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chap_DYN.tex in NEMO/trunk/doc/latex/NEMO/subfiles – NEMO

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1\documentclass[../main/NEMO_manual]{subfiles}
2
3\begin{document}
4
5\chapter{Ocean Dynamics (DYN)}
6\label{chap:DYN}
7
8\thispagestyle{plain}
9
10\chaptertoc
11
12\paragraph{Changes record} ~\\
13
14{\footnotesize
15  \begin{tabularx}{\textwidth}{l||X|X}
16    Release & Author(s) & Modifications \\
17    \hline
18    {\em   4.0} & {\em ...} & {\em ...} \\
19    {\em   3.6} & {\em ...} & {\em ...} \\
20    {\em   3.4} & {\em ...} & {\em ...} \\
21    {\em <=3.4} & {\em ...} & {\em ...}
22  \end{tabularx}
23}
24
25\clearpage
26
27Using the representation described in \autoref{chap:DOM},
28several semi-discrete space forms of the dynamical equations are available depending on
29the vertical coordinate used and on the conservation properties of the vorticity term.
30In all the equations presented here, the masking has been omitted for simplicity.
31One must be aware that all the quantities are masked fields and
32that each time an average or difference operator is used, the resulting field is multiplied by a mask.
33
34The prognostic ocean dynamics equation can be summarized as follows:
35\[
36  \text{NXT} = \dbinom  {\text{VOR} + \text{KEG} + \text {ZAD} }
37  {\text{COR} + \text{ADV}                       }
38  + \text{HPG} + \text{SPG} + \text{LDF} + \text{ZDF}
39\]
40NXT stands for next, referring to the time-stepping.
41The first group of terms on the rhs of this equation corresponds to the Coriolis and advection terms that
42are decomposed into either a vorticity part (VOR), a kinetic energy part (KEG) and
43a vertical advection part (ZAD) in the vector invariant formulation,
44or a Coriolis and advection part (COR+ADV) in the flux formulation.
45The terms following these are the pressure gradient contributions
46(HPG, Hydrostatic Pressure Gradient, and SPG, Surface Pressure Gradient);
47and contributions from lateral diffusion (LDF) and vertical diffusion (ZDF),
48which are added to the rhs in the \mdl{dynldf} and \mdl{dynzdf} modules.
49The vertical diffusion term includes the surface and bottom stresses.
50The external forcings and parameterisations require complex inputs
51(surface wind stress calculation using bulk formulae, estimation of mixing coefficients)
52that are carried out in modules SBC, LDF and ZDF and are described in
53\autoref{chap:SBC}, \autoref{chap:LDF} and \autoref{chap:ZDF}, respectively.
54
55In the present chapter we also describe the diagnostic equations used to compute the horizontal divergence,
56curl of the velocities (\emph{divcur} module) and the vertical velocity (\emph{wzvmod} module).
57
58The different options available to the user are managed by namelist variables.
59For term \textit{ttt} in the momentum equations, the logical namelist variables are \textit{ln\_dynttt\_xxx},
60where \textit{xxx} is a 3 or 4 letter acronym corresponding to each optional scheme.
61%If a CPP key is used for this term its name is \key{ttt}.
62The corresponding code can be found in the \textit{dynttt\_xxx} module in the DYN directory,
63and it is usually computed in the \textit{dyn\_ttt\_xxx} subroutine.
64
65The user has the option of extracting and outputting each tendency term from the 3D momentum equations
66(\texttt{trddyn?} defined), as described in \autoref{chap:MISC}.
67Furthermore, the tendency terms associated with the 2D barotropic vorticity balance (when \texttt{trdvor?} is defined)
68can be derived from the 3D terms.
69%%%
70\gmcomment{STEVEN: not quite sure I've got the sense of the last sentence. does
71MISC correspond to "extracting tendency terms" or "vorticity balance"?}
72
73%% =================================================================================================
74\section{Sea surface height and diagnostic variables ($\eta$, $\zeta$, $\chi$, $w$)}
75\label{sec:DYN_divcur_wzv}
76
77%% =================================================================================================
78\subsection[Horizontal divergence and relative vorticity (\textit{divcur.F90})]{Horizontal divergence and relative vorticity (\protect\mdl{divcur})}
79\label{subsec:DYN_divcur}
80
81The vorticity is defined at an $f$-point (\ie\ corner point) as follows:
82\begin{equation}
83  \label{eq:DYN_divcur_cur}
84  \zeta =\frac{1}{e_{1f}\,e_{2f} }\left( {\;\delta_{i+1/2} \left[ {e_{2v}\;v} \right]
85      -\delta_{j+1/2} \left[ {e_{1u}\;u} \right]\;} \right)
86\end{equation}
87
88The horizontal divergence is defined at a $T$-point.
89It is given by:
90\[
91  % \label{eq:DYN_divcur_div}
92  \chi =\frac{1}{e_{1t}\,e_{2t}\,e_{3t} }
93  \left( {\delta_i \left[ {e_{2u}\,e_{3u}\,u} \right]
94      +\delta_j \left[ {e_{1v}\,e_{3v}\,v} \right]} \right)
95\]
96
97Note that although the vorticity has the same discrete expression in $z$- and $s$-coordinates,
98its physical meaning is not identical.
99$\zeta$ is a pseudo vorticity along $s$-surfaces
100(only pseudo because $(u,v)$ are still defined along geopotential surfaces,
101but are not necessarily defined at the same depth).
102
103The vorticity and divergence at the \textit{before} step are used in the computation of
104the horizontal diffusion of momentum.
105Note that because they have been calculated prior to the Asselin filtering of the \textit{before} velocities,
106the \textit{before} vorticity and divergence arrays must be included in the restart file to
107ensure perfect restartability.
108The vorticity and divergence at the \textit{now} time step are used for the computation of
109the nonlinear advection and of the vertical velocity respectively.
110
111%% =================================================================================================
112\subsection[Horizontal divergence and relative vorticity (\textit{sshwzv.F90})]{Horizontal divergence and relative vorticity (\protect\mdl{sshwzv})}
113\label{subsec:DYN_sshwzv}
114
115The sea surface height is given by:
116\begin{equation}
117  \label{eq:DYN_spg_ssh}
118  \begin{aligned}
119    \frac{\partial \eta }{\partial t}
120    &\equiv    \frac{1}{e_{1t} e_{2t} }\sum\limits_k { \left\{  \delta_i \left[ {e_{2u}\,e_{3u}\;u} \right]
121        +\delta_j \left[ {e_{1v}\,e_{3v}\;v} \right]  \right\} }
122    -    \frac{\textit{emp}}{\rho_w }   \\
123    &\equiv    \sum\limits_k {\chi \ e_{3t}}  -  \frac{\textit{emp}}{\rho_w }
124  \end{aligned}
125\end{equation}
126where \textit{emp} is the surface freshwater budget (evaporation minus precipitation),
127expressed in Kg/m$^2$/s (which is equal to mm/s),
128and $\rho_w$=1,035~Kg/m$^3$ is the reference density of sea water (Boussinesq approximation).
129If river runoff is expressed as a surface freshwater flux (see \autoref{chap:SBC}) then
130\textit{emp} can be written as the evaporation minus precipitation, minus the river runoff.
131The sea-surface height is evaluated using exactly the same time stepping scheme as
132the tracer equation \autoref{eq:TRA_nxt}:
133a leapfrog scheme in combination with an Asselin time filter,
134\ie\ the velocity appearing in \autoref{eq:DYN_spg_ssh} is centred in time (\textit{now} velocity).
135This is of paramount importance.
136Replacing $T$ by the number $1$ in the tracer equation and summing over the water column must lead to
137the sea surface height equation otherwise tracer content will not be conserved
138\citep{griffies.pacanowski.ea_MWR01, leclair.madec_OM09}.
139
140The vertical velocity is computed by an upward integration of the horizontal divergence starting at the bottom,
141taking into account the change of the thickness of the levels:
142\begin{equation}
143  \label{eq:DYN_wzv}
144  \left\{
145    \begin{aligned}
146      &\left. w \right|_{k_b-1/2} \quad= 0    \qquad \text{where } k_b \text{ is the level just above the sea floor }   \\
147      &\left. w \right|_{k+1/2}     = \left. w \right|_{k-1/2}  +  \left. e_{3t} \right|_{k}\;  \left. \chi \right|_k
148      - \frac{1} {2 \rdt} \left\left. e_{3t}^{t+1}\right|_{k} - \left. e_{3t}^{t-1}\right|_{k}\right)
149    \end{aligned}
150  \right.
151\end{equation}
152
153In the case of a non-linear free surface (\texttt{vvl?}), the top vertical velocity is $-\textit{emp}/\rho_w$,
154as changes in the divergence of the barotropic transport are absorbed into the change of the level thicknesses,
155re-orientated downward.
156\gmcomment{not sure of this...  to be modified with the change in emp setting}
157In the case of a linear free surface, the time derivative in \autoref{eq:DYN_wzv} disappears.
158The upper boundary condition applies at a fixed level $z=0$.
159The top vertical velocity is thus equal to the divergence of the barotropic transport
160(\ie\ the first term in the right-hand-side of \autoref{eq:DYN_spg_ssh}).
161
162Note also that whereas the vertical velocity has the same discrete expression in $z$- and $s$-coordinates,
163its physical meaning is not the same:
164in the second case, $w$ is the velocity normal to the $s$-surfaces.
165Note also that the $k$-axis is re-orientated downwards in the \fortran\ code compared to
166the indexing used in the semi-discrete equations such as \autoref{eq:DYN_wzv}
167(see \autoref{subsec:DOM_Num_Index_vertical}).
168
169%% =================================================================================================
170\section{Coriolis and advection: vector invariant form}
171\label{sec:DYN_adv_cor_vect}
172
173\begin{listing}
174  \nlst{namdyn_adv}
175  \caption{\forcode{&namdyn_adv}}
176  \label{lst:namdyn_adv}
177\end{listing}
178
179The vector invariant form of the momentum equations is the one most often used in
180applications of the \NEMO\ ocean model.
181The flux form option (see next section) has been present since version $2$.
182Options are defined through the \nam{dyn_adv}{dyn\_adv} namelist variables Coriolis and
183momentum advection terms are evaluated using a leapfrog scheme,
184\ie\ the velocity appearing in these expressions is centred in time (\textit{now} velocity).
185At the lateral boundaries either free slip, no slip or partial slip boundary conditions are applied following
186\autoref{chap:LBC}.
187
188%% =================================================================================================
189\subsection[Vorticity term (\textit{dynvor.F90})]{Vorticity term (\protect\mdl{dynvor})}
190\label{subsec:DYN_vor}
191
192\begin{listing}
193  \nlst{namdyn_vor}
194  \caption{\forcode{&namdyn_vor}}
195  \label{lst:namdyn_vor}
196\end{listing}
197
198Options are defined through the \nam{dyn_vor}{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[=.true.]{ln_dynvor_con}{ln\_dynvor\_con}).
208The vorticity terms are all computed in dedicated routines that can be found in the \mdl{dynvor} module.
209
210%                 enstrophy conserving scheme
211%% =================================================================================================
212\subsubsection[Enstrophy conserving scheme (\forcode{ln_dynvor_ens})]{Enstrophy conserving scheme (\protect\np{ln_dynvor_ens}{ln\_dynvor\_ens})}
213\label{subsec:DYN_vor_ens}
214
215In the enstrophy conserving case (ENS scheme),
216the discrete formulation of the vorticity term provides a global conservation of the enstrophy
217($ [ (\zeta +f ) / e_{3f} ]^2 $ in $s$-coordinates) for a horizontally non-divergent flow (\ie\ $\chi$=$0$),
218but does not conserve the total kinetic energy.
219It is given by:
220\begin{equation}
221  \label{eq:DYN_vor_ens}
222  \left\{
223    \begin{aligned}
224      {+\frac{1}{e_{1u} } } & {\overline {\left( { \frac{\zeta +f}{e_{3f} }} \right)} }^{\,i}
225      & {\overline{\overline {\left( {e_{1v}\,e_{3v}\;v} \right)}} }^{\,i, j+1/2}    \\
226      {- \frac{1}{e_{2v} } } & {\overline {\left( {\frac{\zeta +f}{e_{3f} }} \right)} }^{\,j}
227      & {\overline{\overline {\left( {e_{2u}\,e_{3u}\;u} \right)}} }^{\,i+1/2, j}
228    \end{aligned}
229  \right.
230\end{equation}
231
232%                 energy conserving scheme
233%% =================================================================================================
234\subsubsection[Energy conserving scheme (\forcode{ln_dynvor_ene})]{Energy conserving scheme (\protect\np{ln_dynvor_ene}{ln\_dynvor\_ene})}
235\label{subsec:DYN_vor_ene}
236
237The kinetic energy conserving scheme (ENE scheme) conserves the global kinetic energy but not the global enstrophy.
