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1\documentclass[../main/NEMO_manual]{subfiles}
2
3\begin{document}
4% ================================================================
5% Chapter ——— Ocean Dynamics (DYN)
6% ================================================================
7\chapter{Ocean Dynamics (DYN)}
8\label{chap:DYN}
9
10\minitoc
11
12Using the representation described in \autoref{chap:DOM},
13several semi-discrete space forms of the dynamical equations are available depending on
14the vertical coordinate used and on the conservation properties of the vorticity term.
15In all the equations presented here, the masking has been omitted for simplicity.
16One must be aware that all the quantities are masked fields and
17that each time an average or difference operator is used, the resulting field is multiplied by a mask.
18
19The prognostic ocean dynamics equation can be summarized as follows:
20\[
21  \text{NXT} = \dbinom  {\text{VOR} + \text{KEG} + \text {ZAD} }
22  {\text{COR} + \text{ADV}                       }
23  + \text{HPG} + \text{SPG} + \text{LDF} + \text{ZDF}
24\]
25NXT stands for next, referring to the time-stepping.
26The first group of terms on the rhs of this equation corresponds to the Coriolis and advection terms that
27are decomposed into either a vorticity part (VOR), a kinetic energy part (KEG) and
28a vertical advection part (ZAD) in the vector invariant formulation,
29or a Coriolis and advection part (COR+ADV) in the flux formulation.
30The terms following these are the pressure gradient contributions
31(HPG, Hydrostatic Pressure Gradient, and SPG, Surface Pressure Gradient);
32and contributions from lateral diffusion (LDF) and vertical diffusion (ZDF),
33which are added to the rhs in the \mdl{dynldf} and \mdl{dynzdf} modules.
34The vertical diffusion term includes the surface and bottom stresses.
35The external forcings and parameterisations require complex inputs
36(surface wind stress calculation using bulk formulae, estimation of mixing coefficients)
37that are carried out in modules SBC, LDF and ZDF and are described in
38\autoref{chap:SBC}, \autoref{chap:LDF} and \autoref{chap:ZDF}, respectively.
39
40In the present chapter we also describe the diagnostic equations used to compute the horizontal divergence,
41curl of the velocities (\emph{divcur} module) and the vertical velocity (\emph{wzvmod} module).
42
43The different options available to the user are managed by namelist variables.
44For term \textit{ttt} in the momentum equations, the logical namelist variables are \textit{ln\_dynttt\_xxx},
45where \textit{xxx} is a 3 or 4 letter acronym corresponding to each optional scheme.
46If a CPP key is used for this term its name is \key{ttt}.
47The corresponding code can be found in the \textit{dynttt\_xxx} module in the DYN directory,
48and it is usually computed in the \textit{dyn\_ttt\_xxx} subroutine.
49
50The user has the option of extracting and outputting each tendency term from the 3D momentum equations
51(\key{trddyn} defined), as described in \autoref{chap:MISC}.
52Furthermore, the tendency terms associated with the 2D barotropic vorticity balance (when \key{trdvor} is defined)
53can be derived from the 3D terms.
54%%%
55\gmcomment{STEVEN: not quite sure I've got the sense of the last sentence. does
56MISC correspond to "extracting tendency terms" or "vorticity balance"?}
57
58% ================================================================
59% Sea Surface Height evolution & Diagnostics variables
60% ================================================================
61\section{Sea surface height and diagnostic variables ($\eta$, $\zeta$, $\chi$, $w$)}
62\label{sec:DYN_divcur_wzv}
63
64%--------------------------------------------------------------------------------------------------------------
65%           Horizontal divergence and relative vorticity
66%--------------------------------------------------------------------------------------------------------------
67\subsection{Horizontal divergence and relative vorticity (\protect\mdl{divcur})}
68\label{subsec:DYN_divcur}
69
70The vorticity is defined at an $f$-point (\ie corner point) as follows:
71\begin{equation}
72  \label{eq:divcur_cur}
73  \zeta =\frac{1}{e_{1f}\,e_{2f} }\left( {\;\delta_{i+1/2} \left[ {e_{2v}\;v} \right]
74      -\delta_{j+1/2} \left[ {e_{1u}\;u} \right]\;} \right)
75\end{equation} 
76
77The horizontal divergence is defined at a $T$-point.
78It is given by:
79\[
80  % \label{eq:divcur_div}
81  \chi =\frac{1}{e_{1t}\,e_{2t}\,e_{3t} }
82  \left( {\delta_i \left[ {e_{2u}\,e_{3u}\,u} \right]
83      +\delta_j \left[ {e_{1v}\,e_{3v}\,v} \right]} \right)
84\]
85
86Note that although the vorticity has the same discrete expression in $z$- and $s$-coordinates,
87its physical meaning is not identical.
88$\zeta$ is a pseudo vorticity along $s$-surfaces
89(only pseudo because $(u,v)$ are still defined along geopotential surfaces,
90but are not necessarily defined at the same depth).
91
92The vorticity and divergence at the \textit{before} step are used in the computation of
93the horizontal diffusion of momentum.
94Note that because they have been calculated prior to the Asselin filtering of the \textit{before} velocities,
95the \textit{before} vorticity and divergence arrays must be included in the restart file to
96ensure perfect restartability.
97The vorticity and divergence at the \textit{now} time step are used for the computation of
98the nonlinear advection and of the vertical velocity respectively.
99
100%--------------------------------------------------------------------------------------------------------------
101%           Sea Surface Height evolution
102%--------------------------------------------------------------------------------------------------------------
103\subsection{Horizontal divergence and relative vorticity (\protect\mdl{sshwzv})}
104\label{subsec:DYN_sshwzv}
105
106The sea surface height is given by:
107\begin{equation}
108  \label{eq:dynspg_ssh}
109  \begin{aligned}
110    \frac{\partial \eta }{\partial t}
111    &\equiv    \frac{1}{e_{1t} e_{2t} }\sum\limits_k { \left\{  \delta_i \left[ {e_{2u}\,e_{3u}\;u} \right]
112        +\delta_j \left[ {e_{1v}\,e_{3v}\;v} \right]  \right\} }
113    -    \frac{\textit{emp}}{\rho_w }   \\
114    &\equiv    \sum\limits_k {\chi \ e_{3t}}  -  \frac{\textit{emp}}{\rho_w }
115  \end{aligned}
116\end{equation}
117where \textit{emp} is the surface freshwater budget (evaporation minus precipitation),
118expressed in Kg/m$^2$/s (which is equal to mm/s),
119and $\rho_w$=1,035~Kg/m$^3$ is the reference density of sea water (Boussinesq approximation).
120If river runoff is expressed as a surface freshwater flux (see \autoref{chap:SBC}) then
121\textit{emp} can be written as the evaporation minus precipitation, minus the river runoff.
122The sea-surface height is evaluated using exactly the same time stepping scheme as
123the tracer equation \autoref{eq:tra_nxt}:
124a leapfrog scheme in combination with an Asselin time filter,
125\ie the velocity appearing in \autoref{eq:dynspg_ssh} is centred in time (\textit{now} velocity).
126This is of paramount importance.
127Replacing $T$ by the number $1$ in the tracer equation and summing over the water column must lead to
128the sea surface height equation otherwise tracer content will not be conserved
129\citep{Griffies_al_MWR01, Leclair_Madec_OM09}.
130
131The vertical velocity is computed by an upward integration of the horizontal divergence starting at the bottom,
132taking into account the change of the thickness of the levels:
133\begin{equation}
134  \label{eq:wzv}
135  \left\{
136    \begin{aligned}
137      &\left. w \right|_{k_b-1/2} \quad= 0    \qquad \text{where } k_b \text{ is the level just above the sea floor }   \\
138      &\left. w \right|_{k+1/2}     = \left. w \right|_{k-1/2}  +  \left. e_{3t} \right|_{k}\;  \left. \chi \right|_k
139      - \frac{1} {2 \rdt} \left\left. e_{3t}^{t+1}\right|_{k} - \left. e_{3t}^{t-1}\right|_{k}\right)
140    \end{aligned}
141  \right.
142\end{equation}
143
144In the case of a non-linear free surface (\key{vvl}), the top vertical velocity is $-\textit{emp}/\rho_w$,
145as changes in the divergence of the barotropic transport are absorbed into the change of the level thicknesses,
146re-orientated downward.
147\gmcomment{not sure of this...  to be modified with the change in emp setting}
148In the case of a linear free surface, the time derivative in \autoref{eq:wzv} disappears.
149The upper boundary condition applies at a fixed level $z=0$.
150The top vertical velocity is thus equal to the divergence of the barotropic transport
151(\ie the first term in the right-hand-side of \autoref{eq:dynspg_ssh}).
152
153Note also that whereas the vertical velocity has the same discrete expression in $z$- and $s$-coordinates,
154its physical meaning is not the same:
155in the second case, $w$ is the velocity normal to the $s$-surfaces.
156Note also that the $k$-axis is re-orientated downwards in the \fortran code compared to
157the indexing used in the semi-discrete equations such as \autoref{eq:wzv}
158(see \autoref{subsec:DOM_Num_Index_vertical}).
159
160
161% ================================================================
162% Coriolis and Advection terms: vector invariant form
163% ================================================================
164\section{Coriolis and advection: vector invariant form}
165\label{sec:DYN_adv_cor_vect}
166%-----------------------------------------nam_dynadv----------------------------------------------------
167
168\nlst{namdyn_adv} 
169%-------------------------------------------------------------------------------------------------------------
170
171The vector invariant form of the momentum equations is the one most often used in
172applications of the \NEMO ocean model.
173The flux form option (see next section) has been present since version $2$.
174Options are defined through the \ngn{namdyn\_adv} namelist variables Coriolis and
175momentum advection terms are evaluated using a leapfrog scheme,
176\ie the velocity appearing in these expressions is centred in time (\textit{now} velocity).
177At the lateral boundaries either free slip, no slip or partial slip boundary conditions are applied following
178\autoref{chap:LBC}.
179
180% -------------------------------------------------------------------------------------------------------------
181%        Vorticity term
182% -------------------------------------------------------------------------------------------------------------
183\subsection{Vorticity term (\protect\mdl{dynvor})}
184\label{subsec:DYN_vor}
185%------------------------------------------nam_dynvor----------------------------------------------------
186
187\nlst{namdyn_vor} 
188%-------------------------------------------------------------------------------------------------------------
189
190Options are defined through the \ngn{namdyn\_vor} namelist variables.
191Four discretisations of the vorticity term (\np{ln\_dynvor\_xxx}\forcode{ = .true.}) are available:
192conserving potential enstrophy of horizontally non-divergent flow (ENS scheme);
193conserving horizontal kinetic energy (ENE scheme);
194conserving potential enstrophy for the relative vorticity term and
195horizontal kinetic energy for the planetary vorticity term (MIX scheme);
196or conserving both the potential enstrophy of horizontally non-divergent flow and horizontal kinetic energy
197(EEN scheme) (see \autoref{subsec:C_vorEEN}).
198In the case of ENS, ENE or MIX schemes the land sea mask may be slightly modified to ensure the consistency of
199vorticity term with analytical equations (\np{ln\_dynvor\_con}\forcode{ = .true.}).
200The vorticity terms are all computed in dedicated routines that can be found in the \mdl{dynvor} module.
201
202%-------------------------------------------------------------
203%                 enstrophy conserving scheme
204%-------------------------------------------------------------
205\subsubsection{Enstrophy conserving scheme (\protect\np{ln\_dynvor\_ens}\forcode{ = .true.})}
206\label{subsec:DYN_vor_ens}
207
208In the enstrophy conserving case (ENS scheme),
209the discrete formulation of the vorticity term provides a global conservation of the enstrophy
210($ [ (\zeta +f ) / e_{3f} ]^2 $ in $s$-coordinates) for a horizontally non-divergent flow (\ie $\chi$=$0$),
211but does not conserve the total kinetic energy.
212It is given by:
213\begin{equation}
214  \label{eq:dynvor_ens}
215  \left\{
216    \begin{aligned}
217      {+\frac{1}{e_{1u} } } & {\overline {\left( { \frac{\zeta +f}{e_{3f} }} \right)} }^{\,i}
218      & {\overline{\overline {\left( {e_{1v}\,e_{3v}\;v} \right)}} }^{\,i, j+1/2}    \\
219      {- \frac{1}{e_{2v} } } & {\overline {\left( {\frac{\zeta +f}{e_{3f} }} \right)} }^{\,j}
220      & {\overline{\overline {\left( {e_{2u}\,e_{3u}\;u} \right)}} }^{\,i+1/2, j}
221    \end{aligned}
222  \right.
