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1\documentclass[NEMO_book]{subfiles}
2\begin{document}
3% ================================================================
4% Chapter 2 ——— Space and Time Domain (DOM)
5% ================================================================
6\chapter{Space Domain (DOM) }
7\label{DOM}
8\minitoc
9
10% Missing things:
11%  - istate: description of the initial state   ==> this has to be put elsewhere..
12%                  perhaps in MISC ?  By the way the initialisation of T S and dynamics
13%                  should be put outside of DOM routine (better with TRC staff and off-line
14%                  tracers)
15%  -geo2ocean:  how to switch from geographic to mesh coordinate
16%     - domclo:  closed sea and lakes.... management of closea sea area : specific to global configuration, both forced and coupled
17
18
19\newpage
20$\ $\newline    % force a new ligne
21
22Having defined the continuous equations in Chap.~\ref{PE} and chosen a time
23discretization Chap.~\ref{STP}, we need to choose a discretization on a grid,
24and numerical algorithms. In the present chapter, we provide a general description
25of the staggered grid used in \NEMO, and other information relevant to the main
26directory routines as well as the DOM (DOMain) directory.
27
28$\ $\newline    % force a new lign
29
30% ================================================================
31% Fundamentals of the Discretisation
32% ================================================================
33\section{Fundamentals of the Discretisation}
34\label{DOM_basics}
35
36% -------------------------------------------------------------------------------------------------------------
37%        Arrangement of Variables
38% -------------------------------------------------------------------------------------------------------------
39\subsection{Arrangement of Variables}
40\label{DOM_cell}
41
42%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
43\begin{figure}[!tb]    \begin{center}
44\includegraphics[width=0.90\textwidth]{Fig_cell}
45\caption{ \label{Fig_cell}   
46Arrangement of variables. $t$ indicates scalar points where temperature,
47salinity, density, pressure and horizontal divergence are defined. ($u$,$v$,$w$)
48indicates vector points, and $f$ indicates vorticity points where both relative and
49planetary vorticities are defined}
50\end{center}   \end{figure}
51%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
52
53The numerical techniques used to solve the Primitive Equations in this model are
54based on the traditional, centred second-order finite difference approximation.
55Special attention has been given to the homogeneity of the solution in the three
56space directions. The arrangement of variables is the same in all directions.
57It consists of cells centred on scalar points ($t$, $S$, $p$, $\rho$) with vector
58points $(u, v, w)$ defined in the centre of each face of the cells (Fig. \ref{Fig_cell}).
59This is the generalisation to three dimensions of the well-known ``C'' grid in
60Arakawa's classification \citep{Mesinger_Arakawa_Bk76}. The relative and
61planetary vorticity, $\zeta$ and $f$, are defined in the centre of each vertical edge
62and the barotropic stream function $\psi$ is defined at horizontal points overlying
63the $\zeta$ and $f$-points.
64
65The ocean mesh ($i.e.$ the position of all the scalar and vector points) is defined
66by the transformation that gives ($\lambda$ ,$\varphi$ ,$z$) as a function of $(i,j,k)$.
67The grid-points are located at integer or integer and a half value of $(i,j,k)$ as
68indicated on Table \ref{Tab_cell}. In all the following, subscripts $u$, $v$, $w$,
69$f$, $uw$, $vw$ or $fw$ indicate the position of the grid-point where the scale
70factors are defined. Each scale factor is defined as the local analytical value
71provided by \eqref{Eq_scale_factors}. As a result, the mesh on which partial
72derivatives $\frac{\partial}{\partial \lambda}, \frac{\partial}{\partial \varphi}$, and
73$\frac{\partial}{\partial z} $ are evaluated is a uniform mesh with a grid size of unity.
74Discrete partial derivatives are formulated by the traditional, centred second order
75finite difference approximation while the scale factors are chosen equal to their
76local analytical value. An important point here is that the partial derivative of the
77scale factors must be evaluated by centred finite difference approximation, not
78from their analytical expression. This preserves the symmetry of the discrete set
79of equations and therefore satisfies many of the continuous properties (see
80Appendix~\ref{Apdx_C}). A similar, related remark can be made about the domain
81size: when needed, an area, volume, or the total ocean depth must be evaluated
82as the sum of the relevant scale factors (see \eqref{DOM_bar}) in the next section).
83
84%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
85\begin{table}[!tb]
86\begin{center} \begin{tabular}{|p{46pt}|p{56pt}|p{56pt}|p{56pt}|}
87\hline
88&$i$     & $j$    & $k$     \\ \hline
89& $i+1/2$   & $j$    & $k$    \\ \hline
90& $i$    & $j+1/2$   & $k$    \\ \hline
91& $i$    & $j$    & $k+1/2$   \\ \hline
92& $i+1/2$   & $j+1/2$   & $k$    \\ \hline
93uw & $i+1/2$   & $j$    & $k+1/2$   \\ \hline
94vw & $i$    & $j+1/2$   & $k+1/2$   \\ \hline
95fw & $i+1/2$   & $j+1/2$   & $k+1/2$   \\ \hline
96\end{tabular}
97\caption{ \label{Tab_cell}
98Location of grid-points as a function of integer or integer and a half value of the column,
99line or level. This indexing is only used for the writing of the semi-discrete equation.
