to zero inside solid area):
162\begin{equation} \label{DOM_bar}
163\bar q   =         \frac{1}{H}    \int_{k^b}^{k^o} {q\;e_{3q} \,dk} 
164      \equiv \frac{1}{H_q }\sum\limits_k {q\;e_{3q} }
165\end{equation}
166where $H_q$  is the ocean depth, which is the masked sum of the vertical scale
167factors at $q$ points, $k^b$ and $k^o$ are the bottom and surface $k$-indices,
168and the symbol $k^o$ refers to a summation over all grid points of the same type
169in the direction indicated by the subscript (here $k$).
170
171In continuous form, the following properties are satisfied:
172\begin{equation} \label{Eq_DOM_curl_grad}
173\nabla \times \nabla q ={\rm {\bf {0}}}
174\end{equation}
175\begin{equation} \label{Eq_DOM_div_curl}
176\nabla \cdot \left( {\nabla \times {\rm {\bf A}}} \right)=0
177\end{equation}
178
179It is straightforward to demonstrate that these properties are verified locally in
180discrete form as soon as the scalar $q$ is taken at $t$-points and the vector
181\textbf{A} has its components defined at vector points $(u,v,w)$.
182
183Let $a$ and $b$ be two fields defined on the mesh, with value zero inside
184continental area. Using integration by parts it can be shown that the differencing
185operators ($\delta_i$, $\delta_j$ and $\delta_k$) are anti-symmetric linear
186operators, and further that the averaging operators $\overline{\,\cdot\,}^{\,i}$,
187$\overline{\,\cdot\,}^{\,k}$ and $\overline{\,\cdot\,}^{\,k}$) are symmetric linear
188operators, $i.e.$
189\begin{align} 
190\label{DOM_di_adj}
191\sum\limits_i { a_i \;\delta _i \left[ b \right]} 
192   &\equiv -\sum\limits_i {\delta _{i+1/2} \left[ a \right]\;b_{i+1/2} }      \\
193\label{DOM_mi_adj}
194\sum\limits_i { a_i \;\overline b^{\,i}} 
195   & \equiv \quad \sum\limits_i {\overline a ^{\,i+1/2}\;b_{i+1/2} } 
196\end{align}
197
198In other words, the adjoint of the differencing and averaging operators are
199$\delta_i^*=\delta_{i+1/2}$ and
200${(\overline{\,\cdot \,}^{\,i})}^*= \overline{\,\cdot\,}^{\,i+1/2}$, respectively.
201These two properties will be used extensively in the Appendix~\ref{Apdx_C} to
202demonstrate integral conservative properties of the discrete formulation chosen.
203
204% -------------------------------------------------------------------------------------------------------------
205%        Numerical Indexing
206% -------------------------------------------------------------------------------------------------------------
207\subsection{Numerical Indexing}
208\label{DOM_Num_Index}
209
210%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
211\begin{figure}[!tb]  \begin{center}
212\includegraphics[width=0.90\textwidth]{./TexFiles/Figures/Fig_index_hor.pdf}
213\caption{   \label{Fig_index_hor}   
214Horizontal integer indexing used in the \textsc{Fortran} code. The dashed area indicates
215the cell in which variables contained in arrays have the same $i$- and $j$-indices}
216\end{center}   \end{figure}
217%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
218
219The array representation used in the \textsc{Fortran} code requires an integer
220indexing while the analytical definition of the mesh (see \S\ref{DOM_cell}) is
221associated with the use of integer values for $t$-points and both integer and
222integer and a half values for all the other points. Therefore a specific integer
223indexing must be defined for points other than $t$-points ($i.e.$ velocity and
224vorticity grid-points). Furthermore, the direction of the vertical indexing has
225been changed so that the surface level is at $k=1$.
226
227% -----------------------------------
228%        Horizontal Indexing
229% -----------------------------------
230\subsubsection{Horizontal Indexing}
231\label{DOM_Num_Index_hor}
232
233The indexing in the horizontal plane has been chosen as shown in Fig.\ref{Fig_index_hor}.
234For an increasing $i$ index ($j$ index), the $t$-point and the eastward $u$-point
235(northward $v$-point) have the same index (see the dashed area in Fig.\ref{Fig_index_hor}).
236A $t$-point and its nearest northeast $f$-point have the same $i$-and $j$-indices.
237
238% -----------------------------------
239%        Vertical indexing
240% -----------------------------------
241\subsubsection{Vertical Indexing}
242\label{DOM_Num_Index_vertical}
243
244In the vertical, the chosen indexing requires special attention since the
245$k$-axis is re-orientated downward in the \textsc{Fortran} code compared
246to the indexing used in the semi-discrete equations and given in \S\ref{DOM_cell}.
247The sea surface corresponds to the $w$-level $k=1$ which is the same index
248as $t$-level just below (Fig.\ref{Fig_index_vert}). The last $w$-level ($k=jpk$)
249either corresponds to the ocean floor or is inside the bathymetry while the last
250$t$-level is always inside the bathymetry (Fig.\ref{Fig_index_vert}). Note that
251for an increasing $k$ index, a $w$-point and the $t$-point just below have the
252same $k$ index, in opposition to what is done in the horizontal plane where
253it is the $t$-point and the nearest velocity points in the direction of the horizontal
254axis that have the same $i$ or $j$ index (compare the dashed area in
255Fig.\ref{Fig_index_hor} and \ref{Fig_index_vert}). Since the scale factors are
256chosen to be strictly positive, a \emph{minus sign} appears in the \textsc{Fortran} 
257code \emph{before all the vertical derivatives} of the discrete equations given in
258this documentation.
259
260%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
261\begin{figure}[!pt]    \begin{center}
262\includegraphics[width=.90\textwidth]{./TexFiles/Figures/Fig_index_vert.pdf}
263\caption{ \label{Fig_index_vert}     
264Vertical integer indexing used in the \textsc{Fortran } code. Note that
265the $k$-axis is orientated downward. The dashed area indicates the cell in
266which variables contained in arrays have the same $k$-index.}
267\end{center}   \end{figure}
268%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
269
270% -----------------------------------
271%        Domain Size
272% -----------------------------------
273\subsubsection{Domain Size}
274\label{DOM_size}
275
276The total size of the computational domain is set by the parameters \np{jpiglo},
277\np{jpjglo} and \np{jpkdta} in the $i$, $j$ and $k$ directions respectively. They are
278given as namelist variables in the \ngn{namcfg} namelist.
