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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\begin{listing}
169  \nlst{namdyn_adv}
170  \caption{\forcode{&namdyn_adv}}
171  \label{lst:namdyn_adv}
172\end{listing}
173%-------------------------------------------------------------------------------------------------------------
174
175The vector invariant form of the momentum equations is the one most often used in
176applications of the \NEMO\ ocean model.
177The flux form option (see next section) has been present since version $2$.
178Options are defined through the \nam{dyn\_adv} namelist variables Coriolis and
179momentum advection terms are evaluated using a leapfrog scheme,
180\ie\ the velocity appearing in these expressions is centred in time (\textit{now} velocity).
181At the lateral boundaries either free slip, no slip or partial slip boundary conditions are applied following
182\autoref{chap:LBC}.
183
184% -------------------------------------------------------------------------------------------------------------
185%        Vorticity term
186% -------------------------------------------------------------------------------------------------------------
187\subsection[Vorticity term (\textit{dynvor.F90})]{Vorticity term (\protect\mdl{dynvor})}
188\label{subsec:DYN_vor}
189%------------------------------------------nam_dynvor----------------------------------------------------
190
191\begin{listing}
192  \nlst{namdyn_vor}
193  \caption{\forcode{&namdyn_vor}}
194  \label{lst:namdyn_vor}
195\end{listing}
196%-------------------------------------------------------------------------------------------------------------
197
198Options are defined through the \nam{dyn\_vor} namelist variables.
199Four discretisations of the vorticity term (\texttt{ln\_dynvor\_xxx}\forcode{=.true.}) are available:
200conserving potential enstrophy of horizontally non-divergent flow (ENS scheme);
201conserving horizontal kinetic energy (ENE scheme);
202conserving potential enstrophy for the relative vorticity term and
203horizontal kinetic energy for the planetary vorticity term (MIX scheme);
204or conserving both the potential enstrophy of horizontally non-divergent flow and horizontal kinetic energy
205(EEN scheme) (see \autoref{subsec:INVARIANTS_vorEEN}).
206In the case of ENS, ENE or MIX schemes the land sea mask may be slightly modified to ensure the consistency of
207vorticity term with analytical equations (\np{ln\_dynvor\_con}\forcode{=.true.}).
208The vorticity terms are all computed in dedicated routines that can be found in the \mdl{dynvor} module.
209
210%-------------------------------------------------------------
211%                 enstrophy conserving scheme
212%-------------------------------------------------------------
213\subsubsection[Enstrophy conserving scheme (\forcode{ln_dynvor_ens})]{Enstrophy conserving scheme (\protect\np{ln\_dynvor\_ens})}
214\label{subsec:DYN_vor_ens}
215
216In the enstrophy conserving case (ENS scheme),
217the discrete formulation of the vorticity term provides a global conservation of the enstrophy
218($ [ (\zeta +f ) / e_{3f} ]^2 $ in $s$-coordinates) for a horizontally non-divergent flow (\ie\ $\chi$=$0$),
219but does not conserve the total kinetic energy.
220It is given by:
221\begin{equation}
222  \label{eq:DYN_vor_ens}
223  \left\{
224    \begin{aligned}
225      {+\frac{1}{e_{1u} } } & {\overline {\left( { \frac{\zeta +f}{e_{3f} }} \right)} }^{\,i}
226      & {\overline{\overline {\left( {e_{1v}\,e_{3v}\;v} \right)}} }^{\,i, j+1/2}    \\
227      {- \frac{1}{e_{2v} } } & {\overline {\left( {\frac{\zeta +f}{e_{3f} }} \right)} }^{\,j}
228      & {\overline{\overline {\left( {e_{2u}\,e_{3u}\;u} \right)}} }^{\,i+1/2, j}
229    \end{aligned}
230  \right.
231\end{equation}
232
233%-------------------------------------------------------------
234%                 energy conserving scheme
235%-------------------------------------------------------------
236\subsubsection[Energy conserving scheme (\forcode{ln_dynvor_ene})]{Energy conserving scheme (\protect\np{ln\_dynvor\_ene})}
237\label{subsec:DYN_vor_ene}
238
239The kinetic energy conserving scheme (ENE scheme) conserves the global kinetic energy but not the global enstrophy.
240It is given by:
241\begin{equation}
242  \label{eq:DYN_vor_ene}
243  \left\{
244    \begin{aligned}
245      {+\frac{1}{e_{1u}}\; {\overline {\left( {\frac{\zeta +f}{e_{3f} }} \right)
246            \;  \overline {\left( {e_{1v}\,e_{3v}\;v} \right)} ^{\,i+1/2}} }^{\,j} }    \\
247      {- \frac{1}{e_{2v}}\; {\overline {\left( {\frac{\zeta +f}{e_{3f} }} \right)
248            \;  \overline {\left( {e_{2u}\,e_{3u}\;u} \right)} ^{\,j+1/2}} }^{\,i} }
249    \end{aligned}
250  \right.
251\end{equation}
252
253%-------------------------------------------------------------
254%                 mix energy/enstrophy conserving scheme
255%-------------------------------------------------------------
256\subsubsection[Mixed energy/enstrophy conserving scheme (\forcode{ln_dynvor_mix})]{Mixed energy/enstrophy conserving scheme (\protect\np{ln\_dynvor\_mix})}
257\label{subsec:DYN_vor_mix}
258
259For the mixed energy/enstrophy conserving scheme (MIX scheme), a mixture of the two previous schemes is used.
260It consists of the ENS scheme (\autoref{eq:DYN_vor_ens}) for the relative vorticity term,
261and of the ENE scheme (\autoref{eq:DYN_vor_ene}) applied to the planetary vorticity term.
262\[
263  % \label{eq:DYN_vor_mix}
264  \left\{ {
265      \begin{aligned}
266        {+\frac{1}{e_{1u} }\; {\overline {\left( {\frac{\zeta }{e_{3f} }} \right)} }^{\,i}
267          \; {\overline{\overline {\left( {e_{1v}\,e_{3v}\;v} \right)}} }^{\,i,j+1/2} -\frac{1}{e_{1u} }
268          \; {\overline {\left( {\frac{f}{e_{3f} }} \right)
269              \;\overline {\left( {e_{1v}\,e_{3v}\;v} \right)} ^{\,i+1/2}} }^{\,j} } \\
270        {-\frac{1}{e_{2v} }\; {\overline {\left( {\frac{\zeta }{e_{3f} }} \right)} }^j
271          \; {\overline{\overline {\left( {e_{2u}\,e_{3u}\;u} \right)}} }^{\,i+1/2,j} +\frac{1}{e_{2v} }
272          \; {\overline {\left( {\frac{f}{e_{3f} }} \right)
273              \;\overline {\left( {e_{2u}\,e_{3u}\;u} \right)} ^{\,j+1/2}} }^{\,i} } \hfill
274      \end{aligned}
275    } \right.
276\]
277
278%-------------------------------------------------------------
279%                 energy and enstrophy conserving scheme
280%-------------------------------------------------------------
281\subsubsection[Energy and enstrophy conserving scheme (\forcode{ln_dynvor_een})]{Energy and enstrophy conserving scheme (\protect\np{ln\_dynvor\_een})}
282\label{subsec:DYN_vor_een}
283
284In both the ENS and ENE schemes,
285it is apparent that the combination of $i$ and $j$ averages of the velocity allows for
286the presence of grid point oscillation structures that will be invisible to the operator.
287These structures are \textit{computational modes} that will be at least partly damped by
288the momentum diffusion operator (\ie\ the subgrid-scale advection), but not by the resolved advection term.
289The ENS and ENE schemes therefore do not contribute to dump any grid point noise in the horizontal velocity field.
290Such noise would result in more noise in the vertical velocity field, an undesirable feature.
291This is a well-known characteristic of $C$-grid discretization where
292$u$ and $v$ are located at different grid points,
293a price worth paying to avoid a double averaging in the pressure gradient term as in the $B$-grid.
294\gmcomment{ To circumvent this, Adcroft (ADD REF HERE)
295Nevertheless, this technique strongly distort the phase and group velocity of Rossby waves....}
296
297A very nice solution to the problem of double averaging was proposed by \citet{arakawa.hsu_MWR90}.
298The idea is to get rid of the double averaging by considering triad combinations of vorticity.
299It is noteworthy that this solution is conceptually quite similar to the one proposed by
300\citep{griffies.gnanadesikan.ea_JPO98} for the discretization of the iso-neutral diffusion operator (see \autoref{apdx:INVARIANTS}).
301
302The \citet{arakawa.hsu_MWR90} vorticity advection scheme for a single layer is modified
303for spherical coordinates as described by \citet{arakawa.lamb_MWR81} to obtain the EEN scheme.
304First consider the discrete expression of the potential vorticity, $q$, defined at an $f$-point:
305\[
306  % \label{eq:DYN_pot_vor}
307  q  = \frac{\zeta +f} {e_{3f} }
308\]
309where the relative vorticity is defined by (\autoref{eq:DYN_divcur_cur}),
310the Coriolis parameter is given by $f=2 \,\Omega \;\sin \varphi _f $ and the layer thickness at $f$-points is:
311\begin{equation}
312  \label{eq:DYN_een_e3f}
313  e_{3f} = \overline{\overline {e_{3t} }} ^{\,i+1/2,j+1/2}
314\end{equation}
315
316%>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
317\begin{figure}[!ht]
318  \centering
319  \includegraphics[width=0.66\textwidth]{Fig_DYN_een_triad}
320  \caption[Triads used in the energy and enstrophy conserving scheme (EEN)]{
321    Triads used in the energy and enstrophy conserving scheme (EEN) for
322    $u$-component (upper panel) and $v$-component (lower panel).}
323  \label{fig:DYN_een_triad}
324\end{figure}
325% >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
326
327A key point in \autoref{eq:DYN_een_e3f} is how the averaging in the \textbf{i}- and \textbf{j}- directions is made.
328It uses the sum of masked t-point vertical scale factor divided either by the sum of the four t-point masks
329(\np{nn\_een\_e3f}\forcode{=1}), or just by $4$ (\np{nn\_een\_e3f}\forcode{=.true.}).
330The latter case preserves the continuity of $e_{3f}$ when one or more of the neighbouring $e_{3t}$ tends to zero and
331extends by continuity the value of $e_{3f}$ into the land areas.
332This case introduces a sub-grid-scale topography at f-points
333(with a systematic reduction of $e_{3f}$ when a model level intercept the bathymetry)
334that tends to reinforce the topostrophy of the flow
335(\ie\ the tendency of the flow to follow the isobaths) \citep{penduff.le-sommer.ea_OS07}.
336
337Next, the vorticity triads, $ {^i_j}\mathbb{Q}^{i_p}_{j_p}$ can be defined at a $T$-point as
338the following triad combinations of the neighbouring potential vorticities defined at f-points
339(\autoref{fig:DYN_een_triad}):
340\begin{equation}
341  \label{eq:DYN_Q_triads}
342  _i^j \mathbb{Q}^{i_p}_{j_p}
343  = \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)
344\end{equation}
345where the indices $i_p$ and $k_p$ take the values: $i_p = -1/2$ or $1/2$ and $j_p = -1/2$ or $1/2$.
346
347Finally, the vorticity terms are represented as:
348\begin{equation}
349  \label{eq:DYN_vor_een}
350  \left\{ {
351      \begin{aligned}
352        +q\,e_3 \, v    &\equiv +\frac{1}{e_{1u} }   \sum_{\substack{i_p,\,k_p}}
353        {^{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}   \\
354        - q\,e_3 \, u     &\equiv -\frac{1}{e_{2v} }    \sum_{\substack{i_p,\,k_p}}
355        {^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}   \\
356      \end{aligned}
357    } \right.
358\end{equation}
359
360This EEN scheme in fact combines the conservation properties of the ENS and ENE schemes.
