source: branches/2017/dev_merge_2017/DOC/tex_sub/chap_DOM.tex @ 9393

Last change on this file since 9393 was 9393, checked in by nicolasmartin, 3 years ago

Cleaning of section headings, reinstating the index by mixing \np and \forcode macros, continued conversion of source code inclusions

File size: 52.5 KB
Line 
1\documentclass[../tex_main/NEMO_manual]{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$\ $\newline    % force a new line
21
22Having defined the continuous equations in Chap.~\ref{PE} and chosen a time
23discretization Chap.~\ref{STP}, we need to choose a discretization on a grid,
24and numerical algorithms. In the present chapter, we provide a general description
25of the staggered grid used in \NEMO, and other information relevant to the main
26directory routines as well as the DOM (DOMain) directory.
27
28$\ $\newline    % force a new line
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{ \protect\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{ \protect\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{   \protect\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{ \protect\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{jpkglo} in the $i$, $j$ and $k$ directions respectively.
271%%%
272%%%
273%%%
274Parameters $jpi$ and $jpj$ refer to the size of each processor subdomain when the code is
275run in parallel using domain decomposition (\key{mpp\_mpi} defined, see \S\ref{LBC_mpp}).
276
277
278$\ $\newline    % force a new line
279
280% ================================================================
281% Domain: List of fields needed
282% ================================================================
283\section{Needed fields}
284\label{DOM_fields}
285The ocean mesh ($i.e.$ the position of all the scalar and vector points) is defined
286by the transformation that gives $(\lambda,\varphi,z)$ as a function of $(i,j,k)$.
287The grid-points are located at integer or integer and a half values of as indicated
288in Table~\ref{Tab_cell}. The associated scale factors are defined using the 
289analytical first derivative of the transformation \eqref{Eq_scale_factors}.
290Necessary fields for configuration definition are: \\
291Geographic position :
292
293longitude : glamt , glamu , glamv and glamf  (at T, U, V and F point)
294
295latitude : gphit , gphiu , gphiv and gphif (at T, U, V and F point)\\
296Coriolis parameter (if domain not on the sphere):
297
298 ff\_f  and  ff\_t (at T and F point)\\
299Scale factors :
300 
301 e1t, e1u, e1v and e1f (on i direction),
302
303 e2t, e2u, e2v and e2f (on j direction)
304
305 and ie1e2u\_v, e1e2u , e1e2v   
306 
307e1e2u , e1e2v are u and v surfaces (if gridsize reduction in some straits)\\
308ie1e2u\_v is a flag to flag set u and  v surfaces are neither read nor computed.\\
309 
310These fields can be read in an domain input file which name is setted in \np{cn\_domcfg} parameter specified in \ngn{namcfg}.
311\forfile{../namelists/namcfg}
312or they can be defined in an analytical way in MY\_SRC directory of the configuration.
313For Reference Configurations of NEMO input domain files are supplied by NEMO System Team. For analytical definition of input fields two routines are supplied: \mdl{userdef\_hgr} and \mdl{userdef\_zgr}. They are an example of GYRE configuration parameters, and they are available in NEMO/OPA\_SRC/USR directory, they provide the horizontal and vertical mesh.
314% -------------------------------------------------------------------------------------------------------------
315%        Needed fields
316% -------------------------------------------------------------------------------------------------------------
317%\subsection{List of needed fields to build DOMAIN}
318%\label{DOM_fields_list}
319
320
321% ================================================================
322% Domain: Horizontal Grid (mesh)
323% ================================================================
324\section{Horizontal grid mesh (\protect\mdl{domhgr})}
325\label{DOM_hgr}
326
327% -------------------------------------------------------------------------------------------------------------
328%        Coordinates and scale factors
329% -------------------------------------------------------------------------------------------------------------
330\subsection{Coordinates and scale factors}
331\label{DOM_hgr_coord_e}
332
333The ocean mesh ($i.e.$ the position of all the scalar and vector points) is defined
334by the transformation that gives $(\lambda,\varphi,z)$ as a function of $(i,j,k)$.
335The grid-points are located at integer or integer and a half values of as indicated
336in Table~\ref{Tab_cell}. The associated scale factors are defined using the
337analytical first derivative of the transformation \eqref{Eq_scale_factors}. These
338definitions are done in two modules, \mdl{domhgr} and \mdl{domzgr}, which
339provide the horizontal and vertical meshes, respectively. This section deals with
340the horizontal mesh parameters.
