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1\documentclass[../main/NEMO_manual]{subfiles}
2
3\begin{document}
4% ================================================================
5% Chapter 2 ——— Space and Time Domain (DOM)
6% ================================================================
7\chapter{Space Domain (DOM)}
8\label{chap:DOM}
9
10\minitoc
11
12% Missing things:
13%  - istate: description of the initial state   ==> this has to be put elsewhere..
14%                  perhaps in MISC ?  By the way the initialisation of T S and dynamics
15%                  should be put outside of DOM routine (better with TRC staff and off-line
16%                  tracers)
17%  -geo2ocean:  how to switch from geographic to mesh coordinate
18%     - domclo:  closed sea and lakes.... management of closea sea area : specific to global configuration, both forced and coupled
19
20\newpage
21
22Having defined the continuous equations in \autoref{chap:PE} and chosen a time discretization \autoref{chap:STP},
23we need to choose a discretization on a grid, and numerical algorithms.
24In the present chapter, we provide a general description of the staggered grid used in \NEMO,
25and other information relevant to the main directory routines as well as the DOM (DOMain) directory.
26
27% ================================================================
28% Fundamentals of the Discretisation
29% ================================================================
30\section{Fundamentals of the discretisation}
31\label{sec:DOM_basics}
32
33% -------------------------------------------------------------------------------------------------------------
34%        Arrangement of Variables
35% -------------------------------------------------------------------------------------------------------------
36\subsection{Arrangement of variables}
37\label{subsec:DOM_cell}
38
39%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
40\begin{figure}[!tb]
41  \begin{center}
42    \includegraphics[]{Fig_cell}
43    \caption{
44      \protect\label{fig:cell}
45      Arrangement of variables.
46      $t$ indicates scalar points where temperature, salinity, density, pressure and
47      horizontal divergence are defined.
48      $(u,v,w)$ indicates vector points, and $f$ indicates vorticity points where both relative and
49      planetary vorticities are defined.
50    }
51  \end{center}
52\end{figure}
53%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
54
55The numerical techniques used to solve the Primitive Equations in this model are based on the traditional,
56centred second-order finite difference approximation.
57Special attention has been given to the homogeneity of the solution in the three space directions.
58The arrangement of variables is the same in all directions.
59It consists of cells centred on scalar points ($t$, $S$, $p$, $\rho$) with vector points $(u, v, w)$ defined in
60the centre of each face of the cells (\autoref{fig:cell}).
61This is the generalisation to three dimensions of the well-known C'' grid in Arakawa's classification
62\citep{mesinger.arakawa_bk76}.
63The relative and planetary vorticity, $\zeta$ and $f$, are defined in the centre of each vertical edge and
64the barotropic stream function $\psi$ is defined at horizontal points overlying the $\zeta$ and $f$-points.
65
66The ocean mesh (\ie the position of all the scalar and vector points) is defined by the transformation that
67gives $(\lambda,\varphi,z)$ as a function of $(i,j,k)$.
68The grid-points are located at integer or integer and a half value of $(i,j,k)$ as indicated on \autoref{tab:cell}.
69In all the following, subscripts $u$, $v$, $w$, $f$, $uw$, $vw$ or $fw$ indicate the position of
70the grid-point where the scale factors are defined.
71Each scale factor is defined as the local analytical value provided by \autoref{eq:scale_factors}.
72As a result, the mesh on which partial derivatives $\pd[]{\lambda}$, $\pd[]{\varphi}$ and
73$\pd[]{z}$ are evaluated in a uniform mesh with a grid size of unity.
74Discrete partial derivatives are formulated by the traditional, centred second order finite difference approximation
75while the scale factors are chosen equal to their local analytical value.
76An important point here is that the partial derivative of the scale factors must be evaluated by
77centred finite difference approximation, not from their analytical expression.
78This preserves the symmetry of the discrete set of equations and therefore satisfies many of
79the continuous properties (see \autoref{apdx:C}).
81when needed, an area, volume, or the total ocean depth must be evaluated as the sum of the relevant scale factors
82(see \autoref{eq:DOM_bar} in the next section).
83
84%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
85\begin{table}[!tb]
86  \begin{center}
87    \begin{tabular}{|p{46pt}|p{56pt}|p{56pt}|p{56pt}|}
88      \hline
89      T  & $i$ & $j$ & $k$ \\
90      \hline
91      u  & $i + 1/2$ & $j$ & $k$ \\
92      \hline
93      v  & $i$ & $j + 1/2$ & $k$ \\
94      \hline
95      w  & $i$ & $j$ & $k + 1/2$ \\
96      \hline
97      f  & $i + 1/2$ & $j + 1/2$ & $k$ \\
98      \hline
99      uw & $i + 1/2$ & $j$ & $k + 1/2$ \\
100      \hline
101      vw & $i$ & $j + 1/2$ & $k + 1/2$ \\
102      \hline
103      fw & $i + 1/2$ & $j + 1/2$ & $k + 1/2$ \\
104      \hline
105    \end{tabular}
106    \caption{
107      \protect\label{tab:cell}
108      Location of grid-points as a function of integer or integer and a half value of the column, line or level.
109      This indexing is only used for the writing of the semi -discrete equation.
110      In the code, the indexing uses integer values only and has a reverse direction in the vertical
111      (see \autoref{subsec:DOM_Num_Index})
112    }
113  \end{center}
114\end{table}
115%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
116
117% -------------------------------------------------------------------------------------------------------------
118%        Vector Invariant Formulation
119% -------------------------------------------------------------------------------------------------------------
120\subsection{Discrete operators}
121\label{subsec:DOM_operators}
122
123Given the values of a variable $q$ at adjacent points, the differencing and averaging operators at
124the midpoint between them are:
125\begin{alignat*}{2}
126  % \label{eq:di_mi}
127  \delta_i [q]      &= &       &q (i + 1/2) - q (i - 1/2) \\
128  \overline q^{\, i} &= &\big\{ &q (i + 1/2) + q (i - 1/2) \big\} / 2
129\end{alignat*}
130
131Similar operators are defined with respect to $i + 1/2$, $j$, $j + 1/2$, $k$, and $k + 1/2$.
132Following \autoref{eq:PE_grad} and \autoref{eq:PE_lap}, the gradient of a variable $q$ defined at
133a $t$-point has its three components defined at $u$-, $v$- and $w$-points while
134its Laplacian is defined at $t$-point.