361It conserves both total energy and potential enstrophy in the limit of horizontally nondivergent flow
362(\ie\ $\chi$=$0$) (see \autoref{subsec:INVARIANTS_vorEEN}).
363Applied to a realistic ocean configuration, it has been shown that it leads to a significant reduction of
364the noise in the vertical velocity field \citep{le-sommer.penduff.ea_OM09}.
365Furthermore, used in combination with a partial steps representation of bottom topography,
366it improves the interaction between current and topography,
367leading to a larger topostrophy of the flow \citep{barnier.madec.ea_OD06, penduff.le-sommer.ea_OS07}.
368
369%--------------------------------------------------------------------------------------------------------------
370%           Kinetic Energy Gradient term
371%--------------------------------------------------------------------------------------------------------------
372\subsection[Kinetic energy gradient term (\textit{dynkeg.F90})]{Kinetic energy gradient term (\protect\mdl{dynkeg})}
373\label{subsec:DYN_keg}
374
375As demonstrated in \autoref{apdx:INVARIANTS},
376there is a single discrete formulation of the kinetic energy gradient term that,
377together with the formulation chosen for the vertical advection (see below),
378conserves the total kinetic energy:
379\[
380  % \label{eq:DYN_keg}
381  \left\{
382    \begin{aligned}
383      -\frac{1}{2 \; e_{1u} }  & \ \delta_{i+1/2} \left[ {\overline {u^2}^{\,i} + \overline{v^2}^{\,j}} \right]   \\
384      -\frac{1}{2 \; e_{2v} }  & \ \delta_{j+1/2} \left[ {\overline {u^2}^{\,i} + \overline{v^2}^{\,j}} \right]
385    \end{aligned}
386  \right.
387\]
388
389%--------------------------------------------------------------------------------------------------------------
390%           Vertical advection term
391%--------------------------------------------------------------------------------------------------------------
392\subsection[Vertical advection term (\textit{dynzad.F90})]{Vertical advection term (\protect\mdl{dynzad})}
393\label{subsec:DYN_zad}
394
395The discrete formulation of the vertical advection, t
396ogether with the formulation chosen for the gradient of kinetic energy (KE) term,
397conserves the total kinetic energy.
398Indeed, the change of KE due to the vertical advection is exactly balanced by
399the change of KE due to the gradient of KE (see \autoref{apdx:INVARIANTS}).
400\[
401  % \label{eq:DYN_zad}
402  \left\{
403    \begin{aligned}
404      -\frac{1} {e_{1u}\,e_{2u}\,e_{3u}} &\ \overline{\ \overline{ e_{1t}\,e_{2t}\;w } ^{\,i+1/2\;\delta_{k+1/2} \left[ u \right]\  }^{\,k}  \\
405      -\frac{1} {e_{1v}\,e_{2v}\,e_{3v}}  &\ \overline{\ \overline{ e_{1t}\,e_{2t}\;w } ^{\,j+1/2\;\delta_{k+1/2} \left[ u \right]\  }^{\,k}
406    \end{aligned}
407  \right.
408\]
409When \np{ln\_dynzad\_zts}\forcode{=.true.},
410a split-explicit time stepping with 5 sub-timesteps is used on the vertical advection term.
411This option can be useful when the value of the timestep is limited by vertical advection \citep{lemarie.debreu.ea_OM15}.
412Note that in this case,
413a similar split-explicit time stepping should be used on vertical advection of tracer to ensure a better stability,
414an option which is only available with a TVD scheme (see \np{ln\_traadv\_tvd\_zts} in \autoref{subsec:TRA_adv_tvd}).
415
416
417% ================================================================
418% Coriolis and Advection : flux form
419% ================================================================
420\section{Coriolis and advection: flux form}
421\label{sec:DYN_adv_cor_flux}
422%------------------------------------------nam_dynadv----------------------------------------------------
423
424%-------------------------------------------------------------------------------------------------------------
425
426Options are defined through the \nam{dyn\_adv} namelist variables.
427In the flux form (as in the vector invariant form),
428the Coriolis and momentum advection terms are evaluated using a leapfrog scheme,
429\ie\ the velocity appearing in their expressions is centred in time (\textit{now} velocity).
430At the lateral boundaries either free slip,
431no slip or partial slip boundary conditions are applied following \autoref{chap:LBC}.
432
433
434%--------------------------------------------------------------------------------------------------------------
435%           Coriolis plus curvature metric terms
436%--------------------------------------------------------------------------------------------------------------
437\subsection[Coriolis plus curvature metric terms (\textit{dynvor.F90})]{Coriolis plus curvature metric terms (\protect\mdl{dynvor})}
438\label{subsec:DYN_cor_flux}
439
440In flux form, the vorticity term reduces to a Coriolis term in which the Coriolis parameter has been modified to account for the "metric" term.
441This altered Coriolis parameter is thus discretised at $f$-points.
442It is given by:
443\begin{multline*}
444  % \label{eq:DYN_cor_metric}
445  f+\frac{1}{e_1 e_2 }\left( {v\frac{\partial e_2 }{\partial i}  -  u\frac{\partial e_1 }{\partial j}} \right\\
446  \equiv   f + \frac{1}{e_{1f} e_{2f} } \left( { \ \overline v ^{i+1/2}\delta_{i+1/2} \left[ {e_{2u} } \right]
447      -  \overline u ^{j+1/2}\delta_{j+1/2} \left[ {e_{1u} } \right]  }  \ \right)
448\end{multline*}
449
450Any of the (\autoref{eq:DYN_vor_ens}), (\autoref{eq:DYN_vor_ene}) and (\autoref{eq:DYN_vor_een}) schemes can be used to
451compute the product of the Coriolis parameter and the vorticity.
452However, the energy-conserving scheme (\autoref{eq:DYN_vor_een}) has exclusively been used to date.
453This term is evaluated using a leapfrog scheme, \ie\ the velocity is centred in time (\textit{now} velocity).
454
455%--------------------------------------------------------------------------------------------------------------
456%           Flux form Advection term
457%--------------------------------------------------------------------------------------------------------------
458\subsection[Flux form advection term (\textit{dynadv.F90})]{Flux form advection term (\protect\mdl{dynadv})}
459\label{subsec:DYN_adv_flux}
460
461The discrete expression of the advection term is given by:
462\[
463  % \label{eq:DYN_adv}
464  \left\{
465    \begin{aligned}
466      \frac{1}{e_{1u}\,e_{2u}\,e_{3u}}
467      \left(      \delta_{i+1/2} \left[ \overline{e_{2u}\,e_{3u}\;u }^{i       }  \ u_t      \right]
468        + \delta_{j       } \left[ \overline{e_{1u}\,e_{3u}\;v }^{i+1/2\ u_f      \right] \right\ \;   \\
469      \left.   + \delta_{k      } \left[ \overline{e_{1w}\,e_{2w}\;w}^{i+1/2\ u_{uw} \right] \right)   \\
470      \\
471      \frac{1}{e_{1v}\,e_{2v}\,e_{3v}}
472      \left(     \delta_{i       } \left[ \overline{e_{2u}\,e_{3u }\;u }^{j+1/2} \ v_f       \right]
473        + \delta_{j+1/2} \left[ \overline{e_{1u}\,e_{3u }\;v }^{i       } \ v_t       \right] \right\ \, \, \\
474      \left+ \delta_{k      } \left[ \overline{e_{1w}\,e_{2w}\;w}^{j+1/2} \ v_{vw}  \right] \right) \\
475    \end{aligned}
476  \right.
477\]
478
479Two advection schemes are available:
480a $2^{nd}$ order centered finite difference scheme, CEN2,
481or a $3^{rd}$ order upstream biased scheme, UBS.
482The latter is described in \citet{shchepetkin.mcwilliams_OM05}.
483The schemes are selected using the namelist logicals \np{ln\_dynadv\_cen2} and \np{ln\_dynadv\_ubs}.
484In flux form, the schemes differ by the choice of a space and time interpolation to define the value of
485$u$ and $v$ at the centre of each face of $u$- and $v$-cells, \ie\ at the $T$-, $f$-,
486and $uw$-points for $u$ and at the $f$-, $T$- and $vw$-points for $v$.
487
488%-------------------------------------------------------------
489%                 2nd order centred scheme
490%-------------------------------------------------------------
491\subsubsection[CEN2: $2^{nd}$ order centred scheme (\forcode{ln_dynadv_cen2})]{CEN2: $2^{nd}$ order centred scheme (\protect\np{ln\_dynadv\_cen2})}
492\label{subsec:DYN_adv_cen2}
493
494In the centered $2^{nd}$ order formulation, the velocity is evaluated as the mean of the two neighbouring points:
495\begin{equation}
496  \label{eq:DYN_adv_cen2}
497  \left\{
498    \begin{aligned}
499      u_T^{cen2} &=\overline u^{i }       \quad &  u_F^{cen2} &=\overline u^{j+1/2}  \quad &  u_{uw}^{cen2} &=\overline u^{k+1/2}   \\
500      v_F^{cen2} &=\overline v ^{i+1/2} \quad & v_F^{cen2} &=\overline v^j    \quad &  v_{vw}^{cen2} &=\overline v ^{k+1/2}  \\
501    \end{aligned}
502  \right.
503\end{equation}
504
505The scheme is non diffusive (\ie\ conserves the kinetic energy) but dispersive (\ie\ it may create false extrema).
506It is therefore notoriously noisy and must be used in conjunction with an explicit diffusion operator to
507produce a sensible solution.
508The associated time-stepping is performed using a leapfrog scheme in conjunction with an Asselin time-filter,
509so $u$ and $v$ are the \emph{now} velocities.
510
511%-------------------------------------------------------------
512%                 UBS scheme
513%-------------------------------------------------------------
514\subsubsection[UBS: Upstream Biased Scheme (\forcode{ln_dynadv_ubs})]{UBS: Upstream Biased Scheme (\protect\np{ln\_dynadv\_ubs})}
515\label{subsec:DYN_adv_ubs}
516
517The UBS advection scheme is an upstream biased third order scheme based on
518an upstream-biased parabolic interpolation.
519For example, the evaluation of $u_T^{ubs} $ is done as follows:
520\begin{equation}
521  \label{eq:DYN_adv_ubs}
522  u_T^{ubs} =\overline u ^i-\;\frac{1}{6}
523  \begin{cases}
524    u"_{i-1/2}&   \text{if $\ \overline{e_{2u}\,e_{3u} \ u}^i  \geqslant 0$ }    \\
525    u"_{i+1/2}&   \text{if $\ \overline{e_{2u}\,e_{3u} \ u}^i  < 0$ }
526  \end{cases}
527\end{equation}
528where $u"_{i+1/2} =\delta_{i+1/2} \left[ {\delta_i \left[ u \right]} \right]$.
529This results in a dissipatively dominant (\ie\ hyper-diffusive) truncation error
530\citep{shchepetkin.mcwilliams_OM05}.
531The overall performance of the advection scheme is similar to that reported in \citet{farrow.stevens_JPO95}.
532It is a relatively good compromise between accuracy and smoothness.
533It is not a \emph{positive} scheme, meaning that false extrema are permitted.
534But the amplitudes of the false extrema are significantly reduced over those in the centred second order method.
535As the scheme already includes a diffusion component, it can be used without explicit lateral diffusion on momentum
536(\ie\ \np{ln\_dynldf\_lap}\forcode{=}\np{ln\_dynldf\_bilap}\forcode{=.false.}),
537and it is recommended to do so.
538
539The UBS scheme is not used in all directions.
540In the vertical, the centred $2^{nd}$ order evaluation of the advection is preferred, \ie\ $u_{uw}^{ubs}$ and
541$u_{vw}^{ubs}$ in \autoref{eq:DYN_adv_cen2} are used.