223\end{equation} 
224
225%-------------------------------------------------------------
226%                 energy conserving scheme
227%-------------------------------------------------------------
228\subsubsection{Energy conserving scheme (\protect\np{ln\_dynvor\_ene}\forcode{ = .true.})}
229\label{subsec:DYN_vor_ene}
230
231The kinetic energy conserving scheme (ENE scheme) conserves the global kinetic energy but not the global enstrophy.
232It is given by:
233\begin{equation}
234  \label{eq:dynvor_ene}
235  \left\{
236    \begin{aligned}
237      {+\frac{1}{e_{1u}}\; {\overline {\left( {\frac{\zeta +f}{e_{3f} }} \right)
238            \;  \overline {\left( {e_{1v}\,e_{3v}\;v} \right)} ^{\,i+1/2}} }^{\,j} }    \\
239      {- \frac{1}{e_{2v}}\; {\overline {\left( {\frac{\zeta +f}{e_{3f} }} \right)
240            \;  \overline {\left( {e_{2u}\,e_{3u}\;u} \right)} ^{\,j+1/2}} }^{\,i} }
241    \end{aligned}
242  \right.
243\end{equation} 
244
245%-------------------------------------------------------------
246%                 mix energy/enstrophy conserving scheme
247%-------------------------------------------------------------
248\subsubsection{Mixed energy/enstrophy conserving scheme (\protect\np{ln\_dynvor\_mix}\forcode{ = .true.}) }
249\label{subsec:DYN_vor_mix}
250
251For the mixed energy/enstrophy conserving scheme (MIX scheme), a mixture of the two previous schemes is used.
252It consists of the ENS scheme (\autoref{eq:dynvor_ens}) for the relative vorticity term,
253and of the ENE scheme (\autoref{eq:dynvor_ene}) applied to the planetary vorticity term.
254\[
255  % \label{eq:dynvor_mix}
256  \left\{ {
257      \begin{aligned}
258        {+\frac{1}{e_{1u} }\; {\overline {\left( {\frac{\zeta }{e_{3f} }} \right)} }^{\,i}
259          \; {\overline{\overline {\left( {e_{1v}\,e_{3v}\;v} \right)}} }^{\,i,j+1/2} -\frac{1}{e_{1u} }
260          \; {\overline {\left( {\frac{f}{e_{3f} }} \right)
261              \;\overline {\left( {e_{1v}\,e_{3v}\;v} \right)} ^{\,i+1/2}} }^{\,j} } \\
262        {-\frac{1}{e_{2v} }\; {\overline {\left( {\frac{\zeta }{e_{3f} }} \right)} }^j
263          \; {\overline{\overline {\left( {e_{2u}\,e_{3u}\;u} \right)}} }^{\,i+1/2,j} +\frac{1}{e_{2v} }
264          \; {\overline {\left( {\frac{f}{e_{3f} }} \right)
265              \;\overline {\left( {e_{2u}\,e_{3u}\;u} \right)} ^{\,j+1/2}} }^{\,i} } \hfill
266      \end{aligned}
267    } \right.
268\]
269
270%-------------------------------------------------------------
271%                 energy and enstrophy conserving scheme
272%-------------------------------------------------------------
273\subsubsection{Energy and enstrophy conserving scheme (\protect\np{ln\_dynvor\_een}\forcode{ = .true.}) }
274\label{subsec:DYN_vor_een}
275
276In both the ENS and ENE schemes,
277it is apparent that the combination of $i$ and $j$ averages of the velocity allows for
278the presence of grid point oscillation structures that will be invisible to the operator.
279These structures are \textit{computational modes} that will be at least partly damped by
280the momentum diffusion operator (\ie the subgrid-scale advection), but not by the resolved advection term.
281The ENS and ENE schemes therefore do not contribute to dump any grid point noise in the horizontal velocity field.
282Such noise would result in more noise in the vertical velocity field, an undesirable feature.
283This is a well-known characteristic of $C$-grid discretization where
284$u$ and $v$ are located at different grid points,
285a price worth paying to avoid a double averaging in the pressure gradient term as in the $B$-grid.
286\gmcomment{ To circumvent this, Adcroft (ADD REF HERE)
287Nevertheless, this technique strongly distort the phase and group velocity of Rossby waves....}
288
289A very nice solution to the problem of double averaging was proposed by \citet{Arakawa_Hsu_MWR90}.
290The idea is to get rid of the double averaging by considering triad combinations of vorticity.
291It is noteworthy that this solution is conceptually quite similar to the one proposed by
292\citep{Griffies_al_JPO98} for the discretization of the iso-neutral diffusion operator (see \autoref{apdx:C}).
293
294The \citet{Arakawa_Hsu_MWR90} vorticity advection scheme for a single layer is modified
295for spherical coordinates as described by \citet{Arakawa_Lamb_MWR81} to obtain the EEN scheme.
296First consider the discrete expression of the potential vorticity, $q$, defined at an $f$-point:
297\[
298  % \label{eq:pot_vor}
299  q  = \frac{\zeta +f} {e_{3f} }
300\]
301where the relative vorticity is defined by (\autoref{eq:divcur_cur}),
302the Coriolis parameter is given by $f=2 \,\Omega \;\sin \varphi _f $ and the layer thickness at $f$-points is:
303\begin{equation}
304  \label{eq:een_e3f}
305  e_{3f} = \overline{\overline {e_{3t} }} ^{\,i+1/2,j+1/2}
306\end{equation}
307
308%>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
309\begin{figure}[!ht]
310  \begin{center}
311    \includegraphics[width=0.70\textwidth]{Fig_DYN_een_triad}
312    \caption{
313      \protect\label{fig:DYN_een_triad}
314      Triads used in the energy and enstrophy conserving scheme (een) for
315      $u$-component (upper panel) and $v$-component (lower panel).
316    }
317  \end{center}
318\end{figure}
319% >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
320
321A key point in \autoref{eq:een_e3f} is how the averaging in the \textbf{i}- and \textbf{j}- directions is made.
322It uses the sum of masked t-point vertical scale factor divided either by the sum of the four t-point masks
323(\np{nn\_een\_e3f}\forcode{ = 1}), or just by $4$ (\np{nn\_een\_e3f}\forcode{ = .true.}).
324The latter case preserves the continuity of $e_{3f}$ when one or more of the neighbouring $e_{3t}$ tends to zero and
325extends by continuity the value of $e_{3f}$ into the land areas.
326This case introduces a sub-grid-scale topography at f-points
327(with a systematic reduction of $e_{3f}$ when a model level intercept the bathymetry)
328that tends to reinforce the topostrophy of the flow
329(\ie the tendency of the flow to follow the isobaths) \citep{Penduff_al_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: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:dynvor_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:C_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_al_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_al_OD06, Penduff_al_OS07}.
362
363%--------------------------------------------------------------------------------------------------------------
364%           Kinetic Energy Gradient term
365%--------------------------------------------------------------------------------------------------------------
366\subsection{Kinetic energy gradient term (\protect\mdl{dynkeg})}
367\label{subsec:DYN_keg}
368
369As demonstrated in \autoref{apdx:C},
370there is a single discrete formulation of the kinetic energy gradient term that,
371together with the formulation chosen for the vertical advection (see below),
372conserves the total kinetic energy:
373\[
374  % \label{eq:dynkeg}
375  \left\{
376    \begin{aligned}
377      -\frac{1}{2 \; e_{1u} }  & \ \delta_{i+1/2} \left[ {\overline {u^2}^{\,i} + \overline{v^2}^{\,j}} \right]   \\
378      -\frac{1}{2 \; e_{2v} }  & \ \delta_{j+1/2} \left[ {\overline {u^2}^{\,i} + \overline{v^2}^{\,j}} \right]
379    \end{aligned}
380  \right.
381\]
382
383%--------------------------------------------------------------------------------------------------------------
384%           Vertical advection term
385%--------------------------------------------------------------------------------------------------------------
386\subsection{Vertical advection term (\protect\mdl{dynzad}) }
387\label{subsec:DYN_zad}
388
389The discrete formulation of the vertical advection, t
390ogether with the formulation chosen for the gradient of kinetic energy (KE) term,
391conserves the total kinetic energy.
392Indeed, the change of KE due to the vertical advection is exactly balanced by
393the change of KE due to the gradient of KE (see \autoref{apdx:C}).
394\[
395  % \label{eq:dynzad}
396  \left\{
397    \begin{aligned}
398      -\frac{1} {e_{1u}\,e_{2u}\,e_{3u}} &\ \overline{\ \overline{ e_{1t}\,e_{2t}\;w } ^{\,i+1/2\;\delta_{k+1/2} \left[ u \right]\  }^{\,k}  \\
399      -\frac{1} {e_{1v}\,e_{2v}\,e_{3v}}  &\ \overline{\ \overline{ e_{1t}\,e_{2t}\;w } ^{\,j+1/2\;\delta_{k+1/2} \left[ u \right]\  }^{\,k}
400    \end{aligned}
401  \right.
402\]
403When \np{ln\_dynzad\_zts}\forcode{ = .true.},
404a split-explicit time stepping with 5 sub-timesteps is used on the vertical advection term.
405This option can be useful when the value of the timestep is limited by vertical advection \citep{Lemarie_OM2015}.
406Note that in this case,
407a similar split-explicit time stepping should be used on vertical advection of tracer to ensure a better stability,
408an option which is only available with a TVD scheme (see \np{ln\_traadv\_tvd\_zts} in \autoref{subsec:TRA_adv_tvd}).
409
410
411% ================================================================
412% Coriolis and Advection : flux form
413% ================================================================
414\section{Coriolis and advection: flux form}
415\label{sec:DYN_adv_cor_flux}
416%------------------------------------------nam_dynadv----------------------------------------------------
417
418\nlst{namdyn_adv} 
419%-------------------------------------------------------------------------------------------------------------
420
421Options are defined through the \ngn{namdyn\_adv} namelist variables.
422In the flux form (as in the vector invariant form),
423the Coriolis and momentum advection terms are evaluated using a leapfrog scheme,
424\ie the velocity appearing in their expressions is centred in time (\textit{now} velocity).
425At the lateral boundaries either free slip,
426no slip or partial slip boundary conditions are applied following \autoref{chap:LBC}.
427
428
429%--------------------------------------------------------------------------------------------------------------
430%           Coriolis plus curvature metric terms
431%--------------------------------------------------------------------------------------------------------------
432\subsection{Coriolis plus curvature metric terms (\protect\mdl{dynvor}) }
433\label{subsec:DYN_cor_flux}
434
435In flux form, the vorticity term reduces to a Coriolis term in which the Coriolis parameter has been modified to account for the "metric" term.
436This altered Coriolis parameter is thus discretised at $f$-points.
437It is given by:
438\begin{multline*}
439  % \label{eq:dyncor_metric}
440  f+\frac{1}{e_1 e_2 }\left( {v\frac{\partial e_2 }{\partial i}  -  u\frac{\partial e_1 }{\partial j}} \right\\
441  \equiv   f + \frac{1}{e_{1f} e_{2f} } \left( { \ \overline v ^{i+1/2}\delta_{i+1/2} \left[ {e_{2u} } \right]
442      -  \overline u ^{j+1/2}\delta_{j+1/2} \left[ {e_{1u} } \right]  }  \ \right)
443\end{multline*} 
444
445Any of the (\autoref{eq:dynvor_ens}), (\autoref{eq:dynvor_ene}) and (\autoref{eq:dynvor_een}) schemes can be used to
446compute the product of the Coriolis parameter and the vorticity.
447However, the energy-conserving scheme (\autoref{eq:dynvor_een}) has exclusively been used to date.
448This term is evaluated using a leapfrog scheme, \ie the velocity is centred in time (\textit{now} velocity).
