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1\documentclass[../main/NEMO_manual]{subfiles}
2
3\begin{document}
4
5\chapter{A brief guide to the DOMAINcfg tool}
6\label{apdx:DOMCFG}
7
8%    {\em 4.0} & {\em Andrew Coward} & {\em Created at v4.0 from materials removed from chap\_DOM that are still relevant to the \forcode{DOMAINcfg} tool and which illustrate and explain the choices to be made by the user when setting up new domains }  \\
9
10\thispagestyle{plain}
11
12\chaptertoc
13
14\paragraph{Changes record} ~\\
15
16{\footnotesize
17  \begin{tabularx}{\textwidth}{l||X|X}
18    Release & Author(s) & Modifications \\
19    \hline
20    {\em   4.0} & {\em ...} & {\em ...} \\
21    {\em   3.6} & {\em ...} & {\em ...} \\
22    {\em   3.4} & {\em ...} & {\em ...} \\
23    {\em <=3.4} & {\em ...} & {\em ...}
24  \end{tabularx}
25}
26
27\clearpage
28
29This appendix briefly describes some of the options available in the
30\forcode{DOMAINcfg} tool mentioned in \autoref{chap:DOM}.
31
32This tool will evolve into an independent utility with its own documentation but its
33current manifestation is mostly a wrapper for \NEMO\ \forcode{DOM} modules more aligned to
34those in the previous versions of \NEMO. These versions allowed the user to define some
35horizontal and vertical grids through additional namelist parameters. Explanations of
36these parameters are retained here for reference pending better documentation for
37\forcode{DOMAINcfg}. Please note that the namelist blocks named in this appendix refer to
38those read by \forcode{DOMAINcfg} via its own \forcode{namelist_ref} and
39\forcode{namelist_cfg} files. Although, due to their origins, these namelists share names
40with those used by \NEMO, they are not interchangeable and should be considered independent
41of those described elsewhere in this manual.
42
43%% =================================================================================================
44\section{Choice of horizontal grid}
45\label{sec:DOMCFG_hor}
46
47\begin{listing}
48  \nlst{namdom_domcfg}
49  \caption{\forcode{&namdom_domcfg}}
50  \label{lst:namdom_domcfg}
51\end{listing}
52
53The user has three options available in defining a horizontal grid, which involve the
54namelist variable \np{jphgr_mesh}{jphgr\_mesh} of the \nam{dom}{dom} (\texttt{DOMAINcfg} variant only)
55namelist.
56
57\begin{description}
58 \item [{\np{jphgr_mesh}{jphgr\_mesh}=0}]  The most general curvilinear orthogonal grids.
59  The coordinates and their first derivatives with respect to $i$ and $j$ are provided
60  in a input file (\ifile{coordinates}), read in \rou{hgr\_read} subroutine of the domhgr module.
61  This is now the only option available within \NEMO\ itself from v4.0 onwards.
62\item [{\np{jphgr_mesh}{jphgr\_mesh}=1 to 5}] A few simple analytical grids are provided (see below).
63  For other analytical grids, the \mdl{domhgr} module (\texttt{DOMAINcfg} variant) must be
64  modified by the user. In most cases, modifying the \mdl{usrdef\_hgr} module of \NEMO\ is
65  a better alternative since this is designed to allow simple analytical domains to be
66  configured and used without the need for external data files.
67\end{description}
68
69There are two simple cases of geographical grids on the sphere. With
70\np{jphgr_mesh}{jphgr\_mesh}=1, the grid (expressed in degrees) is regular in space,
71with grid sizes specified by parameters \np{ppe1_deg}{ppe1\_deg} and \np{ppe2_deg}{ppe2\_deg},
72respectively. Such a geographical grid can be very anisotropic at high latitudes
73because of the convergence of meridians (the zonal scale factors $e_1$
74become much smaller than the meridional scale factors $e_2$). The Mercator
75grid (\np{jphgr_mesh}{jphgr\_mesh}=4) avoids this anisotropy by refining the meridional scale
76factors in the same way as the zonal ones. In this case, meridional scale factors
77and latitudes are calculated analytically using the formulae appropriate for
78a Mercator projection, based on \np{ppe1_deg}{ppe1\_deg} which is a reference grid spacing
79at the equator (this applies even when the geographical equator is situated outside
80the model domain).