279
280Note that are other namelist variables in the \ngn{namcfg} namelist that refer to
281 the domain size.
282The two variables \np{jpidta} and \np{jpjdta} may be larger than \np{jpiglo}, \np{jpjglo}
283when the user wants to use only a sub-region of a given configuration. This is
284the "zoom" capability described in \S\ref{MISC_zoom}. In most applications of
285the model, $jpidta=jpiglo$, $jpjdta=jpjglo$, and $jpizoom=jpjzoom=1$. Parameters
286$jpi$ and $jpj$ refer to the size of each processor subdomain when the code is
287run in parallel using domain decomposition (\key{mpp\_mpi} defined, see
288\S\ref{LBC_mpp}).
289
290
291$\ $\newline    % force a new lign
292
293% ================================================================
294% Domain: Horizontal Grid (mesh)
295% ================================================================
296\section  [Domain: Horizontal Grid (mesh) (\textit{domhgr})]               
297      {Domain: Horizontal Grid (mesh) \small{(\mdl{domhgr} module)} }
298\label{DOM_hgr}
299
300% -------------------------------------------------------------------------------------------------------------
301%        Coordinates and scale factors
302% -------------------------------------------------------------------------------------------------------------
303\subsection{Coordinates and scale factors}
304\label{DOM_hgr_coord_e}
305
306The ocean mesh ($i.e.$ the position of all the scalar and vector points) is defined
307by the transformation that gives $(\lambda,\varphi,z)$ as a function of $(i,j,k)$.
308The grid-points are located at integer or integer and a half values of as indicated
309in Table~\ref{Tab_cell}. The associated scale factors are defined using the
310analytical first derivative of the transformation \eqref{Eq_scale_factors}. These
311definitions are done in two modules, \mdl{domhgr} and \mdl{domzgr}, which
312provide the horizontal and vertical meshes, respectively. This section deals with
313the horizontal mesh parameters.
314
315In a horizontal plane, the location of all the model grid points is defined from the
316analytical expressions of the longitude $\lambda$ and  latitude $\varphi$ as a
317function of  $(i,j)$. The horizontal scale factors are calculated using
318\eqref{Eq_scale_factors}. For example, when the longitude and latitude are
319function of a single value ($i$ and $j$, respectively) (geographical configuration
320of the mesh), the horizontal mesh definition reduces to define the wanted
321$\lambda(i)$, $\varphi(j)$, and their derivatives $\lambda'(i)$ $\varphi'(j)$ in the
322\mdl{domhgr} module. The model computes the grid-point positions and scale
323factors in the horizontal plane as follows:
324\begin{flalign*}
325\lambda_t &\equiv \text{glamt}= \lambda(i)     & \varphi_t &\equiv \text{gphit} = \varphi(j)\\
326\lambda_u &\equiv \text{glamu}= \lambda(i+1/2)& \varphi_u &\equiv \text{gphiu}= \varphi(j)\\
327\lambda_v &\equiv \text{glamv}= \lambda(i)       & \varphi_v &\equiv \text{gphiv} = \varphi(j+1/2)\\
328\lambda_f &\equiv \text{glamf }= \lambda(i+1/2)& \varphi_f &\equiv \text{gphif }= \varphi(j+1/2)
329\end{flalign*}
330\begin{flalign*}
331e_{1t} &\equiv \text{e1t} = r_a |\lambda'(i)    \; \cos\varphi(j)  |&
332e_{2t} &\equiv \text{e2t} = r_a |\varphi'(j)|  \\
333e_{1u} &\equiv \text{e1t} = r_a |\lambda'(i+1/2)   \; \cos\varphi(j)  |&
334e_{2u} &\equiv \text{e2t} = r_a |\varphi'(j)|\\
335e_{1v} &\equiv \text{e1t} = r_a |\lambda'(i)    \; \cos\varphi(j+1/2)  |&
336e_{2v} &\equiv \text{e2t} = r_a |\varphi'(j+1/2)|\\
337e_{1f} &\equiv \text{e1t} = r_a |\lambda'(i+1/2)\; \cos\varphi(j+1/2)  |&
338e_{2f} &\equiv \text{e2t} = r_a |\varphi'(j+1/2)|
339\end{flalign*}
340where the last letter of each computational name indicates the grid point
341considered and $r_a$ is the earth radius (defined in \mdl{phycst} along with
342all universal constants). Note that the horizontal position of and scale factors
343at $w$-points are exactly equal to those of $t$-points, thus no specific arrays
344are defined at $w$-points.
345
346Note that the definition of the scale factors ($i.e.$ as the analytical first derivative
347of the transformation that gives $(\lambda,\varphi,z)$ as a function of $(i,j,k)$) is
348specific to the \NEMO model \citep{Marti_al_JGR92}. As an example, $e_{1t}$ is defined
349locally at a $t$-point, whereas many other models on a C grid choose to define
350such a scale factor as the distance between the $U$-points on each side of the
351$t$-point. Relying on an analytical transformation has two advantages: firstly, there
352is no ambiguity in the scale factors appearing in the discrete equations, since they
353are first introduced in the continuous equations; secondly, analytical transformations
354encourage good practice by the definition of smoothly varying grids (rather than
355allowing the user to set arbitrary jumps in thickness between adjacent layers)
356\citep{Treguier1996}. An example of the effect of such a choice is shown in
357Fig.~\ref{Fig_zgr_e3}.
358%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
359\begin{figure}[!t]     \begin{center}
360\includegraphics[width=0.90\textwidth]{./TexFiles/Figures/Fig_zgr_e3.pdf}
361\caption{ \label{Fig_zgr_e3}   
362Comparison of (a) traditional definitions of grid-point position and grid-size in the vertical,
363and (b) analytically derived grid-point position and scale factors.
