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1\documentclass[../tex_main/NEMO_manual]{subfiles}
2\begin{document}
3% ================================================================
4% Chapter ——— Ocean Dynamics (DYN)
5% ================================================================
6\chapter{Ocean Dynamics (DYN)}
7\label{chap:DYN}
8\minitoc
9
10%\vspace{2.cm}
11$\ $\newline      %force an empty line
12
13Using the representation described in \autoref{chap:DOM}, several semi-discrete
14space forms of the dynamical equations are available depending on the vertical
15coordinate used and on the conservation properties of the vorticity term. In all
16the equations presented here, the masking has been omitted for simplicity.
17One must be aware that all the quantities are masked fields and that each time an
18average or difference operator is used, the resulting field is multiplied by a mask.
19
20The prognostic ocean dynamics equation can be summarized as follows:
21\begin{equation*}
22\text{NXT} = \dbinom {\text{VOR} + \text{KEG} + \text {ZAD} }
23                  {\text{COR} + \text{ADV}                       }
24         + \text{HPG} + \text{SPG} + \text{LDF} + \text{ZDF}
25\end{equation*}
26NXT stands for next, referring to the time-stepping. The first group of terms on
27the rhs of this equation corresponds to the Coriolis and advection
28terms that are decomposed into either a vorticity part (VOR), a kinetic energy part (KEG)
29and a vertical advection part (ZAD) in the vector invariant formulation, or a Coriolis
30and advection part (COR+ADV) in the flux formulation. The terms following these
31are the pressure gradient contributions (HPG, Hydrostatic Pressure Gradient,
32and SPG, Surface Pressure Gradient); and contributions from lateral diffusion
33(LDF) and vertical diffusion (ZDF), which are added to the rhs in the \mdl{dynldf} 
34and \mdl{dynzdf} modules. The vertical diffusion term includes the surface and
35bottom stresses. The external forcings and parameterisations require complex
36inputs (surface wind stress calculation using bulk formulae, estimation of mixing
37coefficients) that are carried out in modules SBC, LDF and ZDF and are described
38in \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
41the horizontal divergence, curl of the velocities (\emph{divcur} module) and
42the vertical velocity (\emph{wzvmod} module).
43
44The different options available to the user are managed by namelist variables.
45For term \textit{ttt} in the momentum equations, the logical namelist variables are \textit{ln\_dynttt\_xxx},
46where \textit{xxx} is a 3 or 4 letter acronym corresponding to each optional scheme.
47If a CPP key is used for this term its name is \key{ttt}. The corresponding
48code can be found in the \textit{dynttt\_xxx} module in the DYN directory, and it is
49usually computed in the \textit{dyn\_ttt\_xxx} subroutine.
50
51The user has the option of extracting and outputting each tendency term from the
523D momentum equations (\key{trddyn} defined), as described in
53\autoref{chap:MISC}.  Furthermore, the tendency terms associated with the 2D
54barotropic vorticity balance (when \key{trdvor} is defined) can be derived from the
553D terms.
56%%%
57\gmcomment{STEVEN: not quite sure I've got the sense of the last sentence. does
58MISC correspond to "extracting tendency terms" or "vorticity balance"?}
59
60$\ $\newline    % force a new ligne
61
62% ================================================================
63% Sea Surface Height evolution & Diagnostics variables
64% ================================================================
65\section{Sea surface height and diagnostic variables ($\eta$, $\zeta$, $\chi$, $w$)}
66\label{sec:DYN_divcur_wzv}
67
68%--------------------------------------------------------------------------------------------------------------
69%           Horizontal divergence and relative vorticity
70%--------------------------------------------------------------------------------------------------------------
71\subsection{Horizontal divergence and relative vorticity (\protect\mdl{divcur})}
72\label{subsec:DYN_divcur}
73
74The vorticity is defined at an $f$-point ($i.e.$ corner point) as follows:
75\begin{equation} \label{eq:divcur_cur}
76\zeta =\frac{1}{e_{1f}\,e_{2f} }\left( {\;\delta _{i+1/2} \left[ {e_{2v}\;v} \right]
77                          -\delta _{j+1/2} \left[ {e_{1u}\;u} \right]\;} \right)
78\end{equation} 
79
80The horizontal divergence is defined at a $T$-point. It is given by:
81\begin{equation} \label{eq:divcur_div}
82\chi =\frac{1}{e_{1t}\,e_{2t}\,e_{3t} }
83      \left( {\delta _i \left[ {e_{2u}\,e_{3u}\,u} \right]
84             +\delta _j \left[ {e_{1v}\,e_{3v}\,v} \right]} \right)
85\end{equation} 
86
87Note that although the vorticity has the same discrete expression in $z$-
88and $s$-coordinates, its physical meaning is not identical. $\zeta$ is a pseudo
89vorticity along $s$-surfaces (only pseudo because $(u,v)$ are still defined along
90geopotential surfaces, but are not necessarily defined at the same depth).