449
450%--------------------------------------------------------------------------------------------------------------
451%           Flux form Advection term
452%--------------------------------------------------------------------------------------------------------------
453\subsection{Flux form advection term (\protect\mdl{dynadv}) }
454\label{subsec:DYN_adv_flux}
455
456The discrete expression of the advection term is given by:
457\[
458  % \label{eq:dynadv}
459  \left\{
460    \begin{aligned}
461      \frac{1}{e_{1u}\,e_{2u}\,e_{3u}}
462      \left(      \delta_{i+1/2} \left[ \overline{e_{2u}\,e_{3u}\;u }^{i       }  \ u_t      \right]
463        + \delta_{j       } \left[ \overline{e_{1u}\,e_{3u}\;v }^{i+1/2\ u_f      \right] \right\ \;   \\
464      \left.   + \delta_{k      } \left[ \overline{e_{1w}\,e_{2w}\;w}^{i+1/2\ u_{uw} \right] \right)   \\
465      \\
466      \frac{1}{e_{1v}\,e_{2v}\,e_{3v}}
467      \left(     \delta_{i       } \left[ \overline{e_{2u}\,e_{3u }\;u }^{j+1/2} \ v_f       \right]
468        + \delta_{j+1/2} \left[ \overline{e_{1u}\,e_{3u }\;v }^{i       } \ v_t       \right] \right\ \, \, \\
469      \left+ \delta_{k      } \left[ \overline{e_{1w}\,e_{2w}\;w}^{j+1/2} \ v_{vw}  \right] \right) \\
470    \end{aligned}
471  \right.
472\]
473
474Two advection schemes are available:
475a $2^{nd}$ order centered finite difference scheme, CEN2,
476or a $3^{rd}$ order upstream biased scheme, UBS.
477The latter is described in \citet{Shchepetkin_McWilliams_OM05}.
478The schemes are selected using the namelist logicals \np{ln\_dynadv\_cen2} and \np{ln\_dynadv\_ubs}.
479In flux form, the schemes differ by the choice of a space and time interpolation to define the value of
480$u$ and $v$ at the centre of each face of $u$- and $v$-cells, \ie at the $T$-, $f$-,
481and $uw$-points for $u$ and at the $f$-, $T$- and $vw$-points for $v$.
482
483%-------------------------------------------------------------
484%                 2nd order centred scheme
485%-------------------------------------------------------------
486\subsubsection{CEN2: $2^{nd}$ order centred scheme (\protect\np{ln\_dynadv\_cen2}\forcode{ = .true.})}
487\label{subsec:DYN_adv_cen2}
488
489In the centered $2^{nd}$ order formulation, the velocity is evaluated as the mean of the two neighbouring points:
490\begin{equation}
491  \label{eq:dynadv_cen2}
492  \left\{
493    \begin{aligned}
494      u_T^{cen2} &=\overline u^{i }       \quad &  u_F^{cen2} &=\overline u^{j+1/2}  \quad &  u_{uw}^{cen2} &=\overline u^{k+1/2}   \\
495      v_F^{cen2} &=\overline v ^{i+1/2} \quad & v_F^{cen2} &=\overline v^j    \quad &  v_{vw}^{cen2} &=\overline v ^{k+1/2}  \\
496    \end{aligned}
497  \right.
498\end{equation} 
499
500The scheme is non diffusive (\ie conserves the kinetic energy) but dispersive (\ie it may create false extrema).
501It is therefore notoriously noisy and must be used in conjunction with an explicit diffusion operator to
502produce a sensible solution.
503The associated time-stepping is performed using a leapfrog scheme in conjunction with an Asselin time-filter,
504so $u$ and $v$ are the \emph{now} velocities.
505
506%-------------------------------------------------------------
507%                 UBS scheme
508%-------------------------------------------------------------
509\subsubsection{UBS: Upstream Biased Scheme (\protect\np{ln\_dynadv\_ubs}\forcode{ = .true.})}
510\label{subsec:DYN_adv_ubs}
511
512The UBS advection scheme is an upstream biased third order scheme based on
513an upstream-biased parabolic interpolation.
514For example, the evaluation of $u_T^{ubs} $ is done as follows:
515\begin{equation}
516  \label{eq:dynadv_ubs}
517  u_T^{ubs} =\overline u ^i-\;\frac{1}{6}
518  \begin{cases}
519    u"_{i-1/2}&   \text{if $\ \overline{e_{2u}\,e_{3u} \ u}^i  \geqslant 0$ }    \\
520    u"_{i+1/2}&   \text{if $\ \overline{e_{2u}\,e_{3u} \ u}^i  < 0$ }
521  \end{cases}
522\end{equation}
523where $u"_{i+1/2} =\delta_{i+1/2} \left[ {\delta_i \left[ u \right]} \right]$.
524This results in a dissipatively dominant (\ie hyper-diffusive) truncation error
525\citep{Shchepetkin_McWilliams_OM05}.
526The overall performance of the advection scheme is similar to that reported in \citet{Farrow1995}.
527It is a relatively good compromise between accuracy and smoothness.
528It is not a \emph{positive} scheme, meaning that false extrema are permitted.
529But the amplitudes of the false extrema are significantly reduced over those in the centred second order method.
530As the scheme already includes a diffusion component, it can be used without explicit lateral diffusion on momentum
531(\ie \np{ln\_dynldf\_lap}\forcode{ = }\np{ln\_dynldf\_bilap}\forcode{ = .false.}),
532and it is recommended to do so.
533
534The UBS scheme is not used in all directions.
535In the vertical, the centred $2^{nd}$ order evaluation of the advection is preferred, \ie $u_{uw}^{ubs}$ and
536$u_{vw}^{ubs}$ in \autoref{eq:dynadv_cen2} are used.
537UBS is diffusive and is associated with vertical mixing of momentum. \gmcomment{ gm  pursue the
538sentence:Since vertical mixing of momentum is a source term of the TKE equation...  }
539
540For stability reasons, the first term in (\autoref{eq:dynadv_ubs}),
541which corresponds to a second order centred scheme, is evaluated using the \textit{now} velocity (centred in time),
542while the second term, which is the diffusion part of the scheme,
543is evaluated using the \textit{before} velocity (forward in time).
544This is discussed by \citet{Webb_al_JAOT98} in the context of the Quick advection scheme.
545
546Note that the UBS and QUICK (Quadratic Upstream Interpolation for Convective Kinematics) schemes only differ by
547one coefficient.
548Replacing $1/6$ by $1/8$ in (\autoref{eq:dynadv_ubs}) leads to the QUICK advection scheme \citep{Webb_al_JAOT98}.
549This option is not available through a namelist parameter, since the $1/6$ coefficient is hard coded.
550Nevertheless it is quite easy to make the substitution in the \mdl{dynadv\_ubs} module and obtain a QUICK scheme.
551
552Note also that in the current version of \mdl{dynadv\_ubs},
553there is also the possibility of using a $4^{th}$ order evaluation of the advective velocity as in ROMS.
554This is an error and should be suppressed soon.
555%%%
556\gmcomment{action :  this have to be done}
557%%%
558
559% ================================================================
560%           Hydrostatic pressure gradient term
561% ================================================================
562\section{Hydrostatic pressure gradient (\protect\mdl{dynhpg})}
563\label{sec:DYN_hpg}
564%------------------------------------------nam_dynhpg---------------------------------------------------
565
566\nlst{namdyn_hpg} 
567%-------------------------------------------------------------------------------------------------------------
568
569Options are defined through the \ngn{namdyn\_hpg} namelist variables.
570The key distinction between the different algorithms used for
571the hydrostatic pressure gradient is the vertical coordinate used,
572since HPG is a \emph{horizontal} pressure gradient, \ie computed along geopotential surfaces.
573As a result, any tilt of the surface of the computational levels will require a specific treatment to
574compute the hydrostatic pressure gradient.
575
576The hydrostatic pressure gradient term is evaluated either using a leapfrog scheme,
577\ie the density appearing in its expression is centred in time (\emph{now} $\rho$),
578or a semi-implcit scheme.
579At the lateral boundaries either free slip, no slip or partial slip boundary conditions are applied.
580
581%--------------------------------------------------------------------------------------------------------------
582%           z-coordinate with full step
583%--------------------------------------------------------------------------------------------------------------
584\subsection{Full step $Z$-coordinate (\protect\np{ln\_dynhpg\_zco}\forcode{ = .true.})}
585\label{subsec:DYN_hpg_zco}
586
587The hydrostatic pressure can be obtained by integrating the hydrostatic equation vertically from the surface.
588However, the pressure is large at great depth while its horizontal gradient is several orders of magnitude smaller.
589This may lead to large truncation errors in the pressure gradient terms.
590Thus, the two horizontal components of the hydrostatic pressure gradient are computed directly as follows:
591
592for $k=km$ (surface layer, $jk=1$ in the code)
593\begin{equation}
594  \label{eq:dynhpg_zco_surf}
595  \left\{
596    \begin{aligned}
597      \left. \delta_{i+1/2} \left[  p^h          \right] \right|_{k=km}
598      &= \frac{1}{2} g \   \left. \delta_{i+1/2} \left[  e_{3w} \ \rho \right] \right|_{k=km}   \\
599      \left. \delta_{j+1/2} \left[  p^h          \right] \right|_{k=km}
600      &= \frac{1}{2} g \   \left. \delta_{j+1/2} \left[  e_{3w} \ \rho \right] \right|_{k=km}   \\
601    \end{aligned}
602  \right.
603\end{equation} 
604
605for $1<k<km$ (interior layer)
606\begin{equation}
607  \label{eq:dynhpg_zco}
608  \left\{
609    \begin{aligned}
610      \left. \delta_{i+1/2} \left[  p^h          \right] \right|_{k}
611      &=             \left. \delta_{i+1/2} \left[  p^h          \right] \right|_{k-1}
612      +    \frac{1}{2}\;g\;   \left. \delta_{i+1/2} \left[  e_{3w} \ \overline {\rho}^{k+1/2} \right] \right|_{k}   \\
613      \left. \delta_{j+1/2} \left[  p^h          \right] \right|_{k}
614      &=                \left. \delta_{j+1/2} \left[  p^h          \right] \right|_{k-1}
615      +    \frac{1}{2}\;g\;   \left. \delta_{j+1/2} \left[  e_{3w} \ \overline {\rho}^{k+1/2} \right] \right|_{k}   \\
616    \end{aligned}
617  \right.
618\end{equation} 
619
620Note that the $1/2$ factor in (\autoref{eq:dynhpg_zco_surf}) is adequate because of the definition of $e_{3w}$ as
621the vertical derivative of the scale factor at the surface level ($z=0$).
622Note also that in case of variable volume level (\key{vvl} defined),
623the surface pressure gradient is included in \autoref{eq:dynhpg_zco_surf} and
624\autoref{eq:dynhpg_zco} through the space and time variations of the vertical scale factor $e_{3w}$.
625
626%--------------------------------------------------------------------------------------------------------------
627%           z-coordinate with partial step
628%--------------------------------------------------------------------------------------------------------------
629\subsection{Partial step $Z$-coordinate (\protect\np{ln\_dynhpg\_zps}\forcode{ = .true.})}
630\label{subsec:DYN_hpg_zps}
631
632With partial bottom cells, tracers in horizontally adjacent cells generally live at different depths.
633Before taking horizontal gradients between these tracer points,
634a linear interpolation is used to approximate the deeper tracer as if
635it actually lived at the depth of the shallower tracer point.
636
637Apart from this modification,
638the horizontal hydrostatic pressure gradient evaluated in the $z$-coordinate with partial step is exactly as in
639the pure $z$-coordinate case.
640As explained in detail in section \autoref{sec:TRA_zpshde},
641the nonlinearity of pressure effects in the equation of state is such that
642it is better to interpolate temperature and salinity vertically before computing the density.
643Horizontal gradients of temperature and salinity are needed for the TRA modules,
644which is the reason why the horizontal gradients of density at the deepest model level are computed in
645module \mdl{zpsdhe} located in the TRA directory and described in \autoref{sec:TRA_zpshde}.
646
647%--------------------------------------------------------------------------------------------------------------
648%           s- and s-z-coordinates
649%--------------------------------------------------------------------------------------------------------------
650\subsection{$S$- and $Z$-$S$-coordinates}
651\label{subsec:DYN_hpg_sco}
652
653Pressure gradient formulations in an $s$-coordinate have been the subject of a vast number of papers
654(\eg, \citet{Song1998, Shchepetkin_McWilliams_OM05}).
655A number of different pressure gradient options are coded but the ROMS-like,
656density Jacobian with cubic polynomial method is currently disabled whilst known bugs are under investigation.