81
82In these two cases (\np{jphgr_mesh}{jphgr\_mesh}=1 or 4), the grid position is defined by the
83longitude and latitude of the south-westernmost point (\np{ppglamt0}
84and \np{ppgphi0}{ppgphi0}). Note that for the Mercator grid the user need only provide
85an approximate starting latitude: the real latitude will be recalculated analytically,
86in order to ensure that the equator corresponds to line passing through $t$-
87and $u$-points.
88
89Rectangular grids ignoring the spherical geometry are defined with
90\np{jphgr_mesh}{jphgr\_mesh} = 2, 3, 5. The domain is either an $f$-plane (\np{jphgr_mesh}{jphgr\_mesh} = 2,
91Coriolis factor is constant) or a beta-plane (\np{jphgr_mesh}{jphgr\_mesh} = 3, the Coriolis factor
92is linear in the $j$-direction). The grid size is uniform in meter in each direction,
93and given by the parameters \np{ppe1_m}{ppe1\_m} and \np{ppe2_m}{ppe2\_m} respectively.
94The zonal grid coordinate (\textit{glam} arrays) is in kilometers, starting at zero
95with the first $t$-point. The meridional coordinate (gphi. arrays) is in kilometers,
96and the second $t$-point corresponds to coordinate $gphit=0$. The input
97variable \np{ppglam0}{ppglam0} is ignored. \np{ppgphi0}{ppgphi0} is used to set the reference
98latitude for computation of the Coriolis parameter. In the case of the beta plane,
99\np{ppgphi0}{ppgphi0} corresponds to the center of the domain. Finally, the special case
100\np{jphgr_mesh}{jphgr\_mesh}=5 corresponds to a beta plane in a rotated domain for the
101GYRE configuration, representing a classical mid-latitude double gyre system.
102The rotation allows us to maximize the jet length relative to the gyre areas
103(and the number of grid points).
104
105%% =================================================================================================
106\section{Vertical grid}
107\label{sec:DOMCFG_vert}
108
109%% =================================================================================================
110\subsection{Vertical reference coordinate}
111\label{sec:DOMCFG_zref}
112
113\begin{figure}[!tb]
114  \centering
115  \includegraphics[width=0.66\textwidth]{DOMCFG_zgr}
116  \caption[DOMAINcfg: default vertical mesh for ORCA2]{
117    Default vertical mesh for ORCA2: 30 ocean levels (L30).
118    Vertical level functions for (a) T-point depth and (b) the associated scale factor for
119    the $z$-coordinate case.}
120  \label{fig:DOMCFG_zgr}
121\end{figure}
122
123The reference coordinate transformation $z_0(k)$ defines the arrays $gdept_0$ and
124$gdepw_0$ for $t$- and $w$-points, respectively. See \autoref{sec:DOMCFG_sco} for the
125S-coordinate options.  As indicated on \autoref{fig:DOM_index_vert} \jp{jpk} is the number of
126$w$-levels.  $gdepw_0(1)$ is the ocean surface.  There are at most \jp{jpk}-1 $t$-points
127inside the ocean, the additional $t$-point at $jk = jpk$ is below the sea floor and is not
128used.  The vertical location of $w$- and $t$-levels is defined from the analytic
129expression of the depth $z_0(k)$ whose analytical derivative with respect to $k$ provides
130the vertical scale factors.  The user must provide the analytical expression of both $z_0$
131and its first derivative with respect to $k$.  This is done in routine \mdl{domzgr}
132through statement functions, using parameters provided in the \nam{dom}{dom} namelist
133(\texttt{DOMAINcfg} variant).
134
135It is possible to define a simple regular vertical grid by giving zero stretching
136(\np[=0]{ppacr}{ppacr}).  In that case, the parameters \jp{jpk} (number of $w$-levels)
137and \np{pphmax}{pphmax} (total ocean depth in meters) fully define the grid.