135These operators have the following discrete forms in the curvilinear $s$-coordinates system:
136$137 % \label{eq:DOM_grad} 138 \nabla q \equiv \frac{1}{e_{1u}} \delta_{i + 1/2} [q] \; \, \vect i 139 + \frac{1}{e_{2v}} \delta_{j + 1/2} [q] \; \, \vect j 140 + \frac{1}{e_{3w}} \delta_{k + 1/2} [q] \; \, \vect k 141$
142\begin{multline*}
143  % \label{eq:DOM_lap}
144  \Delta q \equiv   \frac{1}{e_{1t} \, e_{2t} \, e_{3t}}
145                    \; \lt[   \delta_i \lt( \frac{e_{2u} \, e_{3u}}{e_{1u}} \; \delta_{i + 1/2} [q] \rt)
146                            + \delta_j \lt( \frac{e_{1v} \, e_{3v}}{e_{2v}} \; \delta_{j + 1/2} [q] \rt) \; \rt] \\
147                  + \frac{1}{e_{3t}}
148                              \delta_k \lt[ \frac{1              }{e_{3w}} \; \delta_{k + 1/2} [q] \rt]
149\end{multline*}
150
151Following \autoref{eq:PE_curl} and \autoref{eq:PE_div}, a vector $\vect A = (a_1,a_2,a_3)$ defined at
152vector points $(u,v,w)$ has its three curl components defined at $vw$-, $uw$, and $f$-points, and
153its divergence defined at $t$-points:
154\begin{multline}
155% \label{eq:DOM_curl}
156  \nabla \times \vect A \equiv   \frac{1}{e_{2v} \, e_{3vw}}
157                                 \Big[   \delta_{j + 1/2} (e_{3w} \, a_3)
158                                       - \delta_{k + 1/2} (e_{2v} \, a_2) \Big] \vect i \\
159                               + \frac{1}{e_{2u} \, e_{3uw}}
160                                 \Big[   \delta_{k + 1/2} (e_{1u} \, a_1)
161                                       - \delta_{i + 1/2} (e_{3w} \, a_3) \Big] \vect j \\
162                               + \frac{1}{e_{1f} \, e_{2f}}
163                                 \Big[   \delta_{i + 1/2} (e_{2v} \, a_2)
164                                       - \delta_{j + 1/2} (e_{1u} \, a_1) \Big] \vect k
165\end{multline}
166\begin{equation}
167% \label{eq:DOM_div}
168  \nabla \cdot \vect A \equiv   \frac{1}{e_{1t} \, e_{2t} \, e_{3t}}
169                                \Big[ \delta_i (e_{2u} \, e_{3u} \, a_1) + \delta_j (e_{1v} \, e_{3v} \, a_2) \Big]
170                              + \frac{1}{e_{3t}} \delta_k (a_3)
171\end{equation}
172
173The vertical average over the whole water column denoted by an overbar becomes for a quantity $q$ which
174is a masked field (i.e. equal to zero inside solid area):
175\begin{equation}
176  \label{eq:DOM_bar}
177  \bar q = \frac{1}{H} \int_{k^b}^{k^o} q \; e_{3q} \, dk \equiv \frac{1}{H_q} \sum \limits_k q \; e_{3q}
178\end{equation}
179where $H_q$  is the ocean depth, which is the masked sum of the vertical scale factors at $q$ points,
180$k^b$ and $k^o$ are the bottom and surface $k$-indices, and the symbol $k^o$ refers to a summation over
181all grid points of the same type in the direction indicated by the subscript (here $k$).
182
183In continuous form, the following properties are satisfied:
184\begin{gather}
186  \nabla \times \nabla q = \vect 0 \\
187  \label{eq:DOM_div_curl}
188  \nabla \cdot (\nabla \times \vect A) = 0
189\end{gather}
190
191It is straightforward to demonstrate that these properties are verified locally in discrete form as soon as
192the scalar $q$ is taken at $t$-points and the vector $\vect A$ has its components defined at
193vector points $(u,v,w)$.
194
195Let $a$ and $b$ be two fields defined on the mesh, with value zero inside continental area.
196Using integration by parts it can be shown that the differencing operators ($\delta_i$, $\delta_j$ and $\delta_k$)
197are skew-symmetric linear operators, and further that the averaging operators $\overline{\cdots}^{\, i}$,
198$\overline{\cdots}^{\, j}$ and $\overline{\cdots}^{\, k}$) are symmetric linear operators, \ie
199\begin{alignat}{4}
201  &\sum \limits_i a_i \; \delta_i [b]      &\equiv &- &&\sum \limits_i \delta      _{   i + 1/2} [a] &b_{i + 1/2} \\
203  &\sum \limits_i a_i \; \overline b^{\, i} &\equiv &  &&\sum \limits_i \overline a ^{\, i + 1/2}     &b_{i + 1/2}
204\end{alignat}
205
206In other words, the adjoint of the differencing and averaging operators are $\delta_i^* = \delta_{i + 1/2}$ and
207$(\overline{\cdots}^{\, i})^* = \overline{\cdots}^{\, i + 1/2}$, respectively.
208These two properties will be used extensively in the \autoref{apdx:C} to
209demonstrate integral conservative properties of the discrete formulation chosen.
210
211% -------------------------------------------------------------------------------------------------------------
212%        Numerical Indexing
213% -------------------------------------------------------------------------------------------------------------
214\subsection{Numerical indexing}
215\label{subsec:DOM_Num_Index}
216
217%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
218\begin{figure}[!tb]
219  \begin{center}
220    \includegraphics[]{Fig_index_hor}
221    \caption{
222      \protect\label{fig:index_hor}
223      Horizontal integer indexing used in the \fortran code.
224      The dashed area indicates the cell in which variables contained in arrays have the same $i$- and $j$-indices
225    }
226  \end{center}
227\end{figure}
228%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
229
230The array representation used in the \fortran code requires an integer indexing while
231the analytical definition of the mesh (see \autoref{subsec:DOM_cell}) is associated with the use of
232integer values for $t$-points and both integer and integer and a half values for all the other points.
233Therefore a specific integer indexing must be defined for points other than $t$-points
234(\ie velocity and vorticity grid-points).
235Furthermore, the direction of the vertical indexing has been changed so that the surface level is at $k = 1$.
236
237% -----------------------------------
238%        Horizontal Indexing
239% -----------------------------------
240\subsubsection{Horizontal indexing}
241\label{subsec:DOM_Num_Index_hor}
242
243The indexing in the horizontal plane has been chosen as shown in \autoref{fig:index_hor}.
244For an increasing $i$ index ($j$ index),
245the $t$-point and the eastward $u$-point (northward $v$-point) have the same index
246(see the dashed area in \autoref{fig:index_hor}).
247A $t$-point and its nearest northeast $f$-point have the same $i$-and $j$-indices.
248
249% -----------------------------------
250%        Vertical indexing
251% -----------------------------------
252\subsubsection{Vertical indexing}
253\label{subsec:DOM_Num_Index_vertical}
254
255In the vertical, the chosen indexing requires special attention since the $k$-axis is re-orientated downward in
256the \fortran code compared to the indexing used in the semi -discrete equations and
257given in \autoref{subsec:DOM_cell}.
258The sea surface corresponds to the $w$-level $k = 1$ which is the same index as $t$-level just below
259(\autoref{fig:index_vert}).
260The last $w$-level ($k = jpk$) either corresponds to the ocean floor or is inside the bathymetry while
261the last $t$-level is always inside the bathymetry (\autoref{fig:index_vert}).
262Note that for an increasing $k$ index, a $w$-point and the $t$-point just below have the same $k$ index,
263in opposition to what is done in the horizontal plane where
264it is the $t$-point and the nearest velocity points in the direction of the horizontal axis that
265have the same $i$ or $j$ index
266(compare the dashed area in \autoref{fig:index_hor} and \autoref{fig:index_vert}).
267Since the scale factors are chosen to be strictly positive,
268a \textit{minus sign} appears in the \fortran code \textit{before all the vertical derivatives} of
269the discrete equations given in this documentation.