364For both grids here,  the same $w$-point depth has been chosen but in (a) the
365$t$-points are set half way between $w$-points while in (b) they are defined from
366an analytical function: $z(k)=5\,(i-1/2)^3 - 45\,(i-1/2)^2 + 140\,(i-1/2) - 150$.
367Note the resulting difference between the value of the grid-size $\Delta_k$ and
368those of the scale factor $e_k$. }
369\end{center}   \end{figure}
370%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
371
372% -------------------------------------------------------------------------------------------------------------
373%        Choice of horizontal grid
374% -------------------------------------------------------------------------------------------------------------
375\subsection{Choice of horizontal grid}
376\label{DOM_hgr_msh_choice}
377
378The user has three options available in defining a horizontal grid, which involve
379the namelist variable \np{jphgr\_mesh} of the \ngn{namcfg} namelist.
380\begin{description}
381\item[\np{jphgr\_mesh}=0]  The most general curvilinear orthogonal grids.
382The coordinates and their first derivatives with respect to $i$ and $j$ are provided
383in a input file (\ifile{coordinates}), read in \rou{hgr\_read} subroutine of the domhgr module.
384\item[\np{jphgr\_mesh}=1 to 5] A few simple analytical grids are provided (see below).
385For other analytical grids, the \mdl{domhgr} module must be modified by the user.
386\end{description}
387
388There are two simple cases of geographical grids on the sphere. With
389\np{jphgr\_mesh}=1, the grid (expressed in degrees) is regular in space,
390with grid sizes specified by parameters \np{ppe1\_deg} and \np{ppe2\_deg},
391respectively. Such a geographical grid can be very anisotropic at high latitudes
392because of the convergence of meridians (the zonal scale factors $e_1$ 
393become much smaller than the meridional scale factors $e_2$). The Mercator
394grid (\np{jphgr\_mesh}=4) avoids this anisotropy by refining the meridional scale
395factors in the same way as the zonal ones. In this case, meridional scale factors
396and latitudes are calculated analytically using the formulae appropriate for
397a Mercator projection, based on \np{ppe1\_deg} which is a reference grid spacing
398at the equator (this applies even when the geographical equator is situated outside
399the model domain).
400%%%
401\gmcomment{ give here the analytical expression of the Mercator mesh}
402%%%
403In these two cases (\np{jphgr\_mesh}=1 or 4), the grid position is defined by the
404longitude and latitude of the south-westernmost point (\np{ppglamt0} 
405and \np{ppgphi0}). Note that for the Mercator grid the user need only provide
406an approximate starting latitude: the real latitude will be recalculated analytically,
407in order to ensure that the equator corresponds to line passing through $t$-
408and $u$-points. 
409
410Rectangular grids ignoring the spherical geometry are defined with
411\np{jphgr\_mesh} = 2, 3, 5. The domain is either an $f$-plane (\np{jphgr\_mesh} = 2,
412Coriolis factor is constant) or a beta-plane (\np{jphgr\_mesh} = 3, the Coriolis factor
413is linear in the $j$-direction). The grid size is uniform in meter in each direction,
414and given by the parameters \np{ppe1\_m} and \np{ppe2\_m} respectively.
415The zonal grid coordinate (\textit{glam} arrays) is in kilometers, starting at zero
416with the first $t$-point. The meridional coordinate (gphi. arrays) is in kilometers,
417and the second $t$-point corresponds to coordinate $gphit=0$. The input
418variable \np{ppglam0} is ignored. \np{ppgphi0} is used to set the reference
419latitude for computation of the Coriolis parameter. In the case of the beta plane,
420\np{ppgphi0} corresponds to the center of the domain. Finally, the special case
421\np{jphgr\_mesh}=5 corresponds to a beta plane in a rotated domain for the
422GYRE configuration, representing a classical mid-latitude double gyre system.
423The rotation allows us to maximize the jet length relative to the gyre areas
424(and the number of grid points).
425
426The choice of the grid must be consistent with the boundary conditions specified
427by the parameter \np{jperio} (see {\S\ref{LBC}).
428
429% -------------------------------------------------------------------------------------------------------------
430%        Grid files
431% -------------------------------------------------------------------------------------------------------------
432\subsection{Output Grid files}
433\label{DOM_hgr_files}
434
435All the arrays relating to a particular ocean model configuration (grid-point
436position, scale factors, masks) can be saved in files if $\np{nn\_msh} \not= 0$ 
437(namelist variable in \ngn{namdom}). This can be particularly useful for plots and off-line
438diagnostics. In some cases, the user may choose to make a local modification
439of a scale factor in the code. This is the case in global configurations when
440restricting the width of a specific strait (usually a one-grid-point strait that
441happens to be too wide due to insufficient model resolution). An example
442is Gibraltar Strait in the ORCA2 configuration. When such modifications are done,
443the output grid written when $\np{nn\_msh} \not=0$ is no more equal to the input grid.
444
445$\ $\newline    % force a new lign
446
447% ================================================================
448% Domain: Vertical Grid (domzgr)
449% ================================================================
450\section  [Domain: Vertical Grid (\textit{domzgr})]
451      {Domain: Vertical Grid \small{(\mdl{domzgr} module)} }
452\label{DOM_zgr}
453%-----------------------------------------nam_zgr & namdom-------------------------------------------
454\namdisplay{namzgr} 
455\namdisplay{namdom} 
456%-------------------------------------------------------------------------------------------------------------
457
458Variables are defined through the \ngn{namzgr} and \ngn{namdom} namelists.
459In the vertical, the model mesh is determined by four things:
460(1) the bathymetry given in meters ;
461(2) the number of levels of the model (\jp{jpk}) ;
462(3) the analytical transformation $z(i,j,k)$ and the vertical scale factors
463(derivatives of the transformation) ;
464and (4) the masking system, $i.e.$ the number of wet model levels at each
465$(i,j)$ column of points.