138
139For climate-related studies it is often desirable to concentrate the vertical resolution
140near the ocean surface.  The following function is proposed as a standard for a
141$z$-coordinate (with either full or partial steps):
142\begin{gather}
143  \label{eq:DOMCFG_zgr_ana_1}
144    z_0  (k) = h_{sur} - h_0 \; k - \; h_1 \; \log  \big[ \cosh ((k - h_{th}) / h_{cr}) \big] \\
145    e_3^0(k) = \lt|    - h_0      -    h_1 \; \tanh \big[        (k - h_{th}) / h_{cr}  \big] \rt|
146\end{gather}
147
148where $k = 1$ to \jp{jpk} for $w$-levels and $k = 1$ to $k = 1$ for $t-$levels.  Such an
149expression allows us to define a nearly uniform vertical location of levels at the ocean
150top and bottom with a smooth hyperbolic tangent transition in between (\autoref{fig:DOMCFG_zgr}).
151
152A double hyperbolic tangent version (\np[=.true.]{ldbletanh}{ldbletanh}) is also available
153which permits finer control and is used, typically, to obtain a well resolved upper ocean
154without compromising on resolution at depth using a moderate number of levels.
155
156\begin{gather}
157  \label{eq:DOMCFG_zgr_ana_1b}
158    \begin{split}
159    z_0  (k) = h_{sur} - h_0 \; k &- \; h_1 \; \log  \big[ \cosh ((k - h_{th}) / h_{cr}) \big] \\
160                             \;   &- \; h2_1 \; \log  \big[ \cosh ((k - h2_{th}) / h2_{cr}) \big]
161    \end{split}
162\end{gather}
163\begin{gather}
164    \begin{split}
165    e_3^0(k) = \big|    - h_0    &-   h_1 \; \tanh \big[       (k - h_{th})  / h_{cr}   \big]  \\
166                                 &-  h2_1 \; \tanh \big[       (k - h2_{th}) / h2_{cr}  \big] \big|
167    \end{split}
168\end{gather}
169
170If the ice shelf cavities are opened (\np[=.true.]{ln_isfcav}{ln\_isfcav}), the definition
171of $z_0$ is the same.  However, definition of $e_3^0$ at $t$- and $w$-points is
172respectively changed to:
173\begin{equation}
174  \label{eq:DOMCFG_zgr_ana_2}
175  \begin{split}
176    e_3^T(k) &= z_W (k + 1) - z_W (k    ) \\
177    e_3^W(k) &= z_T (k    ) - z_T (k - 1)
178  \end{split}
179\end{equation}
180
181This formulation decreases the self-generated circulation into the ice shelf cavity
182(which can, in extreme case, leads to numerical instability). This is now the recommended formulation for all configurations using v4.0 onwards. The analytical derivation of thicknesses is maintained for backwards compatibility.
183
184The most used vertical grid for ORCA2 has $10~m$ ($500~m$) resolution in the surface
185(bottom) layers and a depth which varies from 0 at the sea surface to a minimum of
186$-5000~m$.  This leads to the following conditions:
187
188\begin{equation}
189  \label{eq:DOMCFG_zgr_coef}
190  \begin{array}{ll}
191    e_3 (1   + 1/2) =  10. & z(1  ) =     0. \\
192    e_3 (jpk - 1/2) = 500. & z(jpk) = -5000.
193  \end{array}
194\end{equation}
195
196With the choice of the stretching $h_{cr} = 3$ and the number of levels \jp{jpk}~$= 31$,
197the four coefficients $h_{sur}$, $h_0$, $h_1$, and $h_{th}$ in
198\autoref{eq:DOMCFG_zgr_ana_2} have been determined such that \autoref{eq:DOMCFG_zgr_coef}
199is satisfied, through an optimisation procedure using a bisection method.
200For the first standard ORCA2 vertical grid this led to the following values:
201$h_{sur} = 4762.96$, $h_0 = 255.58, h_1 = 245.5813$, and $h_{th} = 21.43336$.
202The resulting depths and scale factors as a function of the model levels are shown in
203\autoref{fig:DOMCFG_zgr} and given in \autoref{tab:DOMCFG_orca_zgr}.
204Those values correspond to the parameters \np{ppsur}{ppsur}, \np{ppa0}{ppa0}, \np{ppa1}{ppa1}, \np{ppkth}{ppkth} in \nam{cfg}{cfg} namelist.