270
271%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
272\begin{figure}[!pt]
273  \begin{center}
274    \includegraphics[]{Fig_index_vert}
275    \caption{
276      \protect\label{fig:index_vert}
277      Vertical integer indexing used in the \fortran code.
278      Note that the $k$-axis is orientated downward.
279      The dashed area indicates the cell in which variables contained in arrays have the same $k$-index.
280    }
281  \end{center}
282\end{figure}
283%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
284
285% -----------------------------------
286%        Domain Size
287% -----------------------------------
288\subsubsection{Domain size}
289\label{subsec:DOM_size}
290
291The total size of the computational domain is set by the parameters \np{jpiglo},
292\np{jpjglo} and \np{jpkglo} in the $i$, $j$ and $k$ directions respectively.
293Parameters $jpi$ and $jpj$ refer to the size of each processor subdomain when
294the code is run in parallel using domain decomposition (\key{mpp\_mpi} defined,
295see \autoref{sec:LBC_mpp}).
296
297% ================================================================
298% Domain: List of fields needed
299% ================================================================
300\section{Needed fields}
301\label{sec:DOM_fields}
302The ocean mesh (\ie the position of all the scalar and vector points) is defined by the transformation that
303gives $(\lambda,\varphi,z)$ as a function of $(i,j,k)$.
304The grid-points are located at integer or integer and a half values of as indicated in \autoref{tab:cell}.
305The associated scale factors are defined using the analytical first derivative of the transformation
306\autoref{eq:scale_factors}.
307Necessary fields for configuration definition are:
308
309\begin{itemize}
310\item
311  Geographic position:
312  longitude with \texttt{glamt}, \texttt{glamu}, \texttt{glamv}, \texttt{glamf} and
313  latitude  with \texttt{gphit}, \texttt{gphiu}, \texttt{gphiv}, \texttt{gphif}
314  (all respectively at T, U, V and F point)
315\item
316  Coriolis parameter (if domain not on the sphere): \texttt{ff\_f} and \texttt{ff\_t}
317  (at T and F point)
318\item
319  Scale factors:
320  \texttt{e1t}, \texttt{e1u}, \texttt{e1v} and \texttt{e1f} (on i direction),
321  \texttt{e2t}, \texttt{e2u}, \texttt{e2v} and \texttt{e2f} (on j direction) and
322  \texttt{ie1e2u\_v}, \texttt{e1e2u}, \texttt{e1e2v}. \\
323  \texttt{e1e2u}, \texttt{e1e2v} are u and v surfaces (if gridsize reduction in some straits),
324  \texttt{ie1e2u\_v} is to flag set u and v surfaces are neither read nor computed.
325\end{itemize}
326
327These fields can be read in an domain input file which name is setted in \np{cn\_domcfg} parameter specified in
328\ngn{namcfg}.
329
330\nlst{namcfg}
331
332Or they can be defined in an analytical way in \path{MY_SRC} directory of the configuration.
333For Reference Configurations of NEMO input domain files are supplied by NEMO System Team.
334For analytical definition of input fields two routines are supplied: \mdl{usrdef\_hgr} and \mdl{usrdef\_zgr}.
335They are an example of GYRE configuration parameters, and they are available in \path{src/OCE/USR} directory,
336they provide the horizontal and vertical mesh.
337% -------------------------------------------------------------------------------------------------------------
338%        Needed fields
339% -------------------------------------------------------------------------------------------------------------
340%\subsection{List of needed fields to build DOMAIN}
341%\label{subsec:DOM_fields_list}
342
343
344% ================================================================
345% Domain: Horizontal Grid (mesh)
346% ================================================================
347\section{Horizontal grid mesh (\protect\mdl{domhgr})}
348\label{sec:DOM_hgr}
349
350% -------------------------------------------------------------------------------------------------------------
351%        Coordinates and scale factors
352% -------------------------------------------------------------------------------------------------------------
353\subsection{Coordinates and scale factors}
354\label{subsec:DOM_hgr_coord_e}
355
356The ocean mesh (\ie the position of all the scalar and vector points) is defined by
357the transformation that gives $(\lambda,\varphi,z)$ as a function of $(i,j,k)$.
358The grid-points are located at integer or integer and a half values of as indicated in \autoref{tab:cell}.
359The associated scale factors are defined using the analytical first derivative of the transformation
360\autoref{eq:scale_factors}.
361These definitions are done in two modules, \mdl{domhgr} and \mdl{domzgr},
362which provide the horizontal and vertical meshes, respectively.
363This section deals with the horizontal mesh parameters.
364
365In a horizontal plane, the location of all the model grid points is defined from
366the analytical expressions of the longitude $\lambda$ and latitude $\varphi$ as a function of $(i,j)$.
367The horizontal scale factors are calculated using \autoref{eq:scale_factors}.
368For example, when the longitude and latitude are function of a single value
369($i$ and $j$, respectively) (geographical configuration of the mesh),
370the horizontal mesh definition reduces to define the wanted $\lambda(i)$, $\varphi(j)$,
371and their derivatives $\lambda'(i) \ \varphi'(j)$ in the \mdl{domhgr} module.
372The model computes the grid-point positions and scale factors in the horizontal plane as follows:
373\begin{align*}
374   \lambda_t &\equiv \text{glamt} =      \lambda (i      )
375  &\varphi_t &\equiv \text{gphit} =      \varphi (j      ) \\
376   \lambda_u &\equiv \text{glamu} =      \lambda (i + 1/2)
377  &\varphi_u &\equiv \text{gphiu} =      \varphi (j      ) \\
378   \lambda_v &\equiv \text{glamv} =      \lambda (i      )
379  &\varphi_v &\equiv \text{gphiv} =      \varphi (j + 1/2) \\
380   \lambda_f &\equiv \text{glamf} =      \lambda (i + 1/2)
381  &\varphi_f &\equiv \text{gphif} =      \varphi (j + 1/2) \\
382   e_{1t}    &\equiv \text{e1t}   = r_a |\lambda'(i      ) \; \cos\varphi(j      ) |
383  &e_{2t}    &\equiv \text{e2t}   = r_a |\varphi'(j      )                         | \\
384   e_{1u}    &\equiv \text{e1t}   = r_a |\lambda'(i + 1/2) \; \cos\varphi(j      ) |
385  &e_{2u}    &\equiv \text{e2t}   = r_a |\varphi'(j      )                         | \\
386   e_{1v}    &\equiv \text{e1t}   = r_a |\lambda'(i      ) \; \cos\varphi(j + 1/2) |
387  &e_{2v}    &\equiv \text{e2t}   = r_a |\varphi'(j + 1/2)                         | \\
388   e_{1f}    &\equiv \text{e1t}   = r_a |\lambda'(i + 1/2) \; \cos\varphi(j + 1/2) |
389  &e_{2f}    &\equiv \text{e2t}   = r_a |\varphi'(j + 1/2)                         |
390\end{align*}
391where the last letter of each computational name indicates the grid point considered and
392$r_a$ is the earth radius (defined in \mdl{phycst} along with all universal constants).
393Note that the horizontal position of and scale factors at $w$-points are exactly equal to those of $t$-points,
394thus no specific arrays are defined at $w$-points.
