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1\documentclass[../main/NEMO_manual]{subfiles}
2
3\begin{document}
4% ================================================================
5% Chapter ——— Ocean Dynamics (DYN)
6% ================================================================
7\chapter{Ocean Dynamics (DYN)}
8\label{chap:DYN}
9
10\chaptertoc
11
12Using the representation described in \autoref{chap:DOM},
13several semi-discrete space forms of the dynamical equations are available depending on
14the vertical coordinate used and on the conservation properties of the vorticity term.
15In all the equations presented here, the masking has been omitted for simplicity.
16One must be aware that all the quantities are masked fields and
17that each time an average or difference operator is used, the resulting field is multiplied by a mask.
18
19The prognostic ocean dynamics equation can be summarized as follows:
20\[
21  \text{NXT} = \dbinom  {\text{VOR} + \text{KEG} + \text {ZAD} }
22  {\text{COR} + \text{ADV}                       }
23  + \text{HPG} + \text{SPG} + \text{LDF} + \text{ZDF}
24\]
25NXT stands for next, referring to the time-stepping.
26The first group of terms on the rhs of this equation corresponds to the Coriolis and advection terms that
27are decomposed into either a vorticity part (VOR), a kinetic energy part (KEG) and
28a vertical advection part (ZAD) in the vector invariant formulation,
29or a Coriolis and advection part (COR+ADV) in the flux formulation.
30The terms following these are the pressure gradient contributions
31(HPG, Hydrostatic Pressure Gradient, and SPG, Surface Pressure Gradient);
32and contributions from lateral diffusion (LDF) and vertical diffusion (ZDF),
33which are added to the rhs in the \mdl{dynldf} and \mdl{dynzdf} modules.
34The vertical diffusion term includes the surface and bottom stresses.
35The external forcings and parameterisations require complex inputs
36(surface wind stress calculation using bulk formulae, estimation of mixing coefficients)
37that are carried out in modules SBC, LDF and ZDF and are described in
38\autoref{chap:SBC}, \autoref{chap:LDF} and \autoref{chap:ZDF}, respectively.
39
40In the present chapter we also describe the diagnostic equations used to compute the horizontal divergence,
41curl of the velocities (\emph{divcur} module) and the vertical velocity (\emph{wzvmod} module).
42
43The different options available to the user are managed by namelist variables.
44For term \textit{ttt} in the momentum equations, the logical namelist variables are \textit{ln\_dynttt\_xxx},
45where \textit{xxx} is a 3 or 4 letter acronym corresponding to each optional scheme.
46%If a CPP key is used for this term its name is \key{ttt}.
47The corresponding code can be found in the \textit{dynttt\_xxx} module in the DYN directory,
48and it is usually computed in the \textit{dyn\_ttt\_xxx} subroutine.
49
50The user has the option of extracting and outputting each tendency term from the 3D momentum equations
51(\texttt{trddyn?} defined), as described in \autoref{chap:MISC}.
52Furthermore, the tendency terms associated with the 2D barotropic vorticity balance (when \texttt{trdvor?} is defined)
53can be derived from the 3D terms.
54%%%
55\gmcomment{STEVEN: not quite sure I've got the sense of the last sentence. does
56MISC correspond to "extracting tendency terms" or "vorticity balance"?}
57
58% ================================================================
59% Sea Surface Height evolution & Diagnostics variables
60% ================================================================
61\section{Sea surface height and diagnostic variables ($\eta$, $\zeta$, $\chi$, $w$)}
62\label{sec:DYN_divcur_wzv}
63
64%--------------------------------------------------------------------------------------------------------------
65%           Horizontal divergence and relative vorticity
66%--------------------------------------------------------------------------------------------------------------
67\subsection[Horizontal divergence and relative vorticity (\textit{divcur.F90})]{Horizontal divergence and relative vorticity (\protect\mdl{divcur})}
68\label{subsec:DYN_divcur}
69
70The vorticity is defined at an $f$-point (\ie\ corner point) as follows:
71\begin{equation}
72  \label{eq:DYN_divcur_cur}
73  \zeta =\frac{1}{e_{1f}\,e_{2f} }\left( {\;\delta_{i+1/2} \left[ {e_{2v}\;v} \right]
74      -\delta_{j+1/2} \left[ {e_{1u}\;u} \right]\;} \right)
75\end{equation}
76
77The horizontal divergence is defined at a $T$-point.
78It is given by:
79\[
80  % \label{eq:DYN_divcur_div}
81  \chi =\frac{1}{e_{1t}\,e_{2t}\,e_{3t} }
82  \left( {\delta_i \left[ {e_{2u}\,e_{3u}\,u} \right]
83      +\delta_j \left[ {e_{1v}\,e_{3v}\,v} \right]} \right)
84\]
85
86Note that although the vorticity has the same discrete expression in $z$- and $s$-coordinates,
87its physical meaning is not identical.
88$\zeta$ is a pseudo vorticity along $s$-surfaces
89(only pseudo because $(u,v)$ are still defined along geopotential surfaces,
90but are not necessarily defined at the same depth).
91
92The vorticity and divergence at the \textit{before} step are used in the computation of
93the horizontal diffusion of momentum.
94Note that because they have been calculated prior to the Asselin filtering of the \textit{before} velocities,
95the \textit{before} vorticity and divergence arrays must be included in the restart file to
96ensure perfect restartability.
97The vorticity and divergence at the \textit{now} time step are used for the computation of
98the nonlinear advection and of the vertical velocity respectively.
99
100%--------------------------------------------------------------------------------------------------------------
101%           Sea Surface Height evolution
102%--------------------------------------------------------------------------------------------------------------
103\subsection[Horizontal divergence and relative vorticity (\textit{sshwzv.F90})]{Horizontal divergence and relative vorticity (\protect\mdl{sshwzv})}
104\label{subsec:DYN_sshwzv}
105
106The sea surface height is given by:
107\begin{equation}
108  \label{eq:DYN_spg_ssh}
109  \begin{aligned}
110    \frac{\partial \eta }{\partial t}
111    &\equiv    \frac{1}{e_{1t} e_{2t} }\sum\limits_k { \left\{  \delta_i \left[ {e_{2u}\,e_{3u}\;u} \right]
112        +\delta_j \left[ {e_{1v}\,e_{3v}\;v} \right]  \right\} }
113    -    \frac{\textit{emp}}{\rho_w }   \\
114    &\equiv    \sum\limits_k {\chi \ e_{3t}}  -  \frac{\textit{emp}}{\rho_w }
115  \end{aligned}
116\end{equation}
117where \textit{emp} is the surface freshwater budget (evaporation minus precipitation),
118expressed in Kg/m$^2$/s (which is equal to mm/s),
119and $\rho_w$=1,035~Kg/m$^3$ is the reference density of sea water (Boussinesq approximation).
120If river runoff is expressed as a surface freshwater flux (see \autoref{chap:SBC}) then
121\textit{emp} can be written as the evaporation minus precipitation, minus the river runoff.
122The sea-surface height is evaluated using exactly the same time stepping scheme as
123the tracer equation \autoref{eq:TRA_nxt}:
124a leapfrog scheme in combination with an Asselin time filter,
125\ie\ the velocity appearing in \autoref{eq:DYN_spg_ssh} is centred in time (\textit{now} velocity).
126This is of paramount importance.
127Replacing $T$ by the number $1$ in the tracer equation and summing over the water column must lead to
128the sea surface height equation otherwise tracer content will not be conserved
129\citep{griffies.pacanowski.ea_MWR01, leclair.madec_OM09}.
130
131The vertical velocity is computed by an upward integration of the horizontal divergence starting at the bottom,
132taking into account the change of the thickness of the levels:
133\begin{equation}
134  \label{eq:DYN_wzv}
135  \left\{
136    \begin{aligned}
137      &\left. w \right|_{k_b-1/2} \quad= 0    \qquad \text{where } k_b \text{ is the level just above the sea floor }   \\
138      &\left. w \right|_{k+1/2}     = \left. w \right|_{k-1/2}  +  \left. e_{3t} \right|_{k}\;  \left. \chi \right|_k
139      - \frac{1} {2 \rdt} \left\left. e_{3t}^{t+1}\right|_{k} - \left. e_{3t}^{t-1}\right|_{k}\right)
140    \end{aligned}
141  \right.
142\end{equation}
143
144In the case of a non-linear free surface (\texttt{vvl?}), the top vertical velocity is $-\textit{emp}/\rho_w$,
145as changes in the divergence of the barotropic transport are absorbed into the change of the level thicknesses,
146re-orientated downward.
147\gmcomment{not sure of this...  to be modified with the change in emp setting}
148In the case of a linear free surface, the time derivative in \autoref{eq:DYN_wzv} disappears.
149The upper boundary condition applies at a fixed level $z=0$.
150The top vertical velocity is thus equal to the divergence of the barotropic transport
151(\ie\ the first term in the right-hand-side of \autoref{eq:DYN_spg_ssh}).
152
153Note also that whereas the vertical velocity has the same discrete expression in $z$- and $s$-coordinates,
154its physical meaning is not the same:
155in the second case, $w$ is the velocity normal to the $s$-surfaces.
156Note also that the $k$-axis is re-orientated downwards in the \fortran code compared to
157the indexing used in the semi-discrete equations such as \autoref{eq:DYN_wzv}
158(see \autoref{subsec:DOM_Num_Index_vertical}).
159
160
161% ================================================================
162% Coriolis and Advection terms: vector invariant form
163% ================================================================
164\section{Coriolis and advection: vector invariant form}
165\label{sec:DYN_adv_cor_vect}
166%-----------------------------------------nam_dynadv----------------------------------------------------
167
168\nlst{namdyn_adv}
169%-------------------------------------------------------------------------------------------------------------
170
171The vector invariant form of the momentum equations is the one most often used in
172applications of the \NEMO\ ocean model.
173The flux form option (see next section) has been present since version $2$.
174Options are defined through the \nam{dyn\_adv} namelist variables Coriolis and
175momentum advection terms are evaluated using a leapfrog scheme,
176\ie\ the velocity appearing in these expressions is centred in time (\textit{now} velocity).
177At the lateral boundaries either free slip, no slip or partial slip boundary conditions are applied following
178\autoref{chap:LBC}.
179
180% -------------------------------------------------------------------------------------------------------------
181%        Vorticity term
182% -------------------------------------------------------------------------------------------------------------
183\subsection[Vorticity term (\textit{dynvor.F90})]{Vorticity term (\protect\mdl{dynvor})}
184\label{subsec:DYN_vor}
185%------------------------------------------nam_dynvor----------------------------------------------------
186
187\nlst{namdyn_vor}
188%-------------------------------------------------------------------------------------------------------------
189
190Options are defined through the \nam{dyn\_vor} namelist variables.
191Four discretisations of the vorticity term (\texttt{ln\_dynvor\_xxx}\forcode{=.true.}) are available:
192conserving potential enstrophy of horizontally non-divergent flow (ENS scheme);
193conserving horizontal kinetic energy (ENE scheme);
194conserving potential enstrophy for the relative vorticity term and
195horizontal kinetic energy for the planetary vorticity term (MIX scheme);
196or conserving both the potential enstrophy of horizontally non-divergent flow and horizontal kinetic energy
197(EEN scheme) (see \autoref{subsec:INVARIANTS_vorEEN}).
198In the case of ENS, ENE or MIX schemes the land sea mask may be slightly modified to ensure the consistency of
199vorticity term with analytical equations (\np{ln\_dynvor\_con}\forcode{=.true.}).
200The vorticity terms are all computed in dedicated routines that can be found in the \mdl{dynvor} module.
201
202%-------------------------------------------------------------
203%                 enstrophy conserving scheme
204%-------------------------------------------------------------
205\subsubsection[Enstrophy conserving scheme (\forcode{ln_dynvor_ens = .true.})]{Enstrophy conserving scheme (\protect\np{ln\_dynvor\_ens}\forcode{ = .true.})}
206\label{subsec:DYN_vor_ens}
207
208In the enstrophy conserving case (ENS scheme),
209the discrete formulation of the vorticity term provides a global conservation of the enstrophy
210($ [ (\zeta +f ) / e_{3f} ]^2 $ in $s$-coordinates) for a horizontally non-divergent flow (\ie\ $\chi$=$0$),
211but does not conserve the total kinetic energy.
212It is given by:
213\begin{equation}
214  \label{eq:DYN_vor_ens}
215  \left\{
216    \begin{aligned}
217      {+\frac{1}{e_{1u} } } & {\overline {\left( { \frac{\zeta +f}{e_{3f} }} \right)} }^{\,i}
218      & {\overline{\overline {\left( {e_{1v}\,e_{3v}\;v} \right)}} }^{\,i, j+1/2}    \\
219      {- \frac{1}{e_{2v} } } & {\overline {\left( {\frac{\zeta +f}{e_{3f} }} \right)} }^{\,j}
220      & {\overline{\overline {\left( {e_{2u}\,e_{3u}\;u} \right)}} }^{\,i+1/2, j}
221    \end{aligned}
222  \right.