205
206Rather than entering parameters $h_{sur}$, $h_0$, and $h_1$ directly, it is possible to
207recalculate them.  In that case the user sets \np{ppsur}{ppsur}~$=$~\np{ppa0}{ppa0}~$=$~\np{ppa1}{ppa1}~$=
208999999$., in \nam{cfg}{cfg} namelist, and specifies instead the four following parameters:
209\begin{itemize}
210\item \np{ppacr}{ppacr}~$= h_{cr}$: stretching factor (nondimensional).
211  The larger \np{ppacr}{ppacr}, the smaller the stretching.
212  Values from $3$ to $10$ are usual.
213\item \np{ppkth}{ppkth}~$= h_{th}$: is approximately the model level at which maximum stretching occurs
214  (nondimensional, usually of order 1/2 or 2/3 of \jp{jpk})
215\item \np{ppdzmin}{ppdzmin}: minimum thickness for the top layer (in meters).
216\item \np{pphmax}{pphmax}: total depth of the ocean (meters).
217\end{itemize}
218
219As an example, for the $45$ layers used in the DRAKKAR configuration those parameters are:
220\jp{jpk}~$= 46$, \np{ppacr}{ppacr}~$= 9$, \np{ppkth}{ppkth}~$= 23.563$, \np{ppdzmin}{ppdzmin}~$= 6~m$,
221\np{pphmax}{pphmax}~$= 5750~m$.
222
223\begin{table}
224  \centering
225  \begin{tabular}{c||r|r|r|r}
226    \hline
227    \textbf{LEVEL} & \textbf{gdept\_1d} & \textbf{gdepw\_1d} & \textbf{e3t\_1d } & \textbf{e3w\_1d} \\
228    \hline
229    1              & \textbf{     5.00} &               0.00 & \textbf{   10.00} &            10.00 \\
230    \hline
231    2              & \textbf{    15.00} &              10.00 & \textbf{   10.00} &            10.00 \\
232    \hline
233    3              & \textbf{    25.00} &              20.00 & \textbf{   10.00} &            10.00 \\
234    \hline
235    4              & \textbf{    35.01} &              30.00 & \textbf{   10.01} &            10.00 \\
236    \hline
237    5              & \textbf{    45.01} &              40.01 & \textbf{   10.01} &            10.01 \\
238    \hline
239    6              & \textbf{    55.03} &              50.02 & \textbf{   10.02} &            10.02 \\
240    \hline
241    7              & \textbf{    65.06} &              60.04 & \textbf{   10.04} &            10.03 \\
242    \hline
243    8              & \textbf{    75.13} &              70.09 & \textbf{   10.09} &            10.06 \\
244    \hline
245    9              & \textbf{    85.25} &              80.18 & \textbf{   10.17} &            10.12 \\
246    \hline
247    10             & \textbf{    95.49} &              90.35 & \textbf{   10.33} &            10.24 \\
248    \hline
249    11             & \textbf{   105.97} &             100.69 & \textbf{   10.65} &            10.47 \\
250    \hline
251    12             & \textbf{   116.90} &             111.36 & \textbf{   11.27} &            10.91 \\
252    \hline
253    13             & \textbf{   128.70} &             122.65 & \textbf{   12.47} &            11.77 \\
254    \hline
255    14             & \textbf{   142.20} &             135.16 & \textbf{   14.78} &            13.43 \\
256    \hline
257    15             & \textbf{   158.96} &             150.03 & \textbf{   19.23} &            16.65 \\
258    \hline
259    16             & \textbf{   181.96} &             169.42 & \textbf{   27.66} &            22.78 \\
260    \hline
261    17             & \textbf{   216.65} &             197.37 & \textbf{   43.26} &            34.30 \\
262    \hline
263    18             & \textbf{   272.48} &             241.13 & \textbf{   70.88} &            55.21 \\
264    \hline
265    19             & \textbf{   364.30} &             312.74 & \textbf{  116.11} &            90.99 \\
266    \hline
267    20             & \textbf{   511.53} &             429.72 & \textbf{  181.55} &           146.43 \\
268    \hline
269    21             & \textbf{   732.20} &             611.89 & \textbf{  261.03} &           220.35 \\
270    \hline
271    22             & \textbf{  1033.22} &             872.87 & \textbf{  339.39} &           301.42 \\
272    \hline
273    23             & \textbf{  1405.70} &            1211.59 & \textbf{  402.26} &           373.31 \\
274    \hline