100In the code, the indexing uses integer values only and has a reverse direction
101in the vertical (see \S\ref{DOM_Num_Index})}
102\end{center}
103\end{table}
104%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
105
106% -------------------------------------------------------------------------------------------------------------
107%        Vector Invariant Formulation
108% -------------------------------------------------------------------------------------------------------------
109\subsection{Discrete Operators}
110\label{DOM_operators}
111
112Given the values of a variable $q$ at adjacent points, the differencing and
113averaging operators at the midpoint between them are:
114\begin{subequations} \label{Eq_di_mi}
115\begin{align}
116 \delta _i [q]       &\  \    q(i+1/2)  - q(i-1/2)    \\
117 \overline q^{\,i} &= \left\{ q(i+1/2) + q(i-1/2) \right\} \; / \; 2
118\end{align}
119\end{subequations}
120
121Similar operators are defined with respect to $i+1/2$, $j$, $j+1/2$, $k$, and
122$k+1/2$. Following \eqref{Eq_PE_grad} and \eqref{Eq_PE_lap}, the gradient of a
123variable $q$ defined at a $t$-point has its three components defined at $u$-, $v$-
124and $w$-points while its Laplacien is defined at $t$-point. These operators have
125the following discrete forms in the curvilinear $s$-coordinate system:
126\begin{equation} \label{Eq_DOM_grad}
127\nabla q\equiv    \frac{1}{e_{1u} } \delta _{i+1/2 } [q] \;\,\mathbf{i}
128      +  \frac{1}{e_{2v} } \delta _{j+1/2 } [q] \;\,\mathbf{j}
129      +  \frac{1}{e_{3w}} \delta _{k+1/2} [q] \;\,\mathbf{k}
130\end{equation}
131\begin{multline} \label{Eq_DOM_lap}
132\Delta q\equiv \frac{1}{e_{1t}\,e_{2t}\,e_{3t} }
133       \;\left(          \delta_\left[ \frac{e_{2u}\,e_{3u}} {e_{1u}} \;\delta_{i+1/2} [q] \right]
134+                        \delta_\left[ \frac{e_{1v}\,e_{3v}}  {e_{2v}} \;\delta_{j+1/2} [q] \right] \;  \right)      \\
135+\frac{1}{e_{3t}} \delta_k \left[ \frac{1}{e_{3w} }                     \;\delta_{k+1/2} [q] \right]
136\end{multline}
137
138Following \eqref{Eq_PE_curl} and \eqref{Eq_PE_div}, a vector ${\rm {\bf A}}=\left( a_1,a_2,a_3\right)$ 
139defined at vector points $(u,v,w)$ has its three curl components defined at $vw$-, $uw$,
140and $f$-points, and its divergence defined at $t$-points:
141\begin{eqnarray}  \label{Eq_DOM_curl}
142 \nabla \times {\rm{\bf A}}\equiv &
143      \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} \\ 
144 +& \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} \\
145 +& \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}
146 \end{eqnarray}
147\begin{eqnarray} \label{Eq_DOM_div}
148\nabla \cdot \rm{\bf A} \equiv 
149    \frac{1}{e_{1t}\,e_{2t}\,e_{3t}} \left( \delta_i \left[e_{2u}\,e_{3u}\,a_1 \right]
150                                           +\delta_j \left[e_{1v}\,e_{3v}\,a_2 \right] \right)+\frac{1}{e_{3t} }\delta_k \left[a_3 \right]
151\end{eqnarray}
152
153The vertical average over the whole water column denoted by an overbar becomes
154for a quantity $q$ which is a masked field (i.e. equal to zero inside solid area):
155\begin{equation} \label{DOM_bar}
156\bar q   =         \frac{1}{H}    \int_{k^b}^{k^o} {q\;e_{3q} \,dk} 
157      \equiv \frac{1}{H_q }\sum\limits_k {q\;e_{3q} }
158\end{equation}
159where $H_q$  is the ocean depth, which is the masked sum of the vertical scale
160factors at $q$ points, $k^b$ and $k^o$ are the bottom and surface $k$-indices,
161and the symbol $k^o$ refers to a summation over all grid points of the same type
162in the direction indicated by the subscript (here $k$).
163
164In continuous form, the following properties are satisfied:
165\begin{equation} \label{Eq_DOM_curl_grad}
166\nabla \times \nabla q ={\rm {\bf {0}}}
167\end{equation}
168\begin{equation} \label{Eq_DOM_div_curl}
169\nabla \cdot \left( {\nabla \times {\rm {\bf A}}} \right)=0
170\end{equation}
171
172It is straightforward to demonstrate that these properties are verified locally in
173discrete form as soon as the scalar $q$ is taken at $t$-points and the vector
174\textbf{A} has its components defined at vector points $(u,v,w)$.
175
176Let $a$ and $b$ be two fields defined on the mesh, with value zero inside
177continental area. Using integration by parts it can be shown that the differencing
178operators ($\delta_i$, $\delta_j$ and $\delta_k$) are skew-symmetric linear operators,
179and further that the averaging operators $\overline{\,\cdot\,}^{\,i}$,
180$\overline{\,\cdot\,}^{\,k}$ and $\overline{\,\cdot\,}^{\,k}$) are symmetric linear
181operators, $i.e.$
182\begin{align} 
183\label{DOM_di_adj}
184\sum\limits_i { a_i \;\delta _i \left[ b \right]} 
185   &\equiv -\sum\limits_i {\delta _{i+1/2} \left[ a \right]\;b_{i+1/2} }      \\
186\label{DOM_mi_adj}
187\sum\limits_i { a_i \;\overline b^{\,i}} 
188   & \equiv \quad \sum\limits_i {\overline a ^{\,i+1/2}\;b_{i+1/2} } 
189\end{align}
190
191In other words, the adjoint of the differencing and averaging operators are
192$\delta_i^*=\delta_{i+1/2}$ and
193${(\overline{\,\cdot \,}^{\,i})}^*= \overline{\,\cdot\,}^{\,i+1/2}$, respectively.
194These two properties will be used extensively in the Appendix~\ref{Apdx_C} to
195demonstrate integral conservative properties of the discrete formulation chosen.
196
197% -------------------------------------------------------------------------------------------------------------
198%        Numerical Indexing
199% -------------------------------------------------------------------------------------------------------------
200\subsection{Numerical Indexing}
201\label{DOM_Num_Index}
202
203%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
204\begin{figure}[!tb]  \begin{center}
205\includegraphics[width=0.90\textwidth]{Fig_index_hor}
206\caption{   \label{Fig_index_hor}   
207Horizontal integer indexing used in the \textsc{Fortran} code. The dashed area indicates
208the cell in which variables contained in arrays have the same $i$- and $j$-indices}
209\end{center}   \end{figure}
210%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
211
212The array representation used in the \textsc{Fortran} code requires an integer
213indexing while the analytical definition of the mesh (see \S\ref{DOM_cell}) is
214associated with the use of integer values for $t$-points and both integer and
215integer and a half values for all the other points. Therefore a specific integer
216indexing must be defined for points other than $t$-points ($i.e.$ velocity and
217vorticity grid-points). Furthermore, the direction of the vertical indexing has
218been changed so that the surface level is at $k=1$.
219
220% -----------------------------------
221%        Horizontal Indexing
222% -----------------------------------
223\subsubsection{Horizontal Indexing}
224\label{DOM_Num_Index_hor}
225
226The indexing in the horizontal plane has been chosen as shown in Fig.\ref{Fig_index_hor}.
227For an increasing $i$ index ($j$ index), the $t$-point and the eastward $u$-point
228(northward $v$-point) have the same index (see the dashed area in Fig.\ref{Fig_index_hor}).
229A $t$-point and its nearest northeast $f$-point have the same $i$-and $j$-indices.
230
231% -----------------------------------
232%        Vertical indexing
233% -----------------------------------
234\subsubsection{Vertical Indexing}
235\label{DOM_Num_Index_vertical}
236
237In the vertical, the chosen indexing requires special attention since the
238$k$-axis is re-orientated downward in the \textsc{Fortran} code compared
239to the indexing used in the semi-discrete equations and given in \S\ref{DOM_cell}.
240The sea surface corresponds to the $w$-level $k=1$ which is the same index
241as $t$-level just below (Fig.\ref{Fig_index_vert}). The last $w$-level ($k=jpk$)
242either corresponds to the ocean floor or is inside the bathymetry while the last
243$t$-level is always inside the bathymetry (Fig.\ref{Fig_index_vert}). Note that
244for an increasing $k$ index, a $w$-point and the $t$-point just below have the
245same $k$ index, in opposition to what is done in the horizontal plane where
246it is the $t$-point and the nearest velocity points in the direction of the horizontal
247axis that have the same $i$ or $j$ index (compare the dashed area in
248Fig.\ref{Fig_index_hor} and \ref{Fig_index_vert}). Since the scale factors are
249chosen to be strictly positive, a \emph{minus sign} appears in the \textsc{Fortran} 
250code \emph{before all the vertical derivatives} of the discrete equations given in
251this documentation.