238It is given by:
239\begin{equation}
240  \label{eq:DYN_vor_ene}
241  \left\{
242    \begin{aligned}
243      {+\frac{1}{e_{1u}}\; {\overline {\left( {\frac{\zeta +f}{e_{3f} }} \right)
244            \;  \overline {\left( {e_{1v}\,e_{3v}\;v} \right)} ^{\,i+1/2}} }^{\,j} }    \\
245      {- \frac{1}{e_{2v}}\; {\overline {\left( {\frac{\zeta +f}{e_{3f} }} \right)
246            \;  \overline {\left( {e_{2u}\,e_{3u}\;u} \right)} ^{\,j+1/2}} }^{\,i} }
247    \end{aligned}
248  \right.
249\end{equation}
250
251%                 mix energy/enstrophy conserving scheme
252%% =================================================================================================
253\subsubsection[Mixed energy/enstrophy conserving scheme (\forcode{ln_dynvor_mix})]{Mixed energy/enstrophy conserving scheme (\protect\np{ln_dynvor_mix}{ln\_dynvor\_mix})}
254\label{subsec:DYN_vor_mix}
255
256For the mixed energy/enstrophy conserving scheme (MIX scheme), a mixture of the two previous schemes is used.
257It consists of the ENS scheme (\autoref{eq:DYN_vor_ens}) for the relative vorticity term,
258and of the ENE scheme (\autoref{eq:DYN_vor_ene}) applied to the planetary vorticity term.
259\[
260  % \label{eq:DYN_vor_mix}
261  \left\{ {
262      \begin{aligned}
263        {+\frac{1}{e_{1u} }\; {\overline {\left( {\frac{\zeta }{e_{3f} }} \right)} }^{\,i}
264          \; {\overline{\overline {\left( {e_{1v}\,e_{3v}\;v} \right)}} }^{\,i,j+1/2} -\frac{1}{e_{1u} }
265          \; {\overline {\left( {\frac{f}{e_{3f} }} \right)
266              \;\overline {\left( {e_{1v}\,e_{3v}\;v} \right)} ^{\,i+1/2}} }^{\,j} } \\
267        {-\frac{1}{e_{2v} }\; {\overline {\left( {\frac{\zeta }{e_{3f} }} \right)} }^j
268          \; {\overline{\overline {\left( {e_{2u}\,e_{3u}\;u} \right)}} }^{\,i+1/2,j} +\frac{1}{e_{2v} }
269          \; {\overline {\left( {\frac{f}{e_{3f} }} \right)
270              \;\overline {\left( {e_{2u}\,e_{3u}\;u} \right)} ^{\,j+1/2}} }^{\,i} } \hfill
271      \end{aligned}
272    } \right.
273\]
274
275%                 energy and enstrophy conserving scheme
276%% =================================================================================================
277\subsubsection[Energy and enstrophy conserving scheme (\forcode{ln_dynvor_een})]{Energy and enstrophy conserving scheme (\protect\np{ln_dynvor_een}{ln\_dynvor\_een})}
278\label{subsec:DYN_vor_een}
279
280In both the ENS and ENE schemes,
281it is apparent that the combination of $i$ and $j$ averages of the velocity allows for
282the presence of grid point oscillation structures that will be invisible to the operator.
283These structures are \textit{computational modes} that will be at least partly damped by
284the momentum diffusion operator (\ie\ the subgrid-scale advection), but not by the resolved advection term.
285The ENS and ENE schemes therefore do not contribute to dump any grid point noise in the horizontal velocity field.
286Such noise would result in more noise in the vertical velocity field, an undesirable feature.
287This is a well-known characteristic of $C$-grid discretization where
288$u$ and $v$ are located at different grid points,
289a price worth paying to avoid a double averaging in the pressure gradient term as in the $B$-grid.
290\gmcomment{ To circumvent this, Adcroft (ADD REF HERE)
291Nevertheless, this technique strongly distort the phase and group velocity of Rossby waves....}
292
293A very nice solution to the problem of double averaging was proposed by \citet{arakawa.hsu_MWR90}.
294The idea is to get rid of the double averaging by considering triad combinations of vorticity.
295It is noteworthy that this solution is conceptually quite similar to the one proposed by
296\citep{griffies.gnanadesikan.ea_JPO98} for the discretization of the iso-neutral diffusion operator (see \autoref{apdx:INVARIANTS}).
297
298The \citet{arakawa.hsu_MWR90} vorticity advection scheme for a single layer is modified
299for spherical coordinates as described by \citet{arakawa.lamb_MWR81} to obtain the EEN scheme.
300First consider the discrete expression of the potential vorticity, $q$, defined at an $f$-point:
301\[
302  % \label{eq:DYN_pot_vor}
303  q  = \frac{\zeta +f} {e_{3f} }
304\]
305where the relative vorticity is defined by (\autoref{eq:DYN_divcur_cur}),
306the Coriolis parameter is given by $f=2 \,\Omega \;\sin \varphi _f $ and the layer thickness at $f$-points is:
307\begin{equation}
308  \label{eq:DYN_een_e3f}
309  e_{3f} = \overline{\overline {e_{3t} }} ^{\,i+1/2,j+1/2}
310\end{equation}
311
312\begin{figure}[!ht]
313  \centering
314  \includegraphics[width=0.66\textwidth]{Fig_DYN_een_triad}
315  \caption[Triads used in the energy and enstrophy conserving scheme (EEN)]{
316    Triads used in the energy and enstrophy conserving scheme (EEN) for
317    $u$-component (upper panel) and $v$-component (lower panel).}
318  \label{fig:DYN_een_triad}
319\end{figure}
320
321A key point in \autoref{eq:DYN_een_e3f} is how the averaging in the \textbf{i}- and \textbf{j}- directions is made.
322It uses the sum of masked t-point vertical scale factor divided either by the sum of the four t-point masks
323(\np[=1]{nn_een_e3f}{nn\_een\_e3f}), or just by $4$ (\np[=.true.]{nn_een_e3f}{nn\_een\_e3f}).
324The latter case preserves the continuity of $e_{3f}$ when one or more of the neighbouring $e_{3t}$ tends to zero and
325extends by continuity the value of $e_{3f}$ into the land areas.
326This case introduces a sub-grid-scale topography at f-points
327(with a systematic reduction of $e_{3f}$ when a model level intercept the bathymetry)
328that tends to reinforce the topostrophy of the flow
329(\ie\ the tendency of the flow to follow the isobaths) \citep{penduff.le-sommer.ea_OS07}.
330
331Next, the vorticity triads, $ {^i_j}\mathbb{Q}^{i_p}_{j_p}$ can be defined at a $T$-point as
332the following triad combinations of the neighbouring potential vorticities defined at f-points
333(\autoref{fig:DYN_een_triad}):
334\begin{equation}
335  \label{eq:DYN_Q_triads}
336  _i^j \mathbb{Q}^{i_p}_{j_p}
337  = \frac{1}{12} \ \left(   q^{i-i_p}_{j+j_p} + q^{i+j_p}_{j+i_p} + q^{i+i_p}_{j-j_p}  \right)
338\end{equation}
339where the indices $i_p$ and $k_p$ take the values: $i_p = -1/2$ or $1/2$ and $j_p = -1/2$ or $1/2$.
340
341Finally, the vorticity terms are represented as:
342\begin{equation}
343  \label{eq:DYN_vor_een}
344  \left\{ {
345      \begin{aligned}
346        +q\,e_3 \, v    &\equiv +\frac{1}{e_{1u} }   \sum_{\substack{i_p,\,k_p}}
347        {^{i+1/2-i_p}_j}  \mathbb{Q}^{i_p}_{j_p}  \left( e_{1v}\,e_{3v} \;\right)^{i+1/2-i_p}_{j+j_p}   \\
348        - q\,e_3 \, u     &\equiv -\frac{1}{e_{2v} }    \sum_{\substack{i_p,\,k_p}}
349        {^i_{j+1/2-j_p}}  \mathbb{Q}^{i_p}_{j_p}  \left( e_{2u}\,e_{3u} \;\right)^{i+i_p}_{j+1/2-j_p}   \\
350      \end{aligned}
351    } \right.
352\end{equation}
353
354This EEN scheme in fact combines the conservation properties of the ENS and ENE schemes.
355It conserves both total energy and potential enstrophy in the limit of horizontally nondivergent flow
356(\ie\ $\chi$=$0$) (see \autoref{subsec:INVARIANTS_vorEEN}).
357Applied to a realistic ocean configuration, it has been shown that it leads to a significant reduction of
358the noise in the vertical velocity field \citep{le-sommer.penduff.ea_OM09}.
359Furthermore, used in combination with a partial steps representation of bottom topography,
360it improves the interaction between current and topography,
361leading to a larger topostrophy of the flow \citep{barnier.madec.ea_OD06, penduff.le-sommer.ea_OS07}.
362
363%% =================================================================================================
364\subsection[Kinetic energy gradient term (\textit{dynkeg.F90})]{Kinetic energy gradient term (\protect\mdl{dynkeg})}
365\label{subsec:DYN_keg}
366
367As demonstrated in \autoref{apdx:INVARIANTS},
368there is a single discrete formulation of the kinetic energy gradient term that,
369together with the formulation chosen for the vertical advection (see below),
370conserves the total kinetic energy:
371\[
372  % \label{eq:DYN_keg}
373  \left\{
374    \begin{aligned}
375      -\frac{1}{2 \; e_{1u} }  & \ \delta_{i+1/2} \left[ {\overline {u^2}^{\,i} + \overline{v^2}^{\,j}} \right]   \\
376      -\frac{1}{2 \; e_{2v} }  & \ \delta_{j+1/2} \left[ {\overline {u^2}^{\,i} + \overline{v^2}^{\,j}} \right]
377    \end{aligned}
378  \right.
379\]
380
381%% =================================================================================================
382\subsection[Vertical advection term (\textit{dynzad.F90})]{Vertical advection term (\protect\mdl{dynzad})}
383\label{subsec:DYN_zad}
384
385The discrete formulation of the vertical advection, t
386ogether with the formulation chosen for the gradient of kinetic energy (KE) term,
387conserves the total kinetic energy.
388Indeed, the change of KE due to the vertical advection is exactly balanced by
389the change of KE due to the gradient of KE (see \autoref{apdx:INVARIANTS}).
390\[
391  % \label{eq:DYN_zad}
392  \left\{
393    \begin{aligned}
394      -\frac{1} {e_{1u}\,e_{2u}\,e_{3u}} &\ \overline{\ \overline{ e_{1t}\,e_{2t}\;w } ^{\,i+1/2\;\delta_{k+1/2} \left[ u \right]\  }^{\,k}  \\
395      -\frac{1} {e_{1v}\,e_{2v}\,e_{3v}}  &\ \overline{\ \overline{ e_{1t}\,e_{2t}\;w } ^{\,j+1/2\;\delta_{k+1/2} \left[ u \right]\  }^{\,k}
396    \end{aligned}
397  \right.
398\]
399When \np[=.true.]{ln_dynzad_zts}{ln\_dynzad\_zts},
400a split-explicit time stepping with 5 sub-timesteps is used on the vertical advection term.
401This option can be useful when the value of the timestep is limited by vertical advection \citep{lemarie.debreu.ea_OM15}.
402Note that in this case,
403a similar split-explicit time stepping should be used on vertical advection of tracer to ensure a better stability,
404an option which is only available with a TVD scheme (see \np{ln_traadv_tvd_zts}{ln\_traadv\_tvd\_zts} in \autoref{subsec:TRA_adv_tvd}).
405
406%% =================================================================================================
407\section{Coriolis and advection: flux form}
408\label{sec:DYN_adv_cor_flux}
409
410Options are defined through the \nam{dyn_adv}{dyn\_adv} namelist variables.
411In the flux form (as in the vector invariant form),
412the Coriolis and momentum advection terms are evaluated using a leapfrog scheme,
413\ie\ the velocity appearing in their expressions is centred in time (\textit{now} velocity).
414At the lateral boundaries either free slip,
415no slip or partial slip boundary conditions are applied following \autoref{chap:LBC}.
416
417%% =================================================================================================
418\subsection[Coriolis plus curvature metric terms (\textit{dynvor.F90})]{Coriolis plus curvature metric terms (\protect\mdl{dynvor})}
419\label{subsec:DYN_cor_flux}
420
421In flux form, the vorticity term reduces to a Coriolis term in which the Coriolis parameter has been modified to account for the "metric" term.
422This altered Coriolis parameter is thus discretised at $f$-points.