542UBS is diffusive and is associated with vertical mixing of momentum. \gmcomment{ gm  pursue the
543sentence:Since vertical mixing of momentum is a source term of the TKE equation...  }
544
545For stability reasons, the first term in (\autoref{eq:DYN_adv_ubs}),
546which corresponds to a second order centred scheme, is evaluated using the \textit{now} velocity (centred in time),
547while the second term, which is the diffusion part of the scheme,
548is evaluated using the \textit{before} velocity (forward in time).
549This is discussed by \citet{webb.de-cuevas.ea_JAOT98} in the context of the Quick advection scheme.
550
551Note that the UBS and QUICK (Quadratic Upstream Interpolation for Convective Kinematics) schemes only differ by
552one coefficient.
553Replacing $1/6$ by $1/8$ in (\autoref{eq:DYN_adv_ubs}) leads to the QUICK advection scheme \citep{webb.de-cuevas.ea_JAOT98}.
554This option is not available through a namelist parameter, since the $1/6$ coefficient is hard coded.
555Nevertheless it is quite easy to make the substitution in the \mdl{dynadv\_ubs} module and obtain a QUICK scheme.
556
557Note also that in the current version of \mdl{dynadv\_ubs},
558there is also the possibility of using a $4^{th}$ order evaluation of the advective velocity as in ROMS.
559This is an error and should be suppressed soon.
560%%%
561\gmcomment{action :  this have to be done}
562%%%
563
564% ================================================================
565%           Hydrostatic pressure gradient term
566% ================================================================
567\section[Hydrostatic pressure gradient (\textit{dynhpg.F90})]{Hydrostatic pressure gradient (\protect\mdl{dynhpg})}
568\label{sec:DYN_hpg}
569%------------------------------------------nam_dynhpg---------------------------------------------------
570
571\begin{listing}
572  \nlst{namdyn_hpg}
573  \caption{\forcode{&namdyn_hpg}}
574  \label{lst:namdyn_hpg}
575\end{listing}
576%-------------------------------------------------------------------------------------------------------------
577
578Options are defined through the \nam{dyn\_hpg} namelist variables.
579The key distinction between the different algorithms used for
580the hydrostatic pressure gradient is the vertical coordinate used,
581since HPG is a \emph{horizontal} pressure gradient, \ie\ computed along geopotential surfaces.
582As a result, any tilt of the surface of the computational levels will require a specific treatment to
583compute the hydrostatic pressure gradient.
584
585The hydrostatic pressure gradient term is evaluated either using a leapfrog scheme,
586\ie\ the density appearing in its expression is centred in time (\emph{now} $\rho$),
587or a semi-implcit scheme.
588At the lateral boundaries either free slip, no slip or partial slip boundary conditions are applied.
589
590%--------------------------------------------------------------------------------------------------------------
591%           z-coordinate with full step
592%--------------------------------------------------------------------------------------------------------------
593\subsection[Full step $Z$-coordinate (\forcode{ln_dynhpg_zco})]{Full step $Z$-coordinate (\protect\np{ln\_dynhpg\_zco})}
594\label{subsec:DYN_hpg_zco}
595
596The hydrostatic pressure can be obtained by integrating the hydrostatic equation vertically from the surface.
597However, the pressure is large at great depth while its horizontal gradient is several orders of magnitude smaller.
598This may lead to large truncation errors in the pressure gradient terms.
599Thus, the two horizontal components of the hydrostatic pressure gradient are computed directly as follows:
600
601for $k=km$ (surface layer, $jk=1$ in the code)
602\begin{equation}
603  \label{eq:DYN_hpg_zco_surf}
604  \left\{
605    \begin{aligned}
606      \left. \delta_{i+1/2} \left[  p^h          \right] \right|_{k=km}
607      &= \frac{1}{2} g \   \left. \delta_{i+1/2} \left[  e_{3w} \ \rho \right] \right|_{k=km}   \\
608      \left. \delta_{j+1/2} \left[  p^h          \right] \right|_{k=km}
609      &= \frac{1}{2} g \   \left. \delta_{j+1/2} \left[  e_{3w} \ \rho \right] \right|_{k=km}   \\
610    \end{aligned}
611  \right.
612\end{equation}
613
614for $1<k<km$ (interior layer)
615\begin{equation}
616  \label{eq:DYN_hpg_zco}
617  \left\{
618    \begin{aligned}
619      \left. \delta_{i+1/2} \left[  p^h          \right] \right|_{k}
620      &=             \left. \delta_{i+1/2} \left[  p^h          \right] \right|_{k-1}
621      +    \frac{1}{2}\;g\;   \left. \delta_{i+1/2} \left[  e_{3w} \ \overline {\rho}^{k+1/2} \right] \right|_{k}   \\
622      \left. \delta_{j+1/2} \left[  p^h          \right] \right|_{k}
623      &=                \left. \delta_{j+1/2} \left[  p^h          \right] \right|_{k-1}
624      +    \frac{1}{2}\;g\;   \left. \delta_{j+1/2} \left[  e_{3w} \ \overline {\rho}^{k+1/2} \right] \right|_{k}   \\
625    \end{aligned}
626  \right.
627\end{equation}
628
629Note that the $1/2$ factor in (\autoref{eq:DYN_hpg_zco_surf}) is adequate because of the definition of $e_{3w}$ as
630the vertical derivative of the scale factor at the surface level ($z=0$).
631Note also that in case of variable volume level (\texttt{vvl?} defined),
632the surface pressure gradient is included in \autoref{eq:DYN_hpg_zco_surf} and
633\autoref{eq:DYN_hpg_zco} through the space and time variations of the vertical scale factor $e_{3w}$.
634
635%--------------------------------------------------------------------------------------------------------------
636%           z-coordinate with partial step
637%--------------------------------------------------------------------------------------------------------------
638\subsection[Partial step $Z$-coordinate (\forcode{ln_dynhpg_zps})]{Partial step $Z$-coordinate (\protect\np{ln\_dynhpg\_zps})}
639\label{subsec:DYN_hpg_zps}
640
641With partial bottom cells, tracers in horizontally adjacent cells generally live at different depths.
642Before taking horizontal gradients between these tracer points,
643a linear interpolation is used to approximate the deeper tracer as if
644it actually lived at the depth of the shallower tracer point.
645
646Apart from this modification,
647the horizontal hydrostatic pressure gradient evaluated in the $z$-coordinate with partial step is exactly as in
648the pure $z$-coordinate case.
649As explained in detail in section \autoref{sec:TRA_zpshde},
650the nonlinearity of pressure effects in the equation of state is such that
651it is better to interpolate temperature and salinity vertically before computing the density.
652Horizontal gradients of temperature and salinity are needed for the TRA modules,
653which is the reason why the horizontal gradients of density at the deepest model level are computed in
654module \mdl{zpsdhe} located in the TRA directory and described in \autoref{sec:TRA_zpshde}.
655
656%--------------------------------------------------------------------------------------------------------------
657%           s- and s-z-coordinates
658%--------------------------------------------------------------------------------------------------------------
659\subsection{$S$- and $Z$-$S$-coordinates}
660\label{subsec:DYN_hpg_sco}
661
662Pressure gradient formulations in an $s$-coordinate have been the subject of a vast number of papers
663(\eg, \citet{song_MWR98, shchepetkin.mcwilliams_OM05}).
664A number of different pressure gradient options are coded but the ROMS-like,
665density Jacobian with cubic polynomial method is currently disabled whilst known bugs are under investigation.
666
667$\bullet$ Traditional coding (see for example \citet{madec.delecluse.ea_JPO96}: (\np{ln\_dynhpg\_sco}\forcode{=.true.})
668\begin{equation}
669  \label{eq:DYN_hpg_sco}
670  \left\{
671    \begin{aligned}
672      - \frac{1}                 {\rho_o \, e_{1u}} \;   \delta_{i+1/2} \left[  p^h  \right]
673      + \frac{g\; \overline {\rho}^{i+1/2}}  {\rho_o \, e_{1u}} \;   \delta_{i+1/2} \left[  z_t   \right]    \\
674      - \frac{1}                 {\rho_o \, e_{2v}} \;   \delta_{j+1/2} \left[  p^h  \right]
675      + \frac{g\; \overline {\rho}^{j+1/2}}  {\rho_o \, e_{2v}} \;   \delta_{j+1/2} \left[  z_t   \right]    \\
676    \end{aligned}
677  \right.
678\end{equation}
679
680Where the first term is the pressure gradient along coordinates,
681computed as in \autoref{eq:DYN_hpg_zco_surf} - \autoref{eq:DYN_hpg_zco},
682and $z_T$ is the depth of the $T$-point evaluated from the sum of the vertical scale factors at the $w$-point
683($e_{3w}$).
684
685$\bullet$ Traditional coding with adaptation for ice shelf cavities (\np{ln\_dynhpg\_isf}\forcode{=.true.}).
686This scheme need the activation of ice shelf cavities (\np{ln\_isfcav}\forcode{=.true.}).
687
688$\bullet$ Pressure Jacobian scheme (prj) (a research paper in preparation) (\np{ln\_dynhpg\_prj}\forcode{=.true.})
689
690$\bullet$ Density Jacobian with cubic polynomial scheme (DJC) \citep{shchepetkin.mcwilliams_OM05}
691(\np{ln\_dynhpg\_djc}\forcode{=.true.}) (currently disabled; under development)
692
693Note that expression \autoref{eq:DYN_hpg_sco} is commonly used when the variable volume formulation is activated
694(\texttt{vvl?}) because in that case, even with a flat bottom,
695the coordinate surfaces are not horizontal but follow the free surface \citep{levier.treguier.ea_rpt07}.
696The pressure jacobian scheme (\np{ln\_dynhpg\_prj}\forcode{=.true.}) is available as
697an improved option to \np{ln\_dynhpg\_sco}\forcode{=.true.} when \texttt{vvl?} is active.
698The pressure Jacobian scheme uses a constrained cubic spline to
699reconstruct the density profile across the water column.
700This method maintains the monotonicity between the density nodes.
701The pressure can be calculated by analytical integration of the density profile and
702a pressure Jacobian method is used to solve the horizontal pressure gradient.
703This method can provide a more accurate calculation of the horizontal pressure gradient than the standard scheme.
704
705\subsection{Ice shelf cavity}
706\label{subsec:DYN_hpg_isf}
707
708Beneath an ice shelf, the total pressure gradient is the sum of the pressure gradient due to the ice shelf load and
709the pressure gradient due to the ocean load (\np{ln\_dynhpg\_isf}\forcode{=.true.}).\\
710
711The main hypothesis to compute the ice shelf load is that the ice shelf is in an isostatic equilibrium.
712The top pressure is computed integrating from surface to the base of the ice shelf a reference density profile
713(prescribed as density of a water at 34.4 PSU and -1.9\deg{C}) and
714corresponds to the water replaced by the ice shelf.
715This top pressure is constant over time.
716A detailed description of this method is described in \citet{losch_JGR08}.\\
717
718The pressure gradient due to ocean load is computed using the expression \autoref{eq:DYN_hpg_sco} described in
719\autoref{subsec:DYN_hpg_sco}.
720
721%--------------------------------------------------------------------------------------------------------------
722%           Time-scheme
723%--------------------------------------------------------------------------------------------------------------
724\subsection[Time-scheme (\forcode{ln_dynhpg_imp})]{Time-scheme (\protect\np{ln\_dynhpg\_imp})}
725\label{subsec:DYN_hpg_imp}
726
727The default time differencing scheme used for the horizontal pressure gradient is a leapfrog scheme and
728therefore the density used in all discrete expressions given above is the  \textit{now} density,
729computed from the \textit{now} temperature and salinity.