223\end{equation}
224
225%-------------------------------------------------------------
226%                 energy conserving scheme
227%-------------------------------------------------------------
228\subsubsection[Energy conserving scheme (\forcode{ln_dynvor_ene = .true.})]{Energy conserving scheme (\protect\np{ln\_dynvor\_ene}\forcode{ = .true.})}
229\label{subsec:DYN_vor_ene}
230
231The kinetic energy conserving scheme (ENE scheme) conserves the global kinetic energy but not the global enstrophy.
232It is given by:
233\begin{equation}
234  \label{eq:DYN_vor_ene}
235  \left\{
236    \begin{aligned}
237      {+\frac{1}{e_{1u}}\; {\overline {\left( {\frac{\zeta +f}{e_{3f} }} \right)
238            \;  \overline {\left( {e_{1v}\,e_{3v}\;v} \right)} ^{\,i+1/2}} }^{\,j} }    \\
239      {- \frac{1}{e_{2v}}\; {\overline {\left( {\frac{\zeta +f}{e_{3f} }} \right)
240            \;  \overline {\left( {e_{2u}\,e_{3u}\;u} \right)} ^{\,j+1/2}} }^{\,i} }
241    \end{aligned}
242  \right.
243\end{equation}
244
245%-------------------------------------------------------------
246%                 mix energy/enstrophy conserving scheme
247%-------------------------------------------------------------
248\subsubsection[Mixed energy/enstrophy conserving scheme (\forcode{ln_dynvor_mix = .true.})]{Mixed energy/enstrophy conserving scheme (\protect\np{ln\_dynvor\_mix}\forcode{ = .true.})}
249\label{subsec:DYN_vor_mix}
250
251For the mixed energy/enstrophy conserving scheme (MIX scheme), a mixture of the two previous schemes is used.
252It consists of the ENS scheme (\autoref{eq:DYN_vor_ens}) for the relative vorticity term,
253and of the ENE scheme (\autoref{eq:DYN_vor_ene}) applied to the planetary vorticity term.
254\[
255  % \label{eq:DYN_vor_mix}
256  \left\{ {
257      \begin{aligned}
258        {+\frac{1}{e_{1u} }\; {\overline {\left( {\frac{\zeta }{e_{3f} }} \right)} }^{\,i}
259          \; {\overline{\overline {\left( {e_{1v}\,e_{3v}\;v} \right)}} }^{\,i,j+1/2} -\frac{1}{e_{1u} }
260          \; {\overline {\left( {\frac{f}{e_{3f} }} \right)
261              \;\overline {\left( {e_{1v}\,e_{3v}\;v} \right)} ^{\,i+1/2}} }^{\,j} } \\
262        {-\frac{1}{e_{2v} }\; {\overline {\left( {\frac{\zeta }{e_{3f} }} \right)} }^j
263          \; {\overline{\overline {\left( {e_{2u}\,e_{3u}\;u} \right)}} }^{\,i+1/2,j} +\frac{1}{e_{2v} }
264          \; {\overline {\left( {\frac{f}{e_{3f} }} \right)
265              \;\overline {\left( {e_{2u}\,e_{3u}\;u} \right)} ^{\,j+1/2}} }^{\,i} } \hfill
266      \end{aligned}
267    } \right.
268\]
269
270%-------------------------------------------------------------
271%                 energy and enstrophy conserving scheme
272%-------------------------------------------------------------
273\subsubsection[Energy and enstrophy conserving scheme (\forcode{ln_dynvor_een = .true.})]{Energy and enstrophy conserving scheme (\protect\np{ln\_dynvor\_een}\forcode{ = .true.})}
274\label{subsec:DYN_vor_een}
275
276In both the ENS and ENE schemes,
277it is apparent that the combination of $i$ and $j$ averages of the velocity allows for
278the presence of grid point oscillation structures that will be invisible to the operator.
279These structures are \textit{computational modes} that will be at least partly damped by
280the momentum diffusion operator (\ie\ the subgrid-scale advection), but not by the resolved advection term.
281The ENS and ENE schemes therefore do not contribute to dump any grid point noise in the horizontal velocity field.
282Such noise would result in more noise in the vertical velocity field, an undesirable feature.
283This is a well-known characteristic of $C$-grid discretization where
284$u$ and $v$ are located at different grid points,
285a price worth paying to avoid a double averaging in the pressure gradient term as in the $B$-grid.
286\gmcomment{ To circumvent this, Adcroft (ADD REF HERE)
287Nevertheless, this technique strongly distort the phase and group velocity of Rossby waves....}
288
289A very nice solution to the problem of double averaging was proposed by \citet{arakawa.hsu_MWR90}.
290The idea is to get rid of the double averaging by considering triad combinations of vorticity.
291It is noteworthy that this solution is conceptually quite similar to the one proposed by
292\citep{griffies.gnanadesikan.ea_JPO98} for the discretization of the iso-neutral diffusion operator (see \autoref{apdx:INVARIANTS}).
293
294The \citet{arakawa.hsu_MWR90} vorticity advection scheme for a single layer is modified
295for spherical coordinates as described by \citet{arakawa.lamb_MWR81} to obtain the EEN scheme.
296First consider the discrete expression of the potential vorticity, $q$, defined at an $f$-point:
297\[
298  % \label{eq:DYN_pot_vor}
299  q  = \frac{\zeta +f} {e_{3f} }
300\]
301where the relative vorticity is defined by (\autoref{eq:DYN_divcur_cur}),
302the Coriolis parameter is given by $f=2 \,\Omega \;\sin \varphi _f $ and the layer thickness at $f$-points is:
303\begin{equation}
304  \label{eq:DYN_een_e3f}
305  e_{3f} = \overline{\overline {e_{3t} }} ^{\,i+1/2,j+1/2}
306\end{equation}
307
308%>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
309\begin{figure}[!ht]
310  \begin{center}
311    \includegraphics[width=\textwidth]{Fig_DYN_een_triad}
312    \caption{
313      \protect\label{fig:DYN_een_triad}
314      Triads used in the energy and enstrophy conserving scheme (een) for
315      $u$-component (upper panel) and $v$-component (lower panel).
316    }
317  \end{center}
318\end{figure}
319% >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
320
321A key point in \autoref{eq:DYN_een_e3f} is how the averaging in the \textbf{i}- and \textbf{j}- directions is made.
322It uses the sum of masked t-point vertical scale factor divided either by the sum of the four t-point masks
323(\np{nn\_een\_e3f}\forcode{=1}), or just by $4$ (\np{nn\_een\_e3f}\forcode{=.true.}).
324The latter case preserves the continuity of $e_{3f}$ when one or more of the neighbouring $e_{3t}$ tends to zero and
325extends by continuity the value of $e_{3f}$ into the land areas.
326This case introduces a sub-grid-scale topography at f-points
327(with a systematic reduction of $e_{3f}$ when a model level intercept the bathymetry)
328that tends to reinforce the topostrophy of the flow
329(\ie\ the tendency of the flow to follow the isobaths) \citep{penduff.le-sommer.ea_OS07}.
330
331Next, the vorticity triads, $ {^i_j}\mathbb{Q}^{i_p}_{j_p}$ can be defined at a $T$-point as
332the following triad combinations of the neighbouring potential vorticities defined at f-points
333(\autoref{fig:DYN_een_triad}):
334\begin{equation}
335  \label{eq:DYN_Q_triads}
336  _i^j \mathbb{Q}^{i_p}_{j_p}
337  = \frac{1}{12} \ \left(   q^{i-i_p}_{j+j_p} + q^{i+j_p}_{j+i_p} + q^{i+i_p}_{j-j_p}  \right)
338\end{equation}
339where the indices $i_p$ and $k_p$ take the values: $i_p = -1/2$ or $1/2$ and $j_p = -1/2$ or $1/2$.
340
341Finally, the vorticity terms are represented as:
342\begin{equation}
343  \label{eq:DYN_vor_een}
344  \left\{ {
345      \begin{aligned}
346        +q\,e_3 \, v    &\equiv +\frac{1}{e_{1u} }   \sum_{\substack{i_p,\,k_p}}
347        {^{i+1/2-i_p}_j}  \mathbb{Q}^{i_p}_{j_p}  \left( e_{1v}\,e_{3v} \;\right)^{i+1/2-i_p}_{j+j_p}   \\
348        - q\,e_3 \, u     &\equiv -\frac{1}{e_{2v} }    \sum_{\substack{i_p,\,k_p}}
349        {^i_{j+1/2-j_p}}  \mathbb{Q}^{i_p}_{j_p}  \left( e_{2u}\,e_{3u} \;\right)^{i+i_p}_{j+1/2-j_p}   \\
350      \end{aligned}
351    } \right.
352\end{equation}
353
354This EEN scheme in fact combines the conservation properties of the ENS and ENE schemes.
355It conserves both total energy and potential enstrophy in the limit of horizontally nondivergent flow
356(\ie\ $\chi$=$0$) (see \autoref{subsec:INVARIANTS_vorEEN}).
357Applied to a realistic ocean configuration, it has been shown that it leads to a significant reduction of
358the noise in the vertical velocity field \citep{le-sommer.penduff.ea_OM09}.
359Furthermore, used in combination with a partial steps representation of bottom topography,
360it improves the interaction between current and topography,
361leading to a larger topostrophy of the flow \citep{barnier.madec.ea_OD06, penduff.le-sommer.ea_OS07}.
362
363%--------------------------------------------------------------------------------------------------------------
364%           Kinetic Energy Gradient term
365%--------------------------------------------------------------------------------------------------------------
366\subsection[Kinetic energy gradient term (\textit{dynkeg.F90})]{Kinetic energy gradient term (\protect\mdl{dynkeg})}
367\label{subsec:DYN_keg}
368
369As demonstrated in \autoref{apdx:INVARIANTS},
370there is a single discrete formulation of the kinetic energy gradient term that,
371together with the formulation chosen for the vertical advection (see below),
372conserves the total kinetic energy:
373\[
374  % \label{eq:DYN_keg}
375  \left\{
376    \begin{aligned}
377      -\frac{1}{2 \; e_{1u} }  & \ \delta_{i+1/2} \left[ {\overline {u^2}^{\,i} + \overline{v^2}^{\,j}} \right]   \\
378      -\frac{1}{2 \; e_{2v} }  & \ \delta_{j+1/2} \left[ {\overline {u^2}^{\,i} + \overline{v^2}^{\,j}} \right]
379    \end{aligned}
380  \right.
381\]
382
383%--------------------------------------------------------------------------------------------------------------
384%           Vertical advection term
385%--------------------------------------------------------------------------------------------------------------
386\subsection[Vertical advection term (\textit{dynzad.F90})]{Vertical advection term (\protect\mdl{dynzad})}
387\label{subsec:DYN_zad}
388
389The discrete formulation of the vertical advection, t
390ogether with the formulation chosen for the gradient of kinetic energy (KE) term,
391conserves the total kinetic energy.
392Indeed, the change of KE due to the vertical advection is exactly balanced by
393the change of KE due to the gradient of KE (see \autoref{apdx:INVARIANTS}).
394\[
395  % \label{eq:DYN_zad}
396  \left\{
397    \begin{aligned}
398      -\frac{1} {e_{1u}\,e_{2u}\,e_{3u}} &\ \overline{\ \overline{ e_{1t}\,e_{2t}\;w } ^{\,i+1/2\;\delta_{k+1/2} \left[ u \right]\  }^{\,k}  \\
399      -\frac{1} {e_{1v}\,e_{2v}\,e_{3v}}  &\ \overline{\ \overline{ e_{1t}\,e_{2t}\;w } ^{\,j+1/2\;\delta_{k+1/2} \left[ u \right]\  }^{\,k}
400    \end{aligned}
401  \right.
402\]
403When \np{ln\_dynzad\_zts}\forcode{=.true.},
404a split-explicit time stepping with 5 sub-timesteps is used on the vertical advection term.
405This option can be useful when the value of the timestep is limited by vertical advection \citep{lemarie.debreu.ea_OM15}.
406Note that in this case,
407a similar split-explicit time stepping should be used on vertical advection of tracer to ensure a better stability,
408an option which is only available with a TVD scheme (see \np{ln\_traadv\_tvd\_zts} in \autoref{subsec:TRA_adv_tvd}).
409
410
411% ================================================================
412% Coriolis and Advection : flux form
413% ================================================================
414\section{Coriolis and advection: flux form}
415\label{sec:DYN_adv_cor_flux}
416%------------------------------------------nam_dynadv----------------------------------------------------
417
418\nlst{namdyn_adv}
419%-------------------------------------------------------------------------------------------------------------
420
421Options are defined through the \nam{dyn\_adv} namelist variables.
422In the flux form (as in the vector invariant form),
423the Coriolis and momentum advection terms are evaluated using a leapfrog scheme,
424\ie\ the velocity appearing in their expressions is centred in time (\textit{now} velocity).
425At the lateral boundaries either free slip,
426no slip or partial slip boundary conditions are applied following \autoref{chap:LBC}.