423It is given by:
424\begin{multline*}
425  % \label{eq:DYN_cor_metric}
426  f+\frac{1}{e_1 e_2 }\left( {v\frac{\partial e_2 }{\partial i}  -  u\frac{\partial e_1 }{\partial j}} \right\\
427  \equiv   f + \frac{1}{e_{1f} e_{2f} } \left( { \ \overline v ^{i+1/2}\delta_{i+1/2} \left[ {e_{2u} } \right]
428      -  \overline u ^{j+1/2}\delta_{j+1/2} \left[ {e_{1u} } \right]  }  \ \right)
429\end{multline*}
430
431Any of the (\autoref{eq:DYN_vor_ens}), (\autoref{eq:DYN_vor_ene}) and (\autoref{eq:DYN_vor_een}) schemes can be used to
432compute the product of the Coriolis parameter and the vorticity.
433However, the energy-conserving scheme (\autoref{eq:DYN_vor_een}) has exclusively been used to date.
434This term is evaluated using a leapfrog scheme, \ie\ the velocity is centred in time (\textit{now} velocity).
435
436%% =================================================================================================
437\subsection[Flux form advection term (\textit{dynadv.F90})]{Flux form advection term (\protect\mdl{dynadv})}
438\label{subsec:DYN_adv_flux}
439
440The discrete expression of the advection term is given by:
441\[
442  % \label{eq:DYN_adv}
443  \left\{
444    \begin{aligned}
445      \frac{1}{e_{1u}\,e_{2u}\,e_{3u}}
446      \left(      \delta_{i+1/2} \left[ \overline{e_{2u}\,e_{3u}\;u }^{i       }  \ u_t      \right]
447        + \delta_{j       } \left[ \overline{e_{1u}\,e_{3u}\;v }^{i+1/2\ u_f      \right] \right\ \;   \\
448      \left.   + \delta_{k      } \left[ \overline{e_{1w}\,e_{2w}\;w}^{i+1/2\ u_{uw} \right] \right)   \\
449      \\
450      \frac{1}{e_{1v}\,e_{2v}\,e_{3v}}
451      \left(     \delta_{i       } \left[ \overline{e_{2u}\,e_{3u }\;u }^{j+1/2} \ v_f       \right]
452        + \delta_{j+1/2} \left[ \overline{e_{1u}\,e_{3u }\;v }^{i       } \ v_t       \right] \right\ \, \, \\
453      \left+ \delta_{k      } \left[ \overline{e_{1w}\,e_{2w}\;w}^{j+1/2} \ v_{vw}  \right] \right) \\
454    \end{aligned}
455  \right.
456\]
457
458Two advection schemes are available:
459a $2^{nd}$ order centered finite difference scheme, CEN2,
460or a $3^{rd}$ order upstream biased scheme, UBS.
461The latter is described in \citet{shchepetkin.mcwilliams_OM05}.
462The schemes are selected using the namelist logicals \np{ln_dynadv_cen2}{ln\_dynadv\_cen2} and \np{ln_dynadv_ubs}{ln\_dynadv\_ubs}.
463In flux form, the schemes differ by the choice of a space and time interpolation to define the value of
464$u$ and $v$ at the centre of each face of $u$- and $v$-cells, \ie\ at the $T$-, $f$-,
465and $uw$-points for $u$ and at the $f$-, $T$- and $vw$-points for $v$.
466
467%                 2nd order centred scheme
468%% =================================================================================================
469\subsubsection[CEN2: $2^{nd}$ order centred scheme (\forcode{ln_dynadv_cen2})]{CEN2: $2^{nd}$ order centred scheme (\protect\np{ln_dynadv_cen2}{ln\_dynadv\_cen2})}
470\label{subsec:DYN_adv_cen2}
471
472In the centered $2^{nd}$ order formulation, the velocity is evaluated as the mean of the two neighbouring points:
473\begin{equation}
474  \label{eq:DYN_adv_cen2}
475  \left\{
476    \begin{aligned}
477      u_T^{cen2} &=\overline u^{i }       \quad &  u_F^{cen2} &=\overline u^{j+1/2}  \quad &  u_{uw}^{cen2} &=\overline u^{k+1/2}   \\
478      v_F^{cen2} &=\overline v ^{i+1/2} \quad & v_F^{cen2} &=\overline v^j    \quad &  v_{vw}^{cen2} &=\overline v ^{k+1/2}  \\
479    \end{aligned}
480  \right.
481\end{equation}
482
483The scheme is non diffusive (\ie\ conserves the kinetic energy) but dispersive (\ie\ it may create false extrema).
484It is therefore notoriously noisy and must be used in conjunction with an explicit diffusion operator to
485produce a sensible solution.
486The associated time-stepping is performed using a leapfrog scheme in conjunction with an Asselin time-filter,
487so $u$ and $v$ are the \emph{now} velocities.
488
489%                 UBS scheme
490%% =================================================================================================
491\subsubsection[UBS: Upstream Biased Scheme (\forcode{ln_dynadv_ubs})]{UBS: Upstream Biased Scheme (\protect\np{ln_dynadv_ubs}{ln\_dynadv\_ubs})}
492\label{subsec:DYN_adv_ubs}
493
494The UBS advection scheme is an upstream biased third order scheme based on
495an upstream-biased parabolic interpolation.
496For example, the evaluation of $u_T^{ubs} $ is done as follows:
497\begin{equation}
498  \label{eq:DYN_adv_ubs}
499  u_T^{ubs} =\overline u ^i-\;\frac{1}{6}
500  \begin{cases}
501    u"_{i-1/2}&   \text{if $\ \overline{e_{2u}\,e_{3u} \ u}^i  \geqslant 0$ }    \\
502    u"_{i+1/2}&   \text{if $\ \overline{e_{2u}\,e_{3u} \ u}^i  < 0$ }
503  \end{cases}
504\end{equation}
505where $u"_{i+1/2} =\delta_{i+1/2} \left[ {\delta_i \left[ u \right]} \right]$.
506This results in a dissipatively dominant (\ie\ hyper-diffusive) truncation error
507\citep{shchepetkin.mcwilliams_OM05}.
508The overall performance of the advection scheme is similar to that reported in \citet{farrow.stevens_JPO95}.
509It is a relatively good compromise between accuracy and smoothness.
510It is not a \emph{positive} scheme, meaning that false extrema are permitted.
511But the amplitudes of the false extrema are significantly reduced over those in the centred second order method.
512As the scheme already includes a diffusion component, it can be used without explicit lateral diffusion on momentum
513(\ie\ \np[=]{ln_dynldf_lap}{ln\_dynldf\_lap}\np[=.false.]{ln_dynldf_bilap}{ln\_dynldf\_bilap}),
514and it is recommended to do so.
515
516The UBS scheme is not used in all directions.
517In the vertical, the centred $2^{nd}$ order evaluation of the advection is preferred, \ie\ $u_{uw}^{ubs}$ and
518$u_{vw}^{ubs}$ in \autoref{eq:DYN_adv_cen2} are used.
519UBS is diffusive and is associated with vertical mixing of momentum. \gmcomment{ gm  pursue the
520sentence:Since vertical mixing of momentum is a source term of the TKE equation...  }
521
522For stability reasons, the first term in (\autoref{eq:DYN_adv_ubs}),
523which corresponds to a second order centred scheme, is evaluated using the \textit{now} velocity (centred in time),
524while the second term, which is the diffusion part of the scheme,
525is evaluated using the \textit{before} velocity (forward in time).
526This is discussed by \citet{webb.de-cuevas.ea_JAOT98} in the context of the Quick advection scheme.
527
528Note that the UBS and QUICK (Quadratic Upstream Interpolation for Convective Kinematics) schemes only differ by
529one coefficient.
530Replacing $1/6$ by $1/8$ in (\autoref{eq:DYN_adv_ubs}) leads to the QUICK advection scheme \citep{webb.de-cuevas.ea_JAOT98}.
531This option is not available through a namelist parameter, since the $1/6$ coefficient is hard coded.
532Nevertheless it is quite easy to make the substitution in the \mdl{dynadv\_ubs} module and obtain a QUICK scheme.
533
534Note also that in the current version of \mdl{dynadv\_ubs},
535there is also the possibility of using a $4^{th}$ order evaluation of the advective velocity as in ROMS.
536This is an error and should be suppressed soon.
537%%%
538\gmcomment{action :  this have to be done}
539%%%
540
541%% =================================================================================================
542\section[Hydrostatic pressure gradient (\textit{dynhpg.F90})]{Hydrostatic pressure gradient (\protect\mdl{dynhpg})}
543\label{sec:DYN_hpg}
544
545\begin{listing}
546  \nlst{namdyn_hpg}
547  \caption{\forcode{&namdyn_hpg}}
548  \label{lst:namdyn_hpg}
549\end{listing}
550
551Options are defined through the \nam{dyn_hpg}{dyn\_hpg} namelist variables.
552The key distinction between the different algorithms used for
553the hydrostatic pressure gradient is the vertical coordinate used,
554since HPG is a \emph{horizontal} pressure gradient, \ie\ computed along geopotential surfaces.
555As a result, any tilt of the surface of the computational levels will require a specific treatment to
556compute the hydrostatic pressure gradient.
557
558The hydrostatic pressure gradient term is evaluated either using a leapfrog scheme,
559\ie\ the density appearing in its expression is centred in time (\emph{now} $\rho$),
560or a semi-implcit scheme.
561At the lateral boundaries either free slip, no slip or partial slip boundary conditions are applied.
562
563%% =================================================================================================
564\subsection[Full step $Z$-coordinate (\forcode{ln_dynhpg_zco})]{Full step $Z$-coordinate (\protect\np{ln_dynhpg_zco}{ln\_dynhpg\_zco})}
565\label{subsec:DYN_hpg_zco}
566
567The hydrostatic pressure can be obtained by integrating the hydrostatic equation vertically from the surface.
568However, the pressure is large at great depth while its horizontal gradient is several orders of magnitude smaller.
569This may lead to large truncation errors in the pressure gradient terms.
570Thus, the two horizontal components of the hydrostatic pressure gradient are computed directly as follows:
571
572for $k=km$ (surface layer, $jk=1$ in the code)
573\begin{equation}
574  \label{eq:DYN_hpg_zco_surf}
575  \left\{
576    \begin{aligned}
577      \left. \delta_{i+1/2} \left[  p^h          \right] \right|_{k=km}
578      &= \frac{1}{2} g \   \left. \delta_{i+1/2} \left[  e_{3w} \ \rho \right] \right|_{k=km}   \\
579      \left. \delta_{j+1/2} \left[  p^h          \right] \right|_{k=km}
580      &= \frac{1}{2} g \   \left. \delta_{j+1/2} \left[  e_{3w} \ \rho \right] \right|_{k=km}   \\
581    \end{aligned}
582  \right.
583\end{equation}
584
585for $1<k<km$ (interior layer)
586\begin{equation}
587  \label{eq:DYN_hpg_zco}
588  \left\{
589    \begin{aligned}
590      \left. \delta_{i+1/2} \left[  p^h          \right] \right|_{k}
591      &=             \left. \delta_{i+1/2} \left[  p^h          \right] \right|_{k-1}
592      +    \frac{1}{2}\;g\;   \left. \delta_{i+1/2} \left[  e_{3w} \ \overline {\rho}^{k+1/2} \right] \right|_{k}   \\
593      \left. \delta_{j+1/2} \left[  p^h          \right] \right|_{k}
594      &=                \left. \delta_{j+1/2} \left[  p^h          \right] \right|_{k-1}
595      +    \frac{1}{2}\;g\;   \left. \delta_{j+1/2} \left[  e_{3w} \ \overline {\rho}^{k+1/2} \right] \right|_{k}   \\
596    \end{aligned}
597  \right.
598\end{equation}
599
600Note that the $1/2$ factor in (\autoref{eq:DYN_hpg_zco_surf}) is adequate because of the definition of $e_{3w}$ as
601the vertical derivative of the scale factor at the surface level ($z=0$).
602Note also that in case of variable volume level (\texttt{vvl?} defined),
603the surface pressure gradient is included in \autoref{eq:DYN_hpg_zco_surf} and
604\autoref{eq:DYN_hpg_zco} through the space and time variations of the vertical scale factor $e_{3w}$.
605
606%% =================================================================================================
607\subsection[Partial step $Z$-coordinate (\forcode{ln_dynhpg_zps})]{Partial step $Z$-coordinate (\protect\np{ln_dynhpg_zps}{ln\_dynhpg\_zps})}
608\label{subsec:DYN_hpg_zps}
609
610With partial bottom cells, tracers in horizontally adjacent cells generally live at different depths.
611Before taking horizontal gradients between these tracer points,
612a linear interpolation is used to approximate the deeper tracer as if
613it actually lived at the depth of the shallower tracer point.