252
253%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
254\begin{figure}[!pt]    \begin{center}
255\includegraphics[width=.90\textwidth]{Fig_index_vert}
256\caption{ \label{Fig_index_vert}     
257Vertical integer indexing used in the \textsc{Fortran } code. Note that
258the $k$-axis is orientated downward. The dashed area indicates the cell in
259which variables contained in arrays have the same $k$-index.}
260\end{center}   \end{figure}
261%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
262
263% -----------------------------------
264%        Domain Size
265% -----------------------------------
266\subsubsection{Domain Size}
267\label{DOM_size}
268
269The total size of the computational domain is set by the parameters \np{jpiglo},
270\np{jpjglo} and \np{jpkdta} in the $i$, $j$ and $k$ directions respectively. They are
271given as namelist variables in the \ngn{namcfg} namelist.
272
273Note that are other namelist variables in the \ngn{namcfg} namelist that refer to
274 the domain size.
275The two variables \np{jpidta} and \np{jpjdta} may be larger than \np{jpiglo}, \np{jpjglo}
276when the user wants to use only a sub-region of a given configuration. This is
277the "zoom" capability described in \S\ref{MISC_zoom}. In most applications of
278the model, $jpidta=jpiglo$, $jpjdta=jpjglo$, and $jpizoom=jpjzoom=1$. Parameters
279$jpi$ and $jpj$ refer to the size of each processor subdomain when the code is
280run in parallel using domain decomposition (\key{mpp\_mpi} defined, see
281\S\ref{LBC_mpp}).
282
283
284$\ $\newline    % force a new lign
285
286% ================================================================
287% Domain: Horizontal Grid (mesh)
288% ================================================================
289\section  [Domain: Horizontal Grid (mesh) (\textit{domhgr})]               
290      {Domain: Horizontal Grid (mesh) \small{(\mdl{domhgr} module)} }
291\label{DOM_hgr}
292
293% -------------------------------------------------------------------------------------------------------------
294%        Coordinates and scale factors
295% -------------------------------------------------------------------------------------------------------------
296\subsection{Coordinates and scale factors}
297\label{DOM_hgr_coord_e}
298
299The ocean mesh ($i.e.$ the position of all the scalar and vector points) is defined
300by the transformation that gives $(\lambda,\varphi,z)$ as a function of $(i,j,k)$.
301The grid-points are located at integer or integer and a half values of as indicated
302in Table~\ref{Tab_cell}. The associated scale factors are defined using the
303analytical first derivative of the transformation \eqref{Eq_scale_factors}. These
304definitions are done in two modules, \mdl{domhgr} and \mdl{domzgr}, which
305provide the horizontal and vertical meshes, respectively. This section deals with
306the horizontal mesh parameters.
307
308In a horizontal plane, the location of all the model grid points is defined from the
309analytical expressions of the longitude $\lambda$ and  latitude $\varphi$ as a
310function of  $(i,j)$. The horizontal scale factors are calculated using
311\eqref{Eq_scale_factors}. For example, when the longitude and latitude are
312function of a single value ($i$ and $j$, respectively) (geographical configuration
313of the mesh), the horizontal mesh definition reduces to define the wanted
314$\lambda(i)$, $\varphi(j)$, and their derivatives $\lambda'(i)$ $\varphi'(j)$ in the
315\mdl{domhgr} module. The model computes the grid-point positions and scale
316factors in the horizontal plane as follows:
317\begin{flalign*}
318\lambda_t &\equiv \text{glamt}= \lambda(i)     & \varphi_t &\equiv \text{gphit} = \varphi(j)\\
319\lambda_u &\equiv \text{glamu}= \lambda(i+1/2)& \varphi_u &\equiv \text{gphiu}= \varphi(j)\\
320\lambda_v &\equiv \text{glamv}= \lambda(i)       & \varphi_v &\equiv \text{gphiv} = \varphi(j+1/2)\\
321\lambda_f &\equiv \text{glamf }= \lambda(i+1/2)& \varphi_f &\equiv \text{gphif }= \varphi(j+1/2)
322\end{flalign*}
323\begin{flalign*}
324e_{1t} &\equiv \text{e1t} = r_a |\lambda'(i)    \; \cos\varphi(j)  |&
325e_{2t} &\equiv \text{e2t} = r_a |\varphi'(j)|  \\
326e_{1u} &\equiv \text{e1t} = r_a |\lambda'(i+1/2)   \; \cos\varphi(j)  |&
327e_{2u} &\equiv \text{e2t} = r_a |\varphi'(j)|\\
328e_{1v} &\equiv \text{e1t} = r_a |\lambda'(i)    \; \cos\varphi(j+1/2)  |&
329e_{2v} &\equiv \text{e2t} = r_a |\varphi'(j+1/2)|\\
330e_{1f} &\equiv \text{e1t} = r_a |\lambda'(i+1/2)\; \cos\varphi(j+1/2)  |&
331e_{2f} &\equiv \text{e2t} = r_a |\varphi'(j+1/2)|
332\end{flalign*}
333where the last letter of each computational name indicates the grid point
334considered and $r_a$ is the earth radius (defined in \mdl{phycst} along with
335all universal constants). Note that the horizontal position of and scale factors
336at $w$-points are exactly equal to those of $t$-points, thus no specific arrays
337are defined at $w$-points.
338
339Note that the definition of the scale factors ($i.e.$ as the analytical first derivative
340of the transformation that gives $(\lambda,\varphi,z)$ as a function of $(i,j,k)$) is
341specific to the \NEMO model \citep{Marti_al_JGR92}. As an example, $e_{1t}$ is defined
342locally at a $t$-point, whereas many other models on a C grid choose to define
343such a scale factor as the distance between the $U$-points on each side of the
344$t$-point. Relying on an analytical transformation has two advantages: firstly, there
345is no ambiguity in the scale factors appearing in the discrete equations, since they
346are first introduced in the continuous equations; secondly, analytical transformations
347encourage good practice by the definition of smoothly varying grids (rather than
348allowing the user to set arbitrary jumps in thickness between adjacent layers)
349\citep{Treguier1996}. An example of the effect of such a choice is shown in
350Fig.~\ref{Fig_zgr_e3}.
351%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
352\begin{figure}[!t]     \begin{center}
353\includegraphics[width=0.90\textwidth]{Fig_zgr_e3}
354\caption{ \label{Fig_zgr_e3}   
355Comparison of (a) traditional definitions of grid-point position and grid-size in the vertical,
356and (b) analytically derived grid-point position and scale factors.
357For both grids here,  the same $w$-point depth has been chosen but in (a) the
358$t$-points are set half way between $w$-points while in (b) they are defined from
359an analytical function: $z(k)=5\,(k-1/2)^3 - 45\,(k-1/2)^2 + 140\,(k-1/2) - 150$.
360Note the resulting difference between the value of the grid-size $\Delta_k$ and
361those of the scale factor $e_k$. }
362\end{center}   \end{figure}
363%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
364
365% -------------------------------------------------------------------------------------------------------------
366%        Choice of horizontal grid
367% -------------------------------------------------------------------------------------------------------------
368\subsection{Choice of horizontal grid}
369\label{DOM_hgr_msh_choice}
370
371The user has three options available in defining a horizontal grid, which involve
372the namelist variable \np{jphgr\_mesh} of the \ngn{namcfg} namelist.