395
396Note that the definition of the scale factors
397(\ie as the analytical first derivative of the transformation that
398gives $(\lambda,\varphi,z)$ as a function of $(i,j,k)$)
399is specific to the \NEMO model \citep{marti.madec.ea_JGR92}.
400As an example, $e_{1t}$ is defined locally at a $t$-point,
401whereas many other models on a C grid choose to define such a scale factor as
402the distance between the $U$-points on each side of the $t$-point.
403Relying on an analytical transformation has two advantages:
404firstly, there is no ambiguity in the scale factors appearing in the discrete equations,
405since they are first introduced in the continuous equations;
406secondly, analytical transformations encourage good practice by the definition of smoothly varying grids
407(rather than allowing the user to set arbitrary jumps in thickness between adjacent layers) \citep{treguier.dukowicz.ea_JGR96}.
408An example of the effect of such a choice is shown in \autoref{fig:zgr_e3}.
409%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
410\begin{figure}[!t]
411  \begin{center}
412    \includegraphics[]{Fig_zgr_e3}
413    \caption{
414      \protect\label{fig:zgr_e3}
415      Comparison of (a) traditional definitions of grid-point position and grid-size in the vertical,
416      and (b) analytically derived grid-point position and scale factors.
417      For both grids here, the same $w$-point depth has been chosen but
418      in (a) the $t$-points are set half way between $w$-points while
419      in (b) they are defined from an analytical function:
420      $z(k) = 5 \, (k - 1/2)^3 - 45 \, (k - 1/2)^2 + 140 \, (k - 1/2) - 150$.
421      Note the resulting difference between the value of the grid-size $\Delta_k$ and
422      those of the scale factor $e_k$.
423    }
424  \end{center}
425\end{figure}
426%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
427
428% -------------------------------------------------------------------------------------------------------------
429%        Choice of horizontal grid
430% -------------------------------------------------------------------------------------------------------------
431\subsection{Choice of horizontal grid}
432\label{subsec:DOM_hgr_msh_choice}
433
434% -------------------------------------------------------------------------------------------------------------
435%        Grid files
436% -------------------------------------------------------------------------------------------------------------
437\subsection{Output grid files}
438\label{subsec:DOM_hgr_files}
439
440All the arrays relating to a particular ocean model configuration (grid-point position, scale factors, masks)
441can be saved in files if \np{nn\_msh} $\not = 0$ (namelist variable in \ngn{namdom}).
442This can be particularly useful for plots and off-line diagnostics.
443In some cases, the user may choose to make a local modification of a scale factor in the code.
444This is the case in global configurations when restricting the width of a specific strait
445(usually a one-grid-point strait that happens to be too wide due to insufficient model resolution).
446An example is Gibraltar Strait in the ORCA2 configuration.
447When such modifications are done,
448the output grid written when \np{nn\_msh} $\not = 0$ is no more equal to the input grid.
449
450% ================================================================
451% Domain: Vertical Grid (domzgr)
452% ================================================================
453\section{Vertical grid (\protect\mdl{domzgr})}
454\label{sec:DOM_zgr}
455%-----------------------------------------nam_zgr & namdom-------------------------------------------
456%
457%\nlst{namzgr}
458
459\nlst{namdom}
460%-------------------------------------------------------------------------------------------------------------
461
462Variables are defined through the \ngn{namzgr} and \ngn{namdom} namelists.
463In the vertical, the model mesh is determined by four things:
464(1) the bathymetry given in meters;
465(2) the number of levels of the model (\jp{jpk});
466(3) the analytical transformation $z(i,j,k)$ and the vertical scale factors (derivatives of the transformation); and
467(4) the masking system, \ie the number of wet model levels at each
468$(i,j)$ column of points.
469
470%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
471\begin{figure}[!tb]
472  \begin{center}
473    \includegraphics[]{Fig_z_zps_s_sps}
474    \caption{
475      \protect\label{fig:z_zps_s_sps}
476      The ocean bottom as seen by the model:
477      (a) $z$-coordinate with full step,
478      (b) $z$-coordinate with partial step,
479      (c) $s$-coordinate: terrain following representation,
480      (d) hybrid $s-z$ coordinate,
481      (e) hybrid $s-z$ coordinate with partial step, and
482      (f) same as (e) but in the non-linear free surface (\protect\np{ln\_linssh}~\forcode{= .false.}).
483      Note that the non-linear free surface can be used with any of the 5 coordinates (a) to (e).
484    }
485  \end{center}
486\end{figure}
487%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
488
489The choice of a vertical coordinate, even if it is made through \ngn{namzgr} namelist parameters,
490must be done once of all at the beginning of an experiment.
491It is not intended as an option which can be enabled or disabled in the middle of an experiment.
492Three main choices are offered (\autoref{fig:z_zps_s_sps}):
493$z$-coordinate with full step bathymetry (\np{ln\_zco}~\forcode{= .true.}),
494$z$-coordinate with partial step bathymetry (\np{ln\_zps}~\forcode{= .true.}),
495or generalized, $s$-coordinate (\np{ln\_sco}~\forcode{= .true.}).
496Hybridation of the three main coordinates are available:
497$s-z$ or $s-zps$ coordinate (\autoref{fig:z_zps_s_sps} and \autoref{fig:z_zps_s_sps}).
498By default a non-linear free surface is used: the coordinate follow the time-variation of the free surface so that
499the transformation is time dependent: $z(i,j,k,t)$ (\autoref{fig:z_zps_s_sps}).
500When a linear free surface is assumed (\np{ln\_linssh}~\forcode{= .true.}),
501the vertical coordinate are fixed in time, but the seawater can move up and down across the $z_0$ surface
502(in other words, the top of the ocean in not a rigid-lid).
503The last choice in terms of vertical coordinate concerns the presence (or not) in
504the model domain of ocean cavities beneath ice shelves.
505Setting \np{ln\_isfcav} to true allows to manage ocean cavities, otherwise they are filled in.
506This option is currently only available in $z$- or $zps$-coordinate,
507and partial step are also applied at the ocean/ice shelf interface.
508
509Contrary to the horizontal grid, the vertical grid is computed in the code and no provision is made for
511The only input file is the bathymetry (in meters) (\ifile{bathy\_meter})
512\footnote{
513  N.B. in full step $z$-coordinate, a \ifile{bathy\_level} file can replace the \ifile{bathy\_meter} file,
514  so that the computation of the number of wet ocean point in each water column is by-passed}.
515If \np{ln\_isfcav}~\forcode{= .true.}, an extra file input file (\ifile{isf\_draft\_meter}) describing
516the ice shelf draft (in meters) is needed.
517
518After reading the bathymetry, the algorithm for vertical grid definition differs between the different options:
519\begin{description}
520\item[\textit{zco}]
521  set a reference coordinate transformation $z_0(k)$, and set $z(i,j,k,t) = z_0(k)$.
522\item[\textit{zps}]
523  set a reference coordinate transformation $z_0(k)$, and calculate the thickness of the deepest level at
524  each $(i,j)$ point using the bathymetry, to obtain the final three-dimensional depth and scale factor arrays.
525\item[\textit{sco}]
526  smooth the bathymetry to fulfill the hydrostatic consistency criteria and
527  set the three-dimensional transformation.