614
615Apart from this modification,
616the horizontal hydrostatic pressure gradient evaluated in the $z$-coordinate with partial step is exactly as in
617the pure $z$-coordinate case.
618As explained in detail in section \autoref{sec:TRA_zpshde},
619the nonlinearity of pressure effects in the equation of state is such that
620it is better to interpolate temperature and salinity vertically before computing the density.
621Horizontal gradients of temperature and salinity are needed for the TRA modules,
622which is the reason why the horizontal gradients of density at the deepest model level are computed in
623module \mdl{zpsdhe} located in the TRA directory and described in \autoref{sec:TRA_zpshde}.
624
625%% =================================================================================================
626\subsection{$S$- and $Z$-$S$-coordinates}
627\label{subsec:DYN_hpg_sco}
628
629Pressure gradient formulations in an $s$-coordinate have been the subject of a vast number of papers
630(\eg, \citet{song_MWR98, shchepetkin.mcwilliams_OM05}).
631A number of different pressure gradient options are coded but the ROMS-like,
632density Jacobian with cubic polynomial method is currently disabled whilst known bugs are under investigation.
633
634$\bullet$ Traditional coding (see for example \citet{madec.delecluse.ea_JPO96}: (\np[=.true.]{ln_dynhpg_sco}{ln\_dynhpg\_sco})
635\begin{equation}
636  \label{eq:DYN_hpg_sco}
637  \left\{
638    \begin{aligned}
639      - \frac{1}                 {\rho_o \, e_{1u}} \;   \delta_{i+1/2} \left[  p^h  \right]
640      + \frac{g\; \overline {\rho}^{i+1/2}}  {\rho_o \, e_{1u}} \;   \delta_{i+1/2} \left[  z_t   \right]    \\
641      - \frac{1}                 {\rho_o \, e_{2v}} \;   \delta_{j+1/2} \left[  p^h  \right]
642      + \frac{g\; \overline {\rho}^{j+1/2}}  {\rho_o \, e_{2v}} \;   \delta_{j+1/2} \left[  z_t   \right]    \\
643    \end{aligned}
644  \right.
645\end{equation}
646
647Where the first term is the pressure gradient along coordinates,
648computed as in \autoref{eq:DYN_hpg_zco_surf} - \autoref{eq:DYN_hpg_zco},
649and $z_T$ is the depth of the $T$-point evaluated from the sum of the vertical scale factors at the $w$-point
650($e_{3w}$).
651
652$\bullet$ Traditional coding with adaptation for ice shelf cavities (\np[=.true.]{ln_dynhpg_isf}{ln\_dynhpg\_isf}).
653This scheme need the activation of ice shelf cavities (\np[=.true.]{ln_isfcav}{ln\_isfcav}).
654
655$\bullet$ Pressure Jacobian scheme (prj) (a research paper in preparation) (\np[=.true.]{ln_dynhpg_prj}{ln\_dynhpg\_prj})
656
657$\bullet$ Density Jacobian with cubic polynomial scheme (DJC) \citep{shchepetkin.mcwilliams_OM05}
658(\np[=.true.]{ln_dynhpg_djc}{ln\_dynhpg\_djc}) (currently disabled; under development)
659
660Note that expression \autoref{eq:DYN_hpg_sco} is commonly used when the variable volume formulation is activated
661(\texttt{vvl?}) because in that case, even with a flat bottom,
662the coordinate surfaces are not horizontal but follow the free surface \citep{levier.treguier.ea_rpt07}.
663The pressure jacobian scheme (\np[=.true.]{ln_dynhpg_prj}{ln\_dynhpg\_prj}) is available as
664an improved option to \np[=.true.]{ln_dynhpg_sco}{ln\_dynhpg\_sco} when \texttt{vvl?} is active.
665The pressure Jacobian scheme uses a constrained cubic spline to
666reconstruct the density profile across the water column.
667This method maintains the monotonicity between the density nodes.
668The pressure can be calculated by analytical integration of the density profile and
669a pressure Jacobian method is used to solve the horizontal pressure gradient.
670This method can provide a more accurate calculation of the horizontal pressure gradient than the standard scheme.
671
672%% =================================================================================================
673\subsection{Ice shelf cavity}
674\label{subsec:DYN_hpg_isf}
675
676Beneath an ice shelf, the total pressure gradient is the sum of the pressure gradient due to the ice shelf load and
677the pressure gradient due to the ocean load (\np[=.true.]{ln_dynhpg_isf}{ln\_dynhpg\_isf}).\\
678
679The main hypothesis to compute the ice shelf load is that the ice shelf is in an isostatic equilibrium.
680The top pressure is computed integrating from surface to the base of the ice shelf a reference density profile
681(prescribed as density of a water at 34.4 PSU and -1.9\deg{C}) and
682corresponds to the water replaced by the ice shelf.
683This top pressure is constant over time.
684A detailed description of this method is described in \citet{losch_JGR08}.\\
685
686The pressure gradient due to ocean load is computed using the expression \autoref{eq:DYN_hpg_sco} described in
687\autoref{subsec:DYN_hpg_sco}.
688
689%% =================================================================================================
690\subsection[Time-scheme (\forcode{ln_dynhpg_imp})]{Time-scheme (\protect\np{ln_dynhpg_imp}{ln\_dynhpg\_imp})}
691\label{subsec:DYN_hpg_imp}
692
693The default time differencing scheme used for the horizontal pressure gradient is a leapfrog scheme and
694therefore the density used in all discrete expressions given above is the  \textit{now} density,
695computed from the \textit{now} temperature and salinity.
696In some specific cases
697(usually high resolution simulations over an ocean domain which includes weakly stratified regions)
698the physical phenomenon that controls the time-step is internal gravity waves (IGWs).
699A semi-implicit scheme for doubling the stability limit associated with IGWs can be used
700\citep{brown.campana_MWR78, maltrud.smith.ea_JGR98}.
701It involves the evaluation of the hydrostatic pressure gradient as
702an average over the three time levels $t-\rdt$, $t$, and $t+\rdt$
703(\ie\ \textit{before}, \textit{now} and  \textit{after} time-steps),
704rather than at the central time level $t$ only, as in the standard leapfrog scheme.
705
706$\bullet$ leapfrog scheme (\np[=.true.]{ln_dynhpg_imp}{ln\_dynhpg\_imp}):
707
708\begin{equation}
709  \label{eq:DYN_hpg_lf}
710  \frac{u^{t+\rdt}-u^{t-\rdt}}{2\rdt} = \;\cdots \;
711  -\frac{1}{\rho_o \,e_{1u} }\delta_{i+1/2} \left[ {p_h^t } \right]
712\end{equation}
713
714$\bullet$ semi-implicit scheme (\np[=.true.]{ln_dynhpg_imp}{ln\_dynhpg\_imp}):
715\begin{equation}
716  \label{eq:DYN_hpg_imp}
717  \frac{u^{t+\rdt}-u^{t-\rdt}}{2\rdt} = \;\cdots \;
718  -\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]
719\end{equation}
720
721The semi-implicit time scheme \autoref{eq:DYN_hpg_imp} is made possible without
722significant additional computation since the density can be updated to time level $t+\rdt$ before
723computing the horizontal hydrostatic pressure gradient.
724It can be easily shown that the stability limit associated with the hydrostatic pressure gradient doubles using
725\autoref{eq:DYN_hpg_imp} compared to that using the standard leapfrog scheme \autoref{eq:DYN_hpg_lf}.
726Note that \autoref{eq:DYN_hpg_imp} is equivalent to applying a time filter to the pressure gradient to
727eliminate high frequency IGWs.
728Obviously, when using \autoref{eq:DYN_hpg_imp},
729the doubling of the time-step is achievable only if no other factors control the time-step,
730such as the stability limits associated with advection or diffusion.
731
732In practice, the semi-implicit scheme is used when \np[=.true.]{ln_dynhpg_imp}{ln\_dynhpg\_imp}.
733In this case, we choose to apply the time filter to temperature and salinity used in the equation of state,
734instead of applying it to the hydrostatic pressure or to the density,
735so that no additional storage array has to be defined.
736The density used to compute the hydrostatic pressure gradient (whatever the formulation) is evaluated as follows:
737\[
738  % \label{eq:DYN_rho_flt}
739  \rho^t = \rho( \widetilde{T},\widetilde {S},z_t)
740  \quad    \text{with}  \quad
741  \widetilde{X} = 1 / 4 \left(  X^{t+\rdt} +2 \,X^t + X^{t-\rdt\right)
742\]
743
744Note that in the semi-implicit case, it is necessary to save the filtered density,
745an extra three-dimensional field, in the restart file to restart the model with exact reproducibility.
746This option is controlled by  \np{nn_dynhpg_rst}{nn\_dynhpg\_rst}, a namelist parameter.
747
748%% =================================================================================================
749\section[Surface pressure gradient (\textit{dynspg.F90})]{Surface pressure gradient (\protect\mdl{dynspg})}
750\label{sec:DYN_spg}
751
752\begin{listing}
753  \nlst{namdyn_spg}
754  \caption{\forcode{&namdyn_spg}}
755  \label{lst:namdyn_spg}
756\end{listing}
757
758Options are defined through the \nam{dyn_spg}{dyn\_spg} namelist variables.
759The surface pressure gradient term is related to the representation of the free surface (\autoref{sec:MB_hor_pg}).
760The main distinction is between the fixed volume case (linear free surface) and
761the variable volume case (nonlinear free surface, \texttt{vvl?} is defined).
762In the linear free surface case (\autoref{subsec:MB_free_surface})
763the vertical scale factors $e_{3}$ are fixed in time,
764while they are time-dependent in the nonlinear case (\autoref{subsec:MB_free_surface}).
765With both linear and nonlinear free surface, external gravity waves are allowed in the equations,
766which imposes a very small time step when an explicit time stepping is used.
767Two methods are proposed to allow a longer time step for the three-dimensional equations:
768the filtered free surface, which is a modification of the continuous equations (see \autoref{eq:MB_flt?}),
769and the split-explicit free surface described below.
770The extra term introduced in the filtered method is calculated implicitly,
771so that the update of the next velocities is done in module \mdl{dynspg\_flt} and not in \mdl{dynnxt}.
772
773The form of the surface pressure gradient term depends on how the user wants to
774handle the fast external gravity waves that are a solution of the analytical equation (\autoref{sec:MB_hor_pg}).
775Three formulations are available, all controlled by a CPP key (ln\_dynspg\_xxx):
776an explicit formulation which requires a small time step;
777a filtered free surface formulation which allows a larger time step by
778adding a filtering term into the momentum equation;
779and a split-explicit free surface formulation, described below, which also allows a larger time step.
780
781The extra term introduced in the filtered method is calculated implicitly, so that a solver is used to compute it.
782As a consequence the update of the $next$ velocities is done in module \mdl{dynspg\_flt} and not in \mdl{dynnxt}.
783
784%% =================================================================================================
785\subsection[Explicit free surface (\forcode{ln_dynspg_exp})]{Explicit free surface (\protect\np{ln_dynspg_exp}{ln\_dynspg\_exp})}
786\label{subsec:DYN_spg_exp}
787
788In the explicit free surface formulation (\np{ln_dynspg_exp}{ln\_dynspg\_exp} set to true),
789the model time step is chosen to be small enough to resolve the external gravity waves
790(typically a few tens of seconds).
791The surface pressure gradient, evaluated using a leap-frog scheme (\ie\ centered in time),
792is thus simply given by :
793\begin{equation}
794  \label{eq:DYN_spg_exp}
795  \left\{
796    \begin{aligned}
797      - \frac{1}{e_{1u}\,\rho_o} \; \delta_{i+1/2} \left[  \,\rho \,\eta\,  \right]    \\
798      - \frac{1}{e_{2v}\,\rho_o} \; \delta_{j+1/2} \left[  \,\rho \,\eta\,  \right]
799    \end{aligned}
800  \right.
801\end{equation}
802
803Note that in the non-linear free surface case (\ie\ \texttt{vvl?} defined),
804the surface pressure gradient is already included in the momentum tendency through
805the level thickness variation allowed in the computation of the hydrostatic pressure gradient.
806Thus, nothing is done in the \mdl{dynspg\_exp} module.
807
808%% =================================================================================================
809\subsection[Split-explicit free surface (\forcode{ln_dynspg_ts})]{Split-explicit free surface (\protect\np{ln_dynspg_ts}{ln\_dynspg\_ts})}
810\label{subsec:DYN_spg_ts}
811%
812%\nlst{namsplit}
813
814The split-explicit free surface formulation used in \NEMO\ (\np{ln_dynspg_ts}{ln\_dynspg\_ts} set to true),
815also called the time-splitting formulation, follows the one proposed by \citet{shchepetkin.mcwilliams_OM05}.