373\begin{description}
374\item[\np{jphgr\_mesh}=0]  The most general curvilinear orthogonal grids.
375The coordinates and their first derivatives with respect to $i$ and $j$ are provided
376in a input file (\ifile{coordinates}), read in \rou{hgr\_read} subroutine of the domhgr module.
377\item[\np{jphgr\_mesh}=1 to 5] A few simple analytical grids are provided (see below).
378For other analytical grids, the \mdl{domhgr} module must be modified by the user.
379\end{description}
380
381There are two simple cases of geographical grids on the sphere. With
382\np{jphgr\_mesh}=1, the grid (expressed in degrees) is regular in space,
383with grid sizes specified by parameters \np{ppe1\_deg} and \np{ppe2\_deg},
384respectively. Such a geographical grid can be very anisotropic at high latitudes
385because of the convergence of meridians (the zonal scale factors $e_1$ 
386become much smaller than the meridional scale factors $e_2$). The Mercator
387grid (\np{jphgr\_mesh}=4) avoids this anisotropy by refining the meridional scale
388factors in the same way as the zonal ones. In this case, meridional scale factors
389and latitudes are calculated analytically using the formulae appropriate for
390a Mercator projection, based on \np{ppe1\_deg} which is a reference grid spacing
391at the equator (this applies even when the geographical equator is situated outside
392the model domain).
393%%%
394\gmcomment{ give here the analytical expression of the Mercator mesh}
395%%%
396In these two cases (\np{jphgr\_mesh}=1 or 4), the grid position is defined by the
397longitude and latitude of the south-westernmost point (\np{ppglamt0} 
398and \np{ppgphi0}). Note that for the Mercator grid the user need only provide
399an approximate starting latitude: the real latitude will be recalculated analytically,
400in order to ensure that the equator corresponds to line passing through $t$-
401and $u$-points. 
402
403Rectangular grids ignoring the spherical geometry are defined with
404\np{jphgr\_mesh} = 2, 3, 5. The domain is either an $f$-plane (\np{jphgr\_mesh} = 2,
405Coriolis factor is constant) or a beta-plane (\np{jphgr\_mesh} = 3, the Coriolis factor
406is linear in the $j$-direction). The grid size is uniform in meter in each direction,
407and given by the parameters \np{ppe1\_m} and \np{ppe2\_m} respectively.
408The zonal grid coordinate (\textit{glam} arrays) is in kilometers, starting at zero
409with the first $t$-point. The meridional coordinate (gphi. arrays) is in kilometers,
410and the second $t$-point corresponds to coordinate $gphit=0$. The input
411variable \np{ppglam0} is ignored. \np{ppgphi0} is used to set the reference
412latitude for computation of the Coriolis parameter. In the case of the beta plane,
413\np{ppgphi0} corresponds to the center of the domain. Finally, the special case
414\np{jphgr\_mesh}=5 corresponds to a beta plane in a rotated domain for the
415GYRE configuration, representing a classical mid-latitude double gyre system.
416The rotation allows us to maximize the jet length relative to the gyre areas
417(and the number of grid points).
418
419The choice of the grid must be consistent with the boundary conditions specified
420by \np{jperio}, a parameter found in \ngn{namcfg} namelist (see {\S\ref{LBC}).
421
422% -------------------------------------------------------------------------------------------------------------
423%        Grid files
424% -------------------------------------------------------------------------------------------------------------
425\subsection{Output Grid files}
426\label{DOM_hgr_files}
427
428All the arrays relating to a particular ocean model configuration (grid-point
429position, scale factors, masks) can be saved in files if $\np{nn\_msh} \not= 0$ 
430(namelist variable in \ngn{namdom}). This can be particularly useful for plots and off-line
431diagnostics. In some cases, the user may choose to make a local modification
432of a scale factor in the code. This is the case in global configurations when
433restricting the width of a specific strait (usually a one-grid-point strait that
434happens to be too wide due to insufficient model resolution). An example
435is Gibraltar Strait in the ORCA2 configuration. When such modifications are done,
436the output grid written when $\np{nn\_msh} \not=0$ is no more equal to the input grid.
437
438$\ $\newline    % force a new lign
439
440% ================================================================
441% Domain: Vertical Grid (domzgr)
442% ================================================================
443\section  [Domain: Vertical Grid (\textit{domzgr})]
444      {Domain: Vertical Grid \small{(\mdl{domzgr} module)} }
445\label{DOM_zgr}
446%-----------------------------------------nam_zgr & namdom-------------------------------------------
447\namdisplay{namzgr} 
448\namdisplay{namdom} 
449%-------------------------------------------------------------------------------------------------------------
450
451Variables are defined through the \ngn{namzgr} and \ngn{namdom} namelists.
452In the vertical, the model mesh is determined by four things:
453(1) the bathymetry given in meters ;
454(2) the number of levels of the model (\jp{jpk}) ;
455(3) the analytical transformation $z(i,j,k)$ and the vertical scale factors
456(derivatives of the transformation) ;
457and (4) the masking system, $i.e.$ the number of wet model levels at each
458$(i,j)$ column of points.
459
460%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
461\begin{figure}[!tb]    \begin{center}
462\includegraphics[width=1.0\textwidth]{Fig_z_zps_s_sps}
463\caption{  \label{Fig_z_zps_s_sps}   
464The ocean bottom as seen by the model:
465(a) $z$-coordinate with full step,
466(b) $z$-coordinate with partial step,
467(c) $s$-coordinate: terrain following representation,
468(d) hybrid $s-z$ coordinate,
469(e) hybrid $s-z$ coordinate with partial step, and
470(f) same as (e) but in the non-linear free surface (\np{ln\_linssh}=false).
471Note that the non-linear free surface can be used with any of the
4725 coordinates (a) to (e).}
473\end{center}   \end{figure}
474%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
475
476The choice of a vertical coordinate, even if it is made through \ngn{namzgr} namelist parameters,
477must be done once of all at the beginning of an experiment. It is not intended as an
478option which can be enabled or disabled in the middle of an experiment. Three main
479choices are offered (Fig.~\ref{Fig_z_zps_s_sps}a to c): $z$-coordinate with full step
480bathymetry (\np{ln\_zco}~=~true), $z$-coordinate with partial step bathymetry
481(\np{ln\_zps}~=~true), or generalized, $s$-coordinate (\np{ln\_sco}~=~true).
482Hybridation of the three main coordinates are available: $s-z$ or $s-zps$ coordinate
483(Fig.~\ref{Fig_z_zps_s_sps}d and \ref{Fig_z_zps_s_sps}e). By default a non-linear free surface is used:
484the coordinate follow the time-variation of the free surface so that the transformation is time dependent:
485$z(i,j,k,t)$ (Fig.~\ref{Fig_z_zps_s_sps}f). When a linear free surface is assumed (\np{ln\_linssh}=true),
486the vertical coordinate are fixed in time, but the seawater can move up and down across the z=0 surface
487(in other words, the top of the ocean in not a rigid-lid).