528\item[\textit{s-z} and \textit{s-zps}]
529  smooth the bathymetry to fulfill the hydrostatic consistency criteria and
530  set the three-dimensional transformation $z(i,j,k)$,
531  and possibly introduce masking of extra land points to better fit the original bathymetry file.
532\end{description}
533%%%
534\gmcomment{   add the description of the smoothing:  envelop topography...}
535%%%
536
537Unless a linear free surface is used (\np{ln\_linssh}~\forcode{= .false.}),
538the arrays describing the grid point depths and vertical scale factors are three set of
539three dimensional arrays $(i,j,k)$ defined at \textit{before}, \textit{now} and \textit{after} time step.
540The time at which they are defined is indicated by a suffix: $\_b$, $\_n$, or $\_a$, respectively.
541They are updated at each model time step using a fixed reference coordinate system which
542computer names have a $\_0$ suffix.
543When the linear free surface option is used (\np{ln\_linssh}~\forcode{= .true.}), \textit{before},
544\textit{now} and \textit{after} arrays are simply set one for all to their reference counterpart.
545
546% -------------------------------------------------------------------------------------------------------------
547%        Meter Bathymetry
548% -------------------------------------------------------------------------------------------------------------
549\subsection{Meter bathymetry}
550\label{subsec:DOM_bathy}
551
552Three options are possible for defining the bathymetry, according to the namelist variable \np{nn\_bathy}
553(found in \ngn{namdom} namelist):
554\begin{description}
555\item[\np{nn\_bathy}~\forcode{= 0}]:
556  a flat-bottom domain is defined.
557  The total depth $z_w (jpk)$ is given by the coordinate transformation.
558  The domain can either be a closed basin or a periodic channel depending on the parameter \np{jperio}.
559\item[\np{nn\_bathy}~\forcode{= -1}]:
560  a domain with a bump of topography one third of the domain width at the central latitude.
561  This is meant for the "EEL-R5" configuration, a periodic or open boundary channel with a seamount.
562\item[\np{nn\_bathy}~\forcode{= 1}]:
563  read a bathymetry and ice shelf draft (if needed).
564  The \ifile{bathy\_meter} file (Netcdf format) provides the ocean depth (positive, in meters) at
565  each grid point of the model grid.
566  The bathymetry is usually built by interpolating a standard bathymetry product (\eg ETOPO2) onto
567  the horizontal ocean mesh.
568  Defining the bathymetry also defines the coastline: where the bathymetry is zero,
569  no model levels are defined (all levels are masked).
570
571  The \ifile{isfdraft\_meter} file (Netcdf format) provides the ice shelf draft (positive, in meters) at
572  each grid point of the model grid.
573  This file is only needed if \np{ln\_isfcav}~\forcode{= .true.}.
574  Defining the ice shelf draft will also define the ice shelf edge and the grounding line position.
575\end{description}
576
577When a global ocean is coupled to an atmospheric model it is better to represent all large water bodies
578(\eg great lakes, Caspian sea...) even if the model resolution does not allow their communication with
579the rest of the ocean.
580This is unnecessary when the ocean is forced by fixed atmospheric conditions,
581so these seas can be removed from the ocean domain.
582The user has the option to set the bathymetry in closed seas to zero (see \autoref{sec:MISC_closea}),
583but the code has to be adapted to the user's configuration.
584
585% -------------------------------------------------------------------------------------------------------------
586%        z-coordinate  and reference coordinate transformation
587% -------------------------------------------------------------------------------------------------------------
588\subsection[$Z$-coordinate (\protect\np{ln\_zco}~\forcode{= .true.}) and ref. coordinate]
589            {$Z$-coordinate (\protect\np{ln\_zco}~\forcode{= .true.}) and reference coordinate}
590\label{subsec:DOM_zco}
591
592%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
593\begin{figure}[!tb]
594  \begin{center}
595    \includegraphics[]{Fig_zgr}
596    \caption{
597      \protect\label{fig:zgr}
598      Default vertical mesh for ORCA2: 30 ocean levels (L30).
599      Vertical level functions for (a) T-point depth and (b) the associated scale factor as computed from
600      \autoref{eq:DOM_zgr_ana_1} using \autoref{eq:DOM_zgr_coef} in $z$-coordinate.
601    }
602  \end{center}
603\end{figure}
604%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
605
606The reference coordinate transformation $z_0(k)$ defines the arrays $gdept_0$ and $gdepw_0$ for $t$- and $w$-points,
607respectively.
608As indicated on \autoref{fig:index_vert} \jp{jpk} is the number of $w$-levels.
609$gdepw_0(1)$ is the ocean surface.
610There are at most \jp{jpk}-1 $t$-points inside the ocean,
611the additional $t$-point at $jk = jpk$ is below the sea floor and is not used.
612The vertical location of $w$- and $t$-levels is defined from the analytic expression of the depth $z_0(k)$ whose
613analytical derivative with respect to $k$ provides the vertical scale factors.
614The user must provide the analytical expression of both $z_0$ and its first derivative with respect to $k$.
615This is done in routine \mdl{domzgr} through statement functions,
616using parameters provided in the \ngn{namcfg} namelist.
617
618It is possible to define a simple regular vertical grid by giving zero stretching (\np{ppacr}~\forcode{= 0}).
619In that case, the parameters \jp{jpk} (number of $w$-levels) and
620\np{pphmax} (total ocean depth in meters) fully define the grid.
621
622For climate-related studies it is often desirable to concentrate the vertical resolution near the ocean surface.
623The following function is proposed as a standard for a $z$-coordinate (with either full or partial steps):
624\begin{gather}
625  \label{eq:DOM_zgr_ana_1}
626    z_0  (k) = h_{sur} - h_0 \; k - \; h_1 \; \log  \big[ \cosh ((k - h_{th}) / h_{cr}) \big] \\
627    e_3^0(k) = \lt|    - h_0      -    h_1 \; \tanh \big[        (k - h_{th}) / h_{cr}  \big] \rt|
628\end{gather}
629where $k = 1$ to \jp{jpk} for $w$-levels and $k = 1$ to $k = 1$ for $T-$levels.
630Such an expression allows us to define a nearly uniform vertical location of levels at the ocean top and bottom with
631a smooth hyperbolic tangent transition in between (\autoref{fig:zgr}).
632
633If the ice shelf cavities are opened (\np{ln\_isfcav}~\forcode{= .true.}), the definition of $z_0$ is the same.
634However, definition of $e_3^0$ at $t$- and $w$-points is respectively changed to:
635\begin{equation}
636  \label{eq:DOM_zgr_ana_2}
637  \begin{split}
638    e_3^T(k) &= z_W (k + 1) - z_W (k    ) \\
639    e_3^W(k) &= z_T (k    ) - z_T (k - 1)
640  \end{split}
641\end{equation}
642This formulation decrease the self-generated circulation into the ice shelf cavity
643(which can, in extreme case, leads to blow up).\\
644
645The most used vertical grid for ORCA2 has $10~m$ ($500~m$) resolution in the surface (bottom) layers and
646a depth which varies from 0 at the sea surface to a minimum of $-5000~m$.
647This leads to the following conditions:
648\begin{equation}
649  \label{eq:DOM_zgr_coef}
650  \begin{array}{ll}
651    e_3 (1   + 1/2) =  10. & z(1  ) =     0. \\
652    e_3 (jpk - 1/2) = 500. & z(jpk) = -5000.