816The general idea is to solve the free surface equation and the associated barotropic velocity equations with
817a smaller time step than $\rdt$, the time step used for the three dimensional prognostic variables
818(\autoref{fig:DYN_spg_ts}).
819The size of the small time step, $\rdt_e$ (the external mode or barotropic time step) is provided through
820the \np{nn_baro}{nn\_baro} namelist parameter as: $\rdt_e = \rdt / nn\_baro$.
821This parameter can be optionally defined automatically (\np[=.true.]{ln_bt_nn_auto}{ln\_bt\_nn\_auto}) considering that
822the stability of the barotropic system is essentially controled by external waves propagation.
823Maximum Courant number is in that case time independent, and easily computed online from the input bathymetry.
824Therefore, $\rdt_e$ is adjusted so that the Maximum allowed Courant number is smaller than \np{rn_bt_cmax}{rn\_bt\_cmax}.
825
826%%%
827The barotropic mode solves the following equations:
828% \begin{subequations}
829%  \label{eq:DYN_BT}
830\begin{equation}
831  \label{eq:DYN_BT_dyn}
832  \frac{\partial {\mathrm \overline{{\mathbf U}}_h} }{\partial t}=
833  -f\;{\mathrm {\mathbf k}}\times {\mathrm \overline{{\mathbf U}}_h}
834  -g\nabla _h \eta -\frac{c_b^{\textbf U}}{H+\eta} \mathrm {\overline{{\mathbf U}}_h} + \mathrm {\overline{\mathbf G}}
835\end{equation}
836\[
837  % \label{eq:DYN_BT_ssh}
838  \frac{\partial \eta }{\partial t}=-\nabla \cdot \left[ {\left( {H+\eta } \right) \; {\mathrm{\mathbf \overline{U}}}_h \,} \right]+P-E
839\]
840% \end{subequations}
841where $\mathrm {\overline{\mathbf G}}$ is a forcing term held constant, containing coupling term between modes,
842surface atmospheric forcing as well as slowly varying barotropic terms not explicitly computed to gain efficiency.
843The third term on the right hand side of \autoref{eq:DYN_BT_dyn} represents the bottom stress
844(see section \autoref{sec:ZDF_drg}), explicitly accounted for at each barotropic iteration.
845Temporal discretization of the system above follows a three-time step Generalized Forward Backward algorithm
846detailed in \citet{shchepetkin.mcwilliams_OM05}.
847AB3-AM4 coefficients used in \NEMO\ follow the second-order accurate,
848"multi-purpose" stability compromise as defined in \citet{shchepetkin.mcwilliams_ibk09}
849(see their figure 12, lower left).
850
851\begin{figure}[!t]
852  \centering
853  \includegraphics[width=0.66\textwidth]{Fig_DYN_dynspg_ts}
854  \caption[Split-explicit time stepping scheme for the external and internal modes]{
855    Schematic of the split-explicit time stepping scheme for the external and internal modes.
856    Time increases to the right.
857    In this particular exemple,
858    a boxcar averaging window over \np{nn_baro}{nn\_baro} barotropic time steps is used
859    (\np[=1]{nn_bt_flt}{nn\_bt\_flt}) and \np[=5]{nn_baro}{nn\_baro}.
860    Internal mode time steps (which are also the model time steps) are denoted by
861    $t-\rdt$, $t$ and $t+\rdt$.
862    Variables with $k$ superscript refer to instantaneous barotropic variables,
863    $< >$ and $<< >>$ operator refer to time filtered variables using respectively primary
864    (red vertical bars) and secondary weights (blue vertical bars).
865    The former are used to obtain time filtered quantities at $t+\rdt$ while
866    the latter are used to obtain time averaged transports to advect tracers.
867    a) Forward time integration:
868    \protect\np[=.true.]{ln_bt_fw}{ln\_bt\_fw}\protect\np[=.true.]{ln_bt_av}{ln\_bt\_av}.
869    b) Centred time integration:
870    \protect\np[=.false.]{ln_bt_fw}{ln\_bt\_fw}, \protect\np[=.true.]{ln_bt_av}{ln\_bt\_av}.
871    c) Forward time integration with no time filtering (POM-like scheme):
872    \protect\np[=.true.]{ln_bt_fw}{ln\_bt\_fw}\protect\np[=.false.]{ln_bt_av}{ln\_bt\_av}.}
873  \label{fig:DYN_spg_ts}
874\end{figure}
875
876In the default case (\np[=.true.]{ln_bt_fw}{ln\_bt\_fw}),
877the external mode is integrated between \textit{now} and \textit{after} baroclinic time-steps
878(\autoref{fig:DYN_spg_ts}a).
879To avoid aliasing of fast barotropic motions into three dimensional equations,
880time filtering is eventually applied on barotropic quantities (\np[=.true.]{ln_bt_av}{ln\_bt\_av}).
881In that case, the integration is extended slightly beyond \textit{after} time step to
882provide time filtered quantities.
883These are used for the subsequent initialization of the barotropic mode in the following baroclinic step.
884Since external mode equations written at baroclinic time steps finally follow a forward time stepping scheme,
885asselin filtering is not applied to barotropic quantities.\\
886Alternatively, one can choose to integrate barotropic equations starting from \textit{before} time step
887(\np[=.false.]{ln_bt_fw}{ln\_bt\_fw}).
888Although more computationaly expensive ( \np{nn_baro}{nn\_baro} additional iterations are indeed necessary),
889the baroclinic to barotropic forcing term given at \textit{now} time step become centred in
890the middle of the integration window.
891It can easily be shown that this property removes part of splitting errors between modes,
892which increases the overall numerical robustness.
893%references to Patrick Marsaleix' work here. Also work done by SHOM group.
894
895%%%
896
897As far as tracer conservation is concerned,
898barotropic velocities used to advect tracers must also be updated at \textit{now} time step.
899This implies to change the traditional order of computations in \NEMO:
900most of momentum trends (including the barotropic mode calculation) updated first, tracers' after.
901This \textit{de facto} makes semi-implicit hydrostatic pressure gradient
902(see section \autoref{subsec:DYN_hpg_imp})
903and time splitting not compatible.
904Advective barotropic velocities are obtained by using a secondary set of filtering weights,
905uniquely defined from the filter coefficients used for the time averaging (\citet{shchepetkin.mcwilliams_OM05}).
906Consistency between the time averaged continuity equation and the time stepping of tracers is here the key to
907obtain exact conservation.
908
909%%%
910
911One can eventually choose to feedback instantaneous values by not using any time filter
912(\np[=.false.]{ln_bt_av}{ln\_bt\_av}).
913In that case, external mode equations are continuous in time,
914\ie\ they are not re-initialized when starting a new sub-stepping sequence.
915This is the method used so far in the POM model, the stability being maintained by
916refreshing at (almost) each barotropic time step advection and horizontal diffusion terms.
917Since the latter terms have not been added in \NEMO\ for computational efficiency,
918removing time filtering is not recommended except for debugging purposes.
919This may be used for instance to appreciate the damping effect of the standard formulation on
920external gravity waves in idealized or weakly non-linear cases.
921Although the damping is lower than for the filtered free surface,
922it is still significant as shown by \citet{levier.treguier.ea_rpt07} in the case of an analytical barotropic Kelvin wave.
923
924\gmcomment{               %%% copy from griffies Book
925
926\textbf{title: Time stepping the barotropic system }
927
928Assume knowledge of the full velocity and tracer fields at baroclinic time $\tau$.
929Hence, we can update the surface height and vertically integrated velocity with a leap-frog scheme using
930the small barotropic time step $\rdt$.
931We have
932
933\[
934  % \label{eq:DYN_spg_ts_eta}
935  \eta^{(b)}(\tau,t_{n+1}) - \eta^{(b)}(\tau,t_{n+1}) (\tau,t_{n-1})
936  = 2 \rdt \left[-\nabla \cdot \textbf{U}^{(b)}(\tau,t_n) + \text{EMP}_w(\tau) \right]
937\]
938\begin{multline*}
939  % \label{eq:DYN_spg_ts_u}
940  \textbf{U}^{(b)}(\tau,t_{n+1}) - \textbf{U}^{(b)}(\tau,t_{n-1}\\
941  = 2\rdt \left[ - f \textbf{k} \times \textbf{U}^{(b)}(\tau,t_{n})
942    - H(\tau) \nabla p_s^{(b)}(\tau,t_{n}) +\textbf{M}(\tau) \right]
943\end{multline*}
944\
945
946In these equations, araised (b) denotes values of surface height and vertically integrated velocity updated with
947the barotropic time steps.
948The $\tau$ time label on $\eta^{(b)}$ and $U^{(b)}$ denotes the baroclinic time at which
949the vertically integrated forcing $\textbf{M}(\tau)$
950(note that this forcing includes the surface freshwater forcing),
951the tracer fields, the freshwater flux $\text{EMP}_w(\tau)$,
952and total depth of the ocean $H(\tau)$ are held for the duration of the barotropic time stepping over
953a single cycle.
954This is also the time that sets the barotropic time steps via
955\[
956  % \label{eq:DYN_spg_ts_t}
957  t_n=\tau+n\rdt
958\]
959with $n$ an integer.
960The density scaled surface pressure is evaluated via
961\[
962  % \label{eq:DYN_spg_ts_ps}
963  p_s^{(b)}(\tau,t_{n}) =
964  \begin{cases}
965    g \;\eta_s^{(b)}(\tau,t_{n}) \;\rho(\tau)_{k=1}) / \rho_o  &      \text{non-linear case} \\
966    g \;\eta_s^{(b)}(\tau,t_{n})  &      \text{linear case}
967  \end{cases}
968\]
969To get started, we assume the following initial conditions
970\[
971  % \label{eq:DYN_spg_ts_eta}
972  \begin{split}
973    \eta^{(b)}(\tau,t_{n=0}) &= \overline{\eta^{(b)}(\tau)}    \\
974    \eta^{(b)}(\tau,t_{n=1}) &= \eta^{(b)}(\tau,t_{n=0}) + \rdt \ \text{RHS}_{n=0}
975  \end{split}
976\]
977with
978\[
979  % \label{eq:DYN_spg_ts_etaF}
980  \overline{\eta^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N \eta^{(b)}(\tau-\rdt,t_{n})
981\]
982the time averaged surface height taken from the previous barotropic cycle.
983Likewise,
984\[
985  % \label{eq:DYN_spg_ts_u}
986  \textbf{U}^{(b)}(\tau,t_{n=0}) = \overline{\textbf{U}^{(b)}(\tau)} \\ \\
987  \textbf{U}(\tau,t_{n=1}) = \textbf{U}^{(b)}(\tau,t_{n=0}) + \rdt \ \text{RHS}_{n=0}
988\]
989with
990\[
991  % \label{eq:DYN_spg_ts_u}
992  \overline{\textbf{U}^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau-\rdt,t_{n})
993\]
994the time averaged vertically integrated transport.
995Notably, there is no Robert-Asselin time filter used in the barotropic portion of the integration.
996
997Upon reaching $t_{n=N} = \tau + 2\rdt \tau$ ,
998the vertically integrated velocity is time averaged to produce the updated vertically integrated velocity at
999baroclinic time $\tau + \rdt \tau$
1000\[
1001  % \label{eq:DYN_spg_ts_u}
1002  \textbf{U}(\tau+\rdt) = \overline{\textbf{U}^{(b)}(\tau+\rdt)} = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau,t_{n})
1003\]
1004The surface height on the new baroclinic time step is then determined via a baroclinic leap-frog using
1005the following form
1006
1007\begin{equation}
1008  \label{eq:DYN_spg_ts_ssh}
1009  \eta(\tau+\Delta) - \eta^{F}(\tau-\Delta) = 2\rdt \ \left[ - \nabla \cdot \textbf{U}(\tau) + \text{EMP}_w \right]
1010\end{equation}
1011
1012The use of this "big-leap-frog" scheme for the surface height ensures compatibility between
1013the mass/volume budgets and the tracer budgets.
1014More discussion of this point is provided in Chapter 10 (see in particular Section 10.2).
1015
1016In general, some form of time filter is needed to maintain integrity of the surface height field due to
1017the leap-frog splitting mode in equation \autoref{eq:DYN_spg_ts_ssh}.
1018We have tried various forms of such filtering,
1019with the following method discussed in \cite{griffies.pacanowski.ea_MWR01} chosen due to
1020its stability and reasonably good maintenance of tracer conservation properties (see ??).