488The last choice in terms of vertical coordinate concerns the presence (or not) in the model domain
489of ocean cavities beneath ice shelves. Setting \np{ln\_isfcav} to true allows to manage ocean cavities,
490otherwise they are filled in. This option is currently only available in $z$- or $zps$-coordinate,
491and partial step are also applied at the ocean/ice shelf interface.
492
493Contrary to the horizontal grid, the vertical grid is computed in the code and no
494provision is made for reading it from a file. The only input file is the bathymetry
495(in meters) (\ifile{bathy\_meter}).
496\footnote{N.B. in full step $z$-coordinate, a \ifile{bathy\_level} file can replace the
497\ifile{bathy\_meter} file, so that the computation of the number of wet ocean point
498in each water column is by-passed}.
499If \np{ln\_isfcav}~=~true, an extra file input file describing the ice shelf draft
500(in meters) (\ifile{isf\_draft\_meter}) is needed.
501
502After reading the bathymetry, the algorithm for vertical grid definition differs
503between the different options:
504\begin{description}
505\item[\textit{zco}] set a reference coordinate transformation $z_0 (k)$, and set $z(i,j,k,t)=z_0 (k)$.
506\item[\textit{zps}] set a reference coordinate transformation $z_0 (k)$, and
507calculate the thickness of the deepest level at each $(i,j)$ point using the
508bathymetry, to obtain the final three-dimensional depth and scale factor arrays.
509\item[\textit{sco}] smooth the bathymetry to fulfil the hydrostatic consistency
510criteria and set the three-dimensional transformation.
511\item[\textit{s-z} and \textit{s-zps}] smooth the bathymetry to fulfil the hydrostatic
512consistency criteria and set the three-dimensional transformation $z(i,j,k)$, and
513possibly introduce masking of extra land points to better fit the original bathymetry file
514\end{description}
515%%%
516\gmcomment{   add the description of the smoothing:  envelop topography...}
517%%%
518
519Unless a linear free surface is used (\np{ln\_linssh}=false), the arrays describing
520the grid point depths and vertical scale factors are three set of three dimensional arrays $(i,j,k)$ 
521defined at \textit{before}, \textit{now} and \textit{after} time step. The time at which they are
522defined is indicated by a suffix:$\_b$, $\_n$, or $\_a$, respectively. They are updated at each model time step
523using a fixed reference coordinate system which computer names have a $\_0$ suffix.
524When the linear free surface option is used (\np{ln\_linssh}=true), \textit{before}, \textit{now} 
525and \textit{after} arrays are simply set one for all to their reference counterpart.
526
527
528% -------------------------------------------------------------------------------------------------------------
529%        Meter Bathymetry
530% -------------------------------------------------------------------------------------------------------------
531\subsection{Meter Bathymetry}
532\label{DOM_bathy}
533
534Three options are possible for defining the bathymetry, according to the
535namelist variable \np{nn\_bathy} (found in \ngn{namdom} namelist):
536\begin{description}
537\item[\np{nn\_bathy} = 0] a flat-bottom domain is defined. The total depth $z_w (jpk)$ 
538is given by the coordinate transformation. The domain can either be a closed
539basin or a periodic channel depending on the parameter \np{jperio}.
540\item[\np{nn\_bathy} = -1] a domain with a bump of topography one third of the
541domain width at the central latitude. This is meant for the "EEL-R5" configuration,
542a periodic or open boundary channel with a seamount.
543\item[\np{nn\_bathy} = 1] read a bathymetry and ice shelf draft (if needed).
544 The \ifile{bathy\_meter} file (Netcdf format) provides the ocean depth (positive, in meters)
545 at each grid point of the model grid. The bathymetry is usually built by interpolating a standard bathymetry product
546($e.g.$ ETOPO2) onto the horizontal ocean mesh. Defining the bathymetry also
547defines the coastline: where the bathymetry is zero, no model levels are defined
548(all levels are masked).
549
550The \ifile{isfdraft\_meter} file (Netcdf format) provides the ice shelf draft (positive, in meters)
551 at each grid point of the model grid. This file is only needed if \np{ln\_isfcav}~=~true.
552Defining the ice shelf draft will also define the ice shelf edge and the grounding line position.
553\end{description}
554
555When a global ocean is coupled to an atmospheric model it is better to represent
556all large water bodies (e.g, great lakes, Caspian sea...) even if the model
557resolution does not allow their communication with the rest of the ocean.
558This is unnecessary when the ocean is forced by fixed atmospheric conditions,
559so these seas can be removed from the ocean domain. The user has the option
560to set the bathymetry in closed seas to zero (see \S\ref{MISC_closea}), but the
561code has to be adapted to the user's configuration.
562
563% -------------------------------------------------------------------------------------------------------------
564%        z-coordinate  and reference coordinate transformation
565% -------------------------------------------------------------------------------------------------------------
566\subsection[$z$-coordinate (\np{ln\_zco}]
567        {$z$-coordinate (\np{ln\_zco}=true) and reference coordinate}
568\label{DOM_zco}
569
570%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
571\begin{figure}[!tb]    \begin{center}
572\includegraphics[width=0.90\textwidth]{Fig_zgr}
573\caption{ \label{Fig_zgr}   
574Default vertical mesh for ORCA2: 30 ocean levels (L30). Vertical level functions for
575(a) T-point depth and (b) the associated scale factor as computed
576from \eqref{DOM_zgr_ana} using \eqref{DOM_zgr_coef} in $z$-coordinate.}
577\end{center}   \end{figure}
578%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
579
580The reference coordinate transformation $z_0 (k)$ defines the arrays $gdept_0$ 
581and $gdepw_0$ for $t$- and $w$-points, respectively. As indicated on
582Fig.\ref{Fig_index_vert} \jp{jpk} is the number of $w$-levels. $gdepw_0(1)$ is the
583ocean surface. There are at most \jp{jpk}-1 $t$-points inside the ocean, the
584additional $t$-point at $jk=jpk$ is below the sea floor and is not used.
585The vertical location of $w$- and $t$-levels is defined from the analytic expression
586of the depth $z_0(k)$ whose analytical derivative with respect to $k$ provides the
587vertical scale factors. The user must provide the analytical expression of both
588$z_0$ and its first derivative with respect to $k$. This is done in routine \mdl{domzgr} 
589through statement functions, using parameters provided in the \ngn{namcfg} namelist.
590
591It is possible to define a simple regular vertical grid by giving zero stretching (\np{ppacr=0}).
592In that case, the parameters \jp{jpk} (number of $w$-levels) and \np{pphmax} 
593(total ocean depth in meters) fully define the grid.