275    24             & \textbf{  1830.89} &            1612.98 & \textbf{  444.87} &           426.00 \\
276    \hline
277    25             & \textbf{  2289.77} &            2057.13 & \textbf{  470.55} &           459.47 \\
278    \hline
279    26             & \textbf{  2768.24} &            2527.22 & \textbf{  484.95} &           478.83 \\
280    \hline
281    27             & \textbf{  3257.48} &            3011.90 & \textbf{  492.70} &           489.44 \\
282    \hline
283    28             & \textbf{  3752.44} &            3504.46 & \textbf{  496.78} &           495.07 \\
284    \hline
285    29             & \textbf{  4250.40} &            4001.16 & \textbf{  498.90} &           498.02 \\
286    \hline
287    30             & \textbf{  4749.91} &            4500.02 & \textbf{  500.00} &           499.54 \\
288    \hline
289    31             & \textbf{  5250.23} &            5000.00 & \textbf{  500.56} &           500.33 \\
290    \hline
291  \end{tabular}
292  \caption[Default vertical mesh in $z$-coordinate for 30 layers ORCA2 configuration]{
293    Default vertical mesh in $z$-coordinate for 30 layers ORCA2 configuration as
294    computed from \autoref{eq:DOMCFG_zgr_ana_2} using
295    the coefficients given in \autoref{eq:DOMCFG_zgr_coef}}
296  \label{tab:DOMCFG_orca_zgr}
297\end{table}
298%%%YY
299%% % -------------------------------------------------------------------------------------------------------------
300%% %        Meter Bathymetry
301%% % -------------------------------------------------------------------------------------------------------------
302%% =================================================================================================
303\subsection{Model bathymetry}
304\label{subsec:DOMCFG_bathy}
305
306Three options are possible for defining the bathymetry, according to the namelist variable
307\np{nn_bathy}{nn\_bathy} (found in \nam{dom}{dom} namelist (\texttt{DOMAINCFG} variant) ):
308\begin{description}
309\item [{\np[=0]{nn_bathy}{nn\_bathy}}]: a flat-bottom domain is defined.
310  The total depth $z_w (jpk)$ is given by the coordinate transformation.
311  The domain can either be a closed basin or a periodic channel depending on the parameter \np{jperio}{jperio}.
312\item [{\np[=-1]{nn_bathy}{nn\_bathy}}]: a domain with a bump of topography one third of the domain width at the central latitude.
313  This is meant for the "EEL-R5" configuration, a periodic or open boundary channel with a seamount.
314\item [{\np[=1]{nn_bathy}{nn\_bathy}}]: read a bathymetry and ice shelf draft (if needed).
315  The \ifile{bathy\_meter} file (Netcdf format) provides the ocean depth (positive, in meters) at
316  each grid point of the model grid.
317  The bathymetry is usually built by interpolating a standard bathymetry product (\eg\ ETOPO2) onto
318  the horizontal ocean mesh.
319  Defining the bathymetry also defines the coastline: where the bathymetry is zero,
320  no wet levels are defined (all levels are masked).
321
322  The \ifile{isfdraft\_meter} file (Netcdf format) provides the ice shelf draft (positive, in meters) at
323  each grid point of the model grid.
324  This file is only needed if \np[=.true.]{ln_isfcav}{ln\_isfcav}.
325  Defining the ice shelf draft will also define the ice shelf edge and the grounding line position.
326\end{description}
327
328%% =================================================================================================
329\subsection{Choice of vertical grid}
330\label{sec:DOMCFG_vgrd}
331
332After reading the bathymetry, the algorithm for vertical grid definition differs between the different options:
333\begin{description}
334\item [\forcode{ln_zco = .true.}] set a reference coordinate transformation $z_0(k)$, and set $z(i,j,k,t) = z_0(k)$ where $z_0(k)$ is the closest match to the depth at $(i,j)$.
335\item [\forcode{ln_zps = .true.}] set a reference coordinate transformation $z_0(k)$, and calculate the thickness of the deepest level at
336  each $(i,j)$ point using the bathymetry, to obtain the final three-dimensional depth and scale factor arrays.