653  \end{array}
654\end{equation}
655
656With the choice of the stretching $h_{cr} = 3$ and the number of levels \jp{jpk}~$= 31$,
657the four coefficients $h_{sur}$, $h_0$, $h_1$, and $h_{th}$ in
658\autoref{eq:DOM_zgr_ana_2} have been determined such that
659\autoref{eq:DOM_zgr_coef} is satisfied, through an optimisation procedure using a bisection method.
660For the first standard ORCA2 vertical grid this led to the following values:
661$h_{sur} = 4762.96$, $h_0 = 255.58, h_1 = 245.5813$, and $h_{th} = 21.43336$.
662The resulting depths and scale factors as a function of the model levels are shown in
663\autoref{fig:zgr} and given in \autoref{tab:orca_zgr}.
664Those values correspond to the parameters \np{ppsur}, \np{ppa0}, \np{ppa1}, \np{ppkth} in \ngn{namcfg} namelist.
665
666Rather than entering parameters $h_{sur}$, $h_0$, and $h_1$ directly, it is possible to recalculate them.
667In that case the user sets \np{ppsur}~$=$~\np{ppa0}~$=$~\np{ppa1}~$= 999999$.,
668in \ngn{namcfg} namelist, and specifies instead the four following parameters:
669\begin{itemize}
670\item
671  \np{ppacr}~$= h_{cr}$: stretching factor (nondimensional).
672  The larger \np{ppacr}, the smaller the stretching.
673  Values from $3$ to $10$ are usual.
674\item
675  \np{ppkth}~$= h_{th}$: is approximately the model level at which maximum stretching occurs
676  (nondimensional, usually of order 1/2 or 2/3 of \jp{jpk})
677\item
678  \np{ppdzmin}: minimum thickness for the top layer (in meters).
679\item
680  \np{pphmax}: total depth of the ocean (meters).
681\end{itemize}
682As an example, for the $45$ layers used in the DRAKKAR configuration those parameters are:
683\jp{jpk}~$= 46$, \np{ppacr}~$= 9$, \np{ppkth}~$= 23.563$, \np{ppdzmin}~$= 6~m$, \np{pphmax}~$= 5750~m$.
684
685%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
686\begin{table}
687  \begin{center}
688    \begin{tabular}{c||r|r|r|r}
689      \hline
690      \textbf{LEVEL} & \textbf{gdept\_1d} & \textbf{gdepw\_1d} & \textbf{e3t\_1d } & \textbf{e3w\_1d} \\
691      \hline
692      1              & \textbf{     5.00} &               0.00 & \textbf{   10.00} &            10.00 \\
693      \hline
694      2              & \textbf{    15.00} &              10.00 & \textbf{   10.00} &            10.00 \\
695      \hline
696      3              & \textbf{    25.00} &              20.00 & \textbf{   10.00} &            10.00 \\
697      \hline
698      4              & \textbf{    35.01} &              30.00 & \textbf{   10.01} &            10.00 \\
699      \hline
700      5              & \textbf{    45.01} &              40.01 & \textbf{   10.01} &            10.01 \\
701      \hline
702      6              & \textbf{    55.03} &              50.02 & \textbf{   10.02} &            10.02 \\
703      \hline
704      7              & \textbf{    65.06} &              60.04 & \textbf{   10.04} &            10.03 \\
705      \hline
706      8              & \textbf{    75.13} &              70.09 & \textbf{   10.09} &            10.06 \\
707      \hline
708      9              & \textbf{    85.25} &              80.18 & \textbf{   10.17} &            10.12 \\
709      \hline
710      10             & \textbf{    95.49} &              90.35 & \textbf{   10.33} &            10.24 \\
711      \hline
712      11             & \textbf{   105.97} &             100.69 & \textbf{   10.65} &            10.47 \\
713      \hline
714      12             & \textbf{   116.90} &             111.36 & \textbf{   11.27} &            10.91 \\
715      \hline
716      13             & \textbf{   128.70} &             122.65 & \textbf{   12.47} &            11.77 \\
717      \hline
718      14             & \textbf{   142.20} &             135.16 & \textbf{   14.78} &            13.43 \\
719      \hline
720      15             & \textbf{   158.96} &             150.03 & \textbf{   19.23} &            16.65 \\
721      \hline
722      16             & \textbf{   181.96} &             169.42 & \textbf{   27.66} &            22.78 \\
723      \hline
724      17             & \textbf{   216.65} &             197.37 & \textbf{   43.26} &            34.30 \\
725      \hline
726      18             & \textbf{   272.48} &             241.13 & \textbf{   70.88} &            55.21 \\
727      \hline
728      19             & \textbf{   364.30} &             312.74 & \textbf{  116.11} &            90.99 \\
729      \hline
730      20             & \textbf{   511.53} &             429.72 & \textbf{  181.55} &           146.43 \\
731      \hline
732      21             & \textbf{   732.20} &             611.89 & \textbf{  261.03} &           220.35 \\
733      \hline
734      22             & \textbf{  1033.22} &             872.87 & \textbf{  339.39} &           301.42 \\
735      \hline
736      23             & \textbf{  1405.70} &            1211.59 & \textbf{  402.26} &           373.31 \\
737      \hline
738      24             & \textbf{  1830.89} &            1612.98 & \textbf{  444.87} &           426.00 \\
739      \hline
740      25             & \textbf{  2289.77} &            2057.13 & \textbf{  470.55} &           459.47 \\
741      \hline
742      26             & \textbf{  2768.24} &            2527.22 & \textbf{  484.95} &           478.83 \\
743      \hline
744      27             & \textbf{  3257.48} &            3011.90 & \textbf{  492.70} &           489.44 \\
745      \hline
746      28             & \textbf{  3752.44} &            3504.46 & \textbf{  496.78} &           495.07 \\
747      \hline
748      29             & \textbf{  4250.40} &            4001.16 & \textbf{  498.90} &           498.02 \\
749      \hline
750      30             & \textbf{  4749.91} &            4500.02 & \textbf{  500.00} &           499.54 \\
751      \hline
752      31             & \textbf{  5250.23} &            5000.00 & \textbf{  500.56} &           500.33 \\
753      \hline
754    \end{tabular}
755  \end{center}
756  \caption{
757    \protect\label{tab:orca_zgr}
758    Default vertical mesh in $z$-coordinate for 30 layers ORCA2 configuration as computed from
759    \autoref{eq:DOM_zgr_ana_2} using the coefficients given in \autoref{eq:DOM_zgr_coef}
760  }
761\end{table}
762%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
763
764% -------------------------------------------------------------------------------------------------------------
765%        z-coordinate with partial step
766% -------------------------------------------------------------------------------------------------------------
767\subsection{$Z$-coordinate with partial step (\protect\np{ln\_zps}~\forcode{= .true.})}
768\label{subsec:DOM_zps}
769%--------------------------------------------namdom-------------------------------------------------------
770
771\nlst{namdom}
772%--------------------------------------------------------------------------------------------------------------
773
774In $z$-coordinate partial step,
775the depths of the model levels are defined by the reference analytical function $z_0(k)$ as described in
776the previous section, \textit{except} in the bottom layer.