466
467%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
468\begin{figure}[!tb]    \begin{center}
469\includegraphics[width=1.0\textwidth]{./TexFiles/Figures/Fig_z_zps_s_sps.pdf}
470\caption{  \label{Fig_z_zps_s_sps}   
471The ocean bottom as seen by the model:
472(a) $z$-coordinate with full step,
473(b) $z$-coordinate with partial step,
474(c) $s$-coordinate: terrain following representation,
475(d) hybrid $s-z$ coordinate,
476(e) hybrid $s-z$ coordinate with partial step, and
477(f) same as (e) but with variable volume associated with the non-linear free surface.
478Note that the variable volume option (\key{vvl}) can be used with any of the
4795 coordinates (a) to (e).}
480\end{center}   \end{figure}
481%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
482
483The choice of a vertical coordinate, even if it is made through a namelist parameter,
484must be done once of all at the beginning of an experiment. It is not intended as an
485option which can be enabled or disabled in the middle of an experiment. Three main
486choices are offered (Fig.~\ref{Fig_z_zps_s_sps}a to c): $z$-coordinate with full step
487bathymetry (\np{ln\_zco}~=~true), $z$-coordinate with partial step bathymetry
488(\np{ln\_zps}~=~true), or generalized, $s$-coordinate (\np{ln\_sco}~=~true).
489Hybridation of the three main coordinates are available: $s-z$ or $s-zps$ coordinate
490(Fig.~\ref{Fig_z_zps_s_sps}d and \ref{Fig_z_zps_s_sps}e). When using the variable
491volume option \key{vvl} ($i.e.$ non-linear free surface), the coordinate follow the
492time-variation of the free surface so that the transformation is time dependent:
493$z(i,j,k,t)$ (Fig.~\ref{Fig_z_zps_s_sps}f). This option can be used with full step
494bathymetry or $s$-coordinate (hybrid and partial step coordinates have not
495yet been tested in NEMO v2.3). If using $z$-coordinate with partial step bathymetry
496(\np{ln\_zps}~=~true), ocean cavity beneath ice shelves can be open (\np{ln\_isfcav}~=~true).
497
498Contrary to the horizontal grid, the vertical grid is computed in the code and no
499provision is made for reading it from a file. The only input file is the bathymetry
500(in meters) (\ifile{bathy\_meter})
501\footnote{N.B. in full step $z$-coordinate, a \ifile{bathy\_level} file can replace the
502\ifile{bathy\_meter} file, so that the computation of the number of wet ocean point
503in each water column is by-passed}.
504After reading the bathymetry, the algorithm for vertical grid definition differs
505between the different options:
506\begin{description}
507\item[\textit{zco}] set a reference coordinate transformation $z_0 (k)$, and set $z(i,j,k,t)=z_0 (k)$.
508\item[\textit{zps}] set a reference coordinate transformation $z_0 (k)$, and
509calculate the thickness of the deepest level at each $(i,j)$ point using the
510bathymetry, to obtain the final three-dimensional depth and scale factor arrays.
511\item[\textit{sco}] smooth the bathymetry to fulfil the hydrostatic consistency
512criteria and set the three-dimensional transformation.
513\item[\textit{s-z} and \textit{s-zps}] smooth the bathymetry to fulfil the hydrostatic
514consistency criteria and set the three-dimensional transformation $z(i,j,k)$, and
515possibly introduce masking of extra land points to better fit the original bathymetry file
516\end{description}
517%%%
518\gmcomment{   add the description of the smoothing:  envelop topography...}
519%%%
520
521The arrays describing the grid point depths and vertical scale factors
522are three dimensional arrays $(i,j,k)$ even in the case of $z$-coordinate with
523full step bottom topography. In non-linear free surface (\key{vvl}), their knowledge
524is required at \textit{before}, \textit{now} and \textit{after} time step, while they
525do not vary in time in linear free surface case.
526To improve the code readability while providing this flexibility, the vertical coordinate
527and scale factors are defined as functions of
528$(i,j,k)$ with "fs" as prefix (examples: \textit{fse3t\_b, fse3t\_n, fse3t\_a,} 
529for the  \textit{before}, \textit{now} and \textit{after} scale factors at $t$-point)
530that can be either three different arrays when \key{vvl} is defined, or a single fixed arrays.
531These functions are defined in the file \hf{domzgr\_substitute} of the DOM directory.
532They are used throughout the code, and replaced by the corresponding arrays at
533the time of pre-processing (CPP capability).
534
535% -------------------------------------------------------------------------------------------------------------
536%        Meter Bathymetry
537% -------------------------------------------------------------------------------------------------------------
538\subsection{Meter Bathymetry}
539\label{DOM_bathy}
540
541Three options are possible for defining the bathymetry, according to the
542namelist variable \np{nn\_bathy}:
543\begin{description}
544\item[\np{nn\_bathy} = 0] a flat-bottom domain is defined. The total depth $z_w (jpk)$ 
545is given by the coordinate transformation. The domain can either be a closed
546basin or a periodic channel depending on the parameter \np{jperio}.
547\item[\np{nn\_bathy} = -1] a domain with a bump of topography one third of the
548domain width at the central latitude. This is meant for the "EEL-R5" configuration,
549a periodic or open boundary channel with a seamount.
550\item[\np{nn\_bathy} = 1] read a bathymetry. The \ifile{bathy\_meter} file (Netcdf format)
551provides the ocean depth (positive, in meters) at each grid point of the model grid.
552The bathymetry is usually built by interpolating a standard bathymetry product
553($e.g.$ ETOPO2) onto the horizontal ocean mesh. Defining the bathymetry also
554defines the coastline: where the bathymetry is zero, no model levels are defined
555(all levels are masked).
556\end{description}
557
558When a global ocean is coupled to an atmospheric model it is better to represent
559all large water bodies (e.g, great lakes, Caspian sea...) even if the model
560resolution does not allow their communication with the rest of the ocean.
561This is unnecessary when the ocean is forced by fixed atmospheric conditions,
562so these seas can be removed from the ocean domain. The user has the option
563to set the bathymetry in closed seas to zero (see \S\ref{MISC_closea}), but the
564code has to be adapted to the user's configuration.