341
342In a horizontal plane, the location of all the model grid points is defined from the
343analytical expressions of the longitude $\lambda$ and  latitude $\varphi$ as a
344function of  $(i,j)$. The horizontal scale factors are calculated using
345\eqref{Eq_scale_factors}. For example, when the longitude and latitude are
346function of a single value ($i$ and $j$, respectively) (geographical configuration
347of the mesh), the horizontal mesh definition reduces to define the wanted
348$\lambda(i)$, $\varphi(j)$, and their derivatives $\lambda'(i)$ $\varphi'(j)$ in the
349\mdl{domhgr} module. The model computes the grid-point positions and scale
350factors in the horizontal plane as follows:
351\begin{flalign*}
352\lambda_t &\equiv \text{glamt}= \lambda(i)     & \varphi_t &\equiv \text{gphit} = \varphi(j)\\
353\lambda_u &\equiv \text{glamu}= \lambda(i+1/2)& \varphi_u &\equiv \text{gphiu}= \varphi(j)\\
354\lambda_v &\equiv \text{glamv}= \lambda(i)       & \varphi_v &\equiv \text{gphiv} = \varphi(j+1/2)\\
355\lambda_f &\equiv \text{glamf }= \lambda(i+1/2)& \varphi_f &\equiv \text{gphif }= \varphi(j+1/2)
356\end{flalign*}
357\begin{flalign*}
358e_{1t} &\equiv \text{e1t} = r_a |\lambda'(i)    \; \cos\varphi(j)  |&
359e_{2t} &\equiv \text{e2t} = r_a |\varphi'(j)|  \\
360e_{1u} &\equiv \text{e1t} = r_a |\lambda'(i+1/2)   \; \cos\varphi(j)  |&
361e_{2u} &\equiv \text{e2t} = r_a |\varphi'(j)|\\
362e_{1v} &\equiv \text{e1t} = r_a |\lambda'(i)    \; \cos\varphi(j+1/2)  |&
363e_{2v} &\equiv \text{e2t} = r_a |\varphi'(j+1/2)|\\
364e_{1f} &\equiv \text{e1t} = r_a |\lambda'(i+1/2)\; \cos\varphi(j+1/2)  |&
365e_{2f} &\equiv \text{e2t} = r_a |\varphi'(j+1/2)|
366\end{flalign*}
367where the last letter of each computational name indicates the grid point
368considered and $r_a$ is the earth radius (defined in \mdl{phycst} along with
369all universal constants). Note that the horizontal position of and scale factors
370at $w$-points are exactly equal to those of $t$-points, thus no specific arrays
371are defined at $w$-points.
372
373Note that the definition of the scale factors ($i.e.$ as the analytical first derivative
374of the transformation that gives $(\lambda,\varphi,z)$ as a function of $(i,j,k)$) is
375specific to the \NEMO model \citep{Marti_al_JGR92}. As an example, $e_{1t}$ is defined
376locally at a $t$-point, whereas many other models on a C grid choose to define
377such a scale factor as the distance between the $U$-points on each side of the
378$t$-point. Relying on an analytical transformation has two advantages: firstly, there
379is no ambiguity in the scale factors appearing in the discrete equations, since they
380are first introduced in the continuous equations; secondly, analytical transformations
381encourage good practice by the definition of smoothly varying grids (rather than
382allowing the user to set arbitrary jumps in thickness between adjacent layers)
383\citep{Treguier1996}. An example of the effect of such a choice is shown in
384Fig.~\ref{Fig_zgr_e3}.
385%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
386\begin{figure}[!t]     \begin{center}
387\includegraphics[width=0.90\textwidth]{Fig_zgr_e3}
388\caption{ \protect\label{Fig_zgr_e3}   
389Comparison of (a) traditional definitions of grid-point position and grid-size in the vertical,
390and (b) analytically derived grid-point position and scale factors.
391For both grids here,  the same $w$-point depth has been chosen but in (a) the
392$t$-points are set half way between $w$-points while in (b) they are defined from
393an analytical function: $z(k)=5\,(k-1/2)^3 - 45\,(k-1/2)^2 + 140\,(k-1/2) - 150$.
394Note the resulting difference between the value of the grid-size $\Delta_k$ and
395those of the scale factor $e_k$. }
396\end{center}   \end{figure}
397%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
398
399% -------------------------------------------------------------------------------------------------------------
400%        Choice of horizontal grid
401% -------------------------------------------------------------------------------------------------------------
402\subsection{Choice of horizontal grid}
403\label{DOM_hgr_msh_choice}
404
405
406% -------------------------------------------------------------------------------------------------------------
407%        Grid files
408% -------------------------------------------------------------------------------------------------------------
409\subsection{Output grid files}
410\label{DOM_hgr_files}
411
412All the arrays relating to a particular ocean model configuration (grid-point
413position, scale factors, masks) can be saved in files if \np{nn\_msh} $\not= 0$ 
414(namelist variable in \ngn{namdom}). This can be particularly useful for plots and off-line
415diagnostics. In some cases, the user may choose to make a local modification
416of a scale factor in the code. This is the case in global configurations when
417restricting the width of a specific strait (usually a one-grid-point strait that
418happens to be too wide due to insufficient model resolution). An example
419is Gibraltar Strait in the ORCA2 configuration. When such modifications are done,
420the output grid written when \np{nn\_msh} $\not= 0$ is no more equal to the input grid.
421
422$\ $\newline    % force a new line
423
424% ================================================================
425% Domain: Vertical Grid (domzgr)
426% ================================================================
427\section{Vertical grid (\protect\mdl{domzgr})}
428\label{DOM_zgr}
429%-----------------------------------------nam_zgr & namdom-------------------------------------------
430%\forfile{../namelists/namzgr}
431\forfile{../namelists/namdom} 
432%-------------------------------------------------------------------------------------------------------------
433
434Variables are defined through the \ngn{namzgr} and \ngn{namdom} namelists.