1021
1022\begin{equation}
1023  \label{eq:DYN_spg_ts_sshf}
1024  \eta^{F}(\tau-\Delta) =  \overline{\eta^{(b)}(\tau)}
1025\end{equation}
1026Another approach tried was
1027
1028\[
1029  % \label{eq:DYN_spg_ts_sshf2}
1030  \eta^{F}(\tau-\Delta) = \eta(\tau)
1031  + (\alpha/2) \left[\overline{\eta^{(b)}}(\tau+\rdt)
1032    + \overline{\eta^{(b)}}(\tau-\rdt) -2 \;\eta(\tau) \right]
1033\]
1034
1035which is useful since it isolates all the time filtering aspects into the term multiplied by $\alpha$.
1036This isolation allows for an easy check that tracer conservation is exact when
1037eliminating tracer and surface height time filtering (see ?? for more complete discussion).
1038However, in the general case with a non-zero $\alpha$,
1039the filter \autoref{eq:DYN_spg_ts_sshf} was found to be more conservative, and so is recommended.
1040
1041}            %%end gm comment (copy of griffies book)
1042
1043%% =================================================================================================
1044\subsection{Filtered free surface (\forcode{dynspg_flt?})}
1045\label{subsec:DYN_spg_fltp}
1046
1047The filtered formulation follows the \citet{roullet.madec_JGR00} implementation.
1048The extra term introduced in the equations (see \autoref{subsec:MB_free_surface}) is solved implicitly.
1049The elliptic solvers available in the code are documented in \autoref{chap:MISC}.
1050
1051%% gm %%======>>>>   given here the discrete eqs provided to the solver
1052\gmcomment{               %%% copy from chap-model basics
1053  \[
1054    % \label{eq:DYN_spg_flt}
1055    \frac{\partial {\mathrm {\mathbf U}}_h }{\partial t}= {\mathrm {\mathbf M}}
1056    - g \nabla \left( \tilde{\rho} \ \eta \right)
1057    - g \ T_c \nabla \left( \widetilde{\rho} \ \partial_t \eta \right)
1058  \]
1059  where $T_c$, is a parameter with dimensions of time which characterizes the force,
1060  $\widetilde{\rho} = \rho / \rho_o$ is the dimensionless density,
1061  and $\mathrm {\mathbf M}$ represents the collected contributions of the Coriolis, hydrostatic pressure gradient,
1062  non-linear and viscous terms in \autoref{eq:MB_dyn}.
1063}   %end gmcomment
1064
1065Note that in the linear free surface formulation (\texttt{vvl?} not defined),
1066the ocean depth is time-independent and so is the matrix to be inverted.
1067It is computed once and for all and applies to all ocean time steps.
1068
1069%% =================================================================================================
1070\section[Lateral diffusion term and operators (\textit{dynldf.F90})]{Lateral diffusion term and operators (\protect\mdl{dynldf})}
1071\label{sec:DYN_ldf}
1072
1073\begin{listing}
1074  \nlst{namdyn_ldf}
1075  \caption{\forcode{&namdyn_ldf}}
1076  \label{lst:namdyn_ldf}
1077\end{listing}
1078
1079Options are defined through the \nam{dyn_ldf}{dyn\_ldf} namelist variables.
1080The options available for lateral diffusion are to use either laplacian (rotated or not) or biharmonic operators.
1081The coefficients may be constant or spatially variable;
1082the description of the coefficients is found in the chapter on lateral physics (\autoref{chap:LDF}).
1083The lateral diffusion of momentum is evaluated using a forward scheme,
1084\ie\ the velocity appearing in its expression is the \textit{before} velocity in time,
1085except for the pure vertical component that appears when a tensor of rotation is used.
1086This latter term is solved implicitly together with the vertical diffusion term (see \autoref{chap:TD}).
1087
1088At the lateral boundaries either free slip,
1089no slip or partial slip boundary conditions are applied according to the user's choice (see \autoref{chap:LBC}).
1090
1091\gmcomment{
1092  Hyperviscous operators are frequently used in the simulation of turbulent flows to
1093  control the dissipation of unresolved small scale features.
1094  Their primary role is to provide strong dissipation at the smallest scale supported by
1095  the grid while minimizing the impact on the larger scale features.
1096  Hyperviscous operators are thus designed to be more scale selective than the traditional,
1097  physically motivated Laplace operator.
1098  In finite difference methods,
1099  the biharmonic operator is frequently the method of choice to achieve this scale selective dissipation since
1100  its damping time (\ie\ its spin down time) scale like $\lambda^{-4}$ for disturbances of wavelength $\lambda$
1101  (so that short waves damped more rapidelly than long ones),
1102  whereas the Laplace operator damping time scales only like $\lambda^{-2}$.
1103}
1104
1105%% =================================================================================================
1106\subsection[Iso-level laplacian (\forcode{ln_dynldf_lap})]{Iso-level laplacian operator (\protect\np{ln_dynldf_lap}{ln\_dynldf\_lap})}
1107\label{subsec:DYN_ldf_lap}
1108
1109For lateral iso-level diffusion, the discrete operator is:
1110\begin{equation}
1111  \label{eq:DYN_ldf_lap}
1112  \left\{
1113    \begin{aligned}
1114      D_u^{l{\mathrm {\mathbf U}}} =\frac{1}{e_{1u} }\delta_{i+1/2} \left[ {A_T^{lm}
1115          \;\chi } \right]-\frac{1}{e_{2u} {\kern 1pt}e_{3u} }\delta_j \left[
1116        {A_f^{lm} \;e_{3f} \zeta } \right] \\ \\
1117      D_v^{l{\mathrm {\mathbf U}}} =\frac{1}{e_{2v} }\delta_{j+1/2} \left[ {A_T^{lm}
1118          \;\chi } \right]+\frac{1}{e_{1v} {\kern 1pt}e_{3v} }\delta_i \left[
1119        {A_f^{lm} \;e_{3f} \zeta } \right]
1120    \end{aligned}
1121  \right.
1122\end{equation}
1123
1124As explained in \autoref{subsec:MB_ldf},
1125this formulation (as the gradient of a divergence and curl of the vorticity) preserves symmetry and
1126ensures a complete separation between the vorticity and divergence parts of the momentum diffusion.
1127
1128%% =================================================================================================
1129\subsection[Rotated laplacian (\forcode{ln_dynldf_iso})]{Rotated laplacian operator (\protect\np{ln_dynldf_iso}{ln\_dynldf\_iso})}
1130\label{subsec:DYN_ldf_iso}
1131
1132A rotation of the lateral momentum diffusion operator is needed in several cases:
1133for iso-neutral diffusion in the $z$-coordinate (\np[=.true.]{ln_dynldf_iso}{ln\_dynldf\_iso}) and
1134for either iso-neutral (\np[=.true.]{ln_dynldf_iso}{ln\_dynldf\_iso}) or
1135geopotential (\np[=.true.]{ln_dynldf_hor}{ln\_dynldf\_hor}) diffusion in the $s$-coordinate.
1136In the partial step case, coordinates are horizontal except at the deepest level and
1137no rotation is performed when \np[=.true.]{ln_dynldf_hor}{ln\_dynldf\_hor}.
1138The diffusion operator is defined simply as the divergence of down gradient momentum fluxes on
1139each momentum component.
1140It must be emphasized that this formulation ignores constraints on the stress tensor such as symmetry.
1141The resulting discrete representation is:
1142\begin{equation}
1143  \label{eq:DYN_ldf_iso}
1144  \begin{split}
1145    D_u^{l\textbf{U}} &= \frac{1}{e_{1u} \, e_{2u} \, e_{3u} } \\
1146    &  \left\{\quad  {\delta_{i+1/2} \left[ {A_T^{lm}  \left(
1147              {\frac{e_{2t} \; e_{3t} }{e_{1t} } \,\delta_{i}[u]
1148                -e_{2t} \; r_{1t} \,\overline{\overline {\delta_{k+1/2}[u]}}^{\,i,\,k}}
1149            \right)} \right]}    \right. \\
1150    & \qquad +\ \delta_j \left[ {A_f^{lm} \left( {\frac{e_{1f}\,e_{3f} }{e_{2f}
1151            }\,\delta_{j+1/2} [u] - e_{1f}\, r_{2f}
1152            \,\overline{\overline {\delta_{k+1/2} [u]}} ^{\,j+1/2,\,k}}
1153        \right)} \right] \\
1154    &\qquad +\ \delta_k \left[ {A_{uw}^{lm} \left( {-e_{2u} \, r_{1uw} \,\overline{\overline
1155              {\delta_{i+1/2} [u]}}^{\,i+1/2,\,k+1/2} }
1156        \right.} \right. \\
1157    &  \ \qquad \qquad \qquad \quad\
1158    - e_{1u} \, r_{2uw} \,\overline{\overline {\delta_{j+1/2} [u]}} ^{\,j,\,k+1/2} \\
1159    & \left. {\left. { \ \qquad \qquad \qquad \ \ \ \left. {\
1160                +\frac{e_{1u}\, e_{2u} }{e_{3uw} }\,\left( {r_{1uw}^2+r_{2uw}^2}
1161                \right)\,\delta_{k+1/2} [u]} \right)} \right]\;\;\;} \right\} \\ \\
1162    D_v^{l\textbf{V}} &= \frac{1}{e_{1v} \, e_{2v} \, e_{3v} } \\
1163    &  \left\{\quad  {\delta_{i+1/2} \left[ {A_f^{lm}  \left(
1164              {\frac{e_{2f} \; e_{3f} }{e_{1f} } \,\delta_{i+1/2}[v]
1165                -e_{2f} \; r_{1f} \,\overline{\overline {\delta_{k+1/2}[v]}}^{\,i+1/2,\,k}}
1166            \right)} \right]}    \right. \\
1167    & \qquad +\ \delta_j \left[ {A_T^{lm} \left( {\frac{e_{1t}\,e_{3t} }{e_{2t}
1168            }\,\delta_{j} [v] - e_{1t}\, r_{2t}
1169            \,\overline{\overline {\delta_{k+1/2} [v]}} ^{\,j,\,k}}
1170        \right)} \right] \\
1171    & \qquad +\ \delta_k \left[ {A_{vw}^{lm} \left( {-e_{2v} \, r_{1vw} \,\overline{\overline
1172              {\delta_{i+1/2} [v]}}^{\,i+1/2,\,k+1/2} }\right.} \right. \\
1173    &  \ \qquad \qquad \qquad \quad\
1174    - e_{1v} \, r_{2vw} \,\overline{\overline {\delta_{j+1/2} [v]}} ^{\,j+1/2,\,k+1/2} \\
1175    & \left. {\left. { \ \qquad \qquad \qquad \ \ \ \left. {\
1176                +\frac{e_{1v}\, e_{2v} }{e_{3vw} }\,\left( {r_{1vw}^2+r_{2vw}^2}
1177                \right)\,\delta_{k+1/2} [v]} \right)} \right]\;\;\;} \right\}
1178  \end{split}
1179\end{equation}
1180where $r_1$ and $r_2$ are the slopes between the surface along which the diffusion operator acts and
1181the surface of computation ($z$- or $s$-surfaces).
1182The way these slopes are evaluated is given in the lateral physics chapter (\autoref{chap:LDF}).
1183
1184%% =================================================================================================
1185\subsection[Iso-level bilaplacian (\forcode{ln_dynldf_bilap})]{Iso-level bilaplacian operator (\protect\np{ln_dynldf_bilap}{ln\_dynldf\_bilap})}
1186\label{subsec:DYN_ldf_bilap}
1187
1188The lateral fourth order operator formulation on momentum is obtained by applying \autoref{eq:DYN_ldf_lap} twice.
1189It requires an additional assumption on boundary conditions:
1190the first derivative term normal to the coast depends on the free or no-slip lateral boundary conditions chosen,
1191while the third derivative terms normal to the coast are set to zero (see \autoref{chap:LBC}).
1192%%%
1193\gmcomment{add a remark on the the change in the position of the coefficient}
1194%%%
1195
1196%% =================================================================================================
1197\section[Vertical diffusion term (\textit{dynzdf.F90})]{Vertical diffusion term (\protect\mdl{dynzdf})}
1198\label{sec:DYN_zdf}
1199
1200Options are defined through the \nam{zdf}{zdf} namelist variables.
1201The 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.
1202Two time stepping schemes can be used for the vertical diffusion term:
1203$(a)$ a forward time differencing scheme
1204(\np[=.true.]{ln_zdfexp}{ln\_zdfexp}) using a time splitting technique (\np{nn_zdfexp}{nn\_zdfexp} $>$ 1) or
1205$(b)$ a backward (or implicit) time differencing scheme (\np[=.false.]{ln_zdfexp}{ln\_zdfexp})
1206(see \autoref{chap:TD}).
1207Note that namelist variables \np{ln_zdfexp}{ln\_zdfexp} and \np{nn_zdfexp}{nn\_zdfexp} apply to both tracers and dynamics.