565
566% -------------------------------------------------------------------------------------------------------------
567%        z-coordinate  and reference coordinate transformation
568% -------------------------------------------------------------------------------------------------------------
569\subsection[$z$-coordinate (\np{ln\_zco}]
570        {$z$-coordinate (\np{ln\_zco}=true) and reference coordinate}
571\label{DOM_zco}
572
573%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
574\begin{figure}[!tb]    \begin{center}
575\includegraphics[width=0.90\textwidth]{./TexFiles/Figures/Fig_zgr.pdf}
576\caption{ \label{Fig_zgr}   
577Default vertical mesh for ORCA2: 30 ocean levels (L30). Vertical level functions for
578(a) T-point depth and (b) the associated scale factor as computed
579from \eqref{DOM_zgr_ana} using \eqref{DOM_zgr_coef} in $z$-coordinate.}
580\end{center}   \end{figure}
581%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
582
583The reference coordinate transformation $z_0 (k)$ defines the arrays $gdept_0$ 
584and $gdepw_0$ for $t$- and $w$-points, respectively. As indicated on
585Fig.\ref{Fig_index_vert} \jp{jpk} is the number of $w$-levels. $gdepw_0(1)$ is the
586ocean surface. There are at most \jp{jpk}-1 $t$-points inside the ocean, the
587additional $t$-point at $jk=jpk$ is below the sea floor and is not used.
588The vertical location of $w$- and $t$-levels is defined from the analytic expression
589of the depth $z_0(k)$ whose analytical derivative with respect to $k$ provides the
590vertical scale factors. The user must provide the analytical expression of both
591$z_0$ and its first derivative with respect to $k$. This is done in routine \mdl{domzgr} 
592through statement functions, using parameters provided in the \ngn{namcfg} namelist.
593
594It is possible to define a simple regular vertical grid by giving zero stretching (\np{ppacr=0}).
595In that case, the parameters \jp{jpk} (number of $w$-levels) and \np{pphmax} 
596(total ocean depth in meters) fully define the grid.
597
598For climate-related studies it is often desirable to concentrate the vertical resolution
599near the ocean surface. The following function is proposed as a standard for a
600$z$-coordinate (with either full or partial steps):
601\begin{equation} \label{DOM_zgr_ana}
602\begin{split}
603 z_0 (k)    &= h_{sur} -h_0 \;k-\;h_1 \;\log \left[ {\,\cosh \left( {{(k-h_{th} )} / {h_{cr} }} \right)\,} \right] \\ 
604 e_3^0 (k)  &= \left| -h_0 -h_1 \;\tanh \left( {{(k-h_{th} )} / {h_{cr} }} \right) \right|
605\end{split}
606\end{equation}
607where $k=1$ to \jp{jpk} for $w$-levels and $k=1$ to $k=1$ for $T-$levels. Such an
608expression allows us to define a nearly uniform vertical location of levels at the
609ocean top and bottom with a smooth hyperbolic tangent transition in between
610(Fig.~\ref{Fig_zgr}).
611
612The most used vertical grid for ORCA2 has $10~m$ ($500~m)$ resolution in the
613surface (bottom) layers and a depth which varies from 0 at the sea surface to a
614minimum of $-5000~m$. This leads to the following conditions:
615\begin{equation} \label{DOM_zgr_coef}
616\begin{split}
617 e_3 (1+1/2)      &=10. \\ 
618 e_3 (jpk-1/2) &=500. \\ 
619 z(1)       &=0. \\ 
620 z(jpk)        &=-5000. \\ 
621\end{split}
622\end{equation}
623
624With the choice of the stretching $h_{cr} =3$ and the number of levels
625\jp{jpk}=$31$, the four coefficients $h_{sur}$, $h_{0}$, $h_{1}$, and $h_{th}$ in
626\eqref{DOM_zgr_ana} have been determined such that \eqref{DOM_zgr_coef} is
627satisfied, through an optimisation procedure using a bisection method. For the first
628standard ORCA2 vertical grid this led to the following values: $h_{sur} =4762.96$,
629$h_0 =255.58, h_1 =245.5813$, and $h_{th} =21.43336$. The resulting depths and
630scale factors as a function of the model levels are shown in Fig.~\ref{Fig_zgr} and
631given in Table \ref{Tab_orca_zgr}. Those values correspond to the parameters
632\np{ppsur}, \np{ppa0}, \np{ppa1}, \np{ppkth} in \ngn{namcfg} namelist.
633
634Rather than entering parameters $h_{sur}$, $h_{0}$, and $h_{1}$ directly, it is
635possible to recalculate them. In that case the user sets
636\np{ppsur}=\np{ppa0}=\np{ppa1}=999999., in \ngn{namcfg} namelist,
637and specifies instead the four following parameters:
638\begin{itemize}
639\item    \np{ppacr}=$h_{cr} $: stretching factor (nondimensional). The larger
640\np{ppacr}, the smaller the stretching. Values from $3$ to $10$ are usual.
641\item    \np{ppkth}=$h_{th} $: is approximately the model level at which maximum
642stretching occurs (nondimensional, usually of order 1/2 or 2/3 of \jp{jpk})
643\item    \np{ppdzmin}: minimum thickness for the top layer (in meters)
644\item    \np{pphmax}: total depth of the ocean (meters).
645\end{itemize}
646As an example, for the $45$ layers used in the DRAKKAR configuration those
647parameters are: \jp{jpk}=46, \np{ppacr}=9, \np{ppkth}=23.563, \np{ppdzmin}=6m,
648\np{pphmax}=5750m.