594
595For climate-related studies it is often desirable to concentrate the vertical resolution
596near the ocean surface. The following function is proposed as a standard for a
597$z$-coordinate (with either full or partial steps):
598\begin{equation} \label{DOM_zgr_ana}
599\begin{split}
600 z_0 (k)    &= h_{sur} -h_0 \;k-\;h_1 \;\log \left[ {\,\cosh \left( {{(k-h_{th} )} / {h_{cr} }} \right)\,} \right] \\ 
601 e_3^0 (k)  &= \left| -h_0 -h_1 \;\tanh \left( {{(k-h_{th} )} / {h_{cr} }} \right) \right|
602\end{split}
603\end{equation}
604where $k=1$ to \jp{jpk} for $w$-levels and $k=1$ to $k=1$ for $T-$levels. Such an
605expression allows us to define a nearly uniform vertical location of levels at the
606ocean top and bottom with a smooth hyperbolic tangent transition in between
607(Fig.~\ref{Fig_zgr}).
608
609If the ice shelf cavities are opened (\np{ln\_isfcav}=~true~}), the definition of $z_0$ is the same.
610However, definition of $e_3^0$ at $t$- and $w$-points is respectively changed to:
611\begin{equation} \label{DOM_zgr_ana}
612\begin{split}
613 e_3^T(k) &= z_W (k+1) - z_W (k)   \\
614 e_3^W(k) &= z_T (k)   - z_T (k-1) \\
615\end{split}
616\end{equation}
617This formulation decrease the self-generated circulation into the ice shelf cavity
618(which can, in extreme case, leads to blow up).\\
619
620 
621The most used vertical grid for ORCA2 has $10~m$ ($500~m)$ resolution in the
622surface (bottom) layers and a depth which varies from 0 at the sea surface to a
623minimum of $-5000~m$. This leads to the following conditions:
624\begin{equation} \label{DOM_zgr_coef}
625\begin{split}
626 e_3 (1+1/2)      &=10. \\ 
627 e_3 (jpk-1/2) &=500. \\ 
628 z(1)       &=0. \\ 
629 z(jpk)        &=-5000. \\ 
630\end{split}
631\end{equation}
632
633With the choice of the stretching $h_{cr} =3$ and the number of levels
634\jp{jpk}=$31$, the four coefficients $h_{sur}$, $h_{0}$, $h_{1}$, and $h_{th}$ in
635\eqref{DOM_zgr_ana} have been determined such that \eqref{DOM_zgr_coef} is
636satisfied, through an optimisation procedure using a bisection method. For the first
637standard ORCA2 vertical grid this led to the following values: $h_{sur} =4762.96$,
638$h_0 =255.58, h_1 =245.5813$, and $h_{th} =21.43336$. The resulting depths and
639scale factors as a function of the model levels are shown in Fig.~\ref{Fig_zgr} and
640given in Table \ref{Tab_orca_zgr}. Those values correspond to the parameters
641\np{ppsur}, \np{ppa0}, \np{ppa1}, \np{ppkth} in \ngn{namcfg} namelist.
642
643Rather than entering parameters $h_{sur}$, $h_{0}$, and $h_{1}$ directly, it is
644possible to recalculate them. In that case the user sets
645\np{ppsur}=\np{ppa0}=\np{ppa1}=999999., in \ngn{namcfg} namelist,
646and specifies instead the four following parameters:
647\begin{itemize}
648\item    \np{ppacr}=$h_{cr} $: stretching factor (nondimensional). The larger
649\np{ppacr}, the smaller the stretching. Values from $3$ to $10$ are usual.
650\item    \np{ppkth}=$h_{th} $: is approximately the model level at which maximum
651stretching occurs (nondimensional, usually of order 1/2 or 2/3 of \jp{jpk})
652\item    \np{ppdzmin}: minimum thickness for the top layer (in meters)
653\item    \np{pphmax}: total depth of the ocean (meters).
654\end{itemize}
655As an example, for the $45$ layers used in the DRAKKAR configuration those
656parameters are: \jp{jpk}=46, \np{ppacr}=9, \np{ppkth}=23.563, \np{ppdzmin}=6m,
657\np{pphmax}=5750m.
658
659%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
660\begin{table}     \begin{center} \begin{tabular}{c||r|r|r|r}
661\hline
662\textbf{LEVEL}& \textbf{gdept}& \textbf{gdepw}& \textbf{e3t }& \textbf{e3w  } \\ \hline
663&  \textbf{  5.00}   &       0.00 & \textbf{ 10.00} &  10.00 \\   \hline
664&  \textbf{15.00} &    10.00 &   \textbf{ 10.00} &  10.00 \\   \hline
665&  \textbf{25.00} &    20.00 &   \textbf{ 10.00} &     10.00 \\   \hline
666&  \textbf{35.01} &    30.00 &   \textbf{ 10.01} &     10.00 \\   \hline
667&  \textbf{45.01} &    40.01 &   \textbf{ 10.01} &  10.01 \\   \hline
668&  \textbf{55.03} &    50.02 &   \textbf{ 10.02} &     10.02 \\   \hline
669&  \textbf{65.06} &    60.04 &   \textbf{ 10.04} &  10.03 \\   \hline
670&  \textbf{75.13} &    70.09 &   \textbf{ 10.09} &  10.06 \\   \hline
671&  \textbf{85.25} &    80.18 &   \textbf{ 10.17} &  10.12 \\   \hline
67210 &  \textbf{95.49} &    90.35 &   \textbf{ 10.33} &  10.24 \\   \hline
67311 &  \textbf{105.97}   &   100.69 &   \textbf{ 10.65} &  10.47 \\   \hline
67412 &  \textbf{116.90}   &   111.36 &   \textbf{ 11.27} &  10.91 \\   \hline
67513 &  \textbf{128.70}   &   122.65 &   \textbf{ 12.47} &  11.77 \\   \hline
67614 &  \textbf{142.20}   &   135.16 &   \textbf{ 14.78} &  13.43 \\   \hline
67715 &  \textbf{158.96}   &   150.03 &   \textbf{ 19.23} &  16.65 \\   \hline
67816 &  \textbf{181.96}   &   169.42 &   \textbf{ 27.66} &  22.78 \\   \hline
67917 &  \textbf{216.65}   &   197.37 &   \textbf{ 43.26} &  34.30 \\ \hline
68018 &  \textbf{272.48}   &   241.13 &   \textbf{ 70.88} &  55.21 \\ \hline
68119 &  \textbf{364.30}   &   312.74 &   \textbf{116.11} &  90.99 \\ \hline
68220 &  \textbf{511.53}   &   429.72 &   \textbf{181.55} &    146.43 \\ \hline
68321 &  \textbf{732.20}   &   611.89 &   \textbf{261.03} &    220.35 \\ \hline
68422 &  \textbf{1033.22}&  872.87 &   \textbf{339.39} &    301.42 \\ \hline
68523 &  \textbf{1405.70}& 1211.59 & \textbf{402.26} &   373.31 \\ \hline
68624 &  \textbf{1830.89}& 1612.98 & \textbf{444.87} &   426.00 \\ \hline
68725 &  \textbf{2289.77}& 2057.13 & \textbf{470.55} &   459.47 \\ \hline
68826 &  \textbf{2768.24}& 2527.22 & \textbf{484.95} &   478.83 \\ \hline
68927 &  \textbf{3257.48}& 3011.90 & \textbf{492.70} &   489.44 \\ \hline
69028 &  \textbf{3752.44}& 3504.46 & \textbf{496.78} &   495.07 \\ \hline
69129 &  \textbf{4250.40}& 4001.16 & \textbf{498.90} &   498.02 \\ \hline
69230 &  \textbf{4749.91}& 4500.02 & \textbf{500.00} &   499.54 \\ \hline
69331 &  \textbf{5250.23}& 5000.00 &   \textbf{500.56} & 500.33 \\ \hline
694\end{tabular} \end{center} 
695\caption{ \label{Tab_orca_zgr}   
696Default vertical mesh in $z$-coordinate for 30 layers ORCA2 configuration as computed
697from \eqref{DOM_zgr_ana} using the coefficients given in \eqref{DOM_zgr_coef}}
698\end{table}
699%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
700
701% -------------------------------------------------------------------------------------------------------------
702%        z-coordinate with partial step