427
428
429%--------------------------------------------------------------------------------------------------------------
430%           Coriolis plus curvature metric terms
431%--------------------------------------------------------------------------------------------------------------
432\subsection[Coriolis plus curvature metric terms (\textit{dynvor.F90})]{Coriolis plus curvature metric terms (\protect\mdl{dynvor})}
433\label{subsec:DYN_cor_flux}
434
435In flux form, the vorticity term reduces to a Coriolis term in which the Coriolis parameter has been modified to account for the "metric" term.
436This altered Coriolis parameter is thus discretised at $f$-points.
437It is given by:
438\begin{multline*}
439  % \label{eq:DYN_cor_metric}
440  f+\frac{1}{e_1 e_2 }\left( {v\frac{\partial e_2 }{\partial i}  -  u\frac{\partial e_1 }{\partial j}} \right\\
441  \equiv   f + \frac{1}{e_{1f} e_{2f} } \left( { \ \overline v ^{i+1/2}\delta_{i+1/2} \left[ {e_{2u} } \right]
442      -  \overline u ^{j+1/2}\delta_{j+1/2} \left[ {e_{1u} } \right]  }  \ \right)
443\end{multline*}
444
445Any of the (\autoref{eq:DYN_vor_ens}), (\autoref{eq:DYN_vor_ene}) and (\autoref{eq:DYN_vor_een}) schemes can be used to
446compute the product of the Coriolis parameter and the vorticity.
447However, the energy-conserving scheme (\autoref{eq:DYN_vor_een}) has exclusively been used to date.
448This term is evaluated using a leapfrog scheme, \ie\ the velocity is centred in time (\textit{now} velocity).
449
450%--------------------------------------------------------------------------------------------------------------
451%           Flux form Advection term
452%--------------------------------------------------------------------------------------------------------------
453\subsection[Flux form advection term (\textit{dynadv.F90})]{Flux form advection term (\protect\mdl{dynadv})}
454\label{subsec:DYN_adv_flux}
455
456The discrete expression of the advection term is given by:
457\[
458  % \label{eq:DYN_adv}
459  \left\{
460    \begin{aligned}
461      \frac{1}{e_{1u}\,e_{2u}\,e_{3u}}
462      \left(      \delta_{i+1/2} \left[ \overline{e_{2u}\,e_{3u}\;u }^{i       }  \ u_t      \right]
463        + \delta_{j       } \left[ \overline{e_{1u}\,e_{3u}\;v }^{i+1/2\ u_f      \right] \right\ \;   \\
464      \left.   + \delta_{k      } \left[ \overline{e_{1w}\,e_{2w}\;w}^{i+1/2\ u_{uw} \right] \right)   \\
465      \\
466      \frac{1}{e_{1v}\,e_{2v}\,e_{3v}}
467      \left(     \delta_{i       } \left[ \overline{e_{2u}\,e_{3u }\;u }^{j+1/2} \ v_f       \right]
468        + \delta_{j+1/2} \left[ \overline{e_{1u}\,e_{3u }\;v }^{i       } \ v_t       \right] \right\ \, \, \\
469      \left+ \delta_{k      } \left[ \overline{e_{1w}\,e_{2w}\;w}^{j+1/2} \ v_{vw}  \right] \right) \\
470    \end{aligned}
471  \right.
472\]
473
474Two advection schemes are available:
475a $2^{nd}$ order centered finite difference scheme, CEN2,
476or a $3^{rd}$ order upstream biased scheme, UBS.
477The latter is described in \citet{shchepetkin.mcwilliams_OM05}.
478The schemes are selected using the namelist logicals \np{ln\_dynadv\_cen2} and \np{ln\_dynadv\_ubs}.
479In flux form, the schemes differ by the choice of a space and time interpolation to define the value of
480$u$ and $v$ at the centre of each face of $u$- and $v$-cells, \ie\ at the $T$-, $f$-,
481and $uw$-points for $u$ and at the $f$-, $T$- and $vw$-points for $v$.
482
483%-------------------------------------------------------------
484%                 2nd order centred scheme
485%-------------------------------------------------------------
486\subsubsection[CEN2: $2^{nd}$ order centred scheme (\forcode{ln_dynadv_cen2 = .true.})]{CEN2: $2^{nd}$ order centred scheme (\protect\np{ln\_dynadv\_cen2}\forcode{ = .true.})}
487\label{subsec:DYN_adv_cen2}
488
489In the centered $2^{nd}$ order formulation, the velocity is evaluated as the mean of the two neighbouring points:
490\begin{equation}
491  \label{eq:DYN_adv_cen2}
492  \left\{
493    \begin{aligned}
494      u_T^{cen2} &=\overline u^{i }       \quad &  u_F^{cen2} &=\overline u^{j+1/2}  \quad &  u_{uw}^{cen2} &=\overline u^{k+1/2}   \\
495      v_F^{cen2} &=\overline v ^{i+1/2} \quad & v_F^{cen2} &=\overline v^j    \quad &  v_{vw}^{cen2} &=\overline v ^{k+1/2}  \\
496    \end{aligned}
497  \right.
498\end{equation}
499
500The scheme is non diffusive (\ie\ conserves the kinetic energy) but dispersive (\ie\ it may create false extrema).
501It is therefore notoriously noisy and must be used in conjunction with an explicit diffusion operator to
502produce a sensible solution.
503The associated time-stepping is performed using a leapfrog scheme in conjunction with an Asselin time-filter,
504so $u$ and $v$ are the \emph{now} velocities.
505
506%-------------------------------------------------------------
507%                 UBS scheme
508%-------------------------------------------------------------
509\subsubsection[UBS: Upstream Biased Scheme (\forcode{ln_dynadv_ubs = .true.})]{UBS: Upstream Biased Scheme (\protect\np{ln\_dynadv\_ubs}\forcode{ = .true.})}
510\label{subsec:DYN_adv_ubs}
511
512The UBS advection scheme is an upstream biased third order scheme based on
513an upstream-biased parabolic interpolation.
514For example, the evaluation of $u_T^{ubs} $ is done as follows:
515\begin{equation}
516  \label{eq:DYN_adv_ubs}
517  u_T^{ubs} =\overline u ^i-\;\frac{1}{6}
518  \begin{cases}
519    u"_{i-1/2}&   \text{if $\ \overline{e_{2u}\,e_{3u} \ u}^i  \geqslant 0$ }    \\
520    u"_{i+1/2}&   \text{if $\ \overline{e_{2u}\,e_{3u} \ u}^i  < 0$ }
521  \end{cases}
522\end{equation}
523where $u"_{i+1/2} =\delta_{i+1/2} \left[ {\delta_i \left[ u \right]} \right]$.
524This results in a dissipatively dominant (\ie\ hyper-diffusive) truncation error
525\citep{shchepetkin.mcwilliams_OM05}.
526The overall performance of the advection scheme is similar to that reported in \citet{farrow.stevens_JPO95}.
527It is a relatively good compromise between accuracy and smoothness.
528It is not a \emph{positive} scheme, meaning that false extrema are permitted.
529But the amplitudes of the false extrema are significantly reduced over those in the centred second order method.
530As the scheme already includes a diffusion component, it can be used without explicit lateral diffusion on momentum
531(\ie\ \np{ln\_dynldf\_lap}\forcode{=}\np{ln\_dynldf\_bilap}\forcode{=.false.}),
532and it is recommended to do so.
533
534The UBS scheme is not used in all directions.
535In the vertical, the centred $2^{nd}$ order evaluation of the advection is preferred, \ie\ $u_{uw}^{ubs}$ and
536$u_{vw}^{ubs}$ in \autoref{eq:DYN_adv_cen2} are used.
537UBS is diffusive and is associated with vertical mixing of momentum. \gmcomment{ gm  pursue the
538sentence:Since vertical mixing of momentum is a source term of the TKE equation...  }
539
540For stability reasons, the first term in (\autoref{eq:DYN_adv_ubs}),
541which corresponds to a second order centred scheme, is evaluated using the \textit{now} velocity (centred in time),
542while the second term, which is the diffusion part of the scheme,
543is evaluated using the \textit{before} velocity (forward in time).
544This is discussed by \citet{webb.de-cuevas.ea_JAOT98} in the context of the Quick advection scheme.
545
546Note that the UBS and QUICK (Quadratic Upstream Interpolation for Convective Kinematics) schemes only differ by
547one coefficient.
548Replacing $1/6$ by $1/8$ in (\autoref{eq:DYN_adv_ubs}) leads to the QUICK advection scheme \citep{webb.de-cuevas.ea_JAOT98}.
549This option is not available through a namelist parameter, since the $1/6$ coefficient is hard coded.
550Nevertheless it is quite easy to make the substitution in the \mdl{dynadv\_ubs} module and obtain a QUICK scheme.
551
552Note also that in the current version of \mdl{dynadv\_ubs},
553there is also the possibility of using a $4^{th}$ order evaluation of the advective velocity as in ROMS.
554This is an error and should be suppressed soon.
555%%%
556\gmcomment{action :  this have to be done}
557%%%
558
559% ================================================================
560%           Hydrostatic pressure gradient term
561% ================================================================
562\section[Hydrostatic pressure gradient (\textit{dynhpg.F90})]{Hydrostatic pressure gradient (\protect\mdl{dynhpg})}
563\label{sec:DYN_hpg}
564%------------------------------------------nam_dynhpg---------------------------------------------------
565
566\nlst{namdyn_hpg}
567%-------------------------------------------------------------------------------------------------------------
568
569Options are defined through the \nam{dyn\_hpg} namelist variables.
570The key distinction between the different algorithms used for
571the hydrostatic pressure gradient is the vertical coordinate used,
572since HPG is a \emph{horizontal} pressure gradient, \ie\ computed along geopotential surfaces.
573As a result, any tilt of the surface of the computational levels will require a specific treatment to
574compute the hydrostatic pressure gradient.
575
576The hydrostatic pressure gradient term is evaluated either using a leapfrog scheme,
577\ie\ the density appearing in its expression is centred in time (\emph{now} $\rho$),
578or a semi-implcit scheme.
579At the lateral boundaries either free slip, no slip or partial slip boundary conditions are applied.
580
581%--------------------------------------------------------------------------------------------------------------
582%           z-coordinate with full step
583%--------------------------------------------------------------------------------------------------------------
584\subsection[Full step $Z$-coordinate (\forcode{ln_dynhpg_zco = .true.})]{Full step $Z$-coordinate (\protect\np{ln\_dynhpg\_zco}\forcode{ = .true.})}
585\label{subsec:DYN_hpg_zco}
586
587The hydrostatic pressure can be obtained by integrating the hydrostatic equation vertically from the surface.
588However, the pressure is large at great depth while its horizontal gradient is several orders of magnitude smaller.
589This may lead to large truncation errors in the pressure gradient terms.
590Thus, the two horizontal components of the hydrostatic pressure gradient are computed directly as follows:
591
592for $k=km$ (surface layer, $jk=1$ in the code)
593\begin{equation}
594  \label{eq:DYN_hpg_zco_surf}
595  \left\{
596    \begin{aligned}
597      \left. \delta_{i+1/2} \left[  p^h          \right] \right|_{k=km}
598      &= \frac{1}{2} g \   \left. \delta_{i+1/2} \left[  e_{3w} \ \rho \right] \right|_{k=km}   \\
599      \left. \delta_{j+1/2} \left[  p^h          \right] \right|_{k=km}
600      &= \frac{1}{2} g \   \left. \delta_{j+1/2} \left[  e_{3w} \ \rho \right] \right|_{k=km}   \\
601    \end{aligned}
602  \right.
603\end{equation}
604
605for $1<k<km$ (interior layer)
606\begin{equation}
607  \label{eq:DYN_hpg_zco}
608  \left\{
609    \begin{aligned}
610      \left. \delta_{i+1/2} \left[  p^h          \right] \right|_{k}
611      &=             \left. \delta_{i+1/2} \left[  p^h          \right] \right|_{k-1}
612      +    \frac{1}{2}\;g\;   \left. \delta_{i+1/2} \left[  e_{3w} \ \overline {\rho}^{k+1/2} \right] \right|_{k}   \\
613      \left. \delta_{j+1/2} \left[  p^h          \right] \right|_{k}
614      &=                \left. \delta_{j+1/2} \left[  p^h          \right] \right|_{k-1}
615      +    \frac{1}{2}\;g\;   \left. \delta_{j+1/2} \left[  e_{3w} \ \overline {\rho}^{k+1/2} \right] \right|_{k}   \\
616    \end{aligned}
617  \right.
618\end{equation}
619
620Note that the $1/2$ factor in (\autoref{eq:DYN_hpg_zco_surf}) is adequate because of the definition of $e_{3w}$ as
621the vertical derivative of the scale factor at the surface level ($z=0$).
622Note also that in case of variable volume level (\texttt{vvl?} defined),
623the surface pressure gradient is included in \autoref{eq:DYN_hpg_zco_surf} and
624\autoref{eq:DYN_hpg_zco} through the space and time variations of the vertical scale factor $e_{3w}$.
625
626%--------------------------------------------------------------------------------------------------------------
627%           z-coordinate with partial step
628%--------------------------------------------------------------------------------------------------------------
629\subsection[Partial step $Z$-coordinate (\forcode{ln_dynhpg_zps = .true.})]{Partial step $Z$-coordinate (\protect\np{ln\_dynhpg\_zps}\forcode{ = .true.})}
630\label{subsec:DYN_hpg_zps}
631
632With partial bottom cells, tracers in horizontally adjacent cells generally live at different depths.