337\item [\forcode{ln_sco = .true.}] smooth the bathymetry to fulfill the hydrostatic consistency criteria and
338  set the three-dimensional transformation.
339\item [\forcode{s-z and s-zps}] smooth the bathymetry to fulfill the hydrostatic consistency criteria and
340  set the three-dimensional transformation $z(i,j,k)$,
341  and possibly introduce masking of extra land points to better fit the original bathymetry file.
342\end{description}
343
344%% =================================================================================================
345\subsubsection[$Z$-coordinate with uniform thickness levels (\forcode{ln_zco})]{$Z$-coordinate with uniform thickness levels (\protect\np{ln_zco}{ln\_zco})}
346\label{subsec:DOMCFG_zco}
347
348With this option the model topography can be fully described by the reference vertical
349coordinate and a 2D integer field giving the number of wet levels at each location
350(\forcode{bathy_level}). The resulting match to the real topography is likely to be poor
351though (especially with thick, deep levels) and slopes poorly represented. This option is
352rarely used in modern simulations but it can be useful for testing purposes.
353
354%% =================================================================================================
355\subsubsection[$Z$-coordinate with partial step (\forcode{ln_zps})]{$Z$-coordinate with partial step (\protect\np{ln_zps}{ln\_zps})}
356\label{subsec:DOMCFG_zps}
357
358In $z$-coordinate partial step, the depths of the model levels are defined by the
359reference analytical function $z_0(k)$ as described in \autoref{sec:DOMCFG_zref},
360\textit{except} in the bottom layer.  The thickness of the bottom layer is allowed to vary
361as a function of geographical location $(\lambda,\varphi)$ to allow a better
362representation of the bathymetry, especially in the case of small slopes (where the
363bathymetry varies by less than one level thickness from one grid point to the next).  The
364reference layer thicknesses $e_{3t}^0$ have been defined in the absence of bathymetry.
365With partial steps, layers from 1 to \jp{jpk}-2 can have a thickness smaller than
366$e_{3t}(jk)$.
367
368The model deepest layer (\jp{jpk}-1) is allowed to have either a smaller or larger
369thickness than $e_{3t}(jpk)$: the maximum thickness allowed is $2*e_{3t}(jpk - 1)$.
370
371This has to be kept in mind when specifying values in \nam{dom}{dom} namelist
372(\texttt{DOMAINCFG} variant), such as the maximum depth \np{pphmax}{pphmax} in partial steps.
373
374For example, with \np{pphmax}{pphmax}~$= 5750~m$ for the DRAKKAR 45 layer grid, the maximum ocean
375depth allowed is actually $6000~m$ (the default thickness $e_{3t}(jpk - 1)$ being
376$250~m$).  Two variables in the namdom namelist are used to define the partial step
377vertical grid.  The mimimum water thickness (in meters) allowed for a cell partially
378filled with bathymetry at level jk is the minimum of \np{rn_e3zps_min}{rn\_e3zps\_min} (thickness in
379meters, usually $20~m$) or $e_{3t}(jk)*$\np{rn_e3zps_rat}{rn\_e3zps\_rat} (a fraction, usually 10\%, of
380the default thickness $e_{3t}(jk)$).
381
382%% =================================================================================================
383\subsubsection[$S$-coordinate (\forcode{ln_sco})]{$S$-coordinate (\protect\np{ln_sco}{ln\_sco})}
384\label{sec:DOMCFG_sco}
385\begin{listing}
386  \nlst{namzgr_sco_domcfg}
387  \caption{\forcode{&namzgr_sco_domcfg}}
388  \label{lst:namzgr_sco_domcfg}
389\end{listing}
390Options are defined in \nam{zgr_sco}{zgr\_sco} (\texttt{DOMAINcfg} only).
391In $s$-coordinate (\np[=.true.]{ln_sco}{ln\_sco}), the depth and thickness of the model levels are defined from
392the product of a depth field and either a stretching function or its derivative, respectively:
393
394\begin{align*}
395  % \label{eq:DOMCFG_sco_ana}
396  z(k)   &= h(i,j) \; z_0 (k) \\
397  e_3(k) &= h(i,j) \; z_0'(k)
398\end{align*}
399
400where $h$ is the depth of the last $w$-level ($z_0(k)$) defined at the $t$-point location in the horizontal and
401$z_0(k)$ is a function which varies from $0$ at the sea surface to $1$ at the ocean bottom.