730In some specific cases
731(usually high resolution simulations over an ocean domain which includes weakly stratified regions)
732the physical phenomenon that controls the time-step is internal gravity waves (IGWs).
733A semi-implicit scheme for doubling the stability limit associated with IGWs can be used
734\citep{brown.campana_MWR78, maltrud.smith.ea_JGR98}.
735It involves the evaluation of the hydrostatic pressure gradient as
736an average over the three time levels $t-\rdt$, $t$, and $t+\rdt$
737(\ie\ \textit{before}, \textit{now} and  \textit{after} time-steps),
738rather than at the central time level $t$ only, as in the standard leapfrog scheme.
739
740$\bullet$ leapfrog scheme (\np{ln\_dynhpg\_imp}\forcode{=.true.}):
741
742\begin{equation}
743  \label{eq:DYN_hpg_lf}
744  \frac{u^{t+\rdt}-u^{t-\rdt}}{2\rdt} = \;\cdots \;
745  -\frac{1}{\rho_o \,e_{1u} }\delta_{i+1/2} \left[ {p_h^t } \right]
746\end{equation}
747
748$\bullet$ semi-implicit scheme (\np{ln\_dynhpg\_imp}\forcode{=.true.}):
749\begin{equation}
750  \label{eq:DYN_hpg_imp}
751  \frac{u^{t+\rdt}-u^{t-\rdt}}{2\rdt} = \;\cdots \;
752  -\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]
753\end{equation}
754
755The semi-implicit time scheme \autoref{eq:DYN_hpg_imp} is made possible without
756significant additional computation since the density can be updated to time level $t+\rdt$ before
757computing the horizontal hydrostatic pressure gradient.
758It can be easily shown that the stability limit associated with the hydrostatic pressure gradient doubles using
759\autoref{eq:DYN_hpg_imp} compared to that using the standard leapfrog scheme \autoref{eq:DYN_hpg_lf}.
760Note that \autoref{eq:DYN_hpg_imp} is equivalent to applying a time filter to the pressure gradient to
761eliminate high frequency IGWs.
762Obviously, when using \autoref{eq:DYN_hpg_imp},
763the doubling of the time-step is achievable only if no other factors control the time-step,
764such as the stability limits associated with advection or diffusion.
765
766In practice, the semi-implicit scheme is used when \np{ln\_dynhpg\_imp}\forcode{=.true.}.
767In this case, we choose to apply the time filter to temperature and salinity used in the equation of state,
768instead of applying it to the hydrostatic pressure or to the density,
769so that no additional storage array has to be defined.
770The density used to compute the hydrostatic pressure gradient (whatever the formulation) is evaluated as follows:
771\[
772  % \label{eq:DYN_rho_flt}
773  \rho^t = \rho( \widetilde{T},\widetilde {S},z_t)
774  \quad    \text{with}  \quad
775  \widetilde{X} = 1 / 4 \left(  X^{t+\rdt} +2 \,X^t + X^{t-\rdt\right)
776\]
777
778Note that in the semi-implicit case, it is necessary to save the filtered density,
779an extra three-dimensional field, in the restart file to restart the model with exact reproducibility.
780This option is controlled by  \np{nn\_dynhpg\_rst}, a namelist parameter.
781
782% ================================================================
783% Surface Pressure Gradient
784% ================================================================
785\section[Surface pressure gradient (\textit{dynspg.F90})]{Surface pressure gradient (\protect\mdl{dynspg})}
786\label{sec:DYN_spg}
787%-----------------------------------------nam_dynspg----------------------------------------------------
788
789\begin{listing}
790  \nlst{namdyn_spg}
791  \caption{\forcode{&namdyn_spg}}
792  \label{lst:namdyn_spg}
793\end{listing}
794%------------------------------------------------------------------------------------------------------------
795
796Options are defined through the \nam{dyn\_spg} namelist variables.
797The surface pressure gradient term is related to the representation of the free surface (\autoref{sec:MB_hor_pg}).
798The main distinction is between the fixed volume case (linear free surface) and
799the variable volume case (nonlinear free surface, \texttt{vvl?} is defined).
800In the linear free surface case (\autoref{subsec:MB_free_surface})
801the vertical scale factors $e_{3}$ are fixed in time,
802while they are time-dependent in the nonlinear case (\autoref{subsec:MB_free_surface}).
803With both linear and nonlinear free surface, external gravity waves are allowed in the equations,
804which imposes a very small time step when an explicit time stepping is used.
805Two methods are proposed to allow a longer time step for the three-dimensional equations:
806the filtered free surface, which is a modification of the continuous equations (see \autoref{eq:MB_flt?}),
807and the split-explicit free surface described below.
808The extra term introduced in the filtered method is calculated implicitly,
809so that the update of the next velocities is done in module \mdl{dynspg\_flt} and not in \mdl{dynnxt}.
810
811
812The form of the surface pressure gradient term depends on how the user wants to
813handle the fast external gravity waves that are a solution of the analytical equation (\autoref{sec:MB_hor_pg}).
814Three formulations are available, all controlled by a CPP key (ln\_dynspg\_xxx):
815an explicit formulation which requires a small time step;
816a filtered free surface formulation which allows a larger time step by
817adding a filtering term into the momentum equation;
818and a split-explicit free surface formulation, described below, which also allows a larger time step.
819
820The extra term introduced in the filtered method is calculated implicitly, so that a solver is used to compute it.
821As a consequence the update of the $next$ velocities is done in module \mdl{dynspg\_flt} and not in \mdl{dynnxt}.
822
823
824%--------------------------------------------------------------------------------------------------------------
825% Explicit free surface formulation
826%--------------------------------------------------------------------------------------------------------------
827\subsection[Explicit free surface (\forcode{ln_dynspg_exp})]{Explicit free surface (\protect\np{ln\_dynspg\_exp})}
828\label{subsec:DYN_spg_exp}
829
830In the explicit free surface formulation (\np{ln\_dynspg\_exp} set to true),
831the model time step is chosen to be small enough to resolve the external gravity waves
832(typically a few tens of seconds).
833The surface pressure gradient, evaluated using a leap-frog scheme (\ie\ centered in time),
834is thus simply given by :
835\begin{equation}
836  \label{eq:DYN_spg_exp}
837  \left\{
838    \begin{aligned}
839      - \frac{1}{e_{1u}\,\rho_o} \; \delta_{i+1/2} \left[  \,\rho \,\eta\,  \right]    \\
840      - \frac{1}{e_{2v}\,\rho_o} \; \delta_{j+1/2} \left[  \,\rho \,\eta\,  \right]
841    \end{aligned}
842  \right.
843\end{equation}
844
845Note that in the non-linear free surface case (\ie\ \texttt{vvl?} defined),
846the surface pressure gradient is already included in the momentum tendency through
847the level thickness variation allowed in the computation of the hydrostatic pressure gradient.
848Thus, nothing is done in the \mdl{dynspg\_exp} module.
849
850%--------------------------------------------------------------------------------------------------------------
851% Split-explict free surface formulation
852%--------------------------------------------------------------------------------------------------------------
853\subsection[Split-explicit free surface (\forcode{ln_dynspg_ts})]{Split-explicit free surface (\protect\np{ln\_dynspg\_ts})}
854\label{subsec:DYN_spg_ts}
855%------------------------------------------namsplit-----------------------------------------------------------
856%
857%\nlst{namsplit}
858%-------------------------------------------------------------------------------------------------------------
859
860The split-explicit free surface formulation used in \NEMO\ (\np{ln\_dynspg\_ts} set to true),
861also called the time-splitting formulation, follows the one proposed by \citet{shchepetkin.mcwilliams_OM05}.
862The general idea is to solve the free surface equation and the associated barotropic velocity equations with
863a smaller time step than $\rdt$, the time step used for the three dimensional prognostic variables
864(\autoref{fig:DYN_spg_ts}).
865The size of the small time step, $\rdt_e$ (the external mode or barotropic time step) is provided through
866the \np{nn\_baro} namelist parameter as: $\rdt_e = \rdt / nn\_baro$.
867This parameter can be optionally defined automatically (\np{ln\_bt\_nn\_auto}\forcode{=.true.}) considering that
868the stability of the barotropic system is essentially controled by external waves propagation.
869Maximum Courant number is in that case time independent, and easily computed online from the input bathymetry.
870Therefore, $\rdt_e$ is adjusted so that the Maximum allowed Courant number is smaller than \np{rn\_bt\_cmax}.
871
872%%%
873The barotropic mode solves the following equations:
874% \begin{subequations}
875%  \label{eq:DYN_BT}
876\begin{equation}
877  \label{eq:DYN_BT_dyn}
878  \frac{\partial {\mathrm \overline{{\mathbf U}}_h} }{\partial t}=
879  -f\;{\mathrm {\mathbf k}}\times {\mathrm \overline{{\mathbf U}}_h}
880  -g\nabla _h \eta -\frac{c_b^{\textbf U}}{H+\eta} \mathrm {\overline{{\mathbf U}}_h} + \mathrm {\overline{\mathbf G}}
881\end{equation}
882\[
883  % \label{eq:DYN_BT_ssh}
884  \frac{\partial \eta }{\partial t}=-\nabla \cdot \left[ {\left( {H+\eta } \right) \; {\mathrm{\mathbf \overline{U}}}_h \,} \right]+P-E
885\]
886% \end{subequations}
887where $\mathrm {\overline{\mathbf G}}$ is a forcing term held constant, containing coupling term between modes,
888surface atmospheric forcing as well as slowly varying barotropic terms not explicitly computed to gain efficiency.
889The third term on the right hand side of \autoref{eq:DYN_BT_dyn} represents the bottom stress
890(see section \autoref{sec:ZDF_drg}), explicitly accounted for at each barotropic iteration.
891Temporal discretization of the system above follows a three-time step Generalized Forward Backward algorithm
892detailed in \citet{shchepetkin.mcwilliams_OM05}.
893AB3-AM4 coefficients used in \NEMO\ follow the second-order accurate,
894"multi-purpose" stability compromise as defined in \citet{shchepetkin.mcwilliams_ibk09}
895(see their figure 12, lower left).
896
897%>   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >
898\begin{figure}[!t]
899  \centering
900  \includegraphics[width=0.66\textwidth]{Fig_DYN_dynspg_ts}
901  \caption[Split-explicit time stepping scheme for the external and internal modes]{
902    Schematic of the split-explicit time stepping scheme for the external and internal modes.
903    Time increases to the right.
904    In this particular exemple,
905    a boxcar averaging window over \np{nn\_baro} barotropic time steps is used
906    (\np{nn\_bt\_flt}\forcode{=1}) and \np{nn\_baro}\forcode{=5}.
907    Internal mode time steps (which are also the model time steps) are denoted by
908    $t-\rdt$, $t$ and $t+\rdt$.
909    Variables with $k$ superscript refer to instantaneous barotropic variables,
910    $< >$ and $<< >>$ operator refer to time filtered variables using respectively primary
911    (red vertical bars) and secondary weights (blue vertical bars).
912    The former are used to obtain time filtered quantities at $t+\rdt$ while
913    the latter are used to obtain time averaged transports to advect tracers.
914    a) Forward time integration:
915    \protect\np{ln\_bt\_fw}\forcode{=.true.}\protect\np{ln\_bt\_av}\forcode{=.true.}.
916    b) Centred time integration:
917    \protect\np{ln\_bt\_fw}\forcode{=.false.}, \protect\np{ln\_bt\_av}\forcode{=.true.}.
918    c) Forward time integration with no time filtering (POM-like scheme):
919    \protect\np{ln\_bt\_fw}\forcode{=.true.}\protect\np{ln\_bt\_av}\forcode{=.false.}.}
920  \label{fig:DYN_spg_ts}
921\end{figure}
922%>   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >
923
924In the default case (\np{ln\_bt\_fw}\forcode{=.true.}),
925the external mode is integrated between \textit{now} and \textit{after} baroclinic time-steps
926(\autoref{fig:DYN_spg_ts}a).