633Before taking horizontal gradients between these tracer points,
634a linear interpolation is used to approximate the deeper tracer as if
635it actually lived at the depth of the shallower tracer point.
636
637Apart from this modification,
638the horizontal hydrostatic pressure gradient evaluated in the $z$-coordinate with partial step is exactly as in
639the pure $z$-coordinate case.
640As explained in detail in section \autoref{sec:TRA_zpshde},
641the nonlinearity of pressure effects in the equation of state is such that
642it is better to interpolate temperature and salinity vertically before computing the density.
643Horizontal gradients of temperature and salinity are needed for the TRA modules,
644which is the reason why the horizontal gradients of density at the deepest model level are computed in
645module \mdl{zpsdhe} located in the TRA directory and described in \autoref{sec:TRA_zpshde}.
646
647%--------------------------------------------------------------------------------------------------------------
648%           s- and s-z-coordinates
649%--------------------------------------------------------------------------------------------------------------
650\subsection{$S$- and $Z$-$S$-coordinates}
651\label{subsec:DYN_hpg_sco}
652
653Pressure gradient formulations in an $s$-coordinate have been the subject of a vast number of papers
654(\eg, \citet{song_MWR98, shchepetkin.mcwilliams_OM05}).
655A number of different pressure gradient options are coded but the ROMS-like,
656density Jacobian with cubic polynomial method is currently disabled whilst known bugs are under investigation.
657
658$\bullet$ Traditional coding (see for example \citet{madec.delecluse.ea_JPO96}: (\np{ln\_dynhpg\_sco}\forcode{=.true.})
659\begin{equation}
660  \label{eq:DYN_hpg_sco}
661  \left\{
662    \begin{aligned}
663      - \frac{1}                 {\rho_o \, e_{1u}} \;   \delta_{i+1/2} \left[  p^h  \right]
664      + \frac{g\; \overline {\rho}^{i+1/2}}  {\rho_o \, e_{1u}} \;   \delta_{i+1/2} \left[  z_t   \right]    \\
665      - \frac{1}                 {\rho_o \, e_{2v}} \;   \delta_{j+1/2} \left[  p^h  \right]
666      + \frac{g\; \overline {\rho}^{j+1/2}}  {\rho_o \, e_{2v}} \;   \delta_{j+1/2} \left[  z_t   \right]    \\
667    \end{aligned}
668  \right.
669\end{equation}
670
671Where the first term is the pressure gradient along coordinates,
672computed as in \autoref{eq:DYN_hpg_zco_surf} - \autoref{eq:DYN_hpg_zco},
673and $z_T$ is the depth of the $T$-point evaluated from the sum of the vertical scale factors at the $w$-point
674($e_{3w}$).
675
676$\bullet$ Traditional coding with adaptation for ice shelf cavities (\np{ln\_dynhpg\_isf}\forcode{=.true.}).
677This scheme need the activation of ice shelf cavities (\np{ln\_isfcav}\forcode{=.true.}).
678
679$\bullet$ Pressure Jacobian scheme (prj) (a research paper in preparation) (\np{ln\_dynhpg\_prj}\forcode{=.true.})
680
681$\bullet$ Density Jacobian with cubic polynomial scheme (DJC) \citep{shchepetkin.mcwilliams_OM05}
682(\np{ln\_dynhpg\_djc}\forcode{=.true.}) (currently disabled; under development)
683
684Note that expression \autoref{eq:DYN_hpg_sco} is commonly used when the variable volume formulation is activated
685(\texttt{vvl?}) because in that case, even with a flat bottom,
686the coordinate surfaces are not horizontal but follow the free surface \citep{levier.treguier.ea_rpt07}.
687The pressure jacobian scheme (\np{ln\_dynhpg\_prj}\forcode{=.true.}) is available as
688an improved option to \np{ln\_dynhpg\_sco}\forcode{=.true.} when \texttt{vvl?} is active.
689The pressure Jacobian scheme uses a constrained cubic spline to
690reconstruct the density profile across the water column.
691This method maintains the monotonicity between the density nodes.
692The pressure can be calculated by analytical integration of the density profile and
693a pressure Jacobian method is used to solve the horizontal pressure gradient.
694This method can provide a more accurate calculation of the horizontal pressure gradient than the standard scheme.
695
696\subsection{Ice shelf cavity}
697\label{subsec:DYN_hpg_isf}
698
699Beneath an ice shelf, the total pressure gradient is the sum of the pressure gradient due to the ice shelf load and
700the pressure gradient due to the ocean load (\np{ln\_dynhpg\_isf}\forcode{=.true.}).\\
701
702The main hypothesis to compute the ice shelf load is that the ice shelf is in an isostatic equilibrium.
703The top pressure is computed integrating from surface to the base of the ice shelf a reference density profile
704(prescribed as density of a water at 34.4 PSU and -1.9\deg{C}) and
705corresponds to the water replaced by the ice shelf.
706This top pressure is constant over time.
707A detailed description of this method is described in \citet{losch_JGR08}.\\
708
709The pressure gradient due to ocean load is computed using the expression \autoref{eq:DYN_hpg_sco} described in
710\autoref{subsec:DYN_hpg_sco}.
711
712%--------------------------------------------------------------------------------------------------------------
713%           Time-scheme
714%--------------------------------------------------------------------------------------------------------------
715\subsection[Time-scheme (\forcode{ln_dynhpg_imp = .{true,false}.})]{Time-scheme (\protect\np{ln\_dynhpg\_imp}\forcode{ = .\{true,false\}}.)}
716\label{subsec:DYN_hpg_imp}
717
718The default time differencing scheme used for the horizontal pressure gradient is a leapfrog scheme and
719therefore the density used in all discrete expressions given above is the  \textit{now} density,
720computed from the \textit{now} temperature and salinity.
721In some specific cases
722(usually high resolution simulations over an ocean domain which includes weakly stratified regions)
723the physical phenomenon that controls the time-step is internal gravity waves (IGWs).
724A semi-implicit scheme for doubling the stability limit associated with IGWs can be used
725\citep{brown.campana_MWR78, maltrud.smith.ea_JGR98}.
726It involves the evaluation of the hydrostatic pressure gradient as
727an average over the three time levels $t-\rdt$, $t$, and $t+\rdt$
728(\ie\ \textit{before}, \textit{now} and  \textit{after} time-steps),
729rather than at the central time level $t$ only, as in the standard leapfrog scheme.
730
731$\bullet$ leapfrog scheme (\np{ln\_dynhpg\_imp}\forcode{=.true.}):
732
733\begin{equation}
734  \label{eq:DYN_hpg_lf}
735  \frac{u^{t+\rdt}-u^{t-\rdt}}{2\rdt} = \;\cdots \;
736  -\frac{1}{\rho_o \,e_{1u} }\delta_{i+1/2} \left[ {p_h^t } \right]
737\end{equation}
738
739$\bullet$ semi-implicit scheme (\np{ln\_dynhpg\_imp}\forcode{=.true.}):
740\begin{equation}
741  \label{eq:DYN_hpg_imp}
742  \frac{u^{t+\rdt}-u^{t-\rdt}}{2\rdt} = \;\cdots \;
743  -\frac{1}{4\,\rho_o \,e_{1u} } \delta_{i+1/2} \left[ p_h^{t+\rdt} +2\,p_h^t +p_h^{t-\rdt}  \right]
744\end{equation}
745
746The semi-implicit time scheme \autoref{eq:DYN_hpg_imp} is made possible without
747significant additional computation since the density can be updated to time level $t+\rdt$ before
748computing the horizontal hydrostatic pressure gradient.
749It can be easily shown that the stability limit associated with the hydrostatic pressure gradient doubles using
750\autoref{eq:DYN_hpg_imp} compared to that using the standard leapfrog scheme \autoref{eq:DYN_hpg_lf}.
751Note that \autoref{eq:DYN_hpg_imp} is equivalent to applying a time filter to the pressure gradient to
752eliminate high frequency IGWs.
753Obviously, when using \autoref{eq:DYN_hpg_imp},
754the doubling of the time-step is achievable only if no other factors control the time-step,
755such as the stability limits associated with advection or diffusion.
756
757In practice, the semi-implicit scheme is used when \np{ln\_dynhpg\_imp}\forcode{=.true.}.
758In this case, we choose to apply the time filter to temperature and salinity used in the equation of state,
759instead of applying it to the hydrostatic pressure or to the density,
760so that no additional storage array has to be defined.
761The density used to compute the hydrostatic pressure gradient (whatever the formulation) is evaluated as follows:
762\[
763  % \label{eq:DYN_rho_flt}
764  \rho^t = \rho( \widetilde{T},\widetilde {S},z_t)
765  \quad    \text{with}  \quad
766  \widetilde{X} = 1 / 4 \left(  X^{t+\rdt} +2 \,X^t + X^{t-\rdt\right)
767\]
768
769Note that in the semi-implicit case, it is necessary to save the filtered density,
770an extra three-dimensional field, in the restart file to restart the model with exact reproducibility.
771This option is controlled by  \np{nn\_dynhpg\_rst}, a namelist parameter.
772
773% ================================================================
774% Surface Pressure Gradient
775% ================================================================
776\section[Surface pressure gradient (\textit{dynspg.F90})]{Surface pressure gradient (\protect\mdl{dynspg})}
777\label{sec:DYN_spg}
778%-----------------------------------------nam_dynspg----------------------------------------------------
779
780\nlst{namdyn_spg}
781%------------------------------------------------------------------------------------------------------------
782
783Options are defined through the \nam{dyn\_spg} namelist variables.
784The surface pressure gradient term is related to the representation of the free surface (\autoref{sec:MB_hor_pg}).
785The main distinction is between the fixed volume case (linear free surface) and
786the variable volume case (nonlinear free surface, \texttt{vvl?} is defined).
787In the linear free surface case (\autoref{subsec:MB_free_surface})
788the vertical scale factors $e_{3}$ are fixed in time,
789while they are time-dependent in the nonlinear case (\autoref{subsec:MB_free_surface}).
790With both linear and nonlinear free surface, external gravity waves are allowed in the equations,
791which imposes a very small time step when an explicit time stepping is used.
792Two methods are proposed to allow a longer time step for the three-dimensional equations:
793the filtered free surface, which is a modification of the continuous equations (see \autoref{eq:MB_flt?}),
794and the split-explicit free surface described below.
795The extra term introduced in the filtered method is calculated implicitly,
796so that the update of the next velocities is done in module \mdl{dynspg\_flt} and not in \mdl{dynnxt}.
797
798
799The form of the surface pressure gradient term depends on how the user wants to
800handle the fast external gravity waves that are a solution of the analytical equation (\autoref{sec:MB_hor_pg}).
801Three formulations are available, all controlled by a CPP key (ln\_dynspg\_xxx):
802an explicit formulation which requires a small time step;
803a filtered free surface formulation which allows a larger time step by
804adding a filtering term into the momentum equation;
805and a split-explicit free surface formulation, described below, which also allows a larger time step.
806
807The extra term introduced in the filtered method is calculated implicitly, so that a solver is used to compute it.
808As a consequence the update of the $next$ velocities is done in module \mdl{dynspg\_flt} and not in \mdl{dynnxt}.
809
810
811%--------------------------------------------------------------------------------------------------------------
812% Explicit free surface formulation
813%--------------------------------------------------------------------------------------------------------------
814\subsection[Explicit free surface (\texttt{ln\_dynspg\_exp}\forcode{ = .true.})]{Explicit free surface (\protect\np{ln\_dynspg\_exp}\forcode{ = .true.})}
815\label{subsec:DYN_spg_exp}
816
817In the explicit free surface formulation (\np{ln\_dynspg\_exp} set to true),
818the model time step is chosen to be small enough to resolve the external gravity waves
819(typically a few tens of seconds).
820The surface pressure gradient, evaluated using a leap-frog scheme (\ie\ centered in time),
821is thus simply given by :
822\begin{equation}
823  \label{eq:DYN_spg_exp}
824  \left\{
825    \begin{aligned}
826      - \frac{1}{e_{1u}\,\rho_o} \; \delta_{i+1/2} \left[  \,\rho \,\eta\,  \right]    \\
827      - \frac{1}{e_{2v}\,\rho_o} \; \delta_{j+1/2} \left[  \,\rho \,\eta\,  \right]
828    \end{aligned}
829  \right.
830\end{equation}
831
832Note that in the non-linear free surface case (\ie\ \texttt{vvl?} defined),
833the surface pressure gradient is already included in the momentum tendency through
834the level thickness variation allowed in the computation of the hydrostatic pressure gradient.
835Thus, nothing is done in the \mdl{dynspg\_exp} module.