657
658$\bullet$ Traditional coding (see for example \citet{Madec_al_JPO96}: (\np{ln\_dynhpg\_sco}\forcode{ = .true.})
659\begin{equation}
660  \label{eq:dynhpg_sco}
661  \left\{
662    \begin{aligned}
663      - \frac{1}                 {\rho_o \, e_{1u}} \;   \delta_{i+1/2} \left[  p^h  \right]
664      + \frac{g\; \overline {\rho}^{i+1/2}}  {\rho_o \, e_{1u}} \;   \delta_{i+1/2} \left[  z_t   \right]    \\
665      - \frac{1}                 {\rho_o \, e_{2v}} \;   \delta_{j+1/2} \left[  p^h  \right]
666      + \frac{g\; \overline {\rho}^{j+1/2}}  {\rho_o \, e_{2v}} \;   \delta_{j+1/2} \left[  z_t   \right]    \\
667    \end{aligned}
668  \right.
669\end{equation} 
670
671Where the first term is the pressure gradient along coordinates,
672computed as in \autoref{eq:dynhpg_zco_surf} - \autoref{eq:dynhpg_zco},
673and $z_T$ is the depth of the $T$-point evaluated from the sum of the vertical scale factors at the $w$-point
674($e_{3w}$).
675 
676$\bullet$ Traditional coding with adaptation for ice shelf cavities (\np{ln\_dynhpg\_isf}\forcode{ = .true.}).
677This scheme need the activation of ice shelf cavities (\np{ln\_isfcav}\forcode{ = .true.}).
678
679$\bullet$ Pressure Jacobian scheme (prj) (a research paper in preparation) (\np{ln\_dynhpg\_prj}\forcode{ = .true.})
680
681$\bullet$ Density Jacobian with cubic polynomial scheme (DJC) \citep{Shchepetkin_McWilliams_OM05} 
682(\np{ln\_dynhpg\_djc}\forcode{ = .true.}) (currently disabled; under development)
683
684Note that expression \autoref{eq:dynhpg_sco} is commonly used when the variable volume formulation is activated
685(\key{vvl}) because in that case, even with a flat bottom,
686the coordinate surfaces are not horizontal but follow the free surface \citep{Levier2007}.
687The pressure jacobian scheme (\np{ln\_dynhpg\_prj}\forcode{ = .true.}) is available as
688an improved option to \np{ln\_dynhpg\_sco}\forcode{ = .true.} when \key{vvl} is active.
689The pressure Jacobian scheme uses a constrained cubic spline to
690reconstruct the density profile across the water column.
691This method maintains the monotonicity between the density nodes.
692The pressure can be calculated by analytical integration of the density profile and
693a pressure Jacobian method is used to solve the horizontal pressure gradient.
694This method can provide a more accurate calculation of the horizontal pressure gradient than the standard scheme.
695
696\subsection{Ice shelf cavity}
697\label{subsec:DYN_hpg_isf}
698Beneath an ice shelf, the total pressure gradient is the sum of the pressure gradient due to the ice shelf load and
699the pressure gradient due to the ocean load (\np{ln\_dynhpg\_isf}\forcode{ = .true.}).\\
700
701The main hypothesis to compute the ice shelf load is that the ice shelf is in an isostatic equilibrium.
702The top pressure is computed integrating from surface to the base of the ice shelf a reference density profile
703(prescribed as density of a water at 34.4 PSU and -1.9\deg{C}) and
704corresponds to the water replaced by the ice shelf.
705This top pressure is constant over time.
706A detailed description of this method is described in \citet{Losch2008}.\\
707
708The pressure gradient due to ocean load is computed using the expression \autoref{eq:dynhpg_sco} described in
709\autoref{subsec:DYN_hpg_sco}.
710
711%--------------------------------------------------------------------------------------------------------------
712%           Time-scheme
713%--------------------------------------------------------------------------------------------------------------
714\subsection{Time-scheme (\protect\np{ln\_dynhpg\_imp}\forcode{ = .true./.false.})}
715\label{subsec:DYN_hpg_imp}
716
717The default time differencing scheme used for the horizontal pressure gradient is a leapfrog scheme and
718therefore the density used in all discrete expressions given above is the  \textit{now} density,
719computed from the \textit{now} temperature and salinity.
720In some specific cases
721(usually high resolution simulations over an ocean domain which includes weakly stratified regions)
722the physical phenomenon that controls the time-step is internal gravity waves (IGWs).
723A semi-implicit scheme for doubling the stability limit associated with IGWs can be used
724\citep{Brown_Campana_MWR78, Maltrud1998}.
725It involves the evaluation of the hydrostatic pressure gradient as
726an average over the three time levels $t-\rdt$, $t$, and $t+\rdt$
727(\ie \textit{before}, \textit{now} and  \textit{after} time-steps),
728rather than at the central time level $t$ only, as in the standard leapfrog scheme.
729
730$\bullet$ leapfrog scheme (\np{ln\_dynhpg\_imp}\forcode{ = .true.}):
731
732\begin{equation}
733  \label{eq:dynhpg_lf}
734  \frac{u^{t+\rdt}-u^{t-\rdt}}{2\rdt} = \;\cdots \;
735  -\frac{1}{\rho_o \,e_{1u} }\delta_{i+1/2} \left[ {p_h^t } \right]
736\end{equation}
737
738$\bullet$ semi-implicit scheme (\np{ln\_dynhpg\_imp}\forcode{ = .true.}):
739\begin{equation}
740  \label{eq:dynhpg_imp}
741  \frac{u^{t+\rdt}-u^{t-\rdt}}{2\rdt} = \;\cdots \;
742  -\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]
743\end{equation}
744
745The semi-implicit time scheme \autoref{eq:dynhpg_imp} is made possible without
746significant additional computation since the density can be updated to time level $t+\rdt$ before
747computing the horizontal hydrostatic pressure gradient.
748It can be easily shown that the stability limit associated with the hydrostatic pressure gradient doubles using
749\autoref{eq:dynhpg_imp} compared to that using the standard leapfrog scheme \autoref{eq:dynhpg_lf}.
750Note that \autoref{eq:dynhpg_imp} is equivalent to applying a time filter to the pressure gradient to
751eliminate high frequency IGWs.
752Obviously, when using \autoref{eq:dynhpg_imp},
753the doubling of the time-step is achievable only if no other factors control the time-step,
754such as the stability limits associated with advection or diffusion.
755
756In practice, the semi-implicit scheme is used when \np{ln\_dynhpg\_imp}\forcode{ = .true.}.
757In this case, we choose to apply the time filter to temperature and salinity used in the equation of state,
758instead of applying it to the hydrostatic pressure or to the density,
759so that no additional storage array has to be defined.
760The density used to compute the hydrostatic pressure gradient (whatever the formulation) is evaluated as follows:
761\[
762  % \label{eq:rho_flt}
763  \rho^t = \rho( \widetilde{T},\widetilde {S},z_t)
764  \quad    \text{with}  \quad
765  \widetilde{X} = 1 / 4 \left(  X^{t+\rdt} +2 \,X^t + X^{t-\rdt\right)
766\]
767
768Note that in the semi-implicit case, it is necessary to save the filtered density,
769an extra three-dimensional field, in the restart file to restart the model with exact reproducibility.
770This option is controlled by  \np{nn\_dynhpg\_rst}, a namelist parameter.
771
772% ================================================================
773% Surface Pressure Gradient
774% ================================================================
775\section{Surface pressure gradient (\protect\mdl{dynspg})}
776\label{sec:DYN_spg}
777%-----------------------------------------nam_dynspg----------------------------------------------------
778
779\nlst{namdyn_spg} 
780%------------------------------------------------------------------------------------------------------------
781
782Options are defined through the \ngn{namdyn\_spg} namelist variables.
783The surface pressure gradient term is related to the representation of the free surface (\autoref{sec:PE_hor_pg}).
784The main distinction is between the fixed volume case (linear free surface) and
785the variable volume case (nonlinear free surface, \key{vvl} is defined).
786In the linear free surface case (\autoref{subsec:PE_free_surface})
787the vertical scale factors $e_{3}$ are fixed in time,
788while they are time-dependent in the nonlinear case (\autoref{subsec:PE_free_surface}).
789With both linear and nonlinear free surface, external gravity waves are allowed in the equations,
790which imposes a very small time step when an explicit time stepping is used.
791Two methods are proposed to allow a longer time step for the three-dimensional equations:
792the filtered free surface, which is a modification of the continuous equations (see \autoref{eq:PE_flt}),
793and the split-explicit free surface described below.
794The extra term introduced in the filtered method is calculated implicitly,
795so that the update of the next velocities is done in module \mdl{dynspg\_flt} and not in \mdl{dynnxt}.
796
797
798The form of the surface pressure gradient term depends on how the user wants to
799handle the fast external gravity waves that are a solution of the analytical equation (\autoref{sec:PE_hor_pg}).
800Three formulations are available, all controlled by a CPP key (ln\_dynspg\_xxx):
801an explicit formulation which requires a small time step;
802a filtered free surface formulation which allows a larger time step by
803adding a filtering term into the momentum equation;
804and a split-explicit free surface formulation, described below, which also allows a larger time step.
805
806The extra term introduced in the filtered method is calculated implicitly, so that a solver is used to compute it.
807As a consequence the update of the $next$ velocities is done in module \mdl{dynspg\_flt} and not in \mdl{dynnxt}.
808
809
810%--------------------------------------------------------------------------------------------------------------
811% Explicit free surface formulation
812%--------------------------------------------------------------------------------------------------------------
813\subsection{Explicit free surface (\protect\key{dynspg\_exp})}
814\label{subsec:DYN_spg_exp}
815
816In the explicit free surface formulation (\key{dynspg\_exp} defined),
817the model time step is chosen to be small enough to resolve the external gravity waves
818(typically a few tens of seconds).
819The surface pressure gradient, evaluated using a leap-frog scheme (\ie centered in time),
820is thus simply given by :
821\begin{equation}
822  \label{eq:dynspg_exp}
823  \left\{
824    \begin{aligned}
825      - \frac{1}{e_{1u}\,\rho_o} \; \delta_{i+1/2} \left[  \,\rho \,\eta\,  \right]    \\
826      - \frac{1}{e_{2v}\,\rho_o} \; \delta_{j+1/2} \left[  \,\rho \,\eta\,  \right]
827    \end{aligned}
828  \right.
829\end{equation} 
830
831Note that in the non-linear free surface case (\ie \key{vvl} defined),
832the surface pressure gradient is already included in the momentum tendency through
833the level thickness variation allowed in the computation of the hydrostatic pressure gradient.
834Thus, nothing is done in the \mdl{dynspg\_exp} module.
835
836%--------------------------------------------------------------------------------------------------------------
837% Split-explict free surface formulation
838%--------------------------------------------------------------------------------------------------------------
839\subsection{Split-explicit free surface (\protect\key{dynspg\_ts})}
840\label{subsec:DYN_spg_ts}
841%------------------------------------------namsplit-----------------------------------------------------------
842%
843%\nlst{namsplit}
844%-------------------------------------------------------------------------------------------------------------
845
846The split-explicit free surface formulation used in \NEMO (\key{dynspg\_ts} defined),
847also called the time-splitting formulation, follows the one proposed by \citet{Shchepetkin_McWilliams_OM05}.
848The general idea is to solve the free surface equation and the associated barotropic velocity equations with
849a smaller time step than $\rdt$, the time step used for the three dimensional prognostic variables
850(\autoref{fig:DYN_dynspg_ts}).
851The size of the small time step, $\rdt_e$ (the external mode or barotropic time step) is provided through
852the \np{nn\_baro} namelist parameter as: $\rdt_e = \rdt / nn\_baro$.
853This parameter can be optionally defined automatically (\np{ln\_bt\_nn\_auto}\forcode{ = .true.}) considering that
854the stability of the barotropic system is essentially controled by external waves propagation.
855Maximum Courant number is in that case time independent, and easily computed online from the input bathymetry.
856Therefore, $\rdt_e$ is adjusted so that the Maximum allowed Courant number is smaller than \np{rn\_bt\_cmax}.