435In the vertical, the model mesh is determined by four things:
436(1) the bathymetry given in meters ;
437(2) the number of levels of the model (\jp{jpk}) ;
438(3) the analytical transformation $z(i,j,k)$ and the vertical scale factors
439(derivatives of the transformation) ;
440and (4) the masking system, $i.e.$ the number of wet model levels at each
441$(i,j)$ column of points.
442
443%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
444\begin{figure}[!tb]    \begin{center}
445\includegraphics[width=1.0\textwidth]{Fig_z_zps_s_sps}
446\caption{  \protect\label{Fig_z_zps_s_sps}   
447The ocean bottom as seen by the model:
448(a) $z$-coordinate with full step,
449(b) $z$-coordinate with partial step,
450(c) $s$-coordinate: terrain following representation,
451(d) hybrid $s-z$ coordinate,
452(e) hybrid $s-z$ coordinate with partial step, and
453(f) same as (e) but in the non-linear free surface (\np{ln\_linssh}\forcode{ = .false.}).
454Note that the non-linear free surface can be used with any of the
4555 coordinates (a) to (e).}
456\end{center}   \end{figure}
457%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
458
459The choice of a vertical coordinate, even if it is made through \ngn{namzgr} namelist parameters,
460must be done once of all at the beginning of an experiment. It is not intended as an
461option which can be enabled or disabled in the middle of an experiment. Three main
462choices are offered (Fig.~\ref{Fig_z_zps_s_sps}a to c): $z$-coordinate with full step
463bathymetry (\np{ln\_zco}\forcode{ = .true.}), $z$-coordinate with partial step bathymetry
464(\np{ln\_zps}\forcode{ = .true.}), or generalized, $s$-coordinate (\np{ln\_sco}\forcode{ = .true.}).
465Hybridation of the three main coordinates are available: $s-z$ or $s-zps$ coordinate
466(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:
467the coordinate follow the time-variation of the free surface so that the transformation is time dependent:
468$z(i,j,k,t)$ (Fig.~\ref{Fig_z_zps_s_sps}f). When a linear free surface is assumed (\np{ln\_linssh}\forcode{ = .true.}),
469the vertical coordinate are fixed in time, but the seawater can move up and down across the z=0 surface
470(in other words, the top of the ocean in not a rigid-lid).
471The last choice in terms of vertical coordinate concerns the presence (or not) in the model domain
472of ocean cavities beneath ice shelves. Setting \np{ln\_isfcav} to true allows to manage ocean cavities,
473otherwise they are filled in. This option is currently only available in $z$- or $zps$-coordinate,
474and partial step are also applied at the ocean/ice shelf interface.
475
476Contrary to the horizontal grid, the vertical grid is computed in the code and no
477provision is made for reading it from a file. The only input file is the bathymetry
478(in meters) (\ifile{bathy\_meter}).
479\footnote{N.B. in full step $z$-coordinate, a \ifile{bathy\_level} file can replace the
480\ifile{bathy\_meter} file, so that the computation of the number of wet ocean point
481in each water column is by-passed}.
482If \np{ln\_isfcav}\forcode{ = .true.}, an extra file input file describing the ice shelf draft
483(in meters) (\ifile{isf\_draft\_meter}) is needed.
484
485After reading the bathymetry, the algorithm for vertical grid definition differs
486between the different options:
487\begin{description}
488\item[\textit{zco}] set a reference coordinate transformation $z_0 (k)$, and set $z(i,j,k,t)=z_0 (k)$.
489\item[\textit{zps}] set a reference coordinate transformation $z_0 (k)$, and
490calculate the thickness of the deepest level at each $(i,j)$ point using the
491bathymetry, to obtain the final three-dimensional depth and scale factor arrays.
492\item[\textit{sco}] smooth the bathymetry to fulfil the hydrostatic consistency
493criteria and set the three-dimensional transformation.
494\item[\textit{s-z} and \textit{s-zps}] smooth the bathymetry to fulfil the hydrostatic
495consistency criteria and set the three-dimensional transformation $z(i,j,k)$, and
496possibly introduce masking of extra land points to better fit the original bathymetry file
497\end{description}
498%%%
499\gmcomment{   add the description of the smoothing:  envelop topography...}
500%%%
501
502Unless a linear free surface is used (\np{ln\_linssh}\forcode{ = .false.}), the arrays describing
503the grid point depths and vertical scale factors are three set of three dimensional arrays $(i,j,k)$ 
504defined at \textit{before}, \textit{now} and \textit{after} time step. The time at which they are
505defined is indicated by a suffix:$\_b$, $\_n$, or $\_a$, respectively. They are updated at each model time step
506using a fixed reference coordinate system which computer names have a $\_0$ suffix.
507When the linear free surface option is used (\np{ln\_linssh}\forcode{ = .true.}), \textit{before}, \textit{now} 
508and \textit{after} arrays are simply set one for all to their reference counterpart.
509
510
511% -------------------------------------------------------------------------------------------------------------
512%        Meter Bathymetry
513% -------------------------------------------------------------------------------------------------------------
514\subsection{Meter bathymetry}
515\label{DOM_bathy}
516
517Three options are possible for defining the bathymetry, according to the
518namelist variable \np{nn\_bathy} (found in \ngn{namdom} namelist):
519\begin{description}
520\item[\np{nn\_bathy}\forcode{ = 0}]: a flat-bottom domain is defined. The total depth $z_w (jpk)$ 
521is given by the coordinate transformation. The domain can either be a closed
522basin or a periodic channel depending on the parameter \np{jperio}.