836
837%--------------------------------------------------------------------------------------------------------------
838% Split-explict free surface formulation
839%--------------------------------------------------------------------------------------------------------------
840\subsection[Split-explicit free surface (\texttt{ln\_dynspg\_ts}\forcode{ = .true.})]{Split-explicit free surface (\protect\np{ln\_dynspg\_ts}\forcode{ = .true.})}
841\label{subsec:DYN_spg_ts}
842%------------------------------------------namsplit-----------------------------------------------------------
843%
844%\nlst{namsplit}
845%-------------------------------------------------------------------------------------------------------------
846
847The split-explicit free surface formulation used in \NEMO\ (\np{ln\_dynspg\_ts} set to true),
848also called the time-splitting formulation, follows the one proposed by \citet{shchepetkin.mcwilliams_OM05}.
849The general idea is to solve the free surface equation and the associated barotropic velocity equations with
850a smaller time step than $\rdt$, the time step used for the three dimensional prognostic variables
851(\autoref{fig:DYN_spg_ts}).
852The size of the small time step, $\rdt_e$ (the external mode or barotropic time step) is provided through
853the \np{nn\_baro} namelist parameter as: $\rdt_e = \rdt / nn\_baro$.
854This parameter can be optionally defined automatically (\np{ln\_bt\_nn\_auto}\forcode{=.true.}) considering that
855the stability of the barotropic system is essentially controled by external waves propagation.
856Maximum Courant number is in that case time independent, and easily computed online from the input bathymetry.
857Therefore, $\rdt_e$ is adjusted so that the Maximum allowed Courant number is smaller than \np{rn\_bt\_cmax}.
858
859%%%
860The barotropic mode solves the following equations:
861% \begin{subequations}
862%  \label{eq:DYN_BT}
863\begin{equation}
864  \label{eq:DYN_BT_dyn}
865  \frac{\partial {\mathrm \overline{{\mathbf U}}_h} }{\partial t}=
866  -f\;{\mathrm {\mathbf k}}\times {\mathrm \overline{{\mathbf U}}_h}
867  -g\nabla _h \eta -\frac{c_b^{\textbf U}}{H+\eta} \mathrm {\overline{{\mathbf U}}_h} + \mathrm {\overline{\mathbf G}}
868\end{equation}
869\[
870  % \label{eq:DYN_BT_ssh}
871  \frac{\partial \eta }{\partial t}=-\nabla \cdot \left[ {\left( {H+\eta } \right) \; {\mathrm{\mathbf \overline{U}}}_h \,} \right]+P-E
872\]
873% \end{subequations}
874where $\mathrm {\overline{\mathbf G}}$ is a forcing term held constant, containing coupling term between modes,
875surface atmospheric forcing as well as slowly varying barotropic terms not explicitly computed to gain efficiency.
876The third term on the right hand side of \autoref{eq:DYN_BT_dyn} represents the bottom stress
877(see section \autoref{sec:ZDF_drg}), explicitly accounted for at each barotropic iteration.
878Temporal discretization of the system above follows a three-time step Generalized Forward Backward algorithm
879detailed in \citet{shchepetkin.mcwilliams_OM05}.
880AB3-AM4 coefficients used in \NEMO\ follow the second-order accurate,
881"multi-purpose" stability compromise as defined in \citet{shchepetkin.mcwilliams_ibk09}
882(see their figure 12, lower left).
883
884%>   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >
885\begin{figure}[!t]
886  \begin{center}
887    \includegraphics[width=\textwidth]{Fig_DYN_dynspg_ts}
888    \caption{
889      \protect\label{fig:DYN_spg_ts}
890      Schematic of the split-explicit time stepping scheme for the external and internal modes.
891      Time increases to the right. In this particular exemple,
892      a boxcar averaging window over $nn\_baro$ barotropic time steps is used ($nn\_bt\_flt=1$) and $nn\_baro=5$.
893      Internal mode time steps (which are also the model time steps) are denoted by $t-\rdt$, $t$ and $t+\rdt$.
894      Variables with $k$ superscript refer to instantaneous barotropic variables,
895      $< >$ and $<< >>$ operator refer to time filtered variables using respectively primary (red vertical bars) and
896      secondary weights (blue vertical bars).
897      The former are used to obtain time filtered quantities at $t+\rdt$ while
898      the latter are used to obtain time averaged transports to advect tracers.
899      a) Forward time integration: \protect\np{ln\_bt\_fw}\forcode{=.true.},
900      \protect\np{ln\_bt\_av}\forcode{=.true.}.
901      b) Centred time integration: \protect\np{ln\_bt\_fw}\forcode{=.false.},
902      \protect\np{ln\_bt\_av}\forcode{=.true.}.
903      c) Forward time integration with no time filtering (POM-like scheme):
904      \protect\np{ln\_bt\_fw}\forcode{=.true.}, \protect\np{ln\_bt\_av}\forcode{=.false.}.
905    }
906  \end{center}
907\end{figure}
908%>   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >
909
910In the default case (\np{ln\_bt\_fw}\forcode{=.true.}),
911the external mode is integrated between \textit{now} and \textit{after} baroclinic time-steps
912(\autoref{fig:DYN_spg_ts}a).
913To avoid aliasing of fast barotropic motions into three dimensional equations,
914time filtering is eventually applied on barotropic quantities (\np{ln\_bt\_av}\forcode{=.true.}).
915In that case, the integration is extended slightly beyond \textit{after} time step to
916provide time filtered quantities.
917These are used for the subsequent initialization of the barotropic mode in the following baroclinic step.
918Since external mode equations written at baroclinic time steps finally follow a forward time stepping scheme,
919asselin filtering is not applied to barotropic quantities.\\
920Alternatively, one can choose to integrate barotropic equations starting from \textit{before} time step
921(\np{ln\_bt\_fw}\forcode{=.false.}).
922Although more computationaly expensive ( \np{nn\_baro} additional iterations are indeed necessary),
923the baroclinic to barotropic forcing term given at \textit{now} time step become centred in
924the middle of the integration window.
925It can easily be shown that this property removes part of splitting errors between modes,
926which increases the overall numerical robustness.
927%references to Patrick Marsaleix' work here. Also work done by SHOM group.
928
929%%%
930
931As far as tracer conservation is concerned,
932barotropic velocities used to advect tracers must also be updated at \textit{now} time step.
933This implies to change the traditional order of computations in \NEMO:
934most of momentum trends (including the barotropic mode calculation) updated first, tracers' after.
935This \textit{de facto} makes semi-implicit hydrostatic pressure gradient
936(see section \autoref{subsec:DYN_hpg_imp})
937and time splitting not compatible.
938Advective barotropic velocities are obtained by using a secondary set of filtering weights,
939uniquely defined from the filter coefficients used for the time averaging (\citet{shchepetkin.mcwilliams_OM05}).
940Consistency between the time averaged continuity equation and the time stepping of tracers is here the key to
941obtain exact conservation.
942
943%%%
944
945One can eventually choose to feedback instantaneous values by not using any time filter
946(\np{ln\_bt\_av}\forcode{=.false.}).
947In that case, external mode equations are continuous in time,
948\ie\ they are not re-initialized when starting a new sub-stepping sequence.
949This is the method used so far in the POM model, the stability being maintained by
950refreshing at (almost) each barotropic time step advection and horizontal diffusion terms.
951Since the latter terms have not been added in \NEMO\ for computational efficiency,
952removing time filtering is not recommended except for debugging purposes.
953This may be used for instance to appreciate the damping effect of the standard formulation on
954external gravity waves in idealized or weakly non-linear cases.
955Although the damping is lower than for the filtered free surface,
956it is still significant as shown by \citet{levier.treguier.ea_rpt07} in the case of an analytical barotropic Kelvin wave.
957
958%>>>>>===============
959\gmcomment{               %%% copy from griffies Book
960
961\textbf{title: Time stepping the barotropic system }
962
963Assume knowledge of the full velocity and tracer fields at baroclinic time $\tau$.
964Hence, we can update the surface height and vertically integrated velocity with a leap-frog scheme using
965the small barotropic time step $\rdt$.
966We have
967
968\[
969  % \label{eq:DYN_spg_ts_eta}
970  \eta^{(b)}(\tau,t_{n+1}) - \eta^{(b)}(\tau,t_{n+1}) (\tau,t_{n-1})
971  = 2 \rdt \left[-\nabla \cdot \textbf{U}^{(b)}(\tau,t_n) + \text{EMP}_w(\tau) \right]
972\]
973\begin{multline*}
974  % \label{eq:DYN_spg_ts_u}
975  \textbf{U}^{(b)}(\tau,t_{n+1}) - \textbf{U}^{(b)}(\tau,t_{n-1}\\
976  = 2\rdt \left[ - f \textbf{k} \times \textbf{U}^{(b)}(\tau,t_{n})
977    - H(\tau) \nabla p_s^{(b)}(\tau,t_{n}) +\textbf{M}(\tau) \right]
978\end{multline*}
979\
980
981In these equations, araised (b) denotes values of surface height and vertically integrated velocity updated with
982the barotropic time steps.
983The $\tau$ time label on $\eta^{(b)}$ and $U^{(b)}$ denotes the baroclinic time at which
984the vertically integrated forcing $\textbf{M}(\tau)$
985(note that this forcing includes the surface freshwater forcing),
986the tracer fields, the freshwater flux $\text{EMP}_w(\tau)$,
987and total depth of the ocean $H(\tau)$ are held for the duration of the barotropic time stepping over
988a single cycle.
989This is also the time that sets the barotropic time steps via
990\[
991  % \label{eq:DYN_spg_ts_t}
992  t_n=\tau+n\rdt
993\]
994with $n$ an integer.
995The density scaled surface pressure is evaluated via
996\[
997  % \label{eq:DYN_spg_ts_ps}
998  p_s^{(b)}(\tau,t_{n}) =
999  \begin{cases}
1000    g \;\eta_s^{(b)}(\tau,t_{n}) \;\rho(\tau)_{k=1}) / \rho_o  &      \text{non-linear case} \\
1001    g \;\eta_s^{(b)}(\tau,t_{n})  &      \text{linear case}
1002  \end{cases}
1003\]
1004To get started, we assume the following initial conditions
1005\[
1006  % \label{eq:DYN_spg_ts_eta}
1007  \begin{split}
1008    \eta^{(b)}(\tau,t_{n=0}) &= \overline{\eta^{(b)}(\tau)}    \\
1009    \eta^{(b)}(\tau,t_{n=1}) &= \eta^{(b)}(\tau,t_{n=0}) + \rdt \ \text{RHS}_{n=0}
1010  \end{split}
1011\]
1012with
1013\[
1014  % \label{eq:DYN_spg_ts_etaF}
1015  \overline{\eta^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N \eta^{(b)}(\tau-\rdt,t_{n})
1016\]
1017the time averaged surface height taken from the previous barotropic cycle.
1018Likewise,
1019\[
1020  % \label{eq:DYN_spg_ts_u}
1021  \textbf{U}^{(b)}(\tau,t_{n=0}) = \overline{\textbf{U}^{(b)}(\tau)} \\ \\
1022  \textbf{U}(\tau,t_{n=1}) = \textbf{U}^{(b)}(\tau,t_{n=0}) + \rdt \ \text{RHS}_{n=0}
1023\]
1024with
1025\[
1026  % \label{eq:DYN_spg_ts_u}
1027  \overline{\textbf{U}^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau-\rdt,t_{n})
1028\]
1029the time averaged vertically integrated transport.
1030Notably, there is no Robert-Asselin time filter used in the barotropic portion of the integration.
1031
1032Upon reaching $t_{n=N} = \tau + 2\rdt \tau$ ,
1033the vertically integrated velocity is time averaged to produce the updated vertically integrated velocity at
1034baroclinic time $\tau + \rdt \tau$
1035\[
1036  % \label{eq:DYN_spg_ts_u}
1037  \textbf{U}(\tau+\rdt) = \overline{\textbf{U}^{(b)}(\tau+\rdt)} = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau,t_{n})
1038\]
1039The surface height on the new baroclinic time step is then determined via a baroclinic leap-frog using
1040the following form
1041
1042\begin{equation}
1043  \label{eq:DYN_spg_ts_ssh}
1044  \eta(\tau+\Delta) - \eta^{F}(\tau-\Delta) = 2\rdt \ \left[ - \nabla \cdot \textbf{U}(\tau) + \text{EMP}_w \right]
1045\end{equation}
1046
1047The use of this "big-leap-frog" scheme for the surface height ensures compatibility between
1048the mass/volume budgets and the tracer budgets.
1049More discussion of this point is provided in Chapter 10 (see in particular Section 10.2).
1050
1051In general, some form of time filter is needed to maintain integrity of the surface height field due to
1052the leap-frog splitting mode in equation \autoref{eq:DYN_spg_ts_ssh}.
1053We have tried various forms of such filtering,
1054with the following method discussed in \cite{griffies.pacanowski.ea_MWR01} chosen due to
1055its stability and reasonably good maintenance of tracer conservation properties (see ??).
1056
1057\begin{equation}
1058  \label{eq:DYN_spg_ts_sshf}
1059  \eta^{F}(\tau-\Delta) =  \overline{\eta^{(b)}(\tau)}
1060\end{equation}
1061Another approach tried was
1062
1063\[
1064  % \label{eq:DYN_spg_ts_sshf2}
1065  \eta^{F}(\tau-\Delta) = \eta(\tau)
1066  + (\alpha/2) \left[\overline{\eta^{(b)}}(\tau+\rdt)
1067    + \overline{\eta^{(b)}}(\tau-\rdt) -2 \;\eta(\tau) \right]
1068\]
1069
1070which is useful since it isolates all the time filtering aspects into the term multiplied by $\alpha$.