649
650%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
651\begin{table}     \begin{center} \begin{tabular}{c||r|r|r|r}
652\hline
653\textbf{LEVEL}& \textbf{gdept}& \textbf{gdepw}& \textbf{e3t }& \textbf{e3w  } \\ \hline
654&  \textbf{  5.00}   &       0.00 & \textbf{ 10.00} &  10.00 \\   \hline
655&  \textbf{15.00} &    10.00 &   \textbf{ 10.00} &  10.00 \\   \hline
656&  \textbf{25.00} &    20.00 &   \textbf{ 10.00} &     10.00 \\   \hline
657&  \textbf{35.01} &    30.00 &   \textbf{ 10.01} &     10.00 \\   \hline
658&  \textbf{45.01} &    40.01 &   \textbf{ 10.01} &  10.01 \\   \hline
659&  \textbf{55.03} &    50.02 &   \textbf{ 10.02} &     10.02 \\   \hline
660&  \textbf{65.06} &    60.04 &   \textbf{ 10.04} &  10.03 \\   \hline
661&  \textbf{75.13} &    70.09 &   \textbf{ 10.09} &  10.06 \\   \hline
662&  \textbf{85.25} &    80.18 &   \textbf{ 10.17} &  10.12 \\   \hline
66310 &  \textbf{95.49} &    90.35 &   \textbf{ 10.33} &  10.24 \\   \hline
66411 &  \textbf{105.97}   &   100.69 &   \textbf{ 10.65} &  10.47 \\   \hline
66512 &  \textbf{116.90}   &   111.36 &   \textbf{ 11.27} &  10.91 \\   \hline
66613 &  \textbf{128.70}   &   122.65 &   \textbf{ 12.47} &  11.77 \\   \hline
66714 &  \textbf{142.20}   &   135.16 &   \textbf{ 14.78} &  13.43 \\   \hline
66815 &  \textbf{158.96}   &   150.03 &   \textbf{ 19.23} &  16.65 \\   \hline
66916 &  \textbf{181.96}   &   169.42 &   \textbf{ 27.66} &  22.78 \\   \hline
67017 &  \textbf{216.65}   &   197.37 &   \textbf{ 43.26} &  34.30 \\ \hline
67118 &  \textbf{272.48}   &   241.13 &   \textbf{ 70.88} &  55.21 \\ \hline
67219 &  \textbf{364.30}   &   312.74 &   \textbf{116.11} &  90.99 \\ \hline
67320 &  \textbf{511.53}   &   429.72 &   \textbf{181.55} &    146.43 \\ \hline
67421 &  \textbf{732.20}   &   611.89 &   \textbf{261.03} &    220.35 \\ \hline
67522 &  \textbf{1033.22}&  872.87 &   \textbf{339.39} &    301.42 \\ \hline
67623 &  \textbf{1405.70}& 1211.59 & \textbf{402.26} &   373.31 \\ \hline
67724 &  \textbf{1830.89}& 1612.98 & \textbf{444.87} &   426.00 \\ \hline
67825 &  \textbf{2289.77}& 2057.13 & \textbf{470.55} &   459.47 \\ \hline
67926 &  \textbf{2768.24}& 2527.22 & \textbf{484.95} &   478.83 \\ \hline
68027 &  \textbf{3257.48}& 3011.90 & \textbf{492.70} &   489.44 \\ \hline
68128 &  \textbf{3752.44}& 3504.46 & \textbf{496.78} &   495.07 \\ \hline
68229 &  \textbf{4250.40}& 4001.16 & \textbf{498.90} &   498.02 \\ \hline
68330 &  \textbf{4749.91}& 4500.02 & \textbf{500.00} &   499.54 \\ \hline
68431 &  \textbf{5250.23}& 5000.00 &   \textbf{500.56} & 500.33 \\ \hline
685\end{tabular} \end{center} 
686\caption{ \label{Tab_orca_zgr}   
687Default vertical mesh in $z$-coordinate for 30 layers ORCA2 configuration as computed
688from \eqref{DOM_zgr_ana} using the coefficients given in \eqref{DOM_zgr_coef}}
689\end{table}
690%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
691
692% -------------------------------------------------------------------------------------------------------------
693%        z-coordinate with partial step
694% -------------------------------------------------------------------------------------------------------------
695\subsection   [$z$-coordinate with partial step (\np{ln\_zps})]
696         {$z$-coordinate with partial step (\np{ln\_zps}=.true.)}
697\label{DOM_zps}
698%--------------------------------------------namdom-------------------------------------------------------
699\namdisplay{namdom} 
700%--------------------------------------------------------------------------------------------------------------
701
702In $z$-coordinate partial step, the depths of the model levels are defined by the
703reference analytical function $z_0 (k)$ as described in the previous
704section, \emph{except} in the bottom layer. The thickness of the bottom layer is
705allowed to vary as a function of geographical location $(\lambda,\varphi)$ to allow a
706better representation of the bathymetry, especially in the case of small
707slopes (where the bathymetry varies by less than one level thickness from
708one grid point to the next). The reference layer thicknesses $e_{3t}^0$ have been
709defined in the absence of bathymetry. With partial steps, layers from 1 to
710\jp{jpk}-2 can have a thickness smaller than $e_{3t}(jk)$. The model deepest layer (\jp{jpk}-1)
711is allowed to have either a smaller or larger thickness than $e_{3t}(jpk)$: the
712maximum thickness allowed is $2*e_{3t}(jpk-1)$. This has to be kept in mind when
713specifying values in \ngn{namdom} namelist, as the maximum depth \np{pphmax} 
714in partial steps: for example, with
715\np{pphmax}$=5750~m$ for the DRAKKAR 45 layer grid, the maximum ocean depth
716allowed is actually $6000~m$ (the default thickness $e_{3t}(jpk-1)$ being $250~m$).
717Two variables in the namdom namelist are used to define the partial step
718vertical grid. The mimimum water thickness (in meters) allowed for a cell
719partially filled with bathymetry at level jk is the minimum of \np{rn\_e3zps\_min} 
720(thickness in meters, usually $20~m$) or $e_{3t}(jk)*\np{rn\_e3zps\_rat}$ (a fraction,
721usually 10\%, of the default thickness $e_{3t}(jk)$).
722
723 \colorbox{yellow}{Add a figure here of pstep especially at last ocean level }
724
725% -------------------------------------------------------------------------------------------------------------
726%        s-coordinate
727% -------------------------------------------------------------------------------------------------------------
728\subsection   [$s$-coordinate (\np{ln\_sco})]
729           {$s$-coordinate (\np{ln\_sco}=true)}
730\label{DOM_sco}
731%------------------------------------------nam_zgr_sco---------------------------------------------------
732\namdisplay{namzgr_sco} 
733%--------------------------------------------------------------------------------------------------------------
734Options are defined in \ngn{namzgr\_sco}.