927To avoid aliasing of fast barotropic motions into three dimensional equations,
928time filtering is eventually applied on barotropic quantities (\np{ln\_bt\_av}\forcode{=.true.}).
929In that case, the integration is extended slightly beyond \textit{after} time step to
930provide time filtered quantities.
931These are used for the subsequent initialization of the barotropic mode in the following baroclinic step.
932Since external mode equations written at baroclinic time steps finally follow a forward time stepping scheme,
933asselin filtering is not applied to barotropic quantities.\\
934Alternatively, one can choose to integrate barotropic equations starting from \textit{before} time step
935(\np{ln\_bt\_fw}\forcode{=.false.}).
936Although more computationaly expensive ( \np{nn\_baro} additional iterations are indeed necessary),
937the baroclinic to barotropic forcing term given at \textit{now} time step become centred in
938the middle of the integration window.
939It can easily be shown that this property removes part of splitting errors between modes,
940which increases the overall numerical robustness.
941%references to Patrick Marsaleix' work here. Also work done by SHOM group.
942
943%%%
944
945As far as tracer conservation is concerned,
946barotropic velocities used to advect tracers must also be updated at \textit{now} time step.
947This implies to change the traditional order of computations in \NEMO:
948most of momentum trends (including the barotropic mode calculation) updated first, tracers' after.
949This \textit{de facto} makes semi-implicit hydrostatic pressure gradient
950(see section \autoref{subsec:DYN_hpg_imp})
951and time splitting not compatible.
952Advective barotropic velocities are obtained by using a secondary set of filtering weights,
953uniquely defined from the filter coefficients used for the time averaging (\citet{shchepetkin.mcwilliams_OM05}).
954Consistency between the time averaged continuity equation and the time stepping of tracers is here the key to
955obtain exact conservation.
956
957%%%
958
959One can eventually choose to feedback instantaneous values by not using any time filter
960(\np{ln\_bt\_av}\forcode{=.false.}).
961In that case, external mode equations are continuous in time,
962\ie\ they are not re-initialized when starting a new sub-stepping sequence.
963This is the method used so far in the POM model, the stability being maintained by
964refreshing at (almost) each barotropic time step advection and horizontal diffusion terms.
965Since the latter terms have not been added in \NEMO\ for computational efficiency,
966removing time filtering is not recommended except for debugging purposes.
967This may be used for instance to appreciate the damping effect of the standard formulation on
968external gravity waves in idealized or weakly non-linear cases.
969Although the damping is lower than for the filtered free surface,
970it is still significant as shown by \citet{levier.treguier.ea_rpt07} in the case of an analytical barotropic Kelvin wave.
971
972%>>>>>===============
973\gmcomment{               %%% copy from griffies Book
974
975\textbf{title: Time stepping the barotropic system }
976
977Assume knowledge of the full velocity and tracer fields at baroclinic time $\tau$.
978Hence, we can update the surface height and vertically integrated velocity with a leap-frog scheme using
979the small barotropic time step $\rdt$.
980We have
981
982\[
983  % \label{eq:DYN_spg_ts_eta}
984  \eta^{(b)}(\tau,t_{n+1}) - \eta^{(b)}(\tau,t_{n+1}) (\tau,t_{n-1})
985  = 2 \rdt \left[-\nabla \cdot \textbf{U}^{(b)}(\tau,t_n) + \text{EMP}_w(\tau) \right]
986\]
987\begin{multline*}
988  % \label{eq:DYN_spg_ts_u}
989  \textbf{U}^{(b)}(\tau,t_{n+1}) - \textbf{U}^{(b)}(\tau,t_{n-1}\\
990  = 2\rdt \left[ - f \textbf{k} \times \textbf{U}^{(b)}(\tau,t_{n})
991    - H(\tau) \nabla p_s^{(b)}(\tau,t_{n}) +\textbf{M}(\tau) \right]
992\end{multline*}
993\
994
995In these equations, araised (b) denotes values of surface height and vertically integrated velocity updated with
996the barotropic time steps.
997The $\tau$ time label on $\eta^{(b)}$ and $U^{(b)}$ denotes the baroclinic time at which
998the vertically integrated forcing $\textbf{M}(\tau)$
999(note that this forcing includes the surface freshwater forcing),
1000the tracer fields, the freshwater flux $\text{EMP}_w(\tau)$,
1001and total depth of the ocean $H(\tau)$ are held for the duration of the barotropic time stepping over
1002a single cycle.
1003This is also the time that sets the barotropic time steps via
1004\[
1005  % \label{eq:DYN_spg_ts_t}
1006  t_n=\tau+n\rdt
1007\]
1008with $n$ an integer.
1009The density scaled surface pressure is evaluated via
1010\[
1011  % \label{eq:DYN_spg_ts_ps}
1012  p_s^{(b)}(\tau,t_{n}) =
1013  \begin{cases}
1014    g \;\eta_s^{(b)}(\tau,t_{n}) \;\rho(\tau)_{k=1}) / \rho_o  &      \text{non-linear case} \\
1015    g \;\eta_s^{(b)}(\tau,t_{n})  &      \text{linear case}
1016  \end{cases}
1017\]
1018To get started, we assume the following initial conditions
1019\[
1020  % \label{eq:DYN_spg_ts_eta}
1021  \begin{split}
1022    \eta^{(b)}(\tau,t_{n=0}) &= \overline{\eta^{(b)}(\tau)}    \\
1023    \eta^{(b)}(\tau,t_{n=1}) &= \eta^{(b)}(\tau,t_{n=0}) + \rdt \ \text{RHS}_{n=0}
1024  \end{split}
1025\]
1026with
1027\[
1028  % \label{eq:DYN_spg_ts_etaF}
1029  \overline{\eta^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N \eta^{(b)}(\tau-\rdt,t_{n})
1030\]
1031the time averaged surface height taken from the previous barotropic cycle.
1032Likewise,
1033\[
1034  % \label{eq:DYN_spg_ts_u}
1035  \textbf{U}^{(b)}(\tau,t_{n=0}) = \overline{\textbf{U}^{(b)}(\tau)} \\ \\
1036  \textbf{U}(\tau,t_{n=1}) = \textbf{U}^{(b)}(\tau,t_{n=0}) + \rdt \ \text{RHS}_{n=0}
1037\]
1038with
1039\[
1040  % \label{eq:DYN_spg_ts_u}
1041  \overline{\textbf{U}^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau-\rdt,t_{n})
1042\]
1043the time averaged vertically integrated transport.
1044Notably, there is no Robert-Asselin time filter used in the barotropic portion of the integration.
1045
1046Upon reaching $t_{n=N} = \tau + 2\rdt \tau$ ,
1047the vertically integrated velocity is time averaged to produce the updated vertically integrated velocity at
1048baroclinic time $\tau + \rdt \tau$
1049\[
1050  % \label{eq:DYN_spg_ts_u}
1051  \textbf{U}(\tau+\rdt) = \overline{\textbf{U}^{(b)}(\tau+\rdt)} = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau,t_{n})
1052\]
1053The surface height on the new baroclinic time step is then determined via a baroclinic leap-frog using
1054the following form
1055
1056\begin{equation}
1057  \label{eq:DYN_spg_ts_ssh}
1058  \eta(\tau+\Delta) - \eta^{F}(\tau-\Delta) = 2\rdt \ \left[ - \nabla \cdot \textbf{U}(\tau) + \text{EMP}_w \right]
1059\end{equation}
1060
1061The use of this "big-leap-frog" scheme for the surface height ensures compatibility between
1062the mass/volume budgets and the tracer budgets.
1063More discussion of this point is provided in Chapter 10 (see in particular Section 10.2).
1064
1065In general, some form of time filter is needed to maintain integrity of the surface height field due to
1066the leap-frog splitting mode in equation \autoref{eq:DYN_spg_ts_ssh}.
1067We have tried various forms of such filtering,
1068with the following method discussed in \cite{griffies.pacanowski.ea_MWR01} chosen due to
1069its stability and reasonably good maintenance of tracer conservation properties (see ??).
1070
1071\begin{equation}
1072  \label{eq:DYN_spg_ts_sshf}
1073  \eta^{F}(\tau-\Delta) =  \overline{\eta^{(b)}(\tau)}
1074\end{equation}
1075Another approach tried was
1076
1077\[
1078  % \label{eq:DYN_spg_ts_sshf2}
1079  \eta^{F}(\tau-\Delta) = \eta(\tau)
1080  + (\alpha/2) \left[\overline{\eta^{(b)}}(\tau+\rdt)
1081    + \overline{\eta^{(b)}}(\tau-\rdt) -2 \;\eta(\tau) \right]
1082\]
1083
1084which is useful since it isolates all the time filtering aspects into the term multiplied by $\alpha$.
1085This isolation allows for an easy check that tracer conservation is exact when
1086eliminating tracer and surface height time filtering (see ?? for more complete discussion).
1087However, in the general case with a non-zero $\alpha$,
1088the filter \autoref{eq:DYN_spg_ts_sshf} was found to be more conservative, and so is recommended.
1089
1090}            %%end gm comment (copy of griffies book)
1091
1092%>>>>>===============
1093
1094
1095%--------------------------------------------------------------------------------------------------------------
1096% Filtered free surface formulation
1097%--------------------------------------------------------------------------------------------------------------
1098\subsection{Filtered free surface (\forcode{dynspg_flt?})}
1099\label{subsec:DYN_spg_fltp}
1100
1101The filtered formulation follows the \citet{roullet.madec_JGR00} implementation.
1102The extra term introduced in the equations (see \autoref{subsec:MB_free_surface}) is solved implicitly.
1103The elliptic solvers available in the code are documented in \autoref{chap:MISC}.
1104
1105%% gm %%======>>>>   given here the discrete eqs provided to the solver
1106\gmcomment{               %%% copy from chap-model basics
1107  \[
1108    % \label{eq:DYN_spg_flt}
1109    \frac{\partial {\mathrm {\mathbf U}}_h }{\partial t}= {\mathrm {\mathbf M}}
1110    - g \nabla \left( \tilde{\rho} \ \eta \right)
1111    - g \ T_c \nabla \left( \widetilde{\rho} \ \partial_t \eta \right)
1112  \]
1113  where $T_c$, is a parameter with dimensions of time which characterizes the force,
1114  $\widetilde{\rho} = \rho / \rho_o$ is the dimensionless density,
1115  and $\mathrm {\mathbf M}$ represents the collected contributions of the Coriolis, hydrostatic pressure gradient,
1116  non-linear and viscous terms in \autoref{eq:MB_dyn}.
1117}   %end gmcomment
1118
1119Note that in the linear free surface formulation (\texttt{vvl?} not defined),
1120the ocean depth is time-independent and so is the matrix to be inverted.
1121It is computed once and for all and applies to all ocean time steps.
1122
1123% ================================================================
1124% Lateral diffusion term
1125% ================================================================
1126\section[Lateral diffusion term and operators (\textit{dynldf.F90})]{Lateral diffusion term and operators (\protect\mdl{dynldf})}
1127\label{sec:DYN_ldf}
1128%------------------------------------------nam_dynldf----------------------------------------------------
1129
1130\begin{listing}
1131  \nlst{namdyn_ldf}
1132  \caption{\forcode{&namdyn_ldf}}
1133  \label{lst:namdyn_ldf}
1134\end{listing}
1135%-------------------------------------------------------------------------------------------------------------
1136
1137Options are defined through the \nam{dyn\_ldf} namelist variables.
1138The options available for lateral diffusion are to use either laplacian (rotated or not) or biharmonic operators.
1139The coefficients may be constant or spatially variable;
1140the description of the coefficients is found in the chapter on lateral physics (\autoref{chap:LDF}).