402The depth field $h$ is not necessary the ocean depth,
403since a mixed step-like and bottom-following representation of the topography can be used
404(\autoref{fig:DOM_z_zps_s_sps}) or an envelop bathymetry can be defined (\autoref{fig:DOM_z_zps_s_sps}).
405The namelist parameter \np{rn_rmax}{rn\_rmax} determines the slope at which
406the terrain-following coordinate intersects the sea bed and becomes a pseudo z-coordinate.
407The coordinate can also be hybridised by specifying \np{rn_sbot_min}{rn\_sbot\_min} and \np{rn_sbot_max}{rn\_sbot\_max} as
408the minimum and maximum depths at which the terrain-following vertical coordinate is calculated.
409
410Options for stretching the coordinate are provided as examples,
411but care must be taken to ensure that the vertical stretch used is appropriate for the application.
412
413The original default \NEMO\ s-coordinate stretching is available if neither of the other options are specified as true
414(\np[=.false.]{ln_s_SH94}{ln\_s\_SH94} and \np[=.false.]{ln_s_SF12}{ln\_s\_SF12}).
415This uses a depth independent $\tanh$ function for the stretching \citep{madec.delecluse.ea_JPO96}:
416
417\[
418  z = s_{min} + C (s) (H - s_{min})
419  % \label{eq:DOMCFG_SH94_1}
420\]
421
422where $s_{min}$ is the depth at which the $s$-coordinate stretching starts and
423allows a $z$-coordinate to placed on top of the stretched coordinate,
424and $z$ is the depth (negative down from the asea surface).
425\begin{gather*}
426  s = - \frac{k}{n - 1} \quad \text{and} \quad 0 \leq k \leq n - 1
427  % \label{eq:DOMCFG_s}
428 \\
429 \label{eq:DOMCFG_sco_function}
430  C(s) = \frac{[\tanh(\theta \, (s + b)) - \tanh(\theta \, b)]}{2 \; \sinh(\theta)}
431\end{gather*}
432
433A stretching function,
434modified from the commonly used \citet{song.haidvogel_JCP94} stretching (\np[=.true.]{ln_s_SH94}{ln\_s\_SH94}),
435is also available and is more commonly used for shelf seas modelling:
436
437\[
438  C(s) =   (1 - b) \frac{\sinh(\theta s)}{\sinh(\theta)}
439         + b       \frac{\tanh \lt[ \theta \lt(s + \frac{1}{2} \rt) \rt] -   \tanh \lt( \frac{\theta}{2} \rt)}
440                        {                                                  2 \tanh \lt( \frac{\theta}{2} \rt)}
441 \label{eq:DOMCFG_SH94_2}
442\]
443
444\begin{figure}[!ht]
445  \centering
446  \includegraphics[width=0.66\textwidth]{DOMCFG_sco_function}
447  \caption[DOMAINcfg: examples of the stretching function applied to a seamount]{
448    Examples of the stretching function applied to a seamount;
449    from left to right: surface, surface and bottom, and bottom intensified resolutions}
450  \label{fig:DOMCFG_sco_function}
451\end{figure}
452
453where $H_c$ is the critical depth (\np{rn_hc}{rn\_hc}) at which the coordinate transitions from pure $\sigma$ to
454the stretched coordinate, and $\theta$ (\np{rn_theta}{rn\_theta}) and $b$ (\np{rn_bb}{rn\_bb}) are the surface and
455bottom control parameters such that $0 \leqslant \theta \leqslant 20$, and $0 \leqslant b \leqslant 1$.
456$b$ has been designed to allow surface and/or bottom increase of the vertical resolution
457(\autoref{fig:DOMCFG_sco_function}).
458
459Another example has been provided at version 3.5 (\np{ln_s_SF12}{ln\_s\_SF12}) that allows a fixed surface resolution in
460an analytical terrain-following stretching \citet{siddorn.furner_OM13}.