1208
1209The formulation of the vertical subgrid scale physics is the same whatever the vertical coordinate is.
1210The vertical diffusion operators given by \autoref{eq:MB_zdf} take the following semi-discrete space form:
1211\[
1212  % \label{eq:DYN_zdf}
1213  \left\{
1214    \begin{aligned}
1215      D_u^{vm} &\equiv \frac{1}{e_{3u}} \ \delta_k \left[ \frac{A_{uw}^{vm} }{e_{3uw} }
1216        \ \delta_{k+1/2} [\,u\,]         \right]     \\
1217      \\
1218      D_v^{vm} &\equiv \frac{1}{e_{3v}} \ \delta_k \left[ \frac{A_{vw}^{vm} }{e_{3vw} }
1219        \ \delta_{k+1/2} [\,v\,]         \right]
1220    \end{aligned}
1221  \right.
1222\]
1223where $A_{uw}^{vm} $ and $A_{vw}^{vm} $ are the vertical eddy viscosity and diffusivity coefficients.
1224The way these coefficients are evaluated depends on the vertical physics used (see \autoref{chap:ZDF}).
1225
1226The surface boundary condition on momentum is the stress exerted by the wind.
1227At the surface, the momentum fluxes are prescribed as the boundary condition on
1228the vertical turbulent momentum fluxes,
1229\begin{equation}
1230  \label{eq:DYN_zdf_sbc}
1231  \left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{z=1}
1232  = \frac{1}{\rho_o} \binom{\tau_u}{\tau_v }
1233\end{equation}
1234where $\left( \tau_u ,\tau_v \right)$ are the two components of the wind stress vector in
1235the (\textbf{i},\textbf{j}) coordinate system.
1236The high mixing coefficients in the surface mixed layer ensure that the surface wind stress is distributed in
1237the vertical over the mixed layer depth.
1238If the vertical mixing coefficient is small (when no mixed layer scheme is used)
1239the surface stress enters only the top model level, as a body force.
1240The surface wind stress is calculated in the surface module routines (SBC, see \autoref{chap:SBC}).
1241
1242The turbulent flux of momentum at the bottom of the ocean is specified through a bottom friction parameterisation
1243(see \autoref{sec:ZDF_drg})
1244
1245%% =================================================================================================
1246\section{External forcings}
1247\label{sec:DYN_forcing}
1248
1249Besides the surface and bottom stresses (see the above section)
1250which are introduced as boundary conditions on the vertical mixing,
1251three other forcings may enter the dynamical equations by affecting the surface pressure gradient.
1252
1253(1) When \np[=.true.]{ln_apr_dyn}{ln\_apr\_dyn} (see \autoref{sec:SBC_apr}),
1254the atmospheric pressure is taken into account when computing the surface pressure gradient.
1255
1256(2) When \np[=.true.]{ln_tide_pot}{ln\_tide\_pot} and \np[=.true.]{ln_tide}{ln\_tide} (see \autoref{sec:SBC_tide}),
1257the tidal potential is taken into account when computing the surface pressure gradient.
1258
1259(3) When \np[=2]{nn_ice_embd}{nn\_ice\_embd} and LIM or CICE is used
1260(\ie\ when the sea-ice is embedded in the ocean),
1261the snow-ice mass is taken into account when computing the surface pressure gradient.
1262
1263\gmcomment{ missing : the lateral boundary condition !!!   another external forcing
1264 }
1265
1266%% =================================================================================================
1267\section{Wetting and drying }
1268\label{sec:DYN_wetdry}
1269
1270There are two main options for wetting and drying code (wd):
1271(a) an iterative limiter (il) and (b) a directional limiter (dl).
1272The directional limiter is based on the scheme developed by \cite{warner.defne.ea_CG13} for RO
1273MS
1274which was in turn based on ideas developed for POM by \cite{oey_OM06}. The iterative
1275limiter is a new scheme.  The iterative limiter is activated by setting $\mathrm{ln\_wd\_il} = \mathrm{.true.}$
1276and $\mathrm{ln\_wd\_dl} = \mathrm{.false.}$. The directional limiter is activated
1277by setting $\mathrm{ln\_wd\_dl} = \mathrm{.true.}$ and $\mathrm{ln\_wd\_il} = \mathrm{.false.}$.
1278
1279\begin{listing}
1280  \nlst{namwad}
1281  \caption{\forcode{&namwad}}
1282  \label{lst:namwad}
1283\end{listing}
1284
1285The following terminology is used. The depth of the topography (positive downwards)
1286at each $(i,j)$ point is the quantity stored in array $\mathrm{ht\_wd}$ in the \NEMO\ code.
1287The height of the free surface (positive upwards) is denoted by $ \mathrm{ssh}$. Given the sign
1288conventions used, the water depth, $h$, is the height of the free surface plus the depth of the
1289topography (i.e. $\mathrm{ssh} + \mathrm{ht\_wd}$).
1290
1291Both wd schemes take all points in the domain below a land elevation of $\mathrm{rn\_wdld}$ to be
1292covered by water. They require the topography specified with a model
1293configuration to have negative depths at points where the land is higher than the
1294topography's reference sea-level. The vertical grid in \NEMO\ is normally computed relative to an
1295initial state with zero sea surface height elevation.
1296The user can choose to compute the vertical grid and heights in the model relative to
1297a non-zero reference height for the free surface. This choice affects the calculation of the metrics and depths
1298(i.e. the $\mathrm{e3t\_0, ht\_0}$ etc. arrays).
1299
1300Points where the water depth is less than $\mathrm{rn\_wdmin1}$ are interpreted as ``dry''.
1301$\mathrm{rn\_wdmin1}$ is usually chosen to be of order $0.05$m but extreme topographies
1302with very steep slopes require larger values for normal choices of time-step. Surface fluxes
1303are also switched off for dry cells to prevent freezing, boiling etc. of very thin water layers.
1304The fluxes are tappered down using a $\mathrm{tanh}$ weighting function
1305to no flux as the dry limit $\mathrm{rn\_wdmin1}$ is approached. Even wet cells can be very shallow.
1306The depth at which to start tapering is controlled by the user by setting $\mathrm{rn\_wd\_sbcdep}$.
1307The fraction $(<1)$ of sufrace fluxes to use at this depth is set by $\mathrm{rn\_wd\_sbcfra}$.
1308
1309Both versions of the code have been tested in six test cases provided in the WAD\_TEST\_CASES configuration
1310and in ``realistic'' configurations covering parts of the north-west European shelf.
1311All these configurations have used pure sigma coordinates. It is expected that
1312the wetting and drying code will work in domains with more general s-coordinates provided
1313the coordinates are pure sigma in the region where wetting and drying actually occurs.
1314
1315The next sub-section descrbies the directional limiter and the following sub-section the iterative limiter.
1316The final sub-section covers some additional considerations that are relevant to both schemes.
1317
1318%   Iterative limiters
1319%% =================================================================================================
1320\subsection[Directional limiter (\textit{wet\_dry.F90})]{Directional limiter (\mdl{wet\_dry})}
1321\label{subsec:DYN_wd_directional_limiter}
1322
1323The principal idea of the directional limiter is that
1324water should not be allowed to flow out of a dry tracer cell (i.e. one whose water depth is less than \np{rn_wdmin1}{rn\_wdmin1}).
1325
1326All the changes associated with this option are made to the barotropic solver for the non-linear
1327free surface code within dynspg\_ts.
1328On each barotropic sub-step the scheme determines the direction of the flow across each face of all the tracer cells
1329and sets the flux across the face to zero when the flux is from a dry tracer cell. This prevents cells
1330whose depth is rn\_wdmin1 or less from drying out further. The scheme does not force $h$ (the water depth) at tracer cells
1331to be at least the minimum depth and hence is able to conserve mass / volume.
1332
1333The flux across each $u$-face of a tracer cell is multiplied by a factor zuwdmask (an array which depends on ji and jj).
1334If the user sets \np[=.false.]{ln_wd_dl_ramp}{ln\_wd\_dl\_ramp} then zuwdmask is 1 when the
1335flux is from a cell with water depth greater than \np{rn_wdmin1}{rn\_wdmin1} and 0 otherwise. If the user sets
1336\np[=.true.]{ln_wd_dl_ramp}{ln\_wd\_dl\_ramp} the flux across the face is ramped down as the water depth decreases
1337from 2 * \np{rn_wdmin1}{rn\_wdmin1} to \np{rn_wdmin1}{rn\_wdmin1}. The use of this ramp reduced grid-scale noise in idealised test cases.
1338
1339At the point where the flux across a $u$-face is multiplied by zuwdmask , we have chosen
1340also to multiply the corresponding velocity on the ``now'' step at that face by zuwdmask. We could have
1341chosen not to do that and to allow fairly large velocities to occur in these ``dry'' cells.
1342The rationale for setting the velocity to zero is that it is the momentum equations that are being solved
1343and the total momentum of the upstream cell (treating it as a finite volume) should be considered
1344to be its depth times its velocity. This depth is considered to be zero at ``dry'' $u$-points consistent with its
1345treatment in the calculation of the flux of mass across the cell face.
1346
1347\cite{warner.defne.ea_CG13} state that in their scheme the velocity masks at the cell faces for the baroclinic
1348timesteps are set to 0 or 1 depending on whether the average of the masks over the barotropic sub-steps is respectively less than
1349or greater than 0.5. That scheme does not conserve tracers in integrations started from constant tracer
1350fields (tracers independent of $x$, $y$ and $z$). Our scheme conserves constant tracers because
1351the velocities used at the tracer cell faces on the baroclinic timesteps are carefully calculated by dynspg\_ts
1352to equal their mean value during the barotropic steps. If the user sets \np[=.true.]{ln_wd_dl_bc}{ln\_wd\_dl\_bc}, the
1353baroclinic velocities are also multiplied by a suitably weighted average of zuwdmask.
1354
1355%   Iterative limiters
1356
1357%% =================================================================================================
1358\subsection[Iterative limiter (\textit{wet\_dry.F90})]{Iterative limiter (\mdl{wet\_dry})}
1359\label{subsec:DYN_wd_iterative_limiter}
1360
1361%% =================================================================================================
1362\subsubsection[Iterative flux limiter (\textit{wet\_dry.F90})]{Iterative flux limiter (\mdl{wet\_dry})}
1363\label{subsec:DYN_wd_il_spg_limiter}
1364
1365The iterative limiter modifies the fluxes across the faces of cells that are either already ``dry''
1366or may become dry within the next time-step using an iterative method.
1367
1368The flux limiter for the barotropic flow (devised by Hedong Liu) can be understood as follows:
1369
1370The continuity equation for the total water depth in a column
1371\begin{equation}
1372  \label{eq:DYN_wd_continuity}
1373  \frac{\partial h}{\partial t} + \mathbf{\nabla.}(h\mathbf{u}) = 0 .
1374\end{equation}
1375can be written in discrete form  as
1376
1377\begin{align}
1378  \label{eq:DYN_wd_continuity_2}
1379  \frac{e_1 e_2}{\Delta t} ( h_{i,j}(t_{n+1}) - h_{i,j}(t_e) )
1380  &= - ( \mathrm{flxu}_{i+1,j} - \mathrm{flxu}_{i,j}  + \mathrm{flxv}_{i,j+1} - \mathrm{flxv}_{i,j} ) \\
1381  &= \mathrm{zzflx}_{i,j} .
1382\end{align}
1383
1384In the above $h$ is the depth of the water in the column at point $(i,j)$,
1385$\mathrm{flxu}_{i+1,j}$ is the flux out of the ``eastern'' face of the cell and
1386$\mathrm{flxv}_{i,j+1}$ the flux out of the ``northern'' face of the cell; $t_{n+1}$ is
1387the new timestep, $t_e$ is the old timestep (either $t_b$ or $t_n$) and $ \Delta t =
1388t_{n+1} - t_e$; $e_1 e_2$ is the area of the tracer cells centred at $(i,j)$ and
1389$\mathrm{zzflx}$ is the sum of the fluxes through all the faces.
1390
1391The flux limiter splits the flux $\mathrm{zzflx}$ into fluxes that are out of the cell
1392(zzflxp) and fluxes that are into the cell (zzflxn).  Clearly
1393
1394\begin{equation}
1395  \label{eq:DYN_wd_zzflx_p_n_1}
1396  \mathrm{zzflx}_{i,j} = \mathrm{zzflxp}_{i,j} + \mathrm{zzflxn}_{i,j} .
1397\end{equation}
1398
1399The flux limiter iteratively adjusts the fluxes $\mathrm{flxu}$ and $\mathrm{flxv}$ until
1400none of the cells will ``dry out''. To be precise the fluxes are limited until none of the
1401cells has water depth less than $\mathrm{rn\_wdmin1}$ on step $n+1$.