1141The lateral diffusion of momentum is evaluated using a forward scheme,
1142\ie\ the velocity appearing in its expression is the \textit{before} velocity in time,
1143except for the pure vertical component that appears when a tensor of rotation is used.
1144This latter term is solved implicitly together with the vertical diffusion term (see \autoref{chap:TD}).
1145
1146At the lateral boundaries either free slip,
1147no slip or partial slip boundary conditions are applied according to the user's choice (see \autoref{chap:LBC}).
1148
1149\gmcomment{
1150  Hyperviscous operators are frequently used in the simulation of turbulent flows to
1151  control the dissipation of unresolved small scale features.
1152  Their primary role is to provide strong dissipation at the smallest scale supported by
1153  the grid while minimizing the impact on the larger scale features.
1154  Hyperviscous operators are thus designed to be more scale selective than the traditional,
1155  physically motivated Laplace operator.
1156  In finite difference methods,
1157  the biharmonic operator is frequently the method of choice to achieve this scale selective dissipation since
1158  its damping time (\ie\ its spin down time) scale like $\lambda^{-4}$ for disturbances of wavelength $\lambda$
1159  (so that short waves damped more rapidelly than long ones),
1160  whereas the Laplace operator damping time scales only like $\lambda^{-2}$.
1161}
1162
1163% ================================================================
1164\subsection[Iso-level laplacian (\forcode{ln_dynldf_lap})]{Iso-level laplacian operator (\protect\np{ln\_dynldf\_lap})}
1165\label{subsec:DYN_ldf_lap}
1166
1167For lateral iso-level diffusion, the discrete operator is:
1168\begin{equation}
1169  \label{eq:DYN_ldf_lap}
1170  \left\{
1171    \begin{aligned}
1172      D_u^{l{\mathrm {\mathbf U}}} =\frac{1}{e_{1u} }\delta_{i+1/2} \left[ {A_T^{lm}
1173          \;\chi } \right]-\frac{1}{e_{2u} {\kern 1pt}e_{3u} }\delta_j \left[
1174        {A_f^{lm} \;e_{3f} \zeta } \right] \\ \\
1175      D_v^{l{\mathrm {\mathbf U}}} =\frac{1}{e_{2v} }\delta_{j+1/2} \left[ {A_T^{lm}
1176          \;\chi } \right]+\frac{1}{e_{1v} {\kern 1pt}e_{3v} }\delta_i \left[
1177        {A_f^{lm} \;e_{3f} \zeta } \right]
1178    \end{aligned}
1179  \right.
1180\end{equation}
1181
1182As explained in \autoref{subsec:MB_ldf},
1183this formulation (as the gradient of a divergence and curl of the vorticity) preserves symmetry and
1184ensures a complete separation between the vorticity and divergence parts of the momentum diffusion.
1185
1186%--------------------------------------------------------------------------------------------------------------
1187%           Rotated laplacian operator
1188%--------------------------------------------------------------------------------------------------------------
1189\subsection[Rotated laplacian (\forcode{ln_dynldf_iso})]{Rotated laplacian operator (\protect\np{ln\_dynldf\_iso})}
1190\label{subsec:DYN_ldf_iso}
1191
1192A rotation of the lateral momentum diffusion operator is needed in several cases:
1193for iso-neutral diffusion in the $z$-coordinate (\np{ln\_dynldf\_iso}\forcode{=.true.}) and
1194for either iso-neutral (\np{ln\_dynldf\_iso}\forcode{=.true.}) or
1195geopotential (\np{ln\_dynldf\_hor}\forcode{=.true.}) diffusion in the $s$-coordinate.
1196In the partial step case, coordinates are horizontal except at the deepest level and
1197no rotation is performed when \np{ln\_dynldf\_hor}\forcode{=.true.}.
1198The diffusion operator is defined simply as the divergence of down gradient momentum fluxes on
1199each momentum component.
1200It must be emphasized that this formulation ignores constraints on the stress tensor such as symmetry.
1201The resulting discrete representation is:
1202\begin{equation}
1203  \label{eq:DYN_ldf_iso}
1204  \begin{split}
1205    D_u^{l\textbf{U}} &= \frac{1}{e_{1u} \, e_{2u} \, e_{3u} } \\
1206    &  \left\{\quad  {\delta_{i+1/2} \left[ {A_T^{lm}  \left(
1207              {\frac{e_{2t} \; e_{3t} }{e_{1t} } \,\delta_{i}[u]
1208                -e_{2t} \; r_{1t} \,\overline{\overline {\delta_{k+1/2}[u]}}^{\,i,\,k}}
1209            \right)} \right]}    \right. \\
1210    & \qquad +\ \delta_j \left[ {A_f^{lm} \left( {\frac{e_{1f}\,e_{3f} }{e_{2f}
1211            }\,\delta_{j+1/2} [u] - e_{1f}\, r_{2f}
1212            \,\overline{\overline {\delta_{k+1/2} [u]}} ^{\,j+1/2,\,k}}
1213        \right)} \right] \\
1214    &\qquad +\ \delta_k \left[ {A_{uw}^{lm} \left( {-e_{2u} \, r_{1uw} \,\overline{\overline
1215              {\delta_{i+1/2} [u]}}^{\,i+1/2,\,k+1/2} }
1216        \right.} \right. \\
1217    &  \ \qquad \qquad \qquad \quad\
1218    - e_{1u} \, r_{2uw} \,\overline{\overline {\delta_{j+1/2} [u]}} ^{\,j,\,k+1/2} \\
1219    & \left. {\left. { \ \qquad \qquad \qquad \ \ \ \left. {\
1220                +\frac{e_{1u}\, e_{2u} }{e_{3uw} }\,\left( {r_{1uw}^2+r_{2uw}^2}
1221                \right)\,\delta_{k+1/2} [u]} \right)} \right]\;\;\;} \right\} \\ \\
1222    D_v^{l\textbf{V}} &= \frac{1}{e_{1v} \, e_{2v} \, e_{3v} } \\
1223    &  \left\{\quad  {\delta_{i+1/2} \left[ {A_f^{lm}  \left(
1224              {\frac{e_{2f} \; e_{3f} }{e_{1f} } \,\delta_{i+1/2}[v]
1225                -e_{2f} \; r_{1f} \,\overline{\overline {\delta_{k+1/2}[v]}}^{\,i+1/2,\,k}}
1226            \right)} \right]}    \right. \\
1227    & \qquad +\ \delta_j \left[ {A_T^{lm} \left( {\frac{e_{1t}\,e_{3t} }{e_{2t}
1228            }\,\delta_{j} [v] - e_{1t}\, r_{2t}
1229            \,\overline{\overline {\delta_{k+1/2} [v]}} ^{\,j,\,k}}
1230        \right)} \right] \\
1231    & \qquad +\ \delta_k \left[ {A_{vw}^{lm} \left( {-e_{2v} \, r_{1vw} \,\overline{\overline
1232              {\delta_{i+1/2} [v]}}^{\,i+1/2,\,k+1/2} }\right.} \right. \\
1233    &  \ \qquad \qquad \qquad \quad\
1234    - e_{1v} \, r_{2vw} \,\overline{\overline {\delta_{j+1/2} [v]}} ^{\,j+1/2,\,k+1/2} \\
1235    & \left. {\left. { \ \qquad \qquad \qquad \ \ \ \left. {\
1236                +\frac{e_{1v}\, e_{2v} }{e_{3vw} }\,\left( {r_{1vw}^2+r_{2vw}^2}
1237                \right)\,\delta_{k+1/2} [v]} \right)} \right]\;\;\;} \right\}
1238  \end{split}
1239\end{equation}
1240where $r_1$ and $r_2$ are the slopes between the surface along which the diffusion operator acts and
1241the surface of computation ($z$- or $s$-surfaces).
1242The way these slopes are evaluated is given in the lateral physics chapter (\autoref{chap:LDF}).
1243
1244%--------------------------------------------------------------------------------------------------------------
1245%           Iso-level bilaplacian operator
1246%--------------------------------------------------------------------------------------------------------------
1247\subsection[Iso-level bilaplacian (\forcode{ln_dynldf_bilap})]{Iso-level bilaplacian operator (\protect\np{ln\_dynldf\_bilap})}
1248\label{subsec:DYN_ldf_bilap}
1249
1250The lateral fourth order operator formulation on momentum is obtained by applying \autoref{eq:DYN_ldf_lap} twice.
1251It requires an additional assumption on boundary conditions:
1252the first derivative term normal to the coast depends on the free or no-slip lateral boundary conditions chosen,
1253while the third derivative terms normal to the coast are set to zero (see \autoref{chap:LBC}).
1254%%%
1255\gmcomment{add a remark on the the change in the position of the coefficient}
1256%%%
1257
1258% ================================================================
1259%           Vertical diffusion term
1260% ================================================================
1261\section[Vertical diffusion term (\textit{dynzdf.F90})]{Vertical diffusion term (\protect\mdl{dynzdf})}
1262\label{sec:DYN_zdf}
1263%----------------------------------------------namzdf------------------------------------------------------
1264
1265%-------------------------------------------------------------------------------------------------------------
1266
1267Options are defined through the \nam{zdf} namelist variables.
1268The 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.
1269Two time stepping schemes can be used for the vertical diffusion term:
1270$(a)$ a forward time differencing scheme
1271(\np{ln\_zdfexp}\forcode{=.true.}) using a time splitting technique (\np{nn\_zdfexp} $>$ 1) or
1272$(b)$ a backward (or implicit) time differencing scheme (\np{ln\_zdfexp}\forcode{=.false.})
1273(see \autoref{chap:TD}).
1274Note that namelist variables \np{ln\_zdfexp} and \np{nn\_zdfexp} apply to both tracers and dynamics.
1275
1276The formulation of the vertical subgrid scale physics is the same whatever the vertical coordinate is.
1277The vertical diffusion operators given by \autoref{eq:MB_zdf} take the following semi-discrete space form:
1278\[
1279  % \label{eq:DYN_zdf}
1280  \left\{
1281    \begin{aligned}
1282      D_u^{vm} &\equiv \frac{1}{e_{3u}} \ \delta_k \left[ \frac{A_{uw}^{vm} }{e_{3uw} }
1283        \ \delta_{k+1/2} [\,u\,]         \right]     \\
1284      \\
1285      D_v^{vm} &\equiv \frac{1}{e_{3v}} \ \delta_k \left[ \frac{A_{vw}^{vm} }{e_{3vw} }
1286        \ \delta_{k+1/2} [\,v\,]         \right]
1287    \end{aligned}
1288  \right.
1289\]
1290where $A_{uw}^{vm} $ and $A_{vw}^{vm} $ are the vertical eddy viscosity and diffusivity coefficients.
1291The way these coefficients are evaluated depends on the vertical physics used (see \autoref{chap:ZDF}).
1292
1293The surface boundary condition on momentum is the stress exerted by the wind.
1294At the surface, the momentum fluxes are prescribed as the boundary condition on
1295the vertical turbulent momentum fluxes,
1296\begin{equation}
1297  \label{eq:DYN_zdf_sbc}
1298  \left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{z=1}
1299  = \frac{1}{\rho_o} \binom{\tau_u}{\tau_v }
1300\end{equation}
1301where $\left( \tau_u ,\tau_v \right)$ are the two components of the wind stress vector in
1302the (\textbf{i},\textbf{j}) coordinate system.
1303The high mixing coefficients in the surface mixed layer ensure that the surface wind stress is distributed in
1304the vertical over the mixed layer depth.
1305If the vertical mixing coefficient is small (when no mixed layer scheme is used)
1306the surface stress enters only the top model level, as a body force.
1307The surface wind stress is calculated in the surface module routines (SBC, see \autoref{chap:SBC}).