857
858%%%
859The barotropic mode solves the following equations:
860% \begin{subequations}
861%  \label{eq:BT}
862\begin{equation}
863  \label{eq:BT_dyn}
864  \frac{\partial {\rm \overline{{\bf U}}_h} }{\partial t}=
865  -f\;{\rm {\bf k}}\times {\rm \overline{{\bf U}}_h}
866  -g\nabla _h \eta -\frac{c_b^{\textbf U}}{H+\eta} \rm {\overline{{\bf U}}_h} + \rm {\overline{\bf G}}
867\end{equation}
868\[
869  % \label{eq:BT_ssh}
870  \frac{\partial \eta }{\partial t}=-\nabla \cdot \left[ {\left( {H+\eta } \right) \; {\rm{\bf \overline{U}}}_h \,} \right]+P-E
871\]
872% \end{subequations}
873where $\rm {\overline{\bf G}}$ is a forcing term held constant, containing coupling term between modes,
874surface atmospheric forcing as well as slowly varying barotropic terms not explicitly computed to gain efficiency.
875The third term on the right hand side of \autoref{eq:BT_dyn} represents the bottom stress
876(see section \autoref{sec:ZDF_bfr}), explicitly accounted for at each barotropic iteration.
877Temporal discretization of the system above follows a three-time step Generalized Forward Backward algorithm
878detailed in \citet{Shchepetkin_McWilliams_OM05}.
879AB3-AM4 coefficients used in \NEMO follow the second-order accurate,
880"multi-purpose" stability compromise as defined in \citet{Shchepetkin_McWilliams_Bk08}
881(see their figure 12, lower left).
882
883%>   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >
884\begin{figure}[!t]
885  \begin{center}
886    \includegraphics[width=0.7\textwidth]{Fig_DYN_dynspg_ts}
887    \caption{
888      \protect\label{fig:DYN_dynspg_ts}
889      Schematic of the split-explicit time stepping scheme for the external and internal modes.
890      Time increases to the right. In this particular exemple,
891      a boxcar averaging window over $nn\_baro$ barotropic time steps is used ($nn\_bt\_flt=1$) and $nn\_baro=5$.
892      Internal mode time steps (which are also the model time steps) are denoted by $t-\rdt$, $t$ and $t+\rdt$.
893      Variables with $k$ superscript refer to instantaneous barotropic variables,
894      $< >$ and $<< >>$ operator refer to time filtered variables using respectively primary (red vertical bars) and
895      secondary weights (blue vertical bars).
896      The former are used to obtain time filtered quantities at $t+\rdt$ while
897      the latter are used to obtain time averaged transports to advect tracers.
898      a) Forward time integration: \protect\np{ln\_bt\_fw}\forcode{ = .true.},
899      \protect\np{ln\_bt\_av}\forcode{ = .true.}.
900      b) Centred time integration: \protect\np{ln\_bt\_fw}\forcode{ = .false.},
901      \protect\np{ln\_bt\_av}\forcode{ = .true.}.
902      c) Forward time integration with no time filtering (POM-like scheme):
903      \protect\np{ln\_bt\_fw}\forcode{ = .true.}, \protect\np{ln\_bt\_av}\forcode{ = .false.}.
904    }
905  \end{center}
906\end{figure}
907%>   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >
908
909In the default case (\np{ln\_bt\_fw}\forcode{ = .true.}),
910the external mode is integrated between \textit{now} and \textit{after} baroclinic time-steps
911(\autoref{fig:DYN_dynspg_ts}a).
912To avoid aliasing of fast barotropic motions into three dimensional equations,
913time filtering is eventually applied on barotropic quantities (\np{ln\_bt\_av}\forcode{ = .true.}).
914In that case, the integration is extended slightly beyond \textit{after} time step to
915provide time filtered quantities.
916These are used for the subsequent initialization of the barotropic mode in the following baroclinic step.
917Since external mode equations written at baroclinic time steps finally follow a forward time stepping scheme,
918asselin filtering is not applied to barotropic quantities.\\
919Alternatively, one can choose to integrate barotropic equations starting from \textit{before} time step
920(\np{ln\_bt\_fw}\forcode{ = .false.}).
921Although more computationaly expensive ( \np{nn\_baro} additional iterations are indeed necessary),
922the baroclinic to barotropic forcing term given at \textit{now} time step become centred in
923the middle of the integration window.
924It can easily be shown that this property removes part of splitting errors between modes,
925which increases the overall numerical robustness.
926%references to Patrick Marsaleix' work here. Also work done by SHOM group.
927
928%%%
929
930As far as tracer conservation is concerned,
931barotropic velocities used to advect tracers must also be updated at \textit{now} time step.
932This implies to change the traditional order of computations in \NEMO:
933most of momentum trends (including the barotropic mode calculation) updated first, tracers' after.
934This \textit{de facto} makes semi-implicit hydrostatic pressure gradient
935(see section \autoref{subsec:DYN_hpg_imp})
936and time splitting not compatible.
937Advective barotropic velocities are obtained by using a secondary set of filtering weights,
938uniquely defined from the filter coefficients used for the time averaging (\citet{Shchepetkin_McWilliams_OM05}).
939Consistency between the time averaged continuity equation and the time stepping of tracers is here the key to
940obtain exact conservation.
941
942%%%
943
944One can eventually choose to feedback instantaneous values by not using any time filter
945(\np{ln\_bt\_av}\forcode{ = .false.}).
946In that case, external mode equations are continuous in time,
947\ie they are not re-initialized when starting a new sub-stepping sequence.
948This is the method used so far in the POM model, the stability being maintained by
949refreshing at (almost) each barotropic time step advection and horizontal diffusion terms.
950Since the latter terms have not been added in \NEMO for computational efficiency,
951removing time filtering is not recommended except for debugging purposes.
952This may be used for instance to appreciate the damping effect of the standard formulation on
953external gravity waves in idealized or weakly non-linear cases.
954Although the damping is lower than for the filtered free surface,
955it is still significant as shown by \citet{Levier2007} in the case of an analytical barotropic Kelvin wave.
956
957%>>>>>===============
958\gmcomment{               %%% copy from griffies Book
959
960\textbf{title: Time stepping the barotropic system }
961
962Assume knowledge of the full velocity and tracer fields at baroclinic time $\tau$.
963Hence, we can update the surface height and vertically integrated velocity with a leap-frog scheme using
964the small barotropic time step $\rdt$.
965We have
966
967\[
968  % \label{eq:DYN_spg_ts_eta}
969  \eta^{(b)}(\tau,t_{n+1}) - \eta^{(b)}(\tau,t_{n+1}) (\tau,t_{n-1})
970  = 2 \rdt \left[-\nabla \cdot \textbf{U}^{(b)}(\tau,t_n) + \text{EMP}_w(\tau) \right]
971\]
972\begin{multline*}
973  % \label{eq:DYN_spg_ts_u}
974  \textbf{U}^{(b)}(\tau,t_{n+1}) - \textbf{U}^{(b)}(\tau,t_{n-1}\\
975  = 2\rdt \left[ - f \textbf{k} \times \textbf{U}^{(b)}(\tau,t_{n})
976    - H(\tau) \nabla p_s^{(b)}(\tau,t_{n}) +\textbf{M}(\tau) \right]
977\end{multline*}
978\
979
980In these equations, araised (b) denotes values of surface height and vertically integrated velocity updated with
981the barotropic time steps.
982The $\tau$ time label on $\eta^{(b)}$ and $U^{(b)}$ denotes the baroclinic time at which
983the vertically integrated forcing $\textbf{M}(\tau)$
984(note that this forcing includes the surface freshwater forcing),
985the tracer fields, the freshwater flux $\text{EMP}_w(\tau)$,
986and total depth of the ocean $H(\tau)$ are held for the duration of the barotropic time stepping over
987a single cycle.
988This is also the time that sets the barotropic time steps via
989\[
990  % \label{eq:DYN_spg_ts_t}
991  t_n=\tau+n\rdt
992\]
993with $n$ an integer.
994The density scaled surface pressure is evaluated via
995\[
996  % \label{eq:DYN_spg_ts_ps}
997  p_s^{(b)}(\tau,t_{n}) =
998  \begin{cases}
999    g \;\eta_s^{(b)}(\tau,t_{n}) \;\rho(\tau)_{k=1}) / \rho_o  &      \text{non-linear case} \\
1000    g \;\eta_s^{(b)}(\tau,t_{n})  &      \text{linear case}
1001  \end{cases}
1002\]
1003To get started, we assume the following initial conditions
1004\[
1005  % \label{eq:DYN_spg_ts_eta}
1006  \begin{split}
1007    \eta^{(b)}(\tau,t_{n=0}) &= \overline{\eta^{(b)}(\tau)}    \\
1008    \eta^{(b)}(\tau,t_{n=1}) &= \eta^{(b)}(\tau,t_{n=0}) + \rdt \ \text{RHS}_{n=0}
1009  \end{split}
1010\]
1011with
1012\[
1013  % \label{eq:DYN_spg_ts_etaF}
1014  \overline{\eta^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N \eta^{(b)}(\tau-\rdt,t_{n})
1015\]
1016the time averaged surface height taken from the previous barotropic cycle.
1017Likewise,
1018\[
1019  % \label{eq:DYN_spg_ts_u}
1020  \textbf{U}^{(b)}(\tau,t_{n=0}) = \overline{\textbf{U}^{(b)}(\tau)} \\ \\
1021  \textbf{U}(\tau,t_{n=1}) = \textbf{U}^{(b)}(\tau,t_{n=0}) + \rdt \ \text{RHS}_{n=0}
1022\]
1023with
1024\[
1025  % \label{eq:DYN_spg_ts_u}
1026  \overline{\textbf{U}^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau-\rdt,t_{n})
1027\]
1028the time averaged vertically integrated transport.
1029Notably, there is no Robert-Asselin time filter used in the barotropic portion of the integration.
1030
1031Upon reaching $t_{n=N} = \tau + 2\rdt \tau$ ,
1032the vertically integrated velocity is time averaged to produce the updated vertically integrated velocity at
1033baroclinic time $\tau + \rdt \tau$ 
1034\[
1035  % \label{eq:DYN_spg_ts_u}
1036  \textbf{U}(\tau+\rdt) = \overline{\textbf{U}^{(b)}(\tau+\rdt)} = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau,t_{n})
1037\]
1038The surface height on the new baroclinic time step is then determined via a baroclinic leap-frog using
1039the following form
1040
1041\begin{equation}
1042  \label{eq:DYN_spg_ts_ssh}
1043  \eta(\tau+\Delta) - \eta^{F}(\tau-\Delta) = 2\rdt \ \left[ - \nabla \cdot \textbf{U}(\tau) + \text{EMP}_w \right] 
1044\end{equation}
1045
1046The use of this "big-leap-frog" scheme for the surface height ensures compatibility between
1047the mass/volume budgets and the tracer budgets.
1048More discussion of this point is provided in Chapter 10 (see in particular Section 10.2).
1049 
1050In general, some form of time filter is needed to maintain integrity of the surface height field due to
1051the leap-frog splitting mode in equation \autoref{eq:DYN_spg_ts_ssh}.
1052We have tried various forms of such filtering,
1053with the following method discussed in \cite{Griffies_al_MWR01} chosen due to
1054its stability and reasonably good maintenance of tracer conservation properties (see ??).
1055
1056\begin{equation}
1057  \label{eq:DYN_spg_ts_sshf}
1058  \eta^{F}(\tau-\Delta) =  \overline{\eta^{(b)}(\tau)}
1059\end{equation}
1060Another approach tried was
1061
1062\[
1063  % \label{eq:DYN_spg_ts_sshf2}
1064  \eta^{F}(\tau-\Delta) = \eta(\tau)
1065  + (\alpha/2) \left[\overline{\eta^{(b)}}(\tau+\rdt)
1066    + \overline{\eta^{(b)}}(\tau-\rdt) -2 \;\eta(\tau) \right]
1067\]
1068
1069which is useful since it isolates all the time filtering aspects into the term multiplied by $\alpha$.
1070This isolation allows for an easy check that tracer conservation is exact when
1071eliminating tracer and surface height time filtering (see ?? for more complete discussion).
1072However, in the general case with a non-zero $\alpha$,
1073the filter \autoref{eq:DYN_spg_ts_sshf} was found to be more conservative, and so is recommended.
1074
1075}            %%end gm comment (copy of griffies book)
1076
1077%>>>>>===============
1078
1079
1080%--------------------------------------------------------------------------------------------------------------
1081% Filtered free surface formulation
1082%--------------------------------------------------------------------------------------------------------------
1083\subsection{Filtered free surface (\protect\key{dynspg\_flt})}
1084\label{subsec:DYN_spg_fltp}
1085
1086The filtered formulation follows the \citet{Roullet_Madec_JGR00} implementation.