523\item[\np{nn\_bathy}\forcode{ = -1}]: a domain with a bump of topography one third of the
524domain width at the central latitude. This is meant for the "EEL-R5" configuration,
525a periodic or open boundary channel with a seamount.
526\item[\np{nn\_bathy}\forcode{ = 1}]: read a bathymetry and ice shelf draft (if needed).
527 The \ifile{bathy\_meter} file (Netcdf format) provides the ocean depth (positive, in meters)
528 at each grid point of the model grid. The bathymetry is usually built by interpolating a standard bathymetry product
529($e.g.$ ETOPO2) onto the horizontal ocean mesh. Defining the bathymetry also
530defines the coastline: where the bathymetry is zero, no model levels are defined
531(all levels are masked).
532
533The \ifile{isfdraft\_meter} file (Netcdf format) provides the ice shelf draft (positive, in meters)
534 at each grid point of the model grid. This file is only needed if \np{ln\_isfcav}\forcode{ = .true.}.
535Defining the ice shelf draft will also define the ice shelf edge and the grounding line position.
536\end{description}
537
538When a global ocean is coupled to an atmospheric model it is better to represent
539all large water bodies (e.g, great lakes, Caspian sea...) even if the model
540resolution does not allow their communication with the rest of the ocean.
541This is unnecessary when the ocean is forced by fixed atmospheric conditions,
542so these seas can be removed from the ocean domain. The user has the option
543to set the bathymetry in closed seas to zero (see \S\ref{MISC_closea}), but the
544code has to be adapted to the user's configuration.
545
546% -------------------------------------------------------------------------------------------------------------
547%        z-coordinate  and reference coordinate transformation
548% -------------------------------------------------------------------------------------------------------------
549\subsection[$Z$-coordinate (\protect\np{ln\_zco}\forcode{ = .true.}) and ref. coordinate]
550            {$Z$-coordinate (\protect\np{ln\_zco}\forcode{ = .true.}) and reference coordinate}
551\label{DOM_zco}
552
553%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
554\begin{figure}[!tb]    \begin{center}
555\includegraphics[width=0.90\textwidth]{Fig_zgr}
556\caption{ \protect\label{Fig_zgr}   
557Default vertical mesh for ORCA2: 30 ocean levels (L30). Vertical level functions for
558(a) T-point depth and (b) the associated scale factor as computed
559from \eqref{DOM_zgr_ana} using \eqref{DOM_zgr_coef} in $z$-coordinate.}
560\end{center}   \end{figure}
561%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
562
563The reference coordinate transformation $z_0 (k)$ defines the arrays $gdept_0$ 
564and $gdepw_0$ for $t$- and $w$-points, respectively. As indicated on
565Fig.\ref{Fig_index_vert} \jp{jpk} is the number of $w$-levels. $gdepw_0(1)$ is the
566ocean surface. There are at most \jp{jpk}-1 $t$-points inside the ocean, the
567additional $t$-point at $jk=jpk$ is below the sea floor and is not used.
568The vertical location of $w$- and $t$-levels is defined from the analytic expression
569of the depth $z_0(k)$ whose analytical derivative with respect to $k$ provides the
570vertical scale factors. The user must provide the analytical expression of both
571$z_0$ and its first derivative with respect to $k$. This is done in routine \mdl{domzgr} 
572through statement functions, using parameters provided in the \ngn{namcfg} namelist.
573
574It is possible to define a simple regular vertical grid by giving zero stretching (\np{ppacr=0}).
575In that case, the parameters \jp{jpk} (number of $w$-levels) and \np{pphmax} 
576(total ocean depth in meters) fully define the grid.
577
578For climate-related studies it is often desirable to concentrate the vertical resolution
579near the ocean surface. The following function is proposed as a standard for a
580$z$-coordinate (with either full or partial steps):
581\begin{equation} \label{DOM_zgr_ana}
582\begin{split}
583 z_0 (k)    &= h_{sur} -h_0 \;k-\;h_1 \;\log \left[ {\,\cosh \left( {{(k-h_{th} )} / {h_{cr} }} \right)\,} \right] \\ 
584 e_3^0 (k)  &= \left| -h_0 -h_1 \;\tanh \left( {{(k-h_{th} )} / {h_{cr} }} \right) \right|
585\end{split}
586\end{equation}
587where $k=1$ to \jp{jpk} for $w$-levels and $k=1$ to $k=1$ for $T-$levels. Such an
588expression allows us to define a nearly uniform vertical location of levels at the
589ocean top and bottom with a smooth hyperbolic tangent transition in between
590(Fig.~\ref{Fig_zgr}).
591
592If the ice shelf cavities are opened (\np{ln\_isfcav}\forcode{ = .true.}), the definition of $z_0$ is the same.