1308
1309The turbulent flux of momentum at the bottom of the ocean is specified through a bottom friction parameterisation
1310(see \autoref{sec:ZDF_drg})
1311
1312% ================================================================
1313% External Forcing
1314% ================================================================
1315\section{External forcings}
1316\label{sec:DYN_forcing}
1317
1318Besides the surface and bottom stresses (see the above section)
1319which are introduced as boundary conditions on the vertical mixing,
1320three other forcings may enter the dynamical equations by affecting the surface pressure gradient.
1321
1322(1) When \np{ln\_apr\_dyn}\forcode{=.true.} (see \autoref{sec:SBC_apr}),
1323the atmospheric pressure is taken into account when computing the surface pressure gradient.
1324
1325(2) When \np{ln\_tide\_pot}\forcode{=.true.} and \np{ln\_tide}\forcode{=.true.} (see \autoref{sec:SBC_tide}),
1326the tidal potential is taken into account when computing the surface pressure gradient.
1327
1328(3) When \np{nn\_ice\_embd}\forcode{=2} and LIM or CICE is used
1329(\ie\ when the sea-ice is embedded in the ocean),
1330the snow-ice mass is taken into account when computing the surface pressure gradient.
1331
1332
1333\gmcomment{ missing : the lateral boundary condition !!!   another external forcing
1334 }
1335
1336% ================================================================
1337% Wetting and drying
1338% ================================================================
1339\section{Wetting and drying }
1340\label{sec:DYN_wetdry}
1341
1342There are two main options for wetting and drying code (wd):
1343(a) an iterative limiter (il) and (b) a directional limiter (dl).
1344The directional limiter is based on the scheme developed by \cite{warner.defne.ea_CG13} for RO
1345MS
1346which was in turn based on ideas developed for POM by \cite{oey_OM06}. The iterative
1347limiter is a new scheme.  The iterative limiter is activated by setting $\mathrm{ln\_wd\_il} = \mathrm{.true.}$
1348and $\mathrm{ln\_wd\_dl} = \mathrm{.false.}$. The directional limiter is activated
1349by setting $\mathrm{ln\_wd\_dl} = \mathrm{.true.}$ and $\mathrm{ln\_wd\_il} = \mathrm{.false.}$.
1350
1351\begin{listing}
1352  \nlst{namwad}
1353  \caption{\forcode{&namwad}}
1354  \label{lst:namwad}
1355\end{listing}
1356
1357The following terminology is used. The depth of the topography (positive downwards)
1358at each $(i,j)$ point is the quantity stored in array $\mathrm{ht\_wd}$ in the \NEMO\ code.
1359The height of the free surface (positive upwards) is denoted by $ \mathrm{ssh}$. Given the sign
1360conventions used, the water depth, $h$, is the height of the free surface plus the depth of the
1361topography (i.e. $\mathrm{ssh} + \mathrm{ht\_wd}$).
1362
1363Both wd schemes take all points in the domain below a land elevation of $\mathrm{rn\_wdld}$ to be
1364covered by water. They require the topography specified with a model
1365configuration to have negative depths at points where the land is higher than the
1366topography's reference sea-level. The vertical grid in \NEMO\ is normally computed relative to an
1367initial state with zero sea surface height elevation.
1368The user can choose to compute the vertical grid and heights in the model relative to
1369a non-zero reference height for the free surface. This choice affects the calculation of the metrics and depths
1370(i.e. the $\mathrm{e3t\_0, ht\_0}$ etc. arrays).
1371
1372Points where the water depth is less than $\mathrm{rn\_wdmin1}$ are interpreted as ``dry''.
1373$\mathrm{rn\_wdmin1}$ is usually chosen to be of order $0.05$m but extreme topographies
1374with very steep slopes require larger values for normal choices of time-step. Surface fluxes
1375are also switched off for dry cells to prevent freezing, boiling etc. of very thin water layers.
1376The fluxes are tappered down using a $\mathrm{tanh}$ weighting function
1377to no flux as the dry limit $\mathrm{rn\_wdmin1}$ is approached. Even wet cells can be very shallow.
1378The depth at which to start tapering is controlled by the user by setting $\mathrm{rn\_wd\_sbcdep}$.
1379The fraction $(<1)$ of sufrace fluxes to use at this depth is set by $\mathrm{rn\_wd\_sbcfra}$.
1380
1381Both versions of the code have been tested in six test cases provided in the WAD\_TEST\_CASES configuration
1382and in ``realistic'' configurations covering parts of the north-west European shelf.
1383All these configurations have used pure sigma coordinates. It is expected that
1384the wetting and drying code will work in domains with more general s-coordinates provided
1385the coordinates are pure sigma in the region where wetting and drying actually occurs.
1386
1387The next sub-section descrbies the directional limiter and the following sub-section the iterative limiter.
1388The final sub-section covers some additional considerations that are relevant to both schemes.
1389
1390
1391%-----------------------------------------------------------------------------------------
1392%   Iterative limiters
1393%-----------------------------------------------------------------------------------------
1394\subsection[Directional limiter (\textit{wet\_dry.F90})]{Directional limiter (\mdl{wet\_dry})}
1395\label{subsec:DYN_wd_directional_limiter}
1396
1397The principal idea of the directional limiter is that
1398water should not be allowed to flow out of a dry tracer cell (i.e. one whose water depth is less than \np{rn\_wdmin1}).
1399
1400All the changes associated with this option are made to the barotropic solver for the non-linear
1401free surface code within dynspg\_ts.
1402On each barotropic sub-step the scheme determines the direction of the flow across each face of all the tracer cells
1403and sets the flux across the face to zero when the flux is from a dry tracer cell. This prevents cells
1404whose depth is rn\_wdmin1 or less from drying out further. The scheme does not force $h$ (the water depth) at tracer cells
1405to be at least the minimum depth and hence is able to conserve mass / volume.
1406
1407The flux across each $u$-face of a tracer cell is multiplied by a factor zuwdmask (an array which depends on ji and jj).
1408If the user sets \np{ln\_wd\_dl\_ramp}\forcode{=.false.} then zuwdmask is 1 when the
1409flux is from a cell with water depth greater than \np{rn\_wdmin1} and 0 otherwise. If the user sets
1410\np{ln\_wd\_dl\_ramp}\forcode{=.true.} the flux across the face is ramped down as the water depth decreases
1411from 2 * \np{rn\_wdmin1} to \np{rn\_wdmin1}. The use of this ramp reduced grid-scale noise in idealised test cases.
1412
1413At the point where the flux across a $u$-face is multiplied by zuwdmask , we have chosen
1414also to multiply the corresponding velocity on the ``now'' step at that face by zuwdmask. We could have
1415chosen not to do that and to allow fairly large velocities to occur in these ``dry'' cells.
1416The rationale for setting the velocity to zero is that it is the momentum equations that are being solved
1417and the total momentum of the upstream cell (treating it as a finite volume) should be considered
1418to be its depth times its velocity. This depth is considered to be zero at ``dry'' $u$-points consistent with its
1419treatment in the calculation of the flux of mass across the cell face.
1420
1421
1422\cite{warner.defne.ea_CG13} state that in their scheme the velocity masks at the cell faces for the baroclinic
1423timesteps are set to 0 or 1 depending on whether the average of the masks over the barotropic sub-steps is respectively less than
1424or greater than 0.5. That scheme does not conserve tracers in integrations started from constant tracer
1425fields (tracers independent of $x$, $y$ and $z$). Our scheme conserves constant tracers because
1426the velocities used at the tracer cell faces on the baroclinic timesteps are carefully calculated by dynspg\_ts
1427to equal their mean value during the barotropic steps. If the user sets \np{ln\_wd\_dl\_bc}\forcode{=.true.}, the
1428baroclinic velocities are also multiplied by a suitably weighted average of zuwdmask.
1429
1430%-----------------------------------------------------------------------------------------
1431%   Iterative limiters
1432%-----------------------------------------------------------------------------------------
1433
1434\subsection[Iterative limiter (\textit{wet\_dry.F90})]{Iterative limiter (\mdl{wet\_dry})}
1435\label{subsec:DYN_wd_iterative_limiter}
1436
1437\subsubsection[Iterative flux limiter (\textit{wet\_dry.F90})]{Iterative flux limiter (\mdl{wet\_dry})}
1438\label{subsec:DYN_wd_il_spg_limiter}
1439
1440The iterative limiter modifies the fluxes across the faces of cells that are either already ``dry''
1441or may become dry within the next time-step using an iterative method.
1442
1443The flux limiter for the barotropic flow (devised by Hedong Liu) can be understood as follows:
1444
1445The continuity equation for the total water depth in a column
1446\begin{equation}
1447  \label{eq:DYN_wd_continuity}
1448  \frac{\partial h}{\partial t} + \mathbf{\nabla.}(h\mathbf{u}) = 0 .
1449\end{equation}
1450can be written in discrete form  as
1451
1452\begin{align}
1453  \label{eq:DYN_wd_continuity_2}
1454  \frac{e_1 e_2}{\Delta t} ( h_{i,j}(t_{n+1}) - h_{i,j}(t_e) )
1455  &= - ( \mathrm{flxu}_{i+1,j} - \mathrm{flxu}_{i,j}  + \mathrm{flxv}_{i,j+1} - \mathrm{flxv}_{i,j} ) \\
1456  &= \mathrm{zzflx}_{i,j} .
1457\end{align}
1458
1459In the above $h$ is the depth of the water in the column at point $(i,j)$,
1460$\mathrm{flxu}_{i+1,j}$ is the flux out of the ``eastern'' face of the cell and
1461$\mathrm{flxv}_{i,j+1}$ the flux out of the ``northern'' face of the cell; $t_{n+1}$ is
1462the new timestep, $t_e$ is the old timestep (either $t_b$ or $t_n$) and $ \Delta t =
1463t_{n+1} - t_e$; $e_1 e_2$ is the area of the tracer cells centred at $(i,j)$ and
1464$\mathrm{zzflx}$ is the sum of the fluxes through all the faces.
1465
1466The flux limiter splits the flux $\mathrm{zzflx}$ into fluxes that are out of the cell
1467(zzflxp) and fluxes that are into the cell (zzflxn).  Clearly
1468
1469\begin{equation}
1470  \label{eq:DYN_wd_zzflx_p_n_1}
1471  \mathrm{zzflx}_{i,j} = \mathrm{zzflxp}_{i,j} + \mathrm{zzflxn}_{i,j} .
1472\end{equation}
1473
1474The flux limiter iteratively adjusts the fluxes $\mathrm{flxu}$ and $\mathrm{flxv}$ until
1475none of the cells will ``dry out''. To be precise the fluxes are limited until none of the
1476cells has water depth less than $\mathrm{rn\_wdmin1}$ on step $n+1$.
1477
1478Let the fluxes on the $m$th iteration step be denoted by $\mathrm{flxu}^{(m)}$ and
1479$\mathrm{flxv}^{(m)}$.  Then the adjustment is achieved by seeking a set of coefficients,
1480$\mathrm{zcoef}_{i,j}^{(m)}$ such that:
1481
1482\begin{equation}
1483  \label{eq:DYN_wd_continuity_coef}
1484  \begin{split}
1485    \mathrm{zzflxp}^{(m)}_{i,j} =& \mathrm{zcoef}_{i,j}^{(m)} \mathrm{zzflxp}^{(0)}_{i,j} \\
1486    \mathrm{zzflxn}^{(m)}_{i,j} =& \mathrm{zcoef}_{i,j}^{(m)} \mathrm{zzflxn}^{(0)}_{i,j}
1487  \end{split}
1488\end{equation}
1489
1490where the coefficients are $1.0$ generally but can vary between $0.0$ and $1.0$ around
1491cells that would otherwise dry.