1087The extra term introduced in the equations (see \autoref{subsec:PE_free_surface}) is solved implicitly.
1088The elliptic solvers available in the code are documented in \autoref{chap:MISC}.
1089
1090%% gm %%======>>>>   given here the discrete eqs provided to the solver
1091\gmcomment{               %%% copy from chap-model basics
1092  \[
1093    % \label{eq:spg_flt}
1094    \frac{\partial {\rm {\bf U}}_h }{\partial t}= {\rm {\bf M}}
1095    - g \nabla \left( \tilde{\rho} \ \eta \right)
1096    - g \ T_c \nabla \left( \widetilde{\rho} \ \partial_t \eta \right)
1097  \]
1098  where $T_c$, is a parameter with dimensions of time which characterizes the force,
1099  $\widetilde{\rho} = \rho / \rho_o$ is the dimensionless density,
1100  and $\rm {\bf M}$ represents the collected contributions of the Coriolis, hydrostatic pressure gradient,
1101  non-linear and viscous terms in \autoref{eq:PE_dyn}.
1102}   %end gmcomment
1103
1104Note that in the linear free surface formulation (\key{vvl} not defined),
1105the ocean depth is time-independent and so is the matrix to be inverted.
1106It is computed once and for all and applies to all ocean time steps.
1107
1108% ================================================================
1109% Lateral diffusion term
1110% ================================================================
1111\section{Lateral diffusion term and operators (\protect\mdl{dynldf})}
1112\label{sec:DYN_ldf}
1113%------------------------------------------nam_dynldf----------------------------------------------------
1114
1115\nlst{namdyn_ldf} 
1116%-------------------------------------------------------------------------------------------------------------
1117
1118Options are defined through the \ngn{namdyn\_ldf} namelist variables.
1119The options available for lateral diffusion are to use either laplacian (rotated or not) or biharmonic operators.
1120The coefficients may be constant or spatially variable;
1121the description of the coefficients is found in the chapter on lateral physics (\autoref{chap:LDF}).
1122The lateral diffusion of momentum is evaluated using a forward scheme,
1123\ie the velocity appearing in its expression is the \textit{before} velocity in time,
1124except for the pure vertical component that appears when a tensor of rotation is used.
1125This latter term is solved implicitly together with the vertical diffusion term (see \autoref{chap:STP}).
1126
1127At the lateral boundaries either free slip,
1128no slip or partial slip boundary conditions are applied according to the user's choice (see \autoref{chap:LBC}).
1129
1130\gmcomment{
1131  Hyperviscous operators are frequently used in the simulation of turbulent flows to
1132  control the dissipation of unresolved small scale features.
1133  Their primary role is to provide strong dissipation at the smallest scale supported by
1134  the grid while minimizing the impact on the larger scale features.
1135  Hyperviscous operators are thus designed to be more scale selective than the traditional,
1136  physically motivated Laplace operator.
1137  In finite difference methods,
1138  the biharmonic operator is frequently the method of choice to achieve this scale selective dissipation since
1139  its damping time (\ie its spin down time) scale like $\lambda^{-4}$ for disturbances of wavelength $\lambda$
1140  (so that short waves damped more rapidelly than long ones),
1141  whereas the Laplace operator damping time scales only like $\lambda^{-2}$.
1142}
1143
1144% ================================================================
1145\subsection[Iso-level laplacian (\protect\np{ln\_dynldf\_lap}\forcode{ = .true.})]
1146            {Iso-level laplacian operator (\protect\np{ln\_dynldf\_lap}\forcode{ = .true.})}
1147\label{subsec:DYN_ldf_lap}
1148
1149For lateral iso-level diffusion, the discrete operator is:
1150\begin{equation}
1151  \label{eq:dynldf_lap}
1152  \left\{
1153    \begin{aligned}
1154      D_u^{l{\rm {\bf U}}} =\frac{1}{e_{1u} }\delta_{i+1/2} \left[ {A_T^{lm}
1155          \;\chi } \right]-\frac{1}{e_{2u} {\kern 1pt}e_{3u} }\delta_j \left[
1156        {A_f^{lm} \;e_{3f} \zeta } \right] \\ \\
1157      D_v^{l{\rm {\bf U}}} =\frac{1}{e_{2v} }\delta_{j+1/2} \left[ {A_T^{lm}
1158          \;\chi } \right]+\frac{1}{e_{1v} {\kern 1pt}e_{3v} }\delta_i \left[
1159        {A_f^{lm} \;e_{3f} \zeta } \right]
1160    \end{aligned}
1161  \right.
1162\end{equation} 
1163
1164As explained in \autoref{subsec:PE_ldf},
1165this formulation (as the gradient of a divergence and curl of the vorticity) preserves symmetry and
1166ensures a complete separation between the vorticity and divergence parts of the momentum diffusion.
1167
1168%--------------------------------------------------------------------------------------------------------------
1169%           Rotated laplacian operator
1170%--------------------------------------------------------------------------------------------------------------
1171\subsection[Rotated laplacian (\protect\np{ln\_dynldf\_iso}\forcode{ = .true.})]
1172            {Rotated laplacian operator (\protect\np{ln\_dynldf\_iso}\forcode{ = .true.})}
1173\label{subsec:DYN_ldf_iso}
1174
1175A rotation of the lateral momentum diffusion operator is needed in several cases:
1176for iso-neutral diffusion in the $z$-coordinate (\np{ln\_dynldf\_iso}\forcode{ = .true.}) and
1177for either iso-neutral (\np{ln\_dynldf\_iso}\forcode{ = .true.}) or
1178geopotential (\np{ln\_dynldf\_hor}\forcode{ = .true.}) diffusion in the $s$-coordinate.
1179In the partial step case, coordinates are horizontal except at the deepest level and
1180no rotation is performed when \np{ln\_dynldf\_hor}\forcode{ = .true.}.
1181The diffusion operator is defined simply as the divergence of down gradient momentum fluxes on
1182each momentum component.
1183It must be emphasized that this formulation ignores constraints on the stress tensor such as symmetry.
1184The resulting discrete representation is:
1185\begin{equation}
1186  \label{eq:dyn_ldf_iso}
1187  \begin{split}
1188    D_u^{l\textbf{U}} &= \frac{1}{e_{1u} \, e_{2u} \, e_{3u} } \\
1189    &  \left\{\quad  {\delta_{i+1/2} \left[ {A_T^{lm}  \left(
1190              {\frac{e_{2t} \; e_{3t} }{e_{1t} } \,\delta_{i}[u]
1191                -e_{2t} \; r_{1t} \,\overline{\overline {\delta_{k+1/2}[u]}}^{\,i,\,k}}
1192            \right)} \right]}    \right. \\
1193    & \qquad +\ \delta_j \left[ {A_f^{lm} \left( {\frac{e_{1f}\,e_{3f} }{e_{2f}
1194            }\,\delta_{j+1/2} [u] - e_{1f}\, r_{2f}
1195            \,\overline{\overline {\delta_{k+1/2} [u]}} ^{\,j+1/2,\,k}}
1196        \right)} \right] \\
1197    &\qquad +\ \delta_k \left[ {A_{uw}^{lm} \left( {-e_{2u} \, r_{1uw} \,\overline{\overline
1198              {\delta_{i+1/2} [u]}}^{\,i+1/2,\,k+1/2} }
1199        \right.} \right. \\
1200    &  \ \qquad \qquad \qquad \quad\
1201    - e_{1u} \, r_{2uw} \,\overline{\overline {\delta_{j+1/2} [u]}} ^{\,j,\,k+1/2} \\
1202    & \left. {\left. { \ \qquad \qquad \qquad \ \ \ \left. {\
1203                +\frac{e_{1u}\, e_{2u} }{e_{3uw} }\,\left( {r_{1uw}^2+r_{2uw}^2}
1204                \right)\,\delta_{k+1/2} [u]} \right)} \right]\;\;\;} \right\} \\ \\
1205    D_v^{l\textbf{V}} &= \frac{1}{e_{1v} \, e_{2v} \, e_{3v} } \\
1206    &  \left\{\quad  {\delta_{i+1/2} \left[ {A_f^{lm}  \left(
1207              {\frac{e_{2f} \; e_{3f} }{e_{1f} } \,\delta_{i+1/2}[v]
1208                -e_{2f} \; r_{1f} \,\overline{\overline {\delta_{k+1/2}[v]}}^{\,i+1/2,\,k}}
1209            \right)} \right]}    \right. \\
1210    & \qquad +\ \delta_j \left[ {A_T^{lm} \left( {\frac{e_{1t}\,e_{3t} }{e_{2t} 
1211            }\,\delta_{j} [v] - e_{1t}\, r_{2t}
1212            \,\overline{\overline {\delta_{k+1/2} [v]}} ^{\,j,\,k}}
1213        \right)} \right] \\
1214    & \qquad +\ \delta_k \left[ {A_{vw}^{lm} \left( {-e_{2v} \, r_{1vw} \,\overline{\overline 
1215              {\delta_{i+1/2} [v]}}^{\,i+1/2,\,k+1/2} }\right.} \right. \\
1216    &  \ \qquad \qquad \qquad \quad\
1217    - e_{1v} \, r_{2vw} \,\overline{\overline {\delta_{j+1/2} [v]}} ^{\,j+1/2,\,k+1/2} \\
1218    & \left. {\left. { \ \qquad \qquad \qquad \ \ \ \left. {\
1219                +\frac{e_{1v}\, e_{2v} }{e_{3vw} }\,\left( {r_{1vw}^2+r_{2vw}^2}
1220                \right)\,\delta_{k+1/2} [v]} \right)} \right]\;\;\;} \right\} 
1221  \end{split}
1222\end{equation}
1223where $r_1$ and $r_2$ are the slopes between the surface along which the diffusion operator acts and
1224the surface of computation ($z$- or $s$-surfaces).
1225The way these slopes are evaluated is given in the lateral physics chapter (\autoref{chap:LDF}).
1226
1227%--------------------------------------------------------------------------------------------------------------
1228%           Iso-level bilaplacian operator
1229%--------------------------------------------------------------------------------------------------------------
1230\subsection[Iso-level bilaplacian (\protect\np{ln\_dynldf\_bilap}\forcode{ = .true.})]
1231            {Iso-level bilaplacian operator (\protect\np{ln\_dynldf\_bilap}\forcode{ = .true.})}
1232\label{subsec:DYN_ldf_bilap}
1233
1234The lateral fourth order operator formulation on momentum is obtained by applying \autoref{eq:dynldf_lap} twice.
1235It requires an additional assumption on boundary conditions:
1236the first derivative term normal to the coast depends on the free or no-slip lateral boundary conditions chosen,
1237while the third derivative terms normal to the coast are set to zero (see \autoref{chap:LBC}).
1238%%%
1239\gmcomment{add a remark on the the change in the position of the coefficient}
1240%%%
1241
1242% ================================================================
1243%           Vertical diffusion term
1244% ================================================================
1245\section{Vertical diffusion term (\protect\mdl{dynzdf})}
1246\label{sec:DYN_zdf}
1247%----------------------------------------------namzdf------------------------------------------------------
1248
1249\nlst{namzdf} 
1250%-------------------------------------------------------------------------------------------------------------
1251
1252Options are defined through the \ngn{namzdf} namelist variables.
1253The 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.
1254Two time stepping schemes can be used for the vertical diffusion term:
1255$(a)$ a forward time differencing scheme
1256(\np{ln\_zdfexp}\forcode{ = .true.}) using a time splitting technique (\np{nn\_zdfexp} $>$ 1) or
1257$(b)$ a backward (or implicit) time differencing scheme (\np{ln\_zdfexp}\forcode{ = .false.})
1258(see \autoref{chap:STP}).
1259Note that namelist variables \np{ln\_zdfexp} and \np{nn\_zdfexp} apply to both tracers and dynamics.
1260
1261The formulation of the vertical subgrid scale physics is the same whatever the vertical coordinate is.