703% -------------------------------------------------------------------------------------------------------------
704\subsection   [$z$-coordinate with partial step (\np{ln\_zps})]
705         {$z$-coordinate with partial step (\np{ln\_zps}=.true.)}
706\label{DOM_zps}
707%--------------------------------------------namdom-------------------------------------------------------
708\namdisplay{namdom} 
709%--------------------------------------------------------------------------------------------------------------
710
711In $z$-coordinate partial step, the depths of the model levels are defined by the
712reference analytical function $z_0 (k)$ as described in the previous
713section, \emph{except} in the bottom layer. The thickness of the bottom layer is
714allowed to vary as a function of geographical location $(\lambda,\varphi)$ to allow a
715better representation of the bathymetry, especially in the case of small
716slopes (where the bathymetry varies by less than one level thickness from
717one grid point to the next). The reference layer thicknesses $e_{3t}^0$ have been
718defined in the absence of bathymetry. With partial steps, layers from 1 to
719\jp{jpk}-2 can have a thickness smaller than $e_{3t}(jk)$. The model deepest layer (\jp{jpk}-1)
720is allowed to have either a smaller or larger thickness than $e_{3t}(jpk)$: the
721maximum thickness allowed is $2*e_{3t}(jpk-1)$. This has to be kept in mind when
722specifying values in \ngn{namdom} namelist, as the maximum depth \np{pphmax} 
723in partial steps: for example, with
724\np{pphmax}$=5750~m$ for the DRAKKAR 45 layer grid, the maximum ocean depth
725allowed is actually $6000~m$ (the default thickness $e_{3t}(jpk-1)$ being $250~m$).
726Two variables in the namdom namelist are used to define the partial step
727vertical grid. The mimimum water thickness (in meters) allowed for a cell
728partially filled with bathymetry at level jk is the minimum of \np{rn\_e3zps\_min} 
729(thickness in meters, usually $20~m$) or $e_{3t}(jk)*\np{rn\_e3zps\_rat}$ (a fraction,
730usually 10\%, of the default thickness $e_{3t}(jk)$).
731
732\gmcomment{ \colorbox{yellow}{Add a figure here of pstep especially at last ocean level }  }
733
734% -------------------------------------------------------------------------------------------------------------
735%        s-coordinate
736% -------------------------------------------------------------------------------------------------------------
737\subsection   [$s$-coordinate (\np{ln\_sco})]
738           {$s$-coordinate (\np{ln\_sco}=true)}
739\label{DOM_sco}
740%------------------------------------------nam_zgr_sco---------------------------------------------------
741\namdisplay{namzgr_sco} 
742%--------------------------------------------------------------------------------------------------------------
743Options are defined in \ngn{namzgr\_sco}.
744In $s$-coordinate (\np{ln\_sco}~=~true), the depth and thickness of the model
745levels are defined from the product of a depth field and either a stretching
746function or its derivative, respectively:
747
748\begin{equation} \label{DOM_sco_ana}
749\begin{split}
750 z(k)       &= h(i,j) \; z_0(k)  \\
751 e_3(k)  &= h(i,j) \; z_0'(k)
752\end{split}
753\end{equation}
754
755where $h$ is the depth of the last $w$-level ($z_0(k)$) defined at the $t$-point
756location in the horizontal and $z_0(k)$ is a function which varies from $0$ at the sea
757surface to $1$ at the ocean bottom. The depth field $h$ is not necessary the ocean
758depth, since a mixed step-like and bottom-following representation of the
759topography 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).
760The namelist parameter \np{rn\_rmax} determines the slope at which the terrain-following coordinate intersects
761the sea bed and becomes a pseudo z-coordinate.
762The coordinate can also be hybridised by specifying \np{rn\_sbot\_min} and \np{rn\_sbot\_max} 
763as the minimum and maximum depths at which the terrain-following vertical coordinate is calculated.
764
765Options for stretching the coordinate are provided as examples, but care must be taken to ensure
766that the vertical stretch used is appropriate for the application.
767
768The original default NEMO s-coordinate stretching is available if neither of the other options
769are specified as true (\np{ln\_s\_SH94}~=~false and \np{ln\_s\_SF12}~=~false).
770This uses a depth independent $\tanh$ function for the stretching \citep{Madec_al_JPO96}:
771
772\begin{equation}
773  z = s_{min}+C\left(s\right)\left(H-s_{min}\right)
774  \label{eq:SH94_1}
775\end{equation}
776
777where $s_{min}$ is the depth at which the $s$-coordinate stretching starts and
778allows a $z$-coordinate to placed on top of the stretched coordinate,
779and $z$ is the depth (negative down from the asea surface).
780
781\begin{equation}
782  s = -\frac{k}{n-1} \quad \text{ and } \quad 0 \leq k \leq n-1
783  \label{eq:s}
784\end{equation}
785
786\begin{equation} \label{DOM_sco_function}
787\begin{split}
788C(s)  &\frac{ \left[   \tanh{ \left( \theta \, (s+b) \right)} 
789               - \tanh{ \left\theta \, b      \right)}  \right]}
790            {2\;\sinh \left( \theta \right)}
791\end{split}
792\end{equation}
793
794A stretching function, modified from the commonly used \citet{Song_Haidvogel_JCP94} 
795stretching (\np{ln\_s\_SH94}~=~true), is also available and is more commonly used for shelf seas modelling:
796
797\begin{equation}
798  C\left(s\right) =   \left(1 - b \right)\frac{ \sinh\left( \theta s\right)}{\sinh\left(\theta\right)} +      \\
799  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)}
800  \label{eq:SH94_2}
801\end{equation}
802
803%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
804\begin{figure}[!ht]    \begin{center}
805\includegraphics[width=1.0\textwidth]{Fig_sco_function}
806\caption{  \label{Fig_sco_function}   
807Examples of the stretching function applied to a seamount; from left to right:
808surface, surface and bottom, and bottom intensified resolutions}
809\end{center}   \end{figure}
810%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
811
812where $H_c$ is the critical depth (\np{rn\_hc}) at which the coordinate transitions from
813pure $\sigma$ to the stretched coordinate,  and $\theta$ (\np{rn\_theta}) and $b$ (\np{rn\_bb})
814are the surface and bottom control parameters such that $0\leqslant \theta \leqslant 20$, and
815$0\leqslant b\leqslant 1$. $b$ has been designed to allow surface and/or bottom
816increase of the vertical resolution (Fig.~\ref{Fig_sco_function}).