1071This isolation allows for an easy check that tracer conservation is exact when
1072eliminating tracer and surface height time filtering (see ?? for more complete discussion).
1073However, in the general case with a non-zero $\alpha$,
1074the filter \autoref{eq:DYN_spg_ts_sshf} was found to be more conservative, and so is recommended.
1075
1076}            %%end gm comment (copy of griffies book)
1077
1078%>>>>>===============
1079
1080
1081%--------------------------------------------------------------------------------------------------------------
1082% Filtered free surface formulation
1083%--------------------------------------------------------------------------------------------------------------
1084\subsection[Filtered free surface (\texttt{dynspg\_flt?})]{Filtered free surface (\protect\texttt{dynspg\_flt?})}
1085\label{subsec:DYN_spg_fltp}
1086
1087The filtered formulation follows the \citet{roullet.madec_JGR00} implementation.
1088The extra term introduced in the equations (see \autoref{subsec:MB_free_surface}) is solved implicitly.
1089The elliptic solvers available in the code are documented in \autoref{chap:MISC}.
1090
1091%% gm %%======>>>>   given here the discrete eqs provided to the solver
1092\gmcomment{               %%% copy from chap-model basics
1093  \[
1094    % \label{eq:DYN_spg_flt}
1095    \frac{\partial {\mathrm {\mathbf U}}_h }{\partial t}= {\mathrm {\mathbf M}}
1096    - g \nabla \left( \tilde{\rho} \ \eta \right)
1097    - g \ T_c \nabla \left( \widetilde{\rho} \ \partial_t \eta \right)
1098  \]
1099  where $T_c$, is a parameter with dimensions of time which characterizes the force,
1100  $\widetilde{\rho} = \rho / \rho_o$ is the dimensionless density,
1101  and $\mathrm {\mathbf M}$ represents the collected contributions of the Coriolis, hydrostatic pressure gradient,
1102  non-linear and viscous terms in \autoref{eq:MB_dyn}.
1103}   %end gmcomment
1104
1105Note that in the linear free surface formulation (\texttt{vvl?} not defined),
1106the ocean depth is time-independent and so is the matrix to be inverted.
1107It is computed once and for all and applies to all ocean time steps.
1108
1109% ================================================================
1110% Lateral diffusion term
1111% ================================================================
1112\section[Lateral diffusion term and operators (\textit{dynldf.F90})]{Lateral diffusion term and operators (\protect\mdl{dynldf})}
1113\label{sec:DYN_ldf}
1114%------------------------------------------nam_dynldf----------------------------------------------------
1115
1116\nlst{namdyn_ldf}
1117%-------------------------------------------------------------------------------------------------------------
1118
1119Options are defined through the \nam{dyn\_ldf} namelist variables.
1120The options available for lateral diffusion are to use either laplacian (rotated or not) or biharmonic operators.
1121The coefficients may be constant or spatially variable;
1122the description of the coefficients is found in the chapter on lateral physics (\autoref{chap:LDF}).
1123The lateral diffusion of momentum is evaluated using a forward scheme,
1124\ie\ the velocity appearing in its expression is the \textit{before} velocity in time,
1125except for the pure vertical component that appears when a tensor of rotation is used.
1126This latter term is solved implicitly together with the vertical diffusion term (see \autoref{chap:TD}).
1127
1128At the lateral boundaries either free slip,
1129no slip or partial slip boundary conditions are applied according to the user's choice (see \autoref{chap:LBC}).
1130
1131\gmcomment{
1132  Hyperviscous operators are frequently used in the simulation of turbulent flows to
1133  control the dissipation of unresolved small scale features.
1134  Their primary role is to provide strong dissipation at the smallest scale supported by
1135  the grid while minimizing the impact on the larger scale features.
1136  Hyperviscous operators are thus designed to be more scale selective than the traditional,
1137  physically motivated Laplace operator.
1138  In finite difference methods,
1139  the biharmonic operator is frequently the method of choice to achieve this scale selective dissipation since
1140  its damping time (\ie\ its spin down time) scale like $\lambda^{-4}$ for disturbances of wavelength $\lambda$
1141  (so that short waves damped more rapidelly than long ones),
1142  whereas the Laplace operator damping time scales only like $\lambda^{-2}$.
1143}
1144
1145% ================================================================
1146\subsection[Iso-level laplacian (\forcode{ln_dynldf_lap = .true.})]{Iso-level laplacian operator (\protect\np{ln\_dynldf\_lap}\forcode{ = .true.})}
1147\label{subsec:DYN_ldf_lap}
1148
1149For lateral iso-level diffusion, the discrete operator is:
1150\begin{equation}
1151  \label{eq:DYN_ldf_lap}
1152  \left\{
1153    \begin{aligned}
1154      D_u^{l{\mathrm {\mathbf U}}} =\frac{1}{e_{1u} }\delta_{i+1/2} \left[ {A_T^{lm}
1155          \;\chi } \right]-\frac{1}{e_{2u} {\kern 1pt}e_{3u} }\delta_j \left[
1156        {A_f^{lm} \;e_{3f} \zeta } \right] \\ \\
1157      D_v^{l{\mathrm {\mathbf U}}} =\frac{1}{e_{2v} }\delta_{j+1/2} \left[ {A_T^{lm}
1158          \;\chi } \right]+\frac{1}{e_{1v} {\kern 1pt}e_{3v} }\delta_i \left[
1159        {A_f^{lm} \;e_{3f} \zeta } \right]
1160    \end{aligned}
1161  \right.
1162\end{equation}
1163
1164As explained in \autoref{subsec:MB_ldf},
1165this formulation (as the gradient of a divergence and curl of the vorticity) preserves symmetry and
1166ensures a complete separation between the vorticity and divergence parts of the momentum diffusion.
1167
1168%--------------------------------------------------------------------------------------------------------------
1169%           Rotated laplacian operator
1170%--------------------------------------------------------------------------------------------------------------
1171\subsection[Rotated laplacian (\forcode{ln_dynldf_iso = .true.})]{Rotated laplacian operator (\protect\np{ln\_dynldf\_iso}\forcode{ = .true.})}
1172\label{subsec:DYN_ldf_iso}
1173
1174A rotation of the lateral momentum diffusion operator is needed in several cases:
1175for iso-neutral diffusion in the $z$-coordinate (\np{ln\_dynldf\_iso}\forcode{=.true.}) and
1176for either iso-neutral (\np{ln\_dynldf\_iso}\forcode{=.true.}) or
1177geopotential (\np{ln\_dynldf\_hor}\forcode{=.true.}) diffusion in the $s$-coordinate.
1178In the partial step case, coordinates are horizontal except at the deepest level and
1179no rotation is performed when \np{ln\_dynldf\_hor}\forcode{=.true.}.
1180The diffusion operator is defined simply as the divergence of down gradient momentum fluxes on
1181each momentum component.
1182It must be emphasized that this formulation ignores constraints on the stress tensor such as symmetry.
1183The resulting discrete representation is:
1184\begin{equation}
1185  \label{eq:DYN_ldf_iso}
1186  \begin{split}
1187    D_u^{l\textbf{U}} &= \frac{1}{e_{1u} \, e_{2u} \, e_{3u} } \\
1188    &  \left\{\quad  {\delta_{i+1/2} \left[ {A_T^{lm}  \left(
1189              {\frac{e_{2t} \; e_{3t} }{e_{1t} } \,\delta_{i}[u]
1190                -e_{2t} \; r_{1t} \,\overline{\overline {\delta_{k+1/2}[u]}}^{\,i,\,k}}
1191            \right)} \right]}    \right. \\
1192    & \qquad +\ \delta_j \left[ {A_f^{lm} \left( {\frac{e_{1f}\,e_{3f} }{e_{2f}
1193            }\,\delta_{j+1/2} [u] - e_{1f}\, r_{2f}
1194            \,\overline{\overline {\delta_{k+1/2} [u]}} ^{\,j+1/2,\,k}}
1195        \right)} \right] \\
1196    &\qquad +\ \delta_k \left[ {A_{uw}^{lm} \left( {-e_{2u} \, r_{1uw} \,\overline{\overline
1197              {\delta_{i+1/2} [u]}}^{\,i+1/2,\,k+1/2} }
1198        \right.} \right. \\
1199    &  \ \qquad \qquad \qquad \quad\
1200    - e_{1u} \, r_{2uw} \,\overline{\overline {\delta_{j+1/2} [u]}} ^{\,j,\,k+1/2} \\
1201    & \left. {\left. { \ \qquad \qquad \qquad \ \ \ \left. {\
1202                +\frac{e_{1u}\, e_{2u} }{e_{3uw} }\,\left( {r_{1uw}^2+r_{2uw}^2}
1203                \right)\,\delta_{k+1/2} [u]} \right)} \right]\;\;\;} \right\} \\ \\
1204    D_v^{l\textbf{V}} &= \frac{1}{e_{1v} \, e_{2v} \, e_{3v} } \\
1205    &  \left\{\quad  {\delta_{i+1/2} \left[ {A_f^{lm}  \left(
1206              {\frac{e_{2f} \; e_{3f} }{e_{1f} } \,\delta_{i+1/2}[v]
1207                -e_{2f} \; r_{1f} \,\overline{\overline {\delta_{k+1/2}[v]}}^{\,i+1/2,\,k}}
1208            \right)} \right]}    \right. \\
1209    & \qquad +\ \delta_j \left[ {A_T^{lm} \left( {\frac{e_{1t}\,e_{3t} }{e_{2t}
1210            }\,\delta_{j} [v] - e_{1t}\, r_{2t}
1211            \,\overline{\overline {\delta_{k+1/2} [v]}} ^{\,j,\,k}}
1212        \right)} \right] \\
1213    & \qquad +\ \delta_k \left[ {A_{vw}^{lm} \left( {-e_{2v} \, r_{1vw} \,\overline{\overline
1214              {\delta_{i+1/2} [v]}}^{\,i+1/2,\,k+1/2} }\right.} \right. \\
1215    &  \ \qquad \qquad \qquad \quad\
1216    - e_{1v} \, r_{2vw} \,\overline{\overline {\delta_{j+1/2} [v]}} ^{\,j+1/2,\,k+1/2} \\
1217    & \left. {\left. { \ \qquad \qquad \qquad \ \ \ \left. {\
1218                +\frac{e_{1v}\, e_{2v} }{e_{3vw} }\,\left( {r_{1vw}^2+r_{2vw}^2}
1219                \right)\,\delta_{k+1/2} [v]} \right)} \right]\;\;\;} \right\}
1220  \end{split}
1221\end{equation}
1222where $r_1$ and $r_2$ are the slopes between the surface along which the diffusion operator acts and
1223the surface of computation ($z$- or $s$-surfaces).
1224The way these slopes are evaluated is given in the lateral physics chapter (\autoref{chap:LDF}).
1225
1226%--------------------------------------------------------------------------------------------------------------
1227%           Iso-level bilaplacian operator
1228%--------------------------------------------------------------------------------------------------------------
1229\subsection[Iso-level bilaplacian (\forcode{ln_dynldf_bilap = .true.})]{Iso-level bilaplacian operator (\protect\np{ln\_dynldf\_bilap}\forcode{ = .true.})}
1230\label{subsec:DYN_ldf_bilap}
1231
1232The lateral fourth order operator formulation on momentum is obtained by applying \autoref{eq:DYN_ldf_lap} twice.
1233It requires an additional assumption on boundary conditions:
1234the first derivative term normal to the coast depends on the free or no-slip lateral boundary conditions chosen,
1235while the third derivative terms normal to the coast are set to zero (see \autoref{chap:LBC}).
1236%%%
1237\gmcomment{add a remark on the the change in the position of the coefficient}
1238%%%
1239
1240% ================================================================
1241%           Vertical diffusion term
1242% ================================================================
1243\section[Vertical diffusion term (\textit{dynzdf.F90})]{Vertical diffusion term (\protect\mdl{dynzdf})}
1244\label{sec:DYN_zdf}
1245%----------------------------------------------namzdf------------------------------------------------------
1246
1247\nlst{namzdf}
1248%-------------------------------------------------------------------------------------------------------------
1249
1250Options are defined through the \nam{zdf} namelist variables.
1251The large vertical diffusion coefficient found in the surface mixed layer together with high vertical resolution implies that in the case of explicit time stepping there would be too restrictive a constraint on the time step.
1252Two time stepping schemes can be used for the vertical diffusion term:
1253$(a)$ a forward time differencing scheme
1254(\np{ln\_zdfexp}\forcode{=.true.}) using a time splitting technique (\np{nn\_zdfexp} $>$ 1) or
1255$(b)$ a backward (or implicit) time differencing scheme (\np{ln\_zdfexp}\forcode{=.false.})
1256(see \autoref{chap:TD}).