461In this case the a stretching function $\gamma$ is defined such that:
462
463\begin{equation}
464  z = - \gamma h \quad \text{with} \quad 0 \leq \gamma \leq 1
465  % \label{eq:DOMCFG_z}
466\end{equation}
467
468The function is defined with respect to $\sigma$, the unstretched terrain-following coordinate:
469
470\begin{gather*}
471  % \label{eq:DOMCFG_gamma_deriv}
472  \gamma =   A \lt( \sigma   - \frac{1}{2} (\sigma^2     + f (\sigma)) \rt)
473           + B \lt( \sigma^3 - f           (\sigma) \rt) + f (\sigma)       \\
474  \intertext{Where:}
475 \label{eq:DOMCFG_gamma}
476  f(\sigma) = (\alpha + 2) \sigma^{\alpha + 1} - (\alpha + 1) \sigma^{\alpha + 2}
477  \quad \text{and} \quad \sigma = \frac{k}{n - 1}
478\end{gather*}
479
480This gives an analytical stretching of $\sigma$ that is solvable in $A$ and $B$ as a function of
481the user prescribed stretching parameter $\alpha$ (\np{rn_alpha}{rn\_alpha}) that stretches towards
482the surface ($\alpha > 1.0$) or the bottom ($\alpha < 1.0$) and
483user prescribed surface (\np{rn_zs}{rn\_zs}) and bottom depths.
484The bottom cell depth in this example is given as a function of water depth:
485
486\[
487  % \label{eq:DOMCFG_zb}
488  Z_b = h a + b
489\]
490
491where the namelist parameters \np{rn_zb_a}{rn\_zb\_a} and \np{rn_zb_b}{rn\_zb\_b} are $a$ and $b$ respectively.
492
493\begin{figure}[!ht]
494  \centering
495  \includegraphics[width=0.66\textwidth]{DOMCFG_compare_coordinates_surface}
496  \caption[DOMAINcfg: comparison of $s$- and $z$-coordinate]{
497    A comparison of the \citet{song.haidvogel_JCP94} $S$-coordinate (solid lines),
498    a 50 level $Z$-coordinate (contoured surfaces) and
499    the \citet{siddorn.furner_OM13} $S$-coordinate (dashed lines) in the surface $100~m$ for
500    a idealised bathymetry that goes from $50~m$ to $5500~m$ depth.
501    For clarity every third coordinate surface is shown.}
502  \label{fig:DOMCFG_fig_compare_coordinates_surface}
503\end{figure}
504 % >>>>>>>>>>>>>>>>>>>>>>>>>>>>
505
506This gives a smooth analytical stretching in computational space that is constrained to
507given specified surface and bottom grid cell thicknesses in real space.
508This is not to be confused with the hybrid schemes that
509superimpose geopotential coordinates on terrain following coordinates thus
510creating a non-analytical vertical coordinate that
511therefore may suffer from large gradients in the vertical resolutions.
512This stretching is less straightforward to implement than the \citet{song.haidvogel_JCP94} stretching,
513but has the advantage of resolving diurnal processes in deep water and has generally flatter slopes.
514
515As with the \citet{song.haidvogel_JCP94} stretching the stretch is only applied at depths greater than
516the critical depth $h_c$.
517In this example two options are available in depths shallower than $h_c$,
518with pure sigma being applied if the \np{ln_sigcrit}{ln\_sigcrit} is true and pure z-coordinates if it is false
519(the z-coordinate being equal to the depths of the stretched coordinate at $h_c$).
520
521Minimising the horizontal slope of the vertical coordinate is important in terrain-following systems as
522large slopes lead to hydrostatic consistency.
523A hydrostatic consistency parameter diagnostic following \citet{haney_JPO91} has been implemented,
524and is output as part of the model mesh file at the start of the run.
525
526%% =================================================================================================
527\subsubsection[\zstar- or \sstar-coordinate (\forcode{ln_linssh})]{\zstar- or \sstar-coordinate (\protect\np{ln_linssh}{ln\_linssh})}
528\label{subsec:DOMCFG_zgr_star}
529
530This option is described in the Report by Levier \textit{et al.} (2007), available on the \NEMO\ web site.
531
532\subinc{\input{../../global/epilogue}}
533
534\end{document}
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