1492
1493The iteration is initialised by setting
1494
1495\begin{equation}
1496  \label{eq:DYN_wd_zzflx_initial}
1497  \mathrm{zzflxp^{(0)}}_{i,j} = \mathrm{zzflxp}_{i,j} , \quad  \mathrm{zzflxn^{(0)}}_{i,j} = \mathrm{zzflxn}_{i,j} .
1498\end{equation}
1499
1500The fluxes out of cell $(i,j)$ are updated at the $m+1$th iteration if the depth of the
1501cell on timestep $t_e$, namely $h_{i,j}(t_e)$, is less than the total flux out of the cell
1502times the timestep divided by the cell area. Using (\autoref{eq:DYN_wd_continuity_2}) this
1503condition is
1504
1505\begin{equation}
1506  \label{eq:DYN_wd_continuity_if}
1507  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} ) .
1508\end{equation}
1509
1510Rearranging (\autoref{eq:DYN_wd_continuity_if}) we can obtain an expression for the maximum
1511outward flux that can be allowed and still maintain the minimum wet depth:
1512
1513\begin{equation}
1514  \label{eq:DYN_wd_max_flux}
1515  \begin{split}
1516    \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{]} \\
1517    \phantom{[} & -  \mathrm{zzflxn}^{(m)}_{i,j} \Big]
1518  \end{split}
1519\end{equation}
1520
1521Note a small tolerance ($\mathrm{rn\_wdmin2}$) has been introduced here {\itshape [Q: Why is
1522this necessary/desirable?]}. Substituting from (\autoref{eq:DYN_wd_continuity_coef}) gives an
1523expression for the coefficient needed to multiply the outward flux at this cell in order
1524to avoid drying.
1525
1526\begin{equation}
1527  \label{eq:DYN_wd_continuity_nxtcoef}
1528  \begin{split}
1529    \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{]} \\
1530    \phantom{[} & -  \mathrm{zzflxn}^{(m)}_{i,j} \Big] \frac{1}{ \mathrm{zzflxp}^{(0)}_{i,j} }
1531  \end{split}
1532\end{equation}
1533
1534Only the outward flux components are altered but, of course, outward fluxes from one cell
1535are inward fluxes to adjacent cells and the balance in these cells may need subsequent
1536adjustment; hence the iterative nature of this scheme.  Note, for example, that the flux
1537across the ``eastern'' face of the $(i,j)$th cell is only updated at the $m+1$th iteration
1538if that flux at the $m$th iteration is out of the $(i,j)$th cell. If that is the case then
1539the flux across that face is into the $(i+1,j)$ cell and that flux will not be updated by
1540the calculation for the $(i+1,j)$th cell. In this sense the updates to the fluxes across
1541the faces of the cells do not ``compete'' (they do not over-write each other) and one
1542would expect the scheme to converge relatively quickly. The scheme is flux based so
1543conserves mass. It also conserves constant tracers for the same reason that the
1544directional limiter does.
1545
1546
1547%----------------------------------------------------------------------------------------
1548%      Surface pressure gradients
1549%----------------------------------------------------------------------------------------
1550\subsubsection[Modification of surface pressure gradients (\textit{dynhpg.F90})]{Modification of surface pressure gradients (\mdl{dynhpg})}
1551\label{subsec:DYN_wd_il_spg}
1552
1553At ``dry'' points the water depth is usually close to $\mathrm{rn\_wdmin1}$. If the
1554topography is sloping at these points the sea-surface will have a similar slope and there
1555will hence be very large horizontal pressure gradients at these points. The WAD modifies
1556the magnitude but not the sign of the surface pressure gradients (zhpi and zhpj) at such
1557points by mulitplying them by positive factors (zcpx and zcpy respectively) that lie
1558between $0$ and $1$.
1559
1560We describe how the scheme works for the ``eastward'' pressure gradient, zhpi, calculated
1561at the $(i,j)$th $u$-point. The scheme uses the ht\_wd depths and surface heights at the
1562neighbouring $(i+1,j)$ and $(i,j)$ tracer points.  zcpx is calculated using two logicals
1563variables, $\mathrm{ll\_tmp1}$ and $\mathrm{ll\_tmp2}$ which are evaluated for each grid
1564column.  The three possible combinations are illustrated in \autoref{fig:DYN_WAD_dynhpg}.
1565
1566%>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
1567\begin{figure}[!ht]
1568  \centering
1569  \includegraphics[width=0.66\textwidth]{Fig_WAD_dynhpg}
1570  \caption[Combinations controlling the limiting of the horizontal pressure gradient in
1571  wetting and drying regimes]{
1572    Three possible combinations of the logical variables controlling the
1573    limiting of the horizontal pressure gradient in wetting and drying regimes}
1574  \label{fig:DYN_WAD_dynhpg}
1575\end{figure}
1576%>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
1577
1578The first logical, $\mathrm{ll\_tmp1}$, is set to true if and only if the water depth at
1579both neighbouring points is greater than $\mathrm{rn\_wdmin1} + \mathrm{rn\_wdmin2}$ and
1580the minimum height of the sea surface at the two points is greater than the maximum height
1581of the topography at the two points:
1582
1583\begin{equation}
1584  \label{eq:DYN_ll_tmp1}
1585  \begin{split}
1586    \mathrm{ll\_tmp1}  = & \mathrm{MIN(sshn(ji,jj), sshn(ji+1,jj))} > \\
1587                     & \quad \mathrm{MAX(-ht\_wd(ji,jj), -ht\_wd(ji+1,jj))\  .and.} \\
1588                     & \mathrm{MAX(sshn(ji,jj) + ht\_wd(ji,jj),} \\
1589                     & \mathrm{\phantom{MAX(}sshn(ji+1,jj) + ht\_wd(ji+1,jj))} >\\
1590                     & \quad\quad\mathrm{rn\_wdmin1 + rn\_wdmin2 }
1591  \end{split}
1592\end{equation}
1593
1594The second logical, $\mathrm{ll\_tmp2}$, is set to true if and only if the maximum height
1595of the sea surface at the two points is greater than the maximum height of the topography
1596at the two points plus $\mathrm{rn\_wdmin1} + \mathrm{rn\_wdmin2}$
1597
1598\begin{equation}
1599  \label{eq:DYN_ll_tmp2}
1600  \begin{split}
1601    \mathrm{ ll\_tmp2 } = & \mathrm{( ABS( sshn(ji,jj) - sshn(ji+1,jj) ) > 1.E-12 )\ .AND.}\\
1602    & \mathrm{( MAX(sshn(ji,jj), sshn(ji+1,jj)) > } \\
1603    & \mathrm{\phantom{(} MAX(-ht\_wd(ji,jj), -ht\_wd(ji+1,jj)) + rn\_wdmin1 + rn\_wdmin2}) .
1604  \end{split}
1605\end{equation}
1606
1607If $\mathrm{ll\_tmp1}$ is true then the surface pressure gradient, zhpi at the $(i,j)$
1608point is unmodified. If both logicals are false zhpi is set to zero.
1609
1610If $\mathrm{ll\_tmp1}$ is true and $\mathrm{ll\_tmp2}$ is false then the surface pressure
1611gradient is multiplied through by zcpx which is the absolute value of the difference in
1612the water depths at the two points divided by the difference in the surface heights at the
1613two points. Thus the sign of the sea surface height gradient is retained but the magnitude
1614of the pressure force is determined by the difference in water depths rather than the
1615difference in surface height between the two points. Note that dividing by the difference
1616between the sea surface heights can be problematic if the heights approach parity. An
1617additional condition is applied to $\mathrm{ ll\_tmp2 }$ to ensure it is .false. in such
1618conditions.
1619
1620\subsubsection[Additional considerations (\textit{usrdef\_zgr.F90})]{Additional considerations (\mdl{usrdef\_zgr})}
1621\label{subsec:DYN_WAD_additional}
1622
1623In the very shallow water where wetting and drying occurs the parametrisation of
1624bottom drag is clearly very important. In order to promote stability
1625it is sometimes useful to calculate the bottom drag using an implicit time-stepping approach.
1626
1627Suitable specifcation of the surface heat flux in wetting and drying domains in forced and
1628coupled simulations needs further consideration. In order to prevent freezing or boiling
1629in uncoupled integrations the net surface heat fluxes need to be appropriately limited.
1630
1631%----------------------------------------------------------------------------------------
1632%      The WAD test cases
1633%----------------------------------------------------------------------------------------
1634\subsection[The WAD test cases (\textit{usrdef\_zgr.F90})]{The WAD test cases (\mdl{usrdef\_zgr})}
1635\label{subsec:DYN_WAD_test_cases}
1636
1637See the WAD tests MY\_DOC documention for details of the WAD test cases.
1638
1639
1640
1641% ================================================================
1642% Time evolution term
1643% ================================================================
1644\section[Time evolution term (\textit{dynnxt.F90})]{Time evolution term (\protect\mdl{dynnxt})}
1645\label{sec:DYN_nxt}
1646
1647%----------------------------------------------namdom----------------------------------------------------
1648
1649%-------------------------------------------------------------------------------------------------------------
1650
1651Options are defined through the \nam{dom} namelist variables.
1652The general framework for dynamics time stepping is a leap-frog scheme,
1653\ie\ a three level centred time scheme associated with an Asselin time filter (cf. \autoref{chap:TD}).
1654The scheme is applied to the velocity, except when
1655using the flux form of momentum advection (cf. \autoref{sec:DYN_adv_cor_flux})
1656in the variable volume case (\texttt{vvl?} defined),
1657where it has to be applied to the thickness weighted velocity (see \autoref{sec:SCOORD_momentum})
1658
1659$\bullet$ vector invariant form or linear free surface
1660(\np{ln\_dynhpg\_vec}\forcode{=.true.} ; \texttt{vvl?} not defined):
1661\[
1662  % \label{eq:DYN_nxt_vec}
1663  \left\{
1664    \begin{aligned}
1665      &u^{t+\rdt} = u_f^{t-\rdt} + 2\rdt  \ \text{RHS}_u^t     \\
1666      &u_f^t \;\quad = u^t+\gamma \,\left[ {u_f^{t-\rdt} -2u^t+u^{t+\rdt}} \right]
1667    \end{aligned}
1668  \right.
1669\]
1670
1671$\bullet$ flux form and nonlinear free surface
1672(\np{ln\_dynhpg\_vec}\forcode{=.false.} ; \texttt{vvl?} defined):
1673\[
1674  % \label{eq:DYN_nxt_flux}
1675  \left\{
1676    \begin{aligned}
1677      &\left(e_{3u}\,u\right)^{t+\rdt} = \left(e_{3u}\,u\right)_f^{t-\rdt} + 2\rdt \; e_{3u} \;\text{RHS}_u^t     \\
1678      &\left(e_{3u}\,u\right)_f^t \;\quad = \left(e_{3u}\,u\right)^t
1679      +\gamma \,\left[ {\left(e_{3u}\,u\right)_f^{t-\rdt} -2\left(e_{3u}\,u\right)^t+\left(e_{3u}\,u\right)^{t+\rdt}} \right]
1680    \end{aligned}
1681  \right.
1682\]
1683where RHS is the right hand side of the momentum equation,
1684the subscript $f$ denotes filtered values and $\gamma$ is the Asselin coefficient.
1685$\gamma$ is initialized as \np{nn\_atfp} (namelist parameter).
1686Its default value is \np{nn\_atfp}\forcode{=10.e-3}.
1687In both cases, the modified Asselin filter is not applied since perfect conservation is not an issue for
1688the momentum equations.
1689
1690Note that with the filtered free surface,
1691the update of the \textit{after} velocities is done in the \mdl{dynsp\_flt} module,
1692and only array swapping and Asselin filtering is done in \mdl{dynnxt}.
1693
1694% ================================================================
1695\biblio
1696
1697\pindex
1698
1699\end{document}
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