1257Note that namelist variables \np{ln\_zdfexp} and \np{nn\_zdfexp} apply to both tracers and dynamics.
1258
1259The formulation of the vertical subgrid scale physics is the same whatever the vertical coordinate is.
1260The vertical diffusion operators given by \autoref{eq:MB_zdf} take the following semi-discrete space form:
1261\[
1262  % \label{eq:DYN_zdf}
1263  \left\{
1264    \begin{aligned}
1265      D_u^{vm} &\equiv \frac{1}{e_{3u}} \ \delta_k \left[ \frac{A_{uw}^{vm} }{e_{3uw} }
1266        \ \delta_{k+1/2} [\,u\,]         \right]     \\
1267      \\
1268      D_v^{vm} &\equiv \frac{1}{e_{3v}} \ \delta_k \left[ \frac{A_{vw}^{vm} }{e_{3vw} }
1269        \ \delta_{k+1/2} [\,v\,]         \right]
1270    \end{aligned}
1271  \right.
1272\]
1273where $A_{uw}^{vm} $ and $A_{vw}^{vm} $ are the vertical eddy viscosity and diffusivity coefficients.
1274The way these coefficients are evaluated depends on the vertical physics used (see \autoref{chap:ZDF}).
1275
1276The surface boundary condition on momentum is the stress exerted by the wind.
1277At the surface, the momentum fluxes are prescribed as the boundary condition on
1278the vertical turbulent momentum fluxes,
1279\begin{equation}
1280  \label{eq:DYN_zdf_sbc}
1281  \left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{z=1}
1282  = \frac{1}{\rho_o} \binom{\tau_u}{\tau_v }
1283\end{equation}
1284where $\left( \tau_u ,\tau_v \right)$ are the two components of the wind stress vector in
1285the (\textbf{i},\textbf{j}) coordinate system.
1286The high mixing coefficients in the surface mixed layer ensure that the surface wind stress is distributed in
1287the vertical over the mixed layer depth.
1288If the vertical mixing coefficient is small (when no mixed layer scheme is used)
1289the surface stress enters only the top model level, as a body force.
1290The surface wind stress is calculated in the surface module routines (SBC, see \autoref{chap:SBC}).
1291
1292The turbulent flux of momentum at the bottom of the ocean is specified through a bottom friction parameterisation
1293(see \autoref{sec:ZDF_drg})
1294
1295% ================================================================
1296% External Forcing
1297% ================================================================
1298\section{External forcings}
1299\label{sec:DYN_forcing}
1300
1301Besides the surface and bottom stresses (see the above section)
1302which are introduced as boundary conditions on the vertical mixing,
1303three other forcings may enter the dynamical equations by affecting the surface pressure gradient.
1304
1305(1) When \np{ln\_apr\_dyn}\forcode{=.true.} (see \autoref{sec:SBC_apr}),
1306the atmospheric pressure is taken into account when computing the surface pressure gradient.
1307
1308(2) When \np{ln\_tide\_pot}\forcode{=.true.} and \np{ln\_tide}\forcode{=.true.} (see \autoref{sec:SBC_tide}),
1309the tidal potential is taken into account when computing the surface pressure gradient.
1310
1311(3) When \np{nn\_ice\_embd}\forcode{=2} and LIM or CICE is used
1312(\ie\ when the sea-ice is embedded in the ocean),
1313the snow-ice mass is taken into account when computing the surface pressure gradient.
1314
1315
1316\gmcomment{ missing : the lateral boundary condition !!!   another external forcing
1317 }
1318
1319% ================================================================
1320% Wetting and drying
1321% ================================================================
1322\section{Wetting and drying }
1323\label{sec:DYN_wetdry}
1324
1325There are two main options for wetting and drying code (wd):
1326(a) an iterative limiter (il) and (b) a directional limiter (dl).
1327The directional limiter is based on the scheme developed by \cite{warner.defne.ea_CG13} for RO
1328MS
1329which was in turn based on ideas developed for POM by \cite{oey_OM06}. The iterative
1330limiter is a new scheme.  The iterative limiter is activated by setting $\mathrm{ln\_wd\_il} = \mathrm{.true.}$
1331and $\mathrm{ln\_wd\_dl} = \mathrm{.false.}$. The directional limiter is activated
1332by setting $\mathrm{ln\_wd\_dl} = \mathrm{.true.}$ and $\mathrm{ln\_wd\_il} = \mathrm{.false.}$.
1333
1334\nlst{namwad}
1335
1336The following terminology is used. The depth of the topography (positive downwards)
1337at each $(i,j)$ point is the quantity stored in array $\mathrm{ht\_wd}$ in the \NEMO\ code.
1338The height of the free surface (positive upwards) is denoted by $ \mathrm{ssh}$. Given the sign
1339conventions used, the water depth, $h$, is the height of the free surface plus the depth of the
1340topography (i.e. $\mathrm{ssh} + \mathrm{ht\_wd}$).
1341
1342Both wd schemes take all points in the domain below a land elevation of $\mathrm{rn\_wdld}$ to be
1343covered by water. They require the topography specified with a model
1344configuration to have negative depths at points where the land is higher than the
1345topography's reference sea-level. The vertical grid in \NEMO\ is normally computed relative to an
1346initial state with zero sea surface height elevation.
1347The user can choose to compute the vertical grid and heights in the model relative to
1348a non-zero reference height for the free surface. This choice affects the calculation of the metrics and depths
1349(i.e. the $\mathrm{e3t\_0, ht\_0}$ etc. arrays).
1350
1351Points where the water depth is less than $\mathrm{rn\_wdmin1}$ are interpreted as ``dry''.
1352$\mathrm{rn\_wdmin1}$ is usually chosen to be of order $0.05$m but extreme topographies
1353with very steep slopes require larger values for normal choices of time-step. Surface fluxes
1354are also switched off for dry cells to prevent freezing, boiling etc. of very thin water layers.
1355The fluxes are tappered down using a $\mathrm{tanh}$ weighting function
1356to no flux as the dry limit $\mathrm{rn\_wdmin1}$ is approached. Even wet cells can be very shallow.
1357The depth at which to start tapering is controlled by the user by setting $\mathrm{rn\_wd\_sbcdep}$.
1358The fraction $(<1)$ of sufrace fluxes to use at this depth is set by $\mathrm{rn\_wd\_sbcfra}$.
1359
1360Both versions of the code have been tested in six test cases provided in the WAD\_TEST\_CASES configuration
1361and in ``realistic'' configurations covering parts of the north-west European shelf.
1362All these configurations have used pure sigma coordinates. It is expected that
1363the wetting and drying code will work in domains with more general s-coordinates provided
1364the coordinates are pure sigma in the region where wetting and drying actually occurs.
1365
1366The next sub-section descrbies the directional limiter and the following sub-section the iterative limiter.
1367The final sub-section covers some additional considerations that are relevant to both schemes.
1368
1369
1370%-----------------------------------------------------------------------------------------
1371%   Iterative limiters
1372%-----------------------------------------------------------------------------------------
1373\subsection[Directional limiter (\textit{wet\_dry.F90})]{Directional limiter (\mdl{wet\_dry})}
1374\label{subsec:DYN_wd_directional_limiter}
1375
1376The principal idea of the directional limiter is that
1377water should not be allowed to flow out of a dry tracer cell (i.e. one whose water depth is less than \np{rn\_wdmin1}).
1378
1379All the changes associated with this option are made to the barotropic solver for the non-linear
1380free surface code within dynspg\_ts.
1381On each barotropic sub-step the scheme determines the direction of the flow across each face of all the tracer cells
1382and sets the flux across the face to zero when the flux is from a dry tracer cell. This prevents cells
1383whose depth is rn\_wdmin1 or less from drying out further. The scheme does not force $h$ (the water depth) at tracer cells
1384to be at least the minimum depth and hence is able to conserve mass / volume.
1385
1386The flux across each $u$-face of a tracer cell is multiplied by a factor zuwdmask (an array which depends on ji and jj).
1387If the user sets \np{ln\_wd\_dl\_ramp}\forcode{=.false.} then zuwdmask is 1 when the
1388flux is from a cell with water depth greater than \np{rn\_wdmin1} and 0 otherwise. If the user sets
1389\np{ln\_wd\_dl\_ramp}\forcode{=.true.} the flux across the face is ramped down as the water depth decreases
1390from 2 * \np{rn\_wdmin1} to \np{rn\_wdmin1}. The use of this ramp reduced grid-scale noise in idealised test cases.
1391
1392At the point where the flux across a $u$-face is multiplied by zuwdmask , we have chosen
1393also to multiply the corresponding velocity on the ``now'' step at that face by zuwdmask. We could have
1394chosen not to do that and to allow fairly large velocities to occur in these ``dry'' cells.
1395The rationale for setting the velocity to zero is that it is the momentum equations that are being solved
1396and the total momentum of the upstream cell (treating it as a finite volume) should be considered
1397to be its depth times its velocity. This depth is considered to be zero at ``dry'' $u$-points consistent with its
1398treatment in the calculation of the flux of mass across the cell face.
1399
1400
1401\cite{warner.defne.ea_CG13} state that in their scheme the velocity masks at the cell faces for the baroclinic
1402timesteps are set to 0 or 1 depending on whether the average of the masks over the barotropic sub-steps is respectively less than
1403or greater than 0.5. That scheme does not conserve tracers in integrations started from constant tracer
1404fields (tracers independent of $x$, $y$ and $z$). Our scheme conserves constant tracers because
1405the velocities used at the tracer cell faces on the baroclinic timesteps are carefully calculated by dynspg\_ts
1406to equal their mean value during the barotropic steps. If the user sets \np{ln\_wd\_dl\_bc}\forcode{=.true.}, the
1407baroclinic velocities are also multiplied by a suitably weighted average of zuwdmask.
1408
1409%-----------------------------------------------------------------------------------------
1410%   Iterative limiters
1411%-----------------------------------------------------------------------------------------
1412
1413\subsection[Iterative limiter (\textit{wet\_dry.F90})]{Iterative limiter (\mdl{wet\_dry})}
1414\label{subsec:DYN_wd_iterative_limiter}
1415
1416\subsubsection[Iterative flux limiter (\textit{wet\_dry.F90})]{Iterative flux limiter (\mdl{wet\_dry})}
1417\label{subsec:DYN_wd_il_spg_limiter}
1418
1419The iterative limiter modifies the fluxes across the faces of cells that are either already ``dry''
1420or may become dry within the next time-step using an iterative method.
1421
1422The flux limiter for the barotropic flow (devised by Hedong Liu) can be understood as follows:
1423
1424The continuity equation for the total water depth in a column
1425\begin{equation}
1426  \label{eq:DYN_wd_continuity}
1427  \frac{\partial h}{\partial t} + \mathbf{\nabla.}(h\mathbf{u}) = 0 .
1428\end{equation}
1429can be written in discrete form  as
1430
1431\begin{align}
1432  \label{eq:DYN_wd_continuity_2}
1433  \frac{e_1 e_2}{\Delta t} ( h_{i,j}(t_{n+1}) - h_{i,j}(t_e) )
1434  &= - ( \mathrm{flxu}_{i+1,j} - \mathrm{flxu}_{i,j}  + \mathrm{flxv}_{i,j+1} - \mathrm{flxv}_{i,j} ) \\
1435  &= \mathrm{zzflx}_{i,j} .
1436\end{align}
1437
1438In the above $h$ is the depth of the water in the column at point $(i,j)$,
1439$\mathrm{flxu}_{i+1,j}$ is the flux out of the ``eastern'' face of the cell and
1440$\mathrm{flxv}_{i,j+1}$ the flux out of the ``northern'' face of the cell; $t_{n+1}$ is
1441the new timestep, $t_e$ is the old timestep (either $t_b$ or $t_n$) and $ \Delta t =
1442t_{n+1} - t_e$; $e_1 e_2$ is the area of the tracer cells centred at $(i,j)$ and
1443$\mathrm{zzflx}$ is the sum of the fluxes through all the faces.
1444
1445The flux limiter splits the flux $\mathrm{zzflx}$ into fluxes that are out of the cell
1446(zzflxp) and fluxes that are into the cell (zzflxn).  Clearly
1447
1448\begin{equation}
1449  \label{eq:DYN_wd_zzflx_p_n_1}
1450  \mathrm{zzflx}_{i,j} = \mathrm{zzflxp}_{i,j} + \mathrm{zzflxn}_{i,j} .
1451\end{equation}
1452
1453The flux limiter iteratively adjusts the fluxes $\mathrm{flxu}$ and $\mathrm{flxv}$ until
1454none of the cells will ``dry out''. To be precise the fluxes are limited until none of the
1455cells has water depth less than $\mathrm{rn\_wdmin1}$ on step $n+1$.