777The thickness of the bottom layer is allowed to vary as a function of geographical location $(\lambda,\varphi)$ to
778allow a better representation of the bathymetry, especially in the case of small slopes
779(where the bathymetry varies by less than one level thickness from one grid point to the next).
780The reference layer thicknesses $e_{3t}^0$ have been defined in the absence of bathymetry.
781With partial steps, layers from 1 to \jp{jpk}-2 can have a thickness smaller than $e_{3t}(jk)$.
782The model deepest layer (\jp{jpk}-1) is allowed to have either a smaller or larger thickness than $e_{3t}(jpk)$:
783the maximum thickness allowed is $2*e_{3t}(jpk - 1)$.
784This has to be kept in mind when specifying values in \ngn{namdom} namelist,
785as the maximum depth \np{pphmax} in partial steps:
786for example, with \np{pphmax}~$= 5750~m$ for the DRAKKAR 45 layer grid,
787the maximum ocean depth allowed is actually $6000~m$ (the default thickness $e_{3t}(jpk - 1)$ being $250~m$).
788Two variables in the namdom namelist are used to define the partial step vertical grid.
789The mimimum water thickness (in meters) allowed for a cell partially filled with bathymetry at level jk is
790the minimum of \np{rn\_e3zps\_min} (thickness in meters, usually $20~m$) or $e_{3t}(jk)*$\np{rn\_e3zps\_rat}
791(a fraction, usually 10\%, of the default thickness $e_{3t}(jk)$).
792
793\gmcomment{ \colorbox{yellow}{Add a figure here of pstep especially at last ocean level }  }
794
795% -------------------------------------------------------------------------------------------------------------
796%        s-coordinate
797% -------------------------------------------------------------------------------------------------------------
798\subsection{$S$-coordinate (\protect\np{ln\_sco}~\forcode{= .true.})}
799\label{subsec:DOM_sco}
800%------------------------------------------nam_zgr_sco---------------------------------------------------
801%
802%\nlst{namzgr_sco}
803%--------------------------------------------------------------------------------------------------------------
804Options are defined in \ngn{namzgr\_sco}.
805In $s$-coordinate (\np{ln\_sco}~\forcode{= .true.}), the depth and thickness of the model levels are defined from
806the product of a depth field and either a stretching function or its derivative, respectively:
807
808\begin{align*}
809  % \label{eq:DOM_sco_ana}
810  z(k)   &= h(i,j) \; z_0 (k) \\
811  e_3(k) &= h(i,j) \; z_0'(k)
812\end{align*}
813
814where $h$ is the depth of the last $w$-level ($z_0(k)$) defined at the $t$-point location in the horizontal and
815$z_0(k)$ is a function which varies from $0$ at the sea surface to $1$ at the ocean bottom.
816The depth field $h$ is not necessary the ocean depth,
817since a mixed step-like and bottom-following representation of the topography can be used
818(\autoref{fig:z_zps_s_sps}) or an envelop bathymetry can be defined (\autoref{fig:z_zps_s_sps}).
819The namelist parameter \np{rn\_rmax} determines the slope at which
820the terrain-following coordinate intersects the sea bed and becomes a pseudo z-coordinate.
821The coordinate can also be hybridised by specifying \np{rn\_sbot\_min} and \np{rn\_sbot\_max} as
822the minimum and maximum depths at which the terrain-following vertical coordinate is calculated.
823
824Options for stretching the coordinate are provided as examples,
825but care must be taken to ensure that the vertical stretch used is appropriate for the application.
826
827The original default NEMO s-coordinate stretching is available if neither of the other options are specified as true
828(\np{ln\_s\_SH94}~\forcode{= .false.} and \np{ln\_s\_SF12}~\forcode{= .false.}).
829This uses a depth independent $\tanh$ function for the stretching \citep{madec.delecluse.ea_JPO96}:
830
831$832 z = s_{min} + C (s) (H - s_{min}) 833 % \label{eq:SH94_1} 834$
835
836where $s_{min}$ is the depth at which the $s$-coordinate stretching starts and
837allows a $z$-coordinate to placed on top of the stretched coordinate,
838and $z$ is the depth (negative down from the asea surface).
839\begin{gather*}
840  s = - \frac{k}{n - 1} \quad \text{and} \quad 0 \leq k \leq n - 1
841  % \label{eq:DOM_s}
842 \\
843  % \label{eq:DOM_sco_function}
844  C(s) = \frac{[\tanh(\theta \, (s + b)) - \tanh(\theta \, b)]}{2 \; \sinh(\theta)}
845\end{gather*}
846
847A stretching function,
848modified from the commonly used \citet{song.haidvogel_JCP94} stretching (\np{ln\_s\_SH94}~\forcode{= .true.}),
849is also available and is more commonly used for shelf seas modelling:
850
851$852 C(s) = (1 - b) \frac{\sinh(\theta s)}{\sinh(\theta)} 853 + b \frac{\tanh \lt[ \theta \lt(s + \frac{1}{2} \rt) \rt] - \tanh \lt( \frac{\theta}{2} \rt)} 854 { 2 \tanh \lt( \frac{\theta}{2} \rt)} 855 % \label{eq:SH94_2} 856$
857
858%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
859\begin{figure}[!ht]
860  \begin{center}
861    \includegraphics[]{Fig_sco_function}
862    \caption{
863      \protect\label{fig:sco_function}
864      Examples of the stretching function applied to a seamount;
865      from left to right: surface, surface and bottom, and bottom intensified resolutions
866    }
867  \end{center}
868\end{figure}
869%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
870
871where $H_c$ is the critical depth (\np{rn\_hc}) at which the coordinate transitions from pure $\sigma$ to
872the stretched coordinate, and $\theta$ (\np{rn\_theta}) and $b$ (\np{rn\_bb}) are the surface and
873bottom control parameters such that $0 \leqslant \theta \leqslant 20$, and $0 \leqslant b \leqslant 1$.
874$b$ has been designed to allow surface and/or bottom increase of the vertical resolution
875(\autoref{fig:sco_function}).
876
877Another example has been provided at version 3.5 (\np{ln\_s\_SF12}) that allows a fixed surface resolution in
878an analytical terrain-following stretching \citet{siddorn.furner_OM13}.
879In this case the a stretching function $\gamma$ is defined such that:
880
881\begin{equation}
882  z = - \gamma h \quad \text{with} \quad 0 \leq \gamma \leq 1
883  % \label{eq:z}
884\end{equation}
885
886The function is defined with respect to $\sigma$, the unstretched terrain-following coordinate:
887
888\begin{gather*}
889  % \label{eq:DOM_gamma_deriv}
890  \gamma =   A \lt( \sigma   - \frac{1}{2} (\sigma^2     + f (\sigma)) \rt)
891           + B \lt( \sigma^3 - f           (\sigma) \rt) + f (\sigma)       \\
892  \intertext{Where:}
893  % \label{eq:DOM_gamma}
894  f(\sigma) = (\alpha + 2) \sigma^{\alpha + 1} - (\alpha + 1) \sigma^{\alpha + 2}
896\end{gather*}
897
898This gives an analytical stretching of $\sigma$ that is solvable in $A$ and $B$ as a function of
899the user prescribed stretching parameter $\alpha$ (\np{rn\_alpha}) that stretches towards
900the surface ($\alpha > 1.0$) or the bottom ($\alpha < 1.0$) and
901user prescribed surface (\np{rn\_zs}) and bottom depths.