817
818Another example has been provided at version 3.5 (\np{ln\_s\_SF12}) that allows
819a fixed surface resolution in an analytical terrain-following stretching \citet{Siddorn_Furner_OM12}.
820In this case the a stretching function $\gamma$ is defined such that:
821
822\begin{equation}
823z = -\gamma h \quad \text{ with } \quad 0 \leq \gamma \leq 1
824\label{eq:z}
825\end{equation}
826
827The function is defined with respect to $\sigma$, the unstretched terrain-following coordinate:
828
829\begin{equation} \label{DOM_gamma_deriv}
830\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)
831\end{equation}
832
833Where:
834\begin{equation} \label{DOM_gamma}
835f\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} 
836\end{equation}
837
838This gives an analytical stretching of $\sigma$ that is solvable in $A$ and $B$ as a function of
839the user prescribed stretching parameter $\alpha$ (\np{rn\_alpha}) that stretches towards
840the surface ($\alpha > 1.0$) or the bottom ($\alpha < 1.0$) and user prescribed surface (\np{rn\_zs})
841and bottom depths. The bottom cell depth in this example is given as a function of water depth:
842
843\begin{equation} \label{DOM_zb}
844Z_b= h a + b
845\end{equation}
846
847where the namelist parameters \np{rn\_zb\_a} and \np{rn\_zb\_b} are $a$ and $b$ respectively.
848
849%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
850\begin{figure}[!ht]
851   \includegraphics[width=1.0\textwidth]{FIG_DOM_compare_coordinates_surface}
852        \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.}
853    \label{fig_compare_coordinates_surface}
854\end{figure}
855%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
856
857This 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.
858
859As 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 $h_c$.
860
861Minimising 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.
862
863% -------------------------------------------------------------------------------------------------------------
864%        z*- or s*-coordinate
865% -------------------------------------------------------------------------------------------------------------
866\subsection{$z^*$- or $s^*$-coordinate (\np{ln\_linssh}=false) }
867\label{DOM_zgr_star}
868
869This option is described in the Report by Levier \textit{et al.} (2007), available on the \NEMO web site.
870
871%gm% key advantage: minimise the diffusion/dispertion associated with advection in response to high frequency surface disturbances
872
873% -------------------------------------------------------------------------------------------------------------
874%        level bathymetry and mask
875% -------------------------------------------------------------------------------------------------------------
876\subsection{level bathymetry and mask}
877\label{DOM_msk}
878
879Whatever the vertical coordinate used, the model offers the possibility of
880representing the bottom topography with steps that follow the face of the
881model cells (step like topography) \citep{Madec_al_JPO96}. The distribution of
882the steps in the horizontal is defined in a 2D integer array, mbathy, which
883gives the number of ocean levels ($i.e.$ those that are not masked) at each
884$t$-point. mbathy is computed from the meter bathymetry using the definiton of
885gdept as the number of $t$-points which gdept $\leq$ bathy.
886
887Modifications of the model bathymetry are performed in the \textit{bat\_ctl} 
888routine (see \mdl{domzgr} module) after mbathy is computed. Isolated grid points
889that do not communicate with another ocean point at the same level are eliminated.
890
891As for the representation of bathymetry, a 2D integer array, misfdep, is created.
892misfdep defines the level of the first wet $t$-point. All the cells between $k=1$ and $misfdep(i,j)-1$ are masked.
893By default, misfdep(:,:)=1 and no cells are masked.
894
895In case of ice shelf cavities, modifications of the model bathymetry and ice shelf draft into
896the cavities are performed in the \textit{zgr\_isf} routine. The compatibility between ice shelf draft and bathymetry is checked.
897All the locations where the isf cavity is thinnest than \np{rn\_isfhmin} meters are grounded ($i.e.$ masked).
898If only one cell on the water column is opened at $t$-, $u$- or $v$-points, the bathymetry or the ice shelf draft is dug to fit this constrain.
899If the incompatibility is too strong (need to dig more than 1 cell), the cell is masked.\\ 
900
901From the \textit{mbathy} and \textit{misfdep} array, the mask fields are defined as follows:
902\begin{align*}
903tmask(i,j,k) &= \begin{cases}   \; 0&   \text{ if $k < misfdep(i,j) $ } \\
904                                \; 1&   \text{ if $misfdep(i,j) \leq k\leq mbathy(i,j)$  }    \\
905                                \; 0&   \text{ if $k > mbathy(i,j)$  }    \end{cases}     \\
906umask(i,j,k) &=         \; tmask(i,j,k) \ * \ tmask(i+1,j,k)   \\
907vmask(i,j,k) &=         \; tmask(i,j,k) \ * \ tmask(i,j+1,k)   \\
908fmask(i,j,k) &=         \; tmask(i,j,k) \ * \ tmask(i+1,j,k)   \\
909             &    \ \ \, * tmask(i,j,k) \ * \ tmask(i+1,j,k) \\
910wmask(i,j,k) &=         \; tmask(i,j,k) \ * \ tmask(i,j,k-1) \text{ with } wmask(i,j,1) = tmask(i,j,1)
911\end{align*}
912
913Note that, without ice shelves cavities, masks at $t-$ and $w-$points are identical with
914the numerical indexing used (\S~\ref{DOM_Num_Index}). Nevertheless, $wmask$ are required
915with oceean cavities to deal with the top boundary (ice shelf/ocean interface)
916exactly in the same way as for the bottom boundary.
917
918The specification of closed lateral boundaries requires that at least the first and last
919rows and columns of the \textit{mbathy} array are set to zero. In the particular
920case of an east-west cyclical boundary condition, \textit{mbathy} has its last
921column equal to the second one and its first column equal to the last but one
922(and so too the mask arrays) (see \S~\ref{LBC_jperio}).
923
924
925% ================================================================
926% Domain: Initial State (dtatsd & istate)
927% ================================================================
928\section  [Domain: Initial State (\textit{istate and dtatsd})]
929      {Domain: Initial State \small{(\mdl{istate} and \mdl{dtatsd} modules)} }
930\label{DTA_tsd}
931%-----------------------------------------namtsd-------------------------------------------
932\namdisplay{namtsd} 
933%------------------------------------------------------------------------------------------
934
935Options are defined in \ngn{namtsd}.
936By default, the ocean start from rest (the velocity field is set to zero) and the initialization of
937temperature and salinity fields is controlled through the \np{ln\_tsd\_ini} namelist parameter.
938\begin{description}
939\item[ln\_tsd\_init = .true.]  use a T and S input files that can be given on the model grid itself or
940on their native input data grid. In the latter case, the data will be interpolated on-the-fly both in the
941horizontal and the vertical to the model grid (see \S~\ref{SBC_iof}). The information relative to the
942input files are given in the \np{sn\_tem} and \np{sn\_sal} structures.
943The computation is done in the \mdl{dtatsd} module.
944\item[ln\_tsd\_init = .false.] use constant salinity value of 35.5 psu and an analytical profile of temperature
945(typical of the tropical ocean), see \rou{istate\_t\_s} subroutine called from \mdl{istate} module.
946\end{description}
947\end{document}
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