1262The vertical diffusion operators given by \autoref{eq:PE_zdf} take the following semi-discrete space form:
1263\[
1264  % \label{eq:dynzdf}
1265  \left\{
1266    \begin{aligned}
1267      D_u^{vm} &\equiv \frac{1}{e_{3u}} \ \delta_k \left[ \frac{A_{uw}^{vm} }{e_{3uw} }
1268        \ \delta_{k+1/2} [\,u\,]         \right]     \\
1269      \\
1270      D_v^{vm} &\equiv \frac{1}{e_{3v}} \ \delta_k \left[ \frac{A_{vw}^{vm} }{e_{3vw} }
1271        \ \delta_{k+1/2} [\,v\,]         \right]
1272    \end{aligned}
1273  \right.
1274\]
1275where $A_{uw}^{vm} $ and $A_{vw}^{vm} $ are the vertical eddy viscosity and diffusivity coefficients.
1276The way these coefficients are evaluated depends on the vertical physics used (see \autoref{chap:ZDF}).
1277
1278The surface boundary condition on momentum is the stress exerted by the wind.
1279At the surface, the momentum fluxes are prescribed as the boundary condition on
1280the vertical turbulent momentum fluxes,
1281\begin{equation}
1282  \label{eq:dynzdf_sbc}
1283  \left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{z=1}
1284  = \frac{1}{\rho_o} \binom{\tau_u}{\tau_v }
1285\end{equation}
1286where $\left( \tau_u ,\tau_v \right)$ are the two components of the wind stress vector in
1287the (\textbf{i},\textbf{j}) coordinate system.
1288The high mixing coefficients in the surface mixed layer ensure that the surface wind stress is distributed in
1289the vertical over the mixed layer depth.
1290If the vertical mixing coefficient is small (when no mixed layer scheme is used)
1291the surface stress enters only the top model level, as a body force.
1292The surface wind stress is calculated in the surface module routines (SBC, see \autoref{chap:SBC}).
1293
1294The turbulent flux of momentum at the bottom of the ocean is specified through a bottom friction parameterisation
1295(see \autoref{sec:ZDF_bfr})
1296
1297% ================================================================
1298% External Forcing
1299% ================================================================
1300\section{External forcings}
1301\label{sec:DYN_forcing}
1302
1303Besides the surface and bottom stresses (see the above section)
1304which are introduced as boundary conditions on the vertical mixing,
1305three other forcings may enter the dynamical equations by affecting the surface pressure gradient.
1306
1307(1) When \np{ln\_apr\_dyn}\forcode{ = .true.} (see \autoref{sec:SBC_apr}),
1308the atmospheric pressure is taken into account when computing the surface pressure gradient.
1309
1310(2) When \np{ln\_tide\_pot}\forcode{ = .true.} and \np{ln\_tide}\forcode{ = .true.} (see \autoref{sec:SBC_tide}),
1311the tidal potential is taken into account when computing the surface pressure gradient.
1312
1313(3) When \np{nn\_ice\_embd}\forcode{ = 2} and LIM or CICE is used
1314(\ie when the sea-ice is embedded in the ocean),
1315the snow-ice mass is taken into account when computing the surface pressure gradient.
1316
1317
1318\gmcomment{ missing : the lateral boundary condition !!!   another external forcing
1319 }
1320
1321% ================================================================
1322% Wetting and drying
1323% ================================================================
1324\section{Wetting and drying }
1325\label{sec:DYN_wetdry}
1326There are two main options for wetting and drying code (wd):
1327(a) an iterative limiter (il) and (b) a directional limiter (dl).
1328The directional limiter is based on the scheme developed by \cite{WarnerEtal13} for RO
1329MS
1330which was in turn based on ideas developed for POM by \cite{Oey06}. The iterative
1331limiter is a new scheme.  The iterative limiter is activated by setting $\mathrm{ln\_wd\_il} = \mathrm{.true.}$
1332and $\mathrm{ln\_wd\_dl} = \mathrm{.false.}$. The directional limiter is activated
1333by setting $\mathrm{ln\_wd\_dl} = \mathrm{.true.}$ and $\mathrm{ln\_wd\_il} = \mathrm{.false.}$.
1334
1335\nlst{namwad}
1336
1337The following terminology is used. The depth of the topography (positive downwards)
1338at each $(i,j)$ point is the quantity stored in array $\mathrm{ht\_wd}$ in the NEMO code.
1339The height of the free surface (positive upwards) is denoted by $ \mathrm{ssh}$. Given the sign
1340conventions used, the water depth, $h$, is the height of the free surface plus the depth of the
1341topography (i.e. $\mathrm{ssh} + \mathrm{ht\_wd}$).
1342
1343Both wd schemes take all points in the domain below a land elevation of $\mathrm{rn\_wdld}$ to be
1344covered by water. They require the topography specified with a model
1345configuration to have negative depths at points where the land is higher than the
1346topography's reference sea-level. The vertical grid in NEMO is normally computed relative to an
1347initial state with zero sea surface height elevation.
1348The user can choose to compute the vertical grid and heights in the model relative to
1349a non-zero reference height for the free surface. This choice affects the calculation of the metrics and depths
1350(i.e. the $\mathrm{e3t\_0, ht\_0}$ etc. arrays).
1351
1352Points where the water depth is less than $\mathrm{rn\_wdmin1}$ are interpreted as ``dry''.
1353$\mathrm{rn\_wdmin1}$ is usually chosen to be of order $0.05$m but extreme topographies
1354with very steep slopes require larger values for normal choices of time-step. Surface fluxes
1355are also switched off for dry cells to prevent freezing, boiling etc. of very thin water layers.
1356The fluxes are tappered down using a $\mathrm{tanh}$ weighting function
1357to no flux as the dry limit $\mathrm{rn\_wdmin1}$ is approached. Even wet cells can be very shallow.
1358The depth at which to start tapering is controlled by the user by setting $\mathrm{rn\_wd\_sbcdep}$.
1359The fraction $(<1)$ of sufrace fluxes to use at this depth is set by $\mathrm{rn\_wd\_sbcfra}$.
1360
1361Both versions of the code have been tested in six test cases provided in the WAD\_TEST\_CASES configuration
1362and in ``realistic'' configurations covering parts of the north-west European shelf.
1363All these configurations have used pure sigma coordinates. It is expected that
1364the wetting and drying code will work in domains with more general s-coordinates provided
1365the coordinates are pure sigma in the region where wetting and drying actually occurs.
1366
1367The next sub-section descrbies the directional limiter and the following sub-section the iterative limiter.
1368The final sub-section covers some additional considerations that are relevant to both schemes.
1369
1370
1371%-----------------------------------------------------------------------------------------
1372%   Iterative limiters
1373%-----------------------------------------------------------------------------------------
1374\subsection   [Directional limiter (\textit{wet\_dry})]
1375         {Directional limiter (\mdl{wet\_dry})}
1376\label{subsec:DYN_wd_directional_limiter}
1377The principal idea of the directional limiter is that
1378water should not be allowed to flow out of a dry tracer cell (i.e. one whose water depth is less than rn\_wdmin1).
1379
1380All the changes associated with this option are made to the barotropic solver for the non-linear
1381free surface code within dynspg\_ts.
1382On each barotropic sub-step the scheme determines the direction of the flow across each face of all the tracer cells
1383and sets the flux across the face to zero when the flux is from a dry tracer cell. This prevents cells
1384whose depth is rn\_wdmin1 or less from drying out further. The scheme does not force $h$ (the water depth) at tracer cells
1385to be at least the minimum depth and hence is able to conserve mass / volume.
1386
1387The flux across each $u$-face of a tracer cell is multiplied by a factor zuwdmask (an array which depends on ji and jj).
1388If the user sets ln\_wd\_dl\_ramp = .False. then zuwdmask is 1 when the
1389flux is from a cell with water depth greater than rn\_wdmin1 and 0 otherwise. If the user sets
1390ln\_wd\_dl\_ramp = .True. the flux across the face is ramped down as the water depth decreases
1391from 2 * rn\_wdmin1 to rn\_wdmin1. The use of this ramp reduced grid-scale noise in idealised test cases.
1392
1393At the point where the flux across a $u$-face is multiplied by zuwdmask , we have chosen
1394also to multiply the corresponding velocity on the ``now'' step at that face by zuwdmask. We could have
1395chosen not to do that and to allow fairly large velocities to occur in these ``dry'' cells.
1396The rationale for setting the velocity to zero is that it is the momentum equations that are being solved
1397and the total momentum of the upstream cell (treating it as a finite volume) should be considered
1398to be its depth times its velocity. This depth is considered to be zero at ``dry'' $u$-points consistent with its
1399treatment in the calculation of the flux of mass across the cell face.
1400
1401
1402\cite{WarnerEtal13} state that in their scheme the velocity masks at the cell faces for the baroclinic
1403timesteps are set to 0 or 1 depending on whether the average of the masks over the barotropic sub-steps is respectively less than
1404or greater than 0.5. That scheme does not conserve tracers in integrations started from constant tracer
1405fields (tracers independent of $x$, $y$ and $z$). Our scheme conserves constant tracers because
1406the velocities used at the tracer cell faces on the baroclinic timesteps are carefully calculated by dynspg\_ts
1407to equal their mean value during the barotropic steps. If the user sets ln\_wd\_dl\_bc = .True., the
1408baroclinic velocities are also multiplied by a suitably weighted average of zuwdmask. 
1409
1410%-----------------------------------------------------------------------------------------
1411%   Iterative limiters
1412%-----------------------------------------------------------------------------------------
1413
1414\subsection   [Iterative limiter (\textit{wet\_dry})]
1415         {Iterative limiter (\mdl{wet\_dry})}
1416\label{subsec:DYN_wd_iterative_limiter}
1417
1418\subsubsection [Iterative flux limiter (\textit{wet\_dry})]
1419         {Iterative flux limiter (\mdl{wet\_dry})}
1420\label{subsubsec:DYN_wd_il_spg_limiter}
1421
1422The iterative limiter modifies the fluxes across the faces of cells that are either already ``dry''
1423or may become dry within the next time-step using an iterative method.
1424
1425The flux limiter for the barotropic flow (devised by Hedong Liu) can be understood as follows:
1426
1427The continuity equation for the total water depth in a column
1428\begin{equation} \label{dyn_wd_continuity}
1429 \frac{\partial h}{\partial t} + \mathbf{\nabla.}(h\mathbf{u}) = 0 .
1430\end{equation}
1431can be written in discrete form  as
1432
1433\begin{align} \label{dyn_wd_continuity_2}
1434\frac{e_1 e_2}{\Delta t} ( h_{i,j}(t_{n+1}) - h_{i,j}(t_e) )
1435&= - ( \mathrm{flxu}_{i+1,j} - \mathrm{flxu}_{i,j}  + \mathrm{flxv}_{i,j+1} - \mathrm{flxv}_{i,j} ) \\
1436&= \mathrm{zzflx}_{i,j} .
1437\end{align}
1438
1439In the above $h$ is the depth of the water in the column at point $(i,j)$,
1440$\mathrm{flxu}_{i+1,j}$ is the flux out of the ``eastern'' face of the cell and
1441$\mathrm{flxv}_{i,j+1}$ the flux out of the ``northern'' face of the cell; $t_{n+1}$ is
1442the new timestep, $t_e$ is the old timestep (either $t_b$ or $t_n$) and $ \Delta t =
1443t_{n+1} - t_e$; $e_1 e_2$ is the area of the tracer cells centred at $(i,j)$ and
1444$\mathrm{zzflx}$ is the sum of the fluxes through all the faces.
1445
1446The flux limiter splits the flux $\mathrm{zzflx}$ into fluxes that are out of the cell
1447(zzflxp) and fluxes that are into the cell (zzflxn).  Clearly
1448
1449\begin{equation} \label{dyn_wd_zzflx_p_n_1}
1450\mathrm{zzflx}_{i,j} = \mathrm{zzflxp}_{i,j} + \mathrm{zzflxn}_{i,j} . 
1451\end{equation}
1452
1453The flux limiter iteratively adjusts the fluxes $\mathrm{flxu}$ and $\mathrm{flxv}$ until
1454none of the cells will ``dry out''. To be precise the fluxes are limited until none of the
1455cells has water depth less than $\mathrm{rn\_wdmin1}$ on step $n+1$.