735In $s$-coordinate (\np{ln\_sco}~=~true), the depth and thickness of the model
736levels are defined from the product of a depth field and either a stretching
737function or its derivative, respectively:
738
739\begin{equation} \label{DOM_sco_ana}
740\begin{split}
741 z(k)       &= h(i,j) \; z_0(k)  \\
742 e_3(k)  &= h(i,j) \; z_0'(k)
743\end{split}
744\end{equation}
745
746where $h$ is the depth of the last $w$-level ($z_0(k)$) defined at the $t$-point
747location in the horizontal and $z_0(k)$ is a function which varies from $0$ at the sea
748surface to $1$ at the ocean bottom. The depth field $h$ is not necessary the ocean
749depth, since a mixed step-like and bottom-following representation of the
750topography can be used (Fig.~\ref{Fig_z_zps_s_sps}d-e) or an envelop bathymetry can be defined (Fig.~\ref{Fig_z_zps_s_sps}f).
751The namelist parameter \np{rn\_rmax} determines the slope at which the terrain-following coordinate intersects the sea bed and becomes a pseudo z-coordinate. The coordinate can also be hybridised by specifying \np{rn\_sbot\_min} and \np{rn\_sbot\_max} as the minimum and maximum depths at which the terrain-following vertical coordinate is calculated.
752
753Options for stretching the coordinate are provided as examples, but care must be taken to ensure that the vertical stretch used is appropriate for the application.
754
755The original default NEMO s-coordinate stretching is available if neither of the other options are specified as true (\np{ln\_sco\_SH94}~=~false and \np{ln\_sco\_SF12}~=~false.) This uses a depth independent $\tanh$ function for the stretching \citep{Madec_al_JPO96}:
756
757\begin{equation}
758  z = s_{min}+C\left(s\right)\left(H-s_{min}\right)
759  \label{eq:SH94_1}
760\end{equation}
761
762where $s_{min}$ is the depth at which the s-coordinate stretching starts and allows a z-coordinate to placed on top of the stretched coordinate, and z is the depth (negative down from the asea surface).
763
764\begin{equation}
765  s = -\frac{k}{n-1} \quad \text{ and } \quad 0 \leq k \leq n-1
766  \label{eq:s}
767\end{equation}
768
769\begin{equation} \label{DOM_sco_function}
770\begin{split}
771C(s)  &\frac{ \left[   \tanh{ \left( \theta \, (s+b) \right)} 
772               - \tanh{ \left\theta \, b      \right)}  \right]}
773            {2\;\sinh \left( \theta \right)}
774\end{split}
775\end{equation}
776
777A stretching function, modified from the commonly used \citet{Song_Haidvogel_JCP94} stretching (\np{ln\_sco\_SH94}~=~true), is also available and is more commonly used for shelf seas modelling:
778
779\begin{equation}
780  C\left(s\right) =   \left(1 - b \right)\frac{ \sinh\left( \theta s\right)}{\sinh\left(\theta\right)} +      \\
781  b\frac{ \tanh \left[ \theta \left(s + \frac{1}{2} \right)\right] - \tanh\left(\frac{\theta}{2}\right)}{ 2\tanh\left (\frac{\theta}{2}\right)}
782  \label{eq:SH94_2}
783\end{equation}
784
785%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
786\begin{figure}[!ht]    \begin{center}
787\includegraphics[width=1.0\textwidth]{./TexFiles/Figures/Fig_sco_function.pdf}
788\caption{  \label{Fig_sco_function}   
789Examples of the stretching function applied to a seamount; from left to right:
790surface, surface and bottom, and bottom intensified resolutions}
791\end{center}   \end{figure}
792%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
793
794where $H_c$ is the critical depth (\np{rn\_hc}) at which the coordinate transitions from pure $\sigma$ to the stretched coordinate,  and $\theta$ (\np{rn\_theta}) and $b$ (\np{rn\_bb}) are the surface and
795bottom control parameters such that $0\leqslant \theta \leqslant 20$, and
796$0\leqslant b\leqslant 1$. $b$ has been designed to allow surface and/or bottom
797increase of the vertical resolution (Fig.~\ref{Fig_sco_function}).
798
799Another example has been provided at version 3.5 (\np{ln\_sco\_SF12}) that allows a fixed surface resolution in an analytical terrain-following stretching \citet{Siddorn_Furner_OM12}. In this case the a stretching function $\gamma$ is defined such that:
800
801\begin{equation}
802z = -\gamma h \quad \text{ with } \quad 0 \leq \gamma \leq 1
803\label{eq:z}
804\end{equation}
805
806The function is defined with respect to $\sigma$, the unstretched terrain-following coordinate:
807
808\begin{equation} \label{DOM_gamma_deriv}
809\gamma= A\left(\sigma-\frac{1}{2}\left(\sigma^{2}+f\left(\sigma\right)\right)\right)+B\left(\sigma^{3}-f\left(\sigma\right)\right)+f\left(\sigma\right)
810\end{equation}
811
812Where:
813\begin{equation} \label{DOM_gamma}
814f\left(\sigma\right)=\left(\alpha+2\right)\sigma^{\alpha+1}-\left(\alpha+1\right)\sigma^{\alpha+2} \quad \text{ and } \quad \sigma = \frac{k}{n-1} 
815\end{equation}
816
817This gives an analytical stretching of $\sigma$ that is solvable in $A$ and $B$ as a function of the user prescribed stretching parameter $\alpha$ (\np{rn\_alpha}) that stretches towards the surface ($\alpha > 1.0$) or the bottom ($\alpha < 1.0$) and user prescribed surface (\np{rn\_zs}) and bottom depths. The bottom cell depth in this example is given as a function of water depth:
818
819\begin{equation} \label{DOM_zb}
820Z_b= h a + b
821\end{equation}
822
823where the namelist parameters \np{rn\_zb\_a} and \np{rn\_zb\_b} are $a$ and $b$ respectively.