1456
1457Let the fluxes on the $m$th iteration step be denoted by $\mathrm{flxu}^{(m)}$ and
1458$\mathrm{flxv}^{(m)}$.  Then the adjustment is achieved by seeking a set of coefficients,
1459$\mathrm{zcoef}_{i,j}^{(m)}$ such that:
1460
1461\begin{equation}
1462  \label{eq:DYN_wd_continuity_coef}
1463  \begin{split}
1464    \mathrm{zzflxp}^{(m)}_{i,j} =& \mathrm{zcoef}_{i,j}^{(m)} \mathrm{zzflxp}^{(0)}_{i,j} \\
1465    \mathrm{zzflxn}^{(m)}_{i,j} =& \mathrm{zcoef}_{i,j}^{(m)} \mathrm{zzflxn}^{(0)}_{i,j}
1466  \end{split}
1467\end{equation}
1468
1469where the coefficients are $1.0$ generally but can vary between $0.0$ and $1.0$ around
1470cells that would otherwise dry.
1471
1472The iteration is initialised by setting
1473
1474\begin{equation}
1475  \label{eq:DYN_wd_zzflx_initial}
1476  \mathrm{zzflxp^{(0)}}_{i,j} = \mathrm{zzflxp}_{i,j} , \quad  \mathrm{zzflxn^{(0)}}_{i,j} = \mathrm{zzflxn}_{i,j} .
1477\end{equation}
1478
1479The fluxes out of cell $(i,j)$ are updated at the $m+1$th iteration if the depth of the
1480cell on timestep $t_e$, namely $h_{i,j}(t_e)$, is less than the total flux out of the cell
1481times the timestep divided by the cell area. Using (\autoref{eq:DYN_wd_continuity_2}) this
1482condition is
1483
1484\begin{equation}
1485  \label{eq:DYN_wd_continuity_if}
1486  h_{i,j}(t_e)  - \mathrm{rn\_wdmin1} <  \frac{\Delta t}{e_1 e_2} ( \mathrm{zzflxp}^{(m)}_{i,j} + \mathrm{zzflxn}^{(m)}_{i,j} ) .
1487\end{equation}
1488
1489Rearranging (\autoref{eq:DYN_wd_continuity_if}) we can obtain an expression for the maximum
1490outward flux that can be allowed and still maintain the minimum wet depth:
1491
1492\begin{equation}
1493  \label{eq:DYN_wd_max_flux}
1494  \begin{split}
1495    \mathrm{zzflxp}^{(m+1)}_{i,j} = \Big[ (h_{i,j}(t_e) & - \mathrm{rn\_wdmin1} - \mathrm{rn\_wdmin2})  \frac{e_1 e_2}{\Delta t} \phantom{]} \\
1496    \phantom{[} & -  \mathrm{zzflxn}^{(m)}_{i,j} \Big]
1497  \end{split}
1498\end{equation}
1499
1500Note a small tolerance ($\mathrm{rn\_wdmin2}$) has been introduced here {\itshape [Q: Why is
1501this necessary/desirable?]}. Substituting from (\autoref{eq:DYN_wd_continuity_coef}) gives an
1502expression for the coefficient needed to multiply the outward flux at this cell in order
1503to avoid drying.
1504
1505\begin{equation}
1506  \label{eq:DYN_wd_continuity_nxtcoef}
1507  \begin{split}
1508    \mathrm{zcoef}^{(m+1)}_{i,j} = \Big[ (h_{i,j}(t_e) & - \mathrm{rn\_wdmin1} - \mathrm{rn\_wdmin2})  \frac{e_1 e_2}{\Delta t} \phantom{]} \\
1509    \phantom{[} & -  \mathrm{zzflxn}^{(m)}_{i,j} \Big] \frac{1}{ \mathrm{zzflxp}^{(0)}_{i,j} }
1510  \end{split}
1511\end{equation}
1512
1513Only the outward flux components are altered but, of course, outward fluxes from one cell
1514are inward fluxes to adjacent cells and the balance in these cells may need subsequent
1515adjustment; hence the iterative nature of this scheme.  Note, for example, that the flux
1516across the ``eastern'' face of the $(i,j)$th cell is only updated at the $m+1$th iteration
1517if that flux at the $m$th iteration is out of the $(i,j)$th cell. If that is the case then
1518the flux across that face is into the $(i+1,j)$ cell and that flux will not be updated by
1519the calculation for the $(i+1,j)$th cell. In this sense the updates to the fluxes across
1520the faces of the cells do not ``compete'' (they do not over-write each other) and one
1521would expect the scheme to converge relatively quickly. The scheme is flux based so
1522conserves mass. It also conserves constant tracers for the same reason that the
1523directional limiter does.
1524
1525
1526%----------------------------------------------------------------------------------------
1527%      Surface pressure gradients
1528%----------------------------------------------------------------------------------------
1529\subsubsection[Modification of surface pressure gradients (\textit{dynhpg.F90})]{Modification of surface pressure gradients (\mdl{dynhpg})}
1530\label{subsec:DYN_wd_il_spg}
1531
1532At ``dry'' points the water depth is usually close to $\mathrm{rn\_wdmin1}$. If the
1533topography is sloping at these points the sea-surface will have a similar slope and there
1534will hence be very large horizontal pressure gradients at these points. The WAD modifies
1535the magnitude but not the sign of the surface pressure gradients (zhpi and zhpj) at such
1536points by mulitplying them by positive factors (zcpx and zcpy respectively) that lie
1537between $0$ and $1$.
1538
1539We describe how the scheme works for the ``eastward'' pressure gradient, zhpi, calculated
1540at the $(i,j)$th $u$-point. The scheme uses the ht\_wd depths and surface heights at the
1541neighbouring $(i+1,j)$ and $(i,j)$ tracer points.  zcpx is calculated using two logicals
1542variables, $\mathrm{ll\_tmp1}$ and $\mathrm{ll\_tmp2}$ which are evaluated for each grid
1543column.  The three possible combinations are illustrated in figure \autoref{fig:DYN_WAD_dynhpg}.
1544
1545%>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
1546\begin{figure}[!ht] \begin{center}
1547\includegraphics[width=\textwidth]{Fig_WAD_dynhpg}
1548\caption{
1549  \label{fig:DYN_WAD_dynhpg}
1550  Illustrations of the three possible combinations of the logical variables controlling the
1551  limiting of the horizontal pressure gradient in wetting and drying regimes}
1552\end{center}
1553\end{figure}
1554%>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
1555
1556The first logical, $\mathrm{ll\_tmp1}$, is set to true if and only if the water depth at
1557both neighbouring points is greater than $\mathrm{rn\_wdmin1} + \mathrm{rn\_wdmin2}$ and
1558the minimum height of the sea surface at the two points is greater than the maximum height
1559of the topography at the two points:
1560
1561\begin{equation}
1562  \label{eq:DYN_ll_tmp1}
1563  \begin{split}
1564    \mathrm{ll\_tmp1}  = & \mathrm{MIN(sshn(ji,jj), sshn(ji+1,jj))} > \\
1565                     & \quad \mathrm{MAX(-ht\_wd(ji,jj), -ht\_wd(ji+1,jj))\  .and.} \\
1566                     & \mathrm{MAX(sshn(ji,jj) + ht\_wd(ji,jj),} \\
1567                     & \mathrm{\phantom{MAX(}sshn(ji+1,jj) + ht\_wd(ji+1,jj))} >\\
1568                     & \quad\quad\mathrm{rn\_wdmin1 + rn\_wdmin2 }
1569  \end{split}
1570\end{equation}
1571
1572The second logical, $\mathrm{ll\_tmp2}$, is set to true if and only if the maximum height
1573of the sea surface at the two points is greater than the maximum height of the topography
1574at the two points plus $\mathrm{rn\_wdmin1} + \mathrm{rn\_wdmin2}$
1575
1576\begin{equation}
1577  \label{eq:DYN_ll_tmp2}
1578  \begin{split}
1579    \mathrm{ ll\_tmp2 } = & \mathrm{( ABS( sshn(ji,jj) - sshn(ji+1,jj) ) > 1.E-12 )\ .AND.}\\
1580    & \mathrm{( MAX(sshn(ji,jj), sshn(ji+1,jj)) > } \\
1581    & \mathrm{\phantom{(} MAX(-ht\_wd(ji,jj), -ht\_wd(ji+1,jj)) + rn\_wdmin1 + rn\_wdmin2}) .
1582  \end{split}
1583\end{equation}
1584
1585If $\mathrm{ll\_tmp1}$ is true then the surface pressure gradient, zhpi at the $(i,j)$
1586point is unmodified. If both logicals are false zhpi is set to zero.
1587
1588If $\mathrm{ll\_tmp1}$ is true and $\mathrm{ll\_tmp2}$ is false then the surface pressure
1589gradient is multiplied through by zcpx which is the absolute value of the difference in
1590the water depths at the two points divided by the difference in the surface heights at the
1591two points. Thus the sign of the sea surface height gradient is retained but the magnitude
1592of the pressure force is determined by the difference in water depths rather than the
1593difference in surface height between the two points. Note that dividing by the difference
1594between the sea surface heights can be problematic if the heights approach parity. An
1595additional condition is applied to $\mathrm{ ll\_tmp2 }$ to ensure it is .false. in such
1596conditions.
1597
1598\subsubsection[Additional considerations (\textit{usrdef\_zgr.F90})]{Additional considerations (\mdl{usrdef\_zgr})}
1599\label{subsec:DYN_WAD_additional}
1600
1601In the very shallow water where wetting and drying occurs the parametrisation of
1602bottom drag is clearly very important. In order to promote stability
1603it is sometimes useful to calculate the bottom drag using an implicit time-stepping approach.
1604
1605Suitable specifcation of the surface heat flux in wetting and drying domains in forced and
1606coupled simulations needs further consideration. In order to prevent freezing or boiling
1607in uncoupled integrations the net surface heat fluxes need to be appropriately limited.
1608
1609%----------------------------------------------------------------------------------------
1610%      The WAD test cases
1611%----------------------------------------------------------------------------------------
1612\subsection[The WAD test cases (\textit{usrdef\_zgr.F90})]{The WAD test cases (\mdl{usrdef\_zgr})}
1613\label{subsec:DYN_WAD_test_cases}
1614
1615See the WAD tests MY\_DOC documention for details of the WAD test cases.
1616
1617
1618
1619% ================================================================
1620% Time evolution term
1621% ================================================================
1622\section[Time evolution term (\textit{dynnxt.F90})]{Time evolution term (\protect\mdl{dynnxt})}
1623\label{sec:DYN_nxt}
1624
1625%----------------------------------------------namdom----------------------------------------------------
1626
1627\nlst{namdom}
1628%-------------------------------------------------------------------------------------------------------------
1629
1630Options are defined through the \nam{dom} namelist variables.
1631The general framework for dynamics time stepping is a leap-frog scheme,
1632\ie\ a three level centred time scheme associated with an Asselin time filter (cf. \autoref{chap:TD}).
1633The scheme is applied to the velocity, except when
1634using the flux form of momentum advection (cf. \autoref{sec:DYN_adv_cor_flux})
1635in the variable volume case (\texttt{vvl?} defined),
1636where it has to be applied to the thickness weighted velocity (see \autoref{sec:SCOORD_momentum})
1637
1638$\bullet$ vector invariant form or linear free surface
1639(\np{ln\_dynhpg\_vec}\forcode{=.true.} ; \texttt{vvl?} not defined):
1640\[
1641  % \label{eq:DYN_nxt_vec}
1642  \left\{
1643    \begin{aligned}
1644      &u^{t+\rdt} = u_f^{t-\rdt} + 2\rdt  \ \text{RHS}_u^t     \\
1645      &u_f^t \;\quad = u^t+\gamma \,\left[ {u_f^{t-\rdt} -2u^t+u^{t+\rdt}} \right]
1646    \end{aligned}
1647  \right.
1648\]
1649
1650$\bullet$ flux form and nonlinear free surface
1651(\np{ln\_dynhpg\_vec}\forcode{=.false.} ; \texttt{vvl?} defined):
1652\[
1653  % \label{eq:DYN_nxt_flux}
1654  \left\{
1655    \begin{aligned}
1656      &\left(e_{3u}\,u\right)^{t+\rdt} = \left(e_{3u}\,u\right)_f^{t-\rdt} + 2\rdt \; e_{3u} \;\text{RHS}_u^t     \\
1657      &\left(e_{3u}\,u\right)_f^t \;\quad = \left(e_{3u}\,u\right)^t
1658      +\gamma \,\left[ {\left(e_{3u}\,u\right)_f^{t-\rdt} -2\left(e_{3u}\,u\right)^t+\left(e_{3u}\,u\right)^{t+\rdt}} \right]
1659    \end{aligned}
1660  \right.
1661\]
1662where RHS is the right hand side of the momentum equation,
1663the subscript $f$ denotes filtered values and $\gamma$ is the Asselin coefficient.
1664$\gamma$ is initialized as \np{nn\_atfp} (namelist parameter).
1665Its default value is \np{nn\_atfp}\forcode{=10.e-3}.
1666In both cases, the modified Asselin filter is not applied since perfect conservation is not an issue for
1667the momentum equations.
1668
1669Note that with the filtered free surface,
1670the update of the \textit{after} velocities is done in the \mdl{dynsp\_flt} module,
1671and only array swapping and Asselin filtering is done in \mdl{dynnxt}.
1672
1673% ================================================================
1674\biblio
1675
1676\pindex
1677
1678\end{document}
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