1456
1457Let the fluxes on the $m$th iteration step be denoted by $\mathrm{flxu}^{(m)}$ and
1458$\mathrm{flxv}^{(m)}$.  Then the adjustment is achieved by seeking a set of coefficients,
1459$\mathrm{zcoef}_{i,j}^{(m)}$ such that:
1460
1461\begin{equation} \label{dyn_wd_continuity_coef}
1462\begin{split}
1463\mathrm{zzflxp}^{(m)}_{i,j} =& \mathrm{zcoef}_{i,j}^{(m)} \mathrm{zzflxp}^{(0)}_{i,j} \\
1464\mathrm{zzflxn}^{(m)}_{i,j} =& \mathrm{zcoef}_{i,j}^{(m)} \mathrm{zzflxn}^{(0)}_{i,j}
1465\end{split}
1466\end{equation}
1467
1468where the coefficients are $1.0$ generally but can vary between $0.0$ and $1.0$ around
1469cells that would otherwise dry.
1470
1471The iteration is initialised by setting
1472
1473\begin{equation} \label{dyn_wd_zzflx_initial}
1474\mathrm{zzflxp^{(0)}}_{i,j} = \mathrm{zzflxp}_{i,j} , \quad  \mathrm{zzflxn^{(0)}}_{i,j} = \mathrm{zzflxn}_{i,j} .
1475\end{equation}
1476
1477The fluxes out of cell $(i,j)$ are updated at the $m+1$th iteration if the depth of the
1478cell on timestep $t_e$, namely $h_{i,j}(t_e)$, is less than the total flux out of the cell
1479times the timestep divided by the cell area. Using (\ref{dyn_wd_continuity_2}) this
1480condition is
1481
1482\begin{equation} \label{dyn_wd_continuity_if}
1483h_{i,j}(t_e)  - \mathrm{rn\_wdmin1} <  \frac{\Delta t}{e_1 e_2} ( \mathrm{zzflxp}^{(m)}_{i,j} + \mathrm{zzflxn}^{(m)}_{i,j} ) .
1484\end{equation}
1485
1486Rearranging (\ref{dyn_wd_continuity_if}) we can obtain an expression for the maximum
1487outward flux that can be allowed and still maintain the minimum wet depth:
1488
1489\begin{equation} \label{dyn_wd_max_flux}
1490\begin{split}
1491\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{]} \\
1492\phantom{[} & -  \mathrm{zzflxn}^{(m)}_{i,j} \Big]
1493\end{split}
1494\end{equation}
1495
1496Note a small tolerance ($\mathrm{rn\_wdmin2}$) has been introduced here {\it [Q: Why is
1497this necessary/desirable?]}. Substituting from (\ref{dyn_wd_continuity_coef}) gives an
1498expression for the coefficient needed to multiply the outward flux at this cell in order
1499to avoid drying.
1500
1501\begin{equation} \label{dyn_wd_continuity_nxtcoef}
1502\begin{split}
1503\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{]} \\
1504\phantom{[} & -  \mathrm{zzflxn}^{(m)}_{i,j} \Big] \frac{1}{ \mathrm{zzflxp}^{(0)}_{i,j} } 
1505\end{split}
1506\end{equation}
1507
1508Only the outward flux components are altered but, of course, outward fluxes from one cell
1509are inward fluxes to adjacent cells and the balance in these cells may need subsequent
1510adjustment; hence the iterative nature of this scheme.  Note, for example, that the flux
1511across the ``eastern'' face of the $(i,j)$th cell is only updated at the $m+1$th iteration
1512if that flux at the $m$th iteration is out of the $(i,j)$th cell. If that is the case then
1513the flux across that face is into the $(i+1,j)$ cell and that flux will not be updated by
1514the calculation for the $(i+1,j)$th cell. In this sense the updates to the fluxes across
1515the faces of the cells do not ``compete'' (they do not over-write each other) and one
1516would expect the scheme to converge relatively quickly. The scheme is flux based so
1517conserves mass. It also conserves constant tracers for the same reason that the
1518directional limiter does.
1519
1520
1521%----------------------------------------------------------------------------------------
1522%      Surface pressure gradients
1523%----------------------------------------------------------------------------------------
1524\subsubsection   [Modification of surface pressure gradients (\textit{dynhpg})]
1525         {Modification of surface pressure gradients (\mdl{dynhpg})}
1526\label{subsubsec:DYN_wd_il_spg}
1527
1528At ``dry'' points the water depth is usually close to $\mathrm{rn\_wdmin1}$. If the
1529topography is sloping at these points the sea-surface will have a similar slope and there
1530will hence be very large horizontal pressure gradients at these points. The WAD modifies
1531the magnitude but not the sign of the surface pressure gradients (zhpi and zhpj) at such
1532points by mulitplying them by positive factors (zcpx and zcpy respectively) that lie
1533between $0$ and $1$.
1534
1535We describe how the scheme works for the ``eastward'' pressure gradient, zhpi, calculated
1536at the $(i,j)$th $u$-point. The scheme uses the ht\_wd depths and surface heights at the
1537neighbouring $(i+1,j)$ and $(i,j)$ tracer points.  zcpx is calculated using two logicals
1538variables, $\mathrm{ll\_tmp1}$ and $\mathrm{ll\_tmp2}$ which are evaluated for each grid
1539column.  The three possible combinations are illustrated in figure \ref{Fig_WAD_dynhpg}.
1540
1541%>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
1542\begin{figure}[!ht] \begin{center}
1543\includegraphics[width=0.8\textwidth]{Fig_WAD_dynhpg}
1544\caption{ \label{Fig_WAD_dynhpg}
1545Illustrations of the three possible combinations of the logical variables controlling the
1546limiting of the horizontal pressure gradient in wetting and drying regimes}
1547\end{center}\end{figure}
1548%>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
1549
1550The first logical, $\mathrm{ll\_tmp1}$, is set to true if and only if the water depth at
1551both neighbouring points is greater than $\mathrm{rn\_wdmin1} + \mathrm{rn\_wdmin2}$ and
1552the minimum height of the sea surface at the two points is greater than the maximum height
1553of the topography at the two points:
1554
1555\begin{equation} \label{dyn_ll_tmp1}
1556\begin{split}
1557\mathrm{ll\_tmp1}  = & \mathrm{MIN(sshn(ji,jj), sshn(ji+1,jj))} > \\
1558                     & \quad \mathrm{MAX(-ht\_wd(ji,jj), -ht\_wd(ji+1,jj))\  .and.} \\
1559& \mathrm{MAX(sshn(ji,jj) + ht\_wd(ji,jj),} \\
1560& \mathrm{\phantom{MAX(}sshn(ji+1,jj) + ht\_wd(ji+1,jj))} >\\
1561& \quad\quad\mathrm{rn\_wdmin1 + rn\_wdmin2 }
1562\end{split}
1563\end{equation}
1564
1565The second logical, $\mathrm{ll\_tmp2}$, is set to true if and only if the maximum height
1566of the sea surface at the two points is greater than the maximum height of the topography
1567at the two points plus $\mathrm{rn\_wdmin1} + \mathrm{rn\_wdmin2}$
1568
1569\begin{equation} \label{dyn_ll_tmp2}
1570\begin{split}
1571\mathrm{ ll\_tmp2 } = & \mathrm{( ABS( sshn(ji,jj) - sshn(ji+1,jj) ) > 1.E-12 )\ .AND.}\\
1572& \mathrm{( MAX(sshn(ji,jj), sshn(ji+1,jj)) > } \\
1573& \mathrm{\phantom{(} MAX(-ht\_wd(ji,jj), -ht\_wd(ji+1,jj)) + rn\_wdmin1 + rn\_wdmin2}) .
1574\end{split}
1575\end{equation}
1576
1577If $\mathrm{ll\_tmp1}$ is true then the surface pressure gradient, zhpi at the $(i,j)$
1578point is unmodified. If both logicals are false zhpi is set to zero.
1579
1580If $\mathrm{ll\_tmp1}$ is true and $\mathrm{ll\_tmp2}$ is false then the surface pressure
1581gradient is multiplied through by zcpx which is the absolute value of the difference in
1582the water depths at the two points divided by the difference in the surface heights at the
1583two points. Thus the sign of the sea surface height gradient is retained but the magnitude
1584of the pressure force is determined by the difference in water depths rather than the
1585difference in surface height between the two points. Note that dividing by the difference
1586between the sea surface heights can be problematic if the heights approach parity. An
1587additional condition is applied to $\mathrm{ ll\_tmp2 }$ to ensure it is .false. in such
1588conditions.
1589
1590\subsubsection   [Additional considerations (\textit{usrdef\_zgr})]
1591         {Additional considerations (\mdl{usrdef\_zgr})}
1592\label{subsubsec:WAD_additional}
1593
1594In the very shallow water where wetting and drying occurs the parametrisation of
1595bottom drag is clearly very important. In order to promote stability
1596it is sometimes useful to calculate the bottom drag using an implicit time-stepping approach.
1597
1598Suitable specifcation of the surface heat flux in wetting and drying domains in forced and
1599coupled simulations needs further consideration. In order to prevent freezing or boiling
1600in uncoupled integrations the net surface heat fluxes need to be appropriately limited.
1601
1602%----------------------------------------------------------------------------------------
1603%      The WAD test cases
1604%----------------------------------------------------------------------------------------
1605\subsection   [The WAD test cases (\textit{usrdef\_zgr})]
1606         {The WAD test cases (\mdl{usrdef\_zgr})}
1607\label{WAD_test_cases}
1608
1609See the WAD tests MY\_DOC documention for details of the WAD test cases.
1610
1611
1612
1613% ================================================================
1614% Time evolution term
1615% ================================================================
1616\section{Time evolution term (\protect\mdl{dynnxt})}
1617\label{sec:DYN_nxt}
1618
1619%----------------------------------------------namdom----------------------------------------------------
1620
1621\nlst{namdom} 
1622%-------------------------------------------------------------------------------------------------------------
1623
1624Options are defined through the \ngn{namdom} namelist variables.
1625The general framework for dynamics time stepping is a leap-frog scheme,
1626\ie a three level centred time scheme associated with an Asselin time filter (cf. \autoref{chap:STP}).
1627The scheme is applied to the velocity, except when
1628using the flux form of momentum advection (cf. \autoref{sec:DYN_adv_cor_flux})
1629in the variable volume case (\key{vvl} defined),
1630where it has to be applied to the thickness weighted velocity (see \autoref{sec:A_momentum}
1631
1632$\bullet$ vector invariant form or linear free surface
1633(\np{ln\_dynhpg\_vec}\forcode{ = .true.} ; \key{vvl} not defined):
1634\[
1635  % \label{eq:dynnxt_vec}
1636  \left\{
1637    \begin{aligned}
1638      &u^{t+\rdt} = u_f^{t-\rdt} + 2\rdt  \ \text{RHS}_u^t     \\
1639      &u_f^t \;\quad = u^t+\gamma \,\left[ {u_f^{t-\rdt} -2u^t+u^{t+\rdt}} \right]
1640    \end{aligned}
1641  \right.
1642\]
1643
1644$\bullet$ flux form and nonlinear free surface
1645(\np{ln\_dynhpg\_vec}\forcode{ = .false.} ; \key{vvl} defined):
1646\[
1647  % \label{eq:dynnxt_flux}
1648  \left\{
1649    \begin{aligned}
1650      &\left(e_{3u}\,u\right)^{t+\rdt} = \left(e_{3u}\,u\right)_f^{t-\rdt} + 2\rdt \; e_{3u} \;\text{RHS}_u^t     \\
1651      &\left(e_{3u}\,u\right)_f^t \;\quad = \left(e_{3u}\,u\right)^t
1652      +\gamma \,\left[ {\left(e_{3u}\,u\right)_f^{t-\rdt} -2\left(e_{3u}\,u\right)^t+\left(e_{3u}\,u\right)^{t+\rdt}} \right]
1653    \end{aligned}
1654  \right.
1655\]
1656where RHS is the right hand side of the momentum equation,
1657the subscript $f$ denotes filtered values and $\gamma$ is the Asselin coefficient.
1658$\gamma$ is initialized as \np{nn\_atfp} (namelist parameter).
1659Its default value is \np{nn\_atfp}\forcode{ = 10.e-3}.
1660In both cases, the modified Asselin filter is not applied since perfect conservation is not an issue for
1661the momentum equations.
1662
1663Note that with the filtered free surface,
1664the update of the \textit{after} velocities is done in the \mdl{dynsp\_flt} module,
1665and only array swapping and Asselin filtering is done in \mdl{dynnxt}.
1666
1667% ================================================================
1668\biblio
1669
1670\pindex
1671
1672\end{document}
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