902The bottom cell depth in this example is given as a function of water depth:
903
904$905 % \label{eq:DOM_zb} 906 Z_b = h a + b 907$
908
909where the namelist parameters \np{rn\_zb\_a} and \np{rn\_zb\_b} are $a$ and $b$ respectively.
910
911%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
912\begin{figure}[!ht]
913  \includegraphics[]{Fig_DOM_compare_coordinates_surface}
914  \caption{
915    A comparison of the \citet{song.haidvogel_JCP94} $S$-coordinate (solid lines),
916    a 50 level $Z$-coordinate (contoured surfaces) and
917    the \citet{siddorn.furner_OM13} $S$-coordinate (dashed lines) in the surface $100~m$ for
918    a idealised bathymetry that goes from $50~m$ to $5500~m$ depth.
919    For clarity every third coordinate surface is shown.
920  }
921  \label{fig:fig_compare_coordinates_surface}
922\end{figure}
923 % >>>>>>>>>>>>>>>>>>>>>>>>>>>>
924
925This gives a smooth analytical stretching in computational space that is constrained to
926given specified surface and bottom grid cell thicknesses in real space.
927This is not to be confused with the hybrid schemes that
928superimpose geopotential coordinates on terrain following coordinates thus
929creating a non-analytical vertical coordinate that
930therefore may suffer from large gradients in the vertical resolutions.
931This stretching is less straightforward to implement than the \citet{song.haidvogel_JCP94} stretching,
932but has the advantage of resolving diurnal processes in deep water and has generally flatter slopes.
933
934As with the \citet{song.haidvogel_JCP94} stretching the stretch is only applied at depths greater than
935the critical depth $h_c$.
936In this example two options are available in depths shallower than $h_c$,
937with pure sigma being applied if the \np{ln\_sigcrit} is true and pure z-coordinates if it is false
938(the z-coordinate being equal to the depths of the stretched coordinate at $h_c$).
939
940Minimising the horizontal slope of the vertical coordinate is important in terrain-following systems as
941large slopes lead to hydrostatic consistency.
942A hydrostatic consistency parameter diagnostic following \citet{haney_JPO91} has been implemented,
943and is output as part of the model mesh file at the start of the run.
944
945% -------------------------------------------------------------------------------------------------------------
946%        z*- or s*-coordinate
947% -------------------------------------------------------------------------------------------------------------
948\subsection{\zstar- or \sstar-coordinate (\protect\np{ln\_linssh}~\forcode{= .false.})}
949\label{subsec:DOM_zgr_star}
950
951This option is described in the Report by Levier \textit{et al.} (2007), available on the \NEMO web site.
952
953%gm% key advantage: minimise the diffusion/dispertion associated with advection in response to high frequency surface disturbances
954
955% -------------------------------------------------------------------------------------------------------------
957% -------------------------------------------------------------------------------------------------------------
959\label{subsec:DOM_msk}
960
961Whatever the vertical coordinate used, the model offers the possibility of representing the bottom topography with
962steps that follow the face of the model cells (step like topography) \citep{madec.delecluse.ea_JPO96}.
963The distribution of the steps in the horizontal is defined in a 2D integer array, mbathy, which
964gives the number of ocean levels (\ie those that are not masked) at each $t$-point.
965mbathy is computed from the meter bathymetry using the definiton of gdept as the number of $t$-points which
966gdept $\leq$ bathy.
967
968Modifications of the model bathymetry are performed in the \textit{bat\_ctl} routine (see \mdl{domzgr} module) after
969mbathy is computed.
970Isolated grid points that do not communicate with another ocean point at the same level are eliminated.
971
972As for the representation of bathymetry, a 2D integer array, misfdep, is created.
973misfdep defines the level of the first wet $t$-point.
974All the cells between $k = 1$ and $misfdep(i,j) - 1$ are masked.
975By default, $misfdep(:,:) = 1$ and no cells are masked.
976
977In case of ice shelf cavities, modifications of the model bathymetry and ice shelf draft into
978the cavities are performed in the \textit{zgr\_isf} routine.
979The compatibility between ice shelf draft and bathymetry is checked.
980All the locations where the isf cavity is thinnest than \np{rn\_isfhmin} meters are grounded (\ie masked).
981If only one cell on the water column is opened at $t$-, $u$- or $v$-points,
982the bathymetry or the ice shelf draft is dug to fit this constrain.
983If the incompatibility is too strong (need to dig more than 1 cell), the cell is masked.
984
985From the \textit{mbathy} and \textit{misfdep} array, the mask fields are defined as follows:
986\begin{alignat*}{2}
988    \begin{cases}
989                  0 &\text{if $k < misfdep(i,j)$} \\
990                  1 &\text{if $misfdep(i,j) \leq k \leq mbathy(i,j)$} \\
991                  0 &\text{if $k > mbathy(i,j)$}
992    \end{cases}
993  \\
1000\end{alignat*}
1001
1002Note that, without ice shelves cavities,
1003masks at $t-$ and $w-$points are identical with the numerical indexing used (\autoref{subsec:DOM_Num_Index}).
1004Nevertheless, $wmask$ are required with ocean cavities to deal with the top boundary (ice shelf/ocean interface)
1005exactly in the same way as for the bottom boundary.
1006
1007The specification of closed lateral boundaries requires that at least
1008the first and last rows and columns of the \textit{mbathy} array are set to zero.
1009In the particular case of an east-west cyclical boundary condition, \textit{mbathy} has its last column equal to
1010the second one and its first column equal to the last but one (and so too the mask arrays)
1011(see \autoref{fig:LBC_jperio}).
1012
1013% ================================================================
1014% Domain: Initial State (dtatsd & istate)
1015% ================================================================
1016\section{Initial state (\protect\mdl{istate} and \protect\mdl{dtatsd})}
1017\label{sec:DTA_tsd}
1018%-----------------------------------------namtsd-------------------------------------------
1019
1020\nlst{namtsd}
1021%------------------------------------------------------------------------------------------
1022
1023Options are defined in \ngn{namtsd}.
1024By default, the ocean start from rest (the velocity field is set to zero) and the initialization of temperature and
1025salinity fields is controlled through the \np{ln\_tsd\_ini} namelist parameter.
1026\begin{description}
1027\item[\np{ln\_tsd\_init}~\forcode{= .true.}]
1028  use a T and S input files that can be given on the model grid itself or on their native input data grid.
1029  In the latter case,
1030  the data will be interpolated on-the-fly both in the horizontal and the vertical to the model grid
1031  (see \autoref{subsec:SBC_iof}).
1032  The information relative to the input files are given in the \np{sn\_tem} and \np{sn\_sal} structures.
1033  The computation is done in the \mdl{dtatsd} module.
1034\item[\np{ln\_tsd\_init}~\forcode{= .false.}]
1035  use constant salinity value of $35.5~psu$ and an analytical profile of temperature
1036  (typical of the tropical ocean), see \rou{istate\_t\_s} subroutine called from \mdl{istate} module.
1037\end{description}
1038
1039\biblio
1040
1041\pindex
1042
1043\end{document}
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