1402
1403Let the fluxes on the $m$th iteration step be denoted by $\mathrm{flxu}^{(m)}$ and
1404$\mathrm{flxv}^{(m)}$.  Then the adjustment is achieved by seeking a set of coefficients,
1405$\mathrm{zcoef}_{i,j}^{(m)}$ such that:
1406
1407\begin{equation}
1408  \label{eq:DYN_wd_continuity_coef}
1409  \begin{split}
1410    \mathrm{zzflxp}^{(m)}_{i,j} =& \mathrm{zcoef}_{i,j}^{(m)} \mathrm{zzflxp}^{(0)}_{i,j} \\
1411    \mathrm{zzflxn}^{(m)}_{i,j} =& \mathrm{zcoef}_{i,j}^{(m)} \mathrm{zzflxn}^{(0)}_{i,j}
1412  \end{split}
1413\end{equation}
1414
1415where the coefficients are $1.0$ generally but can vary between $0.0$ and $1.0$ around
1416cells that would otherwise dry.
1417
1418The iteration is initialised by setting
1419
1420\begin{equation}
1421  \label{eq:DYN_wd_zzflx_initial}
1422  \mathrm{zzflxp^{(0)}}_{i,j} = \mathrm{zzflxp}_{i,j} , \quad  \mathrm{zzflxn^{(0)}}_{i,j} = \mathrm{zzflxn}_{i,j} .
1423\end{equation}
1424
1425The fluxes out of cell $(i,j)$ are updated at the $m+1$th iteration if the depth of the
1426cell on timestep $t_e$, namely $h_{i,j}(t_e)$, is less than the total flux out of the cell
1427times the timestep divided by the cell area. Using (\autoref{eq:DYN_wd_continuity_2}) this
1428condition is
1429
1430\begin{equation}
1431  \label{eq:DYN_wd_continuity_if}
1432  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} ) .
1433\end{equation}
1434
1435Rearranging (\autoref{eq:DYN_wd_continuity_if}) we can obtain an expression for the maximum
1436outward flux that can be allowed and still maintain the minimum wet depth:
1437
1438\begin{equation}
1439  \label{eq:DYN_wd_max_flux}
1440  \begin{split}
1441    \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{]} \\
1442    \phantom{[} & -  \mathrm{zzflxn}^{(m)}_{i,j} \Big]
1443  \end{split}
1444\end{equation}
1445
1446Note a small tolerance ($\mathrm{rn\_wdmin2}$) has been introduced here {\itshape [Q: Why is
1447this necessary/desirable?]}. Substituting from (\autoref{eq:DYN_wd_continuity_coef}) gives an
1448expression for the coefficient needed to multiply the outward flux at this cell in order
1449to avoid drying.
1450
1451\begin{equation}
1452  \label{eq:DYN_wd_continuity_nxtcoef}
1453  \begin{split}
1454    \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{]} \\
1455    \phantom{[} & -  \mathrm{zzflxn}^{(m)}_{i,j} \Big] \frac{1}{ \mathrm{zzflxp}^{(0)}_{i,j} }
1456  \end{split}
1457\end{equation}
1458
1459Only the outward flux components are altered but, of course, outward fluxes from one cell
1460are inward fluxes to adjacent cells and the balance in these cells may need subsequent
1461adjustment; hence the iterative nature of this scheme.  Note, for example, that the flux
1462across the ``eastern'' face of the $(i,j)$th cell is only updated at the $m+1$th iteration
1463if that flux at the $m$th iteration is out of the $(i,j)$th cell. If that is the case then
1464the flux across that face is into the $(i+1,j)$ cell and that flux will not be updated by
1465the calculation for the $(i+1,j)$th cell. In this sense the updates to the fluxes across
1466the faces of the cells do not ``compete'' (they do not over-write each other) and one
1467would expect the scheme to converge relatively quickly. The scheme is flux based so
1468conserves mass. It also conserves constant tracers for the same reason that the
1469directional limiter does.
1470
1471%      Surface pressure gradients
1472%% =================================================================================================
1473\subsubsection[Modification of surface pressure gradients (\textit{dynhpg.F90})]{Modification of surface pressure gradients (\mdl{dynhpg})}
1474\label{subsec:DYN_wd_il_spg}
1475
1476At ``dry'' points the water depth is usually close to $\mathrm{rn\_wdmin1}$. If the
1477topography is sloping at these points the sea-surface will have a similar slope and there
1478will hence be very large horizontal pressure gradients at these points. The WAD modifies
1479the magnitude but not the sign of the surface pressure gradients (zhpi and zhpj) at such
1480points by mulitplying them by positive factors (zcpx and zcpy respectively) that lie
1481between $0$ and $1$.
1482
1483We describe how the scheme works for the ``eastward'' pressure gradient, zhpi, calculated
1484at the $(i,j)$th $u$-point. The scheme uses the ht\_wd depths and surface heights at the
1485neighbouring $(i+1,j)$ and $(i,j)$ tracer points.  zcpx is calculated using two logicals
1486variables, $\mathrm{ll\_tmp1}$ and $\mathrm{ll\_tmp2}$ which are evaluated for each grid
1487column.  The three possible combinations are illustrated in \autoref{fig:DYN_WAD_dynhpg}.
1488
1489\begin{figure}[!ht]
1490  \centering
1491  \includegraphics[width=0.66\textwidth]{Fig_WAD_dynhpg}
1492  \caption[Combinations controlling the limiting of the horizontal pressure gradient in
1493  wetting and drying regimes]{
1494    Three possible combinations of the logical variables controlling the
1495    limiting of the horizontal pressure gradient in wetting and drying regimes}
1496  \label{fig:DYN_WAD_dynhpg}
1497\end{figure}
1498
1499The first logical, $\mathrm{ll\_tmp1}$, is set to true if and only if the water depth at
1500both neighbouring points is greater than $\mathrm{rn\_wdmin1} + \mathrm{rn\_wdmin2}$ and
1501the minimum height of the sea surface at the two points is greater than the maximum height
1502of the topography at the two points:
1503
1504\begin{equation}
1505  \label{eq:DYN_ll_tmp1}
1506  \begin{split}
1507    \mathrm{ll\_tmp1}  = & \mathrm{MIN(sshn(ji,jj), sshn(ji+1,jj))} > \\
1508                     & \quad \mathrm{MAX(-ht\_wd(ji,jj), -ht\_wd(ji+1,jj))\  .and.} \\
1509                     & \mathrm{MAX(sshn(ji,jj) + ht\_wd(ji,jj),} \\
1510                     & \mathrm{\phantom{MAX(}sshn(ji+1,jj) + ht\_wd(ji+1,jj))} >\\
1511                     & \quad\quad\mathrm{rn\_wdmin1 + rn\_wdmin2 }
1512  \end{split}
1513\end{equation}
1514
1515The second logical, $\mathrm{ll\_tmp2}$, is set to true if and only if the maximum height
1516of the sea surface at the two points is greater than the maximum height of the topography
1517at the two points plus $\mathrm{rn\_wdmin1} + \mathrm{rn\_wdmin2}$
1518
1519\begin{equation}
1520  \label{eq:DYN_ll_tmp2}
1521  \begin{split}
1522    \mathrm{ ll\_tmp2 } = & \mathrm{( ABS( sshn(ji,jj) - sshn(ji+1,jj) ) > 1.E-12 )\ .AND.}\\
1523    & \mathrm{( MAX(sshn(ji,jj), sshn(ji+1,jj)) > } \\
1524    & \mathrm{\phantom{(} MAX(-ht\_wd(ji,jj), -ht\_wd(ji+1,jj)) + rn\_wdmin1 + rn\_wdmin2}) .
1525  \end{split}
1526\end{equation}
1527
1528If $\mathrm{ll\_tmp1}$ is true then the surface pressure gradient, zhpi at the $(i,j)$
1529point is unmodified. If both logicals are false zhpi is set to zero.
1530
1531If $\mathrm{ll\_tmp1}$ is true and $\mathrm{ll\_tmp2}$ is false then the surface pressure
1532gradient is multiplied through by zcpx which is the absolute value of the difference in
1533the water depths at the two points divided by the difference in the surface heights at the
1534two points. Thus the sign of the sea surface height gradient is retained but the magnitude
1535of the pressure force is determined by the difference in water depths rather than the
1536difference in surface height between the two points. Note that dividing by the difference
1537between the sea surface heights can be problematic if the heights approach parity. An
1538additional condition is applied to $\mathrm{ ll\_tmp2 }$ to ensure it is .false. in such
1539conditions.
1540
1541%% =================================================================================================
1542\subsubsection[Additional considerations (\textit{usrdef\_zgr.F90})]{Additional considerations (\mdl{usrdef\_zgr})}
1543\label{subsec:DYN_WAD_additional}
1544
1545In the very shallow water where wetting and drying occurs the parametrisation of
1546bottom drag is clearly very important. In order to promote stability
1547it is sometimes useful to calculate the bottom drag using an implicit time-stepping approach.
1548
1549Suitable specifcation of the surface heat flux in wetting and drying domains in forced and
1550coupled simulations needs further consideration. In order to prevent freezing or boiling
1551in uncoupled integrations the net surface heat fluxes need to be appropriately limited.
1552
1553%      The WAD test cases
1554%% =================================================================================================
1555\subsection[The WAD test cases (\textit{usrdef\_zgr.F90})]{The WAD test cases (\mdl{usrdef\_zgr})}
1556\label{subsec:DYN_WAD_test_cases}
1557
1558See the WAD tests MY\_DOC documention for details of the WAD test cases.
1559
1560%% =================================================================================================
1561\section[Time evolution term (\textit{dynnxt.F90})]{Time evolution term (\protect\mdl{dynnxt})}
1562\label{sec:DYN_nxt}
1563
1564Options are defined through the \nam{dom}{dom} namelist variables.
1565The general framework for dynamics time stepping is a leap-frog scheme,
1566\ie\ a three level centred time scheme associated with an Asselin time filter (cf. \autoref{chap:TD}).
1567The scheme is applied to the velocity, except when
1568using the flux form of momentum advection (cf. \autoref{sec:DYN_adv_cor_flux})
1569in the variable volume case (\texttt{vvl?} defined),
1570where it has to be applied to the thickness weighted velocity (see \autoref{sec:SCOORD_momentum})
1571
1572$\bullet$ vector invariant form or linear free surface
1573(\np[=.true.]{ln_dynhpg_vec}{ln\_dynhpg\_vec} ; \texttt{vvl?} not defined):
1574\[
1575  % \label{eq:DYN_nxt_vec}
1576  \left\{
1577    \begin{aligned}
1578      &u^{t+\rdt} = u_f^{t-\rdt} + 2\rdt  \ \text{RHS}_u^t     \\
1579      &u_f^t \;\quad = u^t+\gamma \,\left[ {u_f^{t-\rdt} -2u^t+u^{t+\rdt}} \right]
1580    \end{aligned}
1581  \right.
1582\]
1583
1584$\bullet$ flux form and nonlinear free surface
1585(\np[=.false.]{ln_dynhpg_vec}{ln\_dynhpg\_vec} ; \texttt{vvl?} defined):
1586\[
1587  % \label{eq:DYN_nxt_flux}
1588  \left\{
1589    \begin{aligned}
1590      &\left(e_{3u}\,u\right)^{t+\rdt} = \left(e_{3u}\,u\right)_f^{t-\rdt} + 2\rdt \; e_{3u} \;\text{RHS}_u^t     \\
1591      &\left(e_{3u}\,u\right)_f^t \;\quad = \left(e_{3u}\,u\right)^t
1592      +\gamma \,\left[ {\left(e_{3u}\,u\right)_f^{t-\rdt} -2\left(e_{3u}\,u\right)^t+\left(e_{3u}\,u\right)^{t+\rdt}} \right]
1593    \end{aligned}
1594  \right.
1595\]
1596where RHS is the right hand side of the momentum equation,
1597the subscript $f$ denotes filtered values and $\gamma$ is the Asselin coefficient.
1598$\gamma$ is initialized as \np{nn_atfp}{nn\_atfp} (namelist parameter).
1599Its default value is \np[=10.e-3]{nn_atfp}{nn\_atfp}.
1600In both cases, the modified Asselin filter is not applied since perfect conservation is not an issue for
1601the momentum equations.
1602
1603Note that with the filtered free surface,
1604the update of the \textit{after} velocities is done in the \mdl{dynsp\_flt} module,
1605and only array swapping and Asselin filtering is done in \mdl{dynnxt}.
1606
1607\onlyinsubfile{\input{../../global/epilogue}}
1608
1609\end{document}
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