593However, definition of $e_3^0$ at $t$- and $w$-points is respectively changed to:
594\begin{equation} \label{DOM_zgr_ana}
595\begin{split}
596 e_3^T(k) &= z_W (k+1) - z_W (k)   \\
597 e_3^W(k) &= z_T (k)   - z_T (k-1) \\
598\end{split}
599\end{equation}
600This formulation decrease the self-generated circulation into the ice shelf cavity
601(which can, in extreme case, leads to blow up).\\
602
603 
604The most used vertical grid for ORCA2 has $10~m$ ($500~m)$ resolution in the
605surface (bottom) layers and a depth which varies from 0 at the sea surface to a
606minimum of $-5000~m$. This leads to the following conditions:
607\begin{equation} \label{DOM_zgr_coef}
608\begin{split}
609 e_3 (1+1/2)      &=10. \\ 
610 e_3 (jpk-1/2) &=500. \\ 
611 z(1)       &=0. \\ 
612 z(jpk)        &=-5000. \\ 
613\end{split}
614\end{equation}
615
616With the choice of the stretching $h_{cr} =3$ and the number of levels
617\jp{jpk}=$31$, the four coefficients $h_{sur}$, $h_{0}$, $h_{1}$, and $h_{th}$ in
618\eqref{DOM_zgr_ana} have been determined such that \eqref{DOM_zgr_coef} is
619satisfied, through an optimisation procedure using a bisection method. For the first
620standard ORCA2 vertical grid this led to the following values: $h_{sur} =4762.96$,
621$h_0 =255.58, h_1 =245.5813$, and $h_{th} =21.43336$. The resulting depths and
622scale factors as a function of the model levels are shown in Fig.~\ref{Fig_zgr} and
623given in Table \ref{Tab_orca_zgr}. Those values correspond to the parameters
624\np{ppsur}, \np{ppa0}, \np{ppa1}, \np{ppkth} in \ngn{namcfg} namelist.
625
626Rather than entering parameters $h_{sur}$, $h_{0}$, and $h_{1}$ directly, it is
627possible to recalculate them. In that case the user sets
628\np{ppsur}\forcode{ = }\np{ppa0}\forcode{ = }\np{ppa1}\forcode{ = 999999}., in \ngn{namcfg} namelist,
629and specifies instead the four following parameters:
630\begin{itemize}
631\item    \np{ppacr}=$h_{cr} $: stretching factor (nondimensional). The larger
632\np{ppacr}, the smaller the stretching. Values from $3$ to $10$ are usual.
633\item    \np{ppkth}=$h_{th} $: is approximately the model level at which maximum
634stretching occurs (nondimensional, usually of order 1/2 or 2/3 of \jp{jpk})
635\item    \np{ppdzmin}: minimum thickness for the top layer (in meters)
636\item    \np{pphmax}: total depth of the ocean (meters).
637\end{itemize}
638As an example, for the $45$ layers used in the DRAKKAR configuration those
639parameters are: \jp{jpk}\forcode{ = 46}, \np{ppacr}\forcode{ = 9}, \np{ppkth}\forcode{ = 23.563}, \np{ppdzmin}\forcode{ = 6}m, \np{pphmax}\forcode{ = 5750}m.
640
641%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
642\begin{table}     \begin{center} \begin{tabular}{c||r|r|r|r}
643\hline
644\textbf{LEVEL}& \textbf{gdept\_1d}& \textbf{gdepw\_1d}& \textbf{e3t\_1d }& \textbf{e3w\_1d  } \\ \hline
645&  \textbf{  5.00}   &       0.00 & \textbf{ 10.00} &  10.00 \\   \hline
646&  \textbf{15.00} &    10.00 &   \textbf{ 10.00} &  10.00 \\   \hline
647&  \textbf{25.00} &    20.00 &   \textbf{ 10.00} &     10.00 \\   \hline
648&  \textbf{35.01} &    30.00 &   \textbf{ 10.01} &     10.00 \\   \hline
649&  \textbf{45.01} &    40.01 &   \textbf{ 10.01} &  10.01 \\   \hline
650&  \textbf{55.03} &    50.02 &   \textbf{ 10.02} &     10.02 \\   \hline
651&  \textbf{65.06} &    60.04 &   \textbf{ 10.04} &  10.03 \\   \hline
652&  \textbf{75.13} &    70.09 &   \textbf{ 10.09} &  10.06 \\   \hline
653&  \textbf{85.25} &    80.18 &   \textbf{ 10.17} &  10.12 \\   \hline
65410 &  \textbf{95.49} &    90.35 &   \textbf{ 10.33} &  10.24 \\   \hline
65511 &  \textbf{105.97}   &   100.69 &   \textbf{ 10.65} &  10.47 \\   \hline
65612 &  \textbf{116.90}   &   111.36 &   \textbf{ 11.27} &  10.91 \\   \hline
65713 &  \textbf{128.70}   &   122.65 &   \textbf{ 12.47} &  11.77 \\   \hline
65814 &  \textbf{142.20}   &   135.16 &   \textbf{ 14.78} &  13.43 \\   \hline
65915 &  \textbf{158.96}   &   150.03 &   \textbf{ 19.23} &  16.65 \\   \hline
66016 &  \textbf{181.96}   &   169.42 &   \textbf{ 27.66} &  22.78 \\   \hline
66117 &  \textbf{216.65}   &   197.37 &   \textbf{ 43.26} &  34.30 \\ \hline
66218 &  \textbf{272.48}   &   241.13 &   \textbf{ 70.88} &  55.21 \\ \hline
66319 &  \textbf{364.30}   &   312.74 &   \textbf{116.11} &  90.99 \\ \hline
66420 &  \textbf{511.53}   &   429.72 &   \textbf{181.55} &    146.43 \\ \hline
66521 &  \textbf{732.20}   &   611.89 &   \textbf{261.03} &    220.35 \\ \hline
66622 &  \textbf{1033.22}&  872.87 &   \textbf{339.39} &    301.42 \\ \hline
66723 &  \textbf{1405.70}& 1211.59 & \textbf{402.26} &   373.31 \\ \hline
66824 &  \textbf{1830.89}& 1612.98 & \textbf{444.87} &   426.00 \\ \hline