824
825%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
826\begin{figure}[!ht]
827   \includegraphics[width=1.0\textwidth]{./TexFiles/Figures/FIG_DOM_compare_coordinates_surface.pdf}
828        \caption{A comparison of the \citet{Song_Haidvogel_JCP94} $S$-coordinate (solid lines), a 50 level $Z$-coordinate (contoured surfaces) and the \citet{Siddorn_Furner_OM12} $S$-coordinate (dashed lines) in the surface 100m for a idealised bathymetry that goes from 50m to 5500m depth. For clarity every third coordinate surface is shown.}
829    \label{fig_compare_coordinates_surface}
830\end{figure}
831%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
832
833This gives a smooth analytical stretching in computational space that is constrained to given specified surface and bottom grid cell thicknesses in real space. This is not to be confused with the hybrid schemes that superimpose geopotential coordinates on terrain following coordinates thus creating a non-analytical vertical coordinate that therefore may suffer from large gradients in the vertical resolutions. This stretching is less straightforward to implement than the \citet{Song_Haidvogel_JCP94} stretching, but has the advantage of resolving diurnal processes in deep water and has generally flatter slopes.
834
835As with the \citet{Song_Haidvogel_JCP94} stretching the stretch is only applied at depths greater than the critical depth $h_c$. In this example two options are available in depths shallower than $h_c$, with pure sigma being applied if the \np{ln\_sigcrit} is true and pure z-coordinates if it is false (the z-coordinate being equal to the depths of the stretched coordinate at $h_c$.
836
837Minimising the horizontal slope of the vertical coordinate is important in terrain-following systems as large slopes lead to hydrostatic consistency. A hydrostatic consistency parameter diagnostic following \citet{Haney1991} has been implemented, and is output as part of the model mesh file at the start of the run.
838
839% -------------------------------------------------------------------------------------------------------------
840%        z*- or s*-coordinate
841% -------------------------------------------------------------------------------------------------------------
842\subsection{$z^*$- or $s^*$-coordinate (add \key{vvl}) }
843\label{DOM_zgr_vvl}
844
845This option is described in the Report by Levier \textit{et al.} (2007), available on
846the \NEMO web site.
847
848%gm% key advantage: minimise the diffusion/dispertion associated with advection in response to high frequency surface disturbances
849
850% -------------------------------------------------------------------------------------------------------------
851%        level bathymetry and mask
852% -------------------------------------------------------------------------------------------------------------
853\subsection{level bathymetry and mask}
854\label{DOM_msk}
855
856Whatever the vertical coordinate used, the model offers the possibility of
857representing the bottom topography with steps that follow the face of the
858model cells (step like topography) \citep{Madec_al_JPO96}. The distribution of
859the steps in the horizontal is defined in a 2D integer array, mbathy, which
860gives the number of ocean levels ($i.e.$ those that are not masked) at each
861$t$-point. mbathy is computed from the meter bathymetry using the definiton of
862gdept as the number of $t$-points which gdept $\leq$ bathy.
863
864Modifications of the model bathymetry are performed in the \textit{bat\_ctl} 
865routine (see \mdl{domzgr} module) after mbathy is computed. Isolated grid points
866that do not communicate with another ocean point at the same level are eliminated.
867
868From the \textit{mbathy} array, the mask fields are defined as follows:
869\begin{align*}
870tmask(i,j,k) &= \begin{cases}   \; 1&   \text{ if $k\leq mbathy(i,j)$  }    \\
871                                                \; 0&   \text{ if $k\leq mbathy(i,j)$  }    \end{cases}     \\
872umask(i,j,k) &=         \; tmask(i,j,k) \ * \ tmask(i+1,j,k)   \\
873vmask(i,j,k) &=         \; tmask(i,j,k) \ * \ tmask(i,j+1,k)   \\
874fmask(i,j,k) &=         \; tmask(i,j,k) \ * \ tmask(i+1,j,k)   \\
875                   & \ \ \, * tmask(i,j,k) \ * \ tmask(i+1,j,k)
876\end{align*}
877
878Note that \textit{wmask} is not defined as it is exactly equal to \textit{tmask} with
879the numerical indexing used (\S~\ref{DOM_Num_Index}). Moreover, the
880specification of closed lateral boundaries requires that at least the first and last
881rows and columns of the \textit{mbathy} array are set to zero. In the particular
882case of an east-west cyclical boundary condition, \textit{mbathy} has its last
883column equal to the second one and its first column equal to the last but one
884(and so too the mask arrays) (see \S~\ref{LBC_jperio}).
885
886%%%
887\gmcomment{   \colorbox{yellow}{Add one word on tricky trick !} mbathy in further modified in zdfbfr{\ldots}}
888%%%
889
890% ================================================================
891% Domain: Initial State (dtatsd & istate)
892% ================================================================
893\section  [Domain: Initial State (\textit{istate and dtatsd})]
894      {Domain: Initial State \small{(\mdl{istate} and \mdl{dtatsd} modules)} }
895\label{DTA_tsd}
896%-----------------------------------------namtsd-------------------------------------------
897\namdisplay{namtsd} 
898%------------------------------------------------------------------------------------------
899
900Options are defined in \ngn{namtsd}.
901By default, the ocean start from rest (the velocity field is set to zero) and the initialization of
902temperature and salinity fields is controlled through the \np{ln\_tsd\_ini} namelist parameter.
903\begin{description}
904\item[ln\_tsd\_init = .true.]  use a T and S input files that can be given on the model grid itself or
905on their native input data grid. In the latter case, the data will be interpolated on-the-fly both in the
906horizontal and the vertical to the model grid (see \S~\ref{SBC_iof}). The information relative to the
907input files are given in the \np{sn\_tem} and \np{sn\_sal} structures.
908The computation is done in the \mdl{dtatsd} module.
909\item[ln\_tsd\_init = .false.] use constant salinity value of 35.5 psu and an analytical profile of temperature
910(typical of the tropical ocean), see \rou{istate\_t\_s} subroutine called from \mdl{istate} module.
911\end{description}
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