66925 &  \textbf{2289.77}& 2057.13 & \textbf{470.55} &   459.47 \\ \hline
67026 &  \textbf{2768.24}& 2527.22 & \textbf{484.95} &   478.83 \\ \hline
67127 &  \textbf{3257.48}& 3011.90 & \textbf{492.70} &   489.44 \\ \hline
67228 &  \textbf{3752.44}& 3504.46 & \textbf{496.78} &   495.07 \\ \hline
67329 &  \textbf{4250.40}& 4001.16 & \textbf{498.90} &   498.02 \\ \hline
67430 &  \textbf{4749.91}& 4500.02 & \textbf{500.00} &   499.54 \\ \hline
67531 &  \textbf{5250.23}& 5000.00 &   \textbf{500.56} & 500.33 \\ \hline
676\end{tabular} \end{center} 
677\caption{ \protect\label{Tab_orca_zgr}   
678Default vertical mesh in $z$-coordinate for 30 layers ORCA2 configuration as computed
679from \eqref{DOM_zgr_ana} using the coefficients given in \eqref{DOM_zgr_coef}}
680\end{table}
681%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
682
683% -------------------------------------------------------------------------------------------------------------
684%        z-coordinate with partial step
685% -------------------------------------------------------------------------------------------------------------
686\subsection{$Z$-coordinate with partial step (\protect\np{ln\_zps}\forcode{ = .true.})}
687\label{DOM_zps}
688%--------------------------------------------namdom-------------------------------------------------------
689\forfile{../namelists/namdom} 
690%--------------------------------------------------------------------------------------------------------------
691
692In $z$-coordinate partial step, the depths of the model levels are defined by the
693reference analytical function $z_0 (k)$ as described in the previous
694section, \emph{except} in the bottom layer. The thickness of the bottom layer is
695allowed to vary as a function of geographical location $(\lambda,\varphi)$ to allow a
696better representation of the bathymetry, especially in the case of small
697slopes (where the bathymetry varies by less than one level thickness from
698one grid point to the next). The reference layer thicknesses $e_{3t}^0$ have been
699defined in the absence of bathymetry. With partial steps, layers from 1 to
700\jp{jpk}-2 can have a thickness smaller than $e_{3t}(jk)$. The model deepest layer (\jp{jpk}-1)
701is allowed to have either a smaller or larger thickness than $e_{3t}(jpk)$: the
702maximum thickness allowed is $2*e_{3t}(jpk-1)$. This has to be kept in mind when
703specifying values in \ngn{namdom} namelist, as the maximum depth \np{pphmax} 
704in partial steps: for example, with
705\np{pphmax}$=5750~m$ for the DRAKKAR 45 layer grid, the maximum ocean depth
706allowed is actually $6000~m$ (the default thickness $e_{3t}(jpk-1)$ being $250~m$).
707Two variables in the namdom namelist are used to define the partial step
708vertical grid. The mimimum water thickness (in meters) allowed for a cell
709partially filled with bathymetry at level jk is the minimum of \np{rn\_e3zps\_min} 
710(thickness in meters, usually $20~m$) or $e_{3t}(jk)*$\np{rn\_e3zps\_rat} (a fraction,
711usually 10\%, of the default thickness $e_{3t}(jk)$).
712
713\gmcomment{ \colorbox{yellow}{Add a figure here of pstep especially at last ocean level }  }
714
715% -------------------------------------------------------------------------------------------------------------
716%        s-coordinate
717% -------------------------------------------------------------------------------------------------------------
718\subsection{$S$-coordinate (\protect\np{ln\_sco}\forcode{ = .true.})}
719\label{DOM_sco}
720%------------------------------------------nam_zgr_sco---------------------------------------------------
721%\forfile{../namelists/namzgr_sco}
722%--------------------------------------------------------------------------------------------------------------
723Options are defined in \ngn{namzgr\_sco}.
724In $s$-coordinate (\np{ln\_sco}\forcode{ = .true.}), the depth and thickness of the model
725levels are defined from the product of a depth field and either a stretching
726function or its derivative, respectively:
727
728\begin{equation} \label{DOM_sco_ana}
729\begin{split}
730 z(k)       &= h(i,j) \; z_0(k)  \\
731 e_3(k)  &= h(i,j) \; z_0'(k)
732\end{split}
733\end{equation}
734
735where $h$ is the depth of the last $w$-level ($z_0(k)$) defined at the $t$-point
736location in the horizontal and $z_0(k)$ is a function which varies from $0$ at the sea
737surface to $1$ at the ocean bottom. The depth field $h$ is not necessary the ocean
738depth, since a mixed step-like and bottom-following representation of the
739topography 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).
740The namelist parameter \np{rn\_rmax} determines the slope at which the terrain-following coordinate intersects
741the sea bed and becomes a pseudo z-coordinate.
742The coordinate can also be hybridised by specifying \np{rn\_sbot\_min} and \np{rn\_sbot\_max} 
743as the minimum and maximum depths at which the terrain-following vertical coordinate is calculated.
744
745Options for stretching the coordinate are provided as examples, but care must be taken to ensure
746that the vertical stretch used is appropriate for the application.
747
748The original default NEMO s-coordinate stretching is available if neither of the other options
749are specified as true (\np{ln\_s\_SH94}\forcode{ = .false.} and \np{ln\_s\_SF12}\forcode{ = .false.}).
750This uses a depth independent $\tanh$ function for the stretching \citep{Madec_al_JPO96}:
751
752\begin{equation}
753  z = s_{min}+C\left(s\right)\left(H-s_{min}\right)
754  \label{eq:SH94_1}
755\end{equation}
756
757where $s_{min}$ is the depth at which the $s$-coordinate stretching starts and
758allows a $z$-coordinate to placed on top of the stretched coordinate,
759and $z$ is the depth (negative down from the asea surface).
760
761\begin{equation}
762  s = -\frac{k}{n-1} \quad \text{ and } \quad 0 \leq k \leq n-1
763  \label{eq:s}
764\end{equation}
765
766\begin{equation} \label{DOM_sco_function}
767\begin{split}
768C(s)  &\frac{ \left[   \tanh{ \left( \theta \, (s+b) \right)} 
769               - \tanh{ \left\theta \, b      \right)}  \right]}
770            {2\;\sinh \left( \theta \right)}
771\end{split}
772\end{equation}
773
774A stretching function, modified from the commonly used \citet{Song_Haidvogel_JCP94} 
775stretching (\np{ln\_s\_SH94}\forcode{ = .true.}), is also available and is more commonly used for shelf seas modelling:
776
777\begin{equation}
778  C\left(s\right) =   \left(1 - b \right)\frac{ \sinh\left( \theta s\right)}{\sinh\left(\theta\right)} +      \\
779  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)}
780  \label{eq:SH94_2}
781\end{equation}
782
783%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
784\begin{figure}[!ht]    \begin{center}
785\includegraphics[width=1.0\textwidth]{Fig_sco_function}
786\caption{  \protect\label{Fig_sco_function}   
787Examples of the stretching function applied to a seamount; from left to right:
788surface, surface and bottom, and bottom intensified resolutions}
789\end{center}   \end{figure}
790%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
791
792where $H_c$ is the critical depth (\np{rn\_hc}) at which the coordinate transitions from
793pure $\sigma$ to the stretched coordinate,  and $\theta$ (\np{rn\_theta}) and $b$ (\np{rn\_bb})
794are the surface and bottom control parameters such that $0\leqslant \theta \leqslant 20$, and
795$0\leqslant b\leqslant 1$. $b$ has been designed to allow surface and/or bottom
796increase of the vertical resolution (Fig.~\ref{Fig_sco_function}).
797
798Another example has been provided at version 3.5 (\np{ln\_s\_SF12}) that allows
799a fixed surface resolution in an analytical terrain-following stretching \citet{Siddorn_Furner_OM12}.
800In this case the a stretching function $\gamma$ is defined such that:
801
802\begin{equation}
803z = -\gamma h \quad \text{ with } \quad 0 \leq \gamma \leq 1
804\label{eq:z}
805\end{equation}
806
807The function is defined with respect to $\sigma$, the unstretched terrain-following coordinate:
808
809\begin{equation} \label{DOM_gamma_deriv}
810\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)
811\end{equation}
812
813Where:
814\begin{equation} \label{DOM_gamma}
815f\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} 
816\end{equation}
817
818This gives an analytical stretching of $\sigma$ that is solvable in $A$ and $B$ as a function of
819the user prescribed stretching parameter $\alpha$ (\np{rn\_alpha}) that stretches towards
820the surface ($\alpha > 1.0$) or the bottom ($\alpha < 1.0$) and user prescribed surface (\np{rn\_zs})
821and bottom depths. The bottom cell depth in this example is given as a function of water depth:
822
823\begin{equation} \label{DOM_zb}
824Z_b= h a + b
825\end{equation}
826
827where the namelist parameters \np{rn\_zb\_a} and \np{rn\_zb\_b} are $a$ and $b$ respectively.
828
829%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
830\begin{figure}[!ht]
831   \includegraphics[width=1.0\textwidth]{FIG_DOM_compare_coordinates_surface}
832        \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.}
833    \label{fig_compare_coordinates_surface}
834\end{figure}
835%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
836
837This 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.
838
839As 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 $h_c$.
840
841Minimising 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.
842
843% -------------------------------------------------------------------------------------------------------------
844%        z*- or s*-coordinate
845% -------------------------------------------------------------------------------------------------------------
846\subsection{$Z^*$- or $S^*$-coordinate (\protect\np{ln\_linssh}\forcode{ = .false.}) }
847\label{DOM_zgr_star}
848
849This option is described in the Report by Levier \textit{et al.} (2007), available on the \NEMO web site.
850
851%gm% key advantage: minimise the diffusion/dispertion associated with advection in response to high frequency surface disturbances
852
853% -------------------------------------------------------------------------------------------------------------
854%        level bathymetry and mask
855% -------------------------------------------------------------------------------------------------------------
856\subsection{Level bathymetry and mask}
857\label{DOM_msk}
858
859Whatever the vertical coordinate used, the model offers the possibility of
860representing the bottom topography with steps that follow the face of the
861model cells (step like topography) \citep{Madec_al_JPO96}. The distribution of
862the steps in the horizontal is defined in a 2D integer array, mbathy, which
863gives the number of ocean levels ($i.e.$ those that are not masked) at each
864$t$-point. mbathy is computed from the meter bathymetry using the definiton of
865gdept as the number of $t$-points which gdept $\leq$ bathy.
866
867Modifications of the model bathymetry are performed in the \textit{bat\_ctl} 
868routine (see \mdl{domzgr} module) after mbathy is computed. Isolated grid points
869that do not communicate with another ocean point at the same level are eliminated.
870
871As for the representation of bathymetry, a 2D integer array, misfdep, is created.
872misfdep defines the level of the first wet $t$-point. All the cells between $k=1$ and $misfdep(i,j)-1$ are masked.
873By default, misfdep(:,:)=1 and no cells are masked.
874
875In case of ice shelf cavities, modifications of the model bathymetry and ice shelf draft into
876the cavities are performed in the \textit{zgr\_isf} routine. The compatibility between ice shelf draft and bathymetry is checked.
877All the locations where the isf cavity is thinnest than \np{rn\_isfhmin} meters are grounded ($i.e.$ masked).
878If 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.
879If the incompatibility is too strong (need to dig more than 1 cell), the cell is masked.\\ 
880
881From the \textit{mbathy} and \textit{misfdep} array, the mask fields are defined as follows:
882\begin{align*}
883tmask(i,j,k) &= \begin{cases}   \; 0&   \text{ if $k < misfdep(i,j) $ } \\
884                                \; 1&   \text{ if $misfdep(i,j) \leq k\leq mbathy(i,j)$  }    \\
885                                \; 0&   \text{ if $k > mbathy(i,j)$  }    \end{cases}     \\
886umask(i,j,k) &=         \; tmask(i,j,k) \ * \ tmask(i+1,j,k)   \\
887vmask(i,j,k) &=         \; tmask(i,j,k) \ * \ tmask(i,j+1,k)   \\
888fmask(i,j,k) &=         \; tmask(i,j,k) \ * \ tmask(i+1,j,k)   \\
889             &    \ \ \, * tmask(i,j,k) \ * \ tmask(i+1,j,k) \\
890wmask(i,j,k) &=         \; tmask(i,j,k) \ * \ tmask(i,j,k-1) \text{ with } wmask(i,j,1) = tmask(i,j,1)
891\end{align*}
892
893Note that, without ice shelves cavities, masks at $t-$ and $w-$points are identical with
894the numerical indexing used (\S~\ref{DOM_Num_Index}). Nevertheless, $wmask$ are required
895with oceean cavities to deal with the top boundary (ice shelf/ocean interface)
896exactly in the same way as for the bottom boundary.
897
898The specification of closed lateral boundaries requires that at least the first and last
899rows and columns of the \textit{mbathy} array are set to zero. In the particular
900case of an east-west cyclical boundary condition, \textit{mbathy} has its last
901column equal to the second one and its first column equal to the last but one
902(and so too the mask arrays) (see \S~\ref{LBC_jperio}).
903
904
905% ================================================================
906% Domain: Initial State (dtatsd & istate)
907% ================================================================
908\section{Initial state (\protect\mdl{istate} and \protect\mdl{dtatsd})}
909\label{DTA_tsd}
910%-----------------------------------------namtsd-------------------------------------------
911\forfile{../namelists/namtsd} 
912%------------------------------------------------------------------------------------------
913
914Options are defined in \ngn{namtsd}.
915By default, the ocean start from rest (the velocity field is set to zero) and the initialization of
916temperature and salinity fields is controlled through the \np{ln\_tsd\_ini} namelist parameter.
917\begin{description}
918\item[\np{ln\_tsd\_init}\forcode{ = .true.}] use a T and S input files that can be given on the model grid itself or
919on their native input data grid. In the latter case, the data will be interpolated on-the-fly both in the
920horizontal and the vertical to the model grid (see \S~\ref{SBC_iof}). The information relative to the
921input files are given in the \np{sn\_tem} and \np{sn\_sal} structures.
922The computation is done in the \mdl{dtatsd} module.
923\item[\np{ln\_tsd\_init}\forcode{ = .false.}] use constant salinity value of 35.5 psu and an analytical profile of temperature
924(typical of the tropical ocean), see \rou{istate\_t\_s} subroutine called from \mdl{istate} module.
925\end{description}
926\end{document}
Note: See TracBrowser for help on using the repository browser.