91
92The vorticity and divergence at the \textit{before} step are used in the computation
93of the horizontal diffusion of momentum. Note that because they have been
94calculated prior to the Asselin filtering of the \textit{before} velocities, the
95\textit{before} vorticity and divergence arrays must be included in the restart file
96to ensure perfect restartability. The vorticity and divergence at the \textit{now} 
97time step are used for the computation of the nonlinear advection and of the
98vertical 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} \label{eq:dynspg_ssh}
108\begin{aligned}
109\frac{\partial \eta }{\partial t}
110&\equiv    \frac{1}{e_{1t} e_{2t} }\sum\limits_k { \left\{  \delta _i \left[ {e_{2u}\,e_{3u}\;u} \right]
111                                                                                  +\delta _j \left[ {e_{1v}\,e_{3v}\;v} \right]  \right\} } 
112           -    \frac{\textit{emp}}{\rho _w }   \\
113&\equiv    \sum\limits_k {\chi \ e_{3t}}  -  \frac{\textit{emp}}{\rho _w }
114\end{aligned}
115\end{equation}
116where \textit{emp} is the surface freshwater budget (evaporation minus precipitation),
117expressed in Kg/m$^2$/s (which is equal to mm/s), and $\rho _w$=1,035~Kg/m$^3$ 
118is the reference density of sea water (Boussinesq approximation). If river runoff is
119expressed as a surface freshwater flux (see \autoref{chap:SBC}) then \textit{emp} can be
120written as the evaporation minus precipitation, minus the river runoff.
121The sea-surface height is evaluated using exactly the same time stepping scheme
122as the tracer equation \autoref{eq:tra_nxt}:
123a leapfrog scheme in combination with an Asselin time filter, $i.e.$ the velocity appearing
124in \autoref{eq:dynspg_ssh} is centred in time (\textit{now} velocity).
125This is of paramount importance. Replacing $T$ by the number $1$ in the tracer equation and summing
126over the water column must lead to the sea surface height equation otherwise tracer content
127will not be conserved \citep{Griffies_al_MWR01, Leclair_Madec_OM09}.
128
129The vertical velocity is computed by an upward integration of the horizontal
130divergence starting at the bottom, taking into account the change of the thickness of the levels :
131\begin{equation} \label{eq:wzv}
132\left\{   \begin{aligned}
133&\left. w \right|_{k_b-1/2} \quad= 0    \qquad \text{where } k_b \text{ is the level just above the sea floor }   \\
134&\left. w \right|_{k+1/2}     = \left. w \right|_{k-1/2}  +  \left. e_{3t} \right|_{k}\;  \left. \chi \right|_
135                                         - \frac{1} {2 \rdt} \left\left. e_{3t}^{t+1}\right|_{k} - \left. e_{3t}^{t-1}\right|_{k}\right)
136\end{aligned}   \right.
137\end{equation}
138
139In the case of a non-linear free surface (\key{vvl}), the top vertical velocity is $-\textit{emp}/\rho_w$,
140as changes in the divergence of the barotropic transport are absorbed into the change
141of the level thicknesses, re-orientated downward.
142\gmcomment{not sure of this...  to be modified with the change in emp setting}
143In the case of a linear free surface, the time derivative in \autoref{eq:wzv} disappears.
144The upper boundary condition applies at a fixed level $z=0$. The top vertical velocity
145is thus equal to the divergence of the barotropic transport ($i.e.$ the first term in the
146right-hand-side of \autoref{eq:dynspg_ssh}).
147
148Note also that whereas the vertical velocity has the same discrete
149expression in $z$- and $s$-coordinates, its physical meaning is not the same:
150in the second case, $w$ is the velocity normal to the $s$-surfaces.
151Note also that the $k$-axis is re-orientated downwards in the \textsc{fortran} code compared
152to the indexing used in the semi-discrete equations such as \autoref{eq:wzv} 
153(see \autoref{subsec:DOM_Num_Index_vertical}).
154
155
156% ================================================================
157% Coriolis and Advection terms: vector invariant form
158% ================================================================
159\section{Coriolis and advection: vector invariant form}
160\label{sec:DYN_adv_cor_vect}
161%-----------------------------------------nam_dynadv----------------------------------------------------
162\forfile{../namelists/namdyn_adv} 
163%-------------------------------------------------------------------------------------------------------------
164
165The vector invariant form of the momentum equations is the one most
166often used in applications of the \NEMO ocean model. The flux form option
167(see next section) has been present since version $2$. Options are defined
168through the \ngn{namdyn\_adv} namelist variables
169Coriolis and momentum advection terms are evaluated using a leapfrog
170scheme, $i.e.$ the velocity appearing in these expressions is centred in
171time (\textit{now} velocity).
172At the lateral boundaries either free slip, no slip or partial slip boundary
173conditions are applied following \autoref{chap:LBC}.
174
175% -------------------------------------------------------------------------------------------------------------
176%        Vorticity term
177% -------------------------------------------------------------------------------------------------------------
178\subsection{Vorticity term (\protect\mdl{dynvor})}
179\label{subsec:DYN_vor}
180%------------------------------------------nam_dynvor----------------------------------------------------
181\forfile{../namelists/namdyn_vor} 
182%-------------------------------------------------------------------------------------------------------------
183
184Options are defined through the \ngn{namdyn\_vor} namelist variables.
185Four discretisations of the vorticity term (\np{ln\_dynvor\_xxx}\forcode{ = .true.}) are available:
186conserving potential enstrophy of horizontally non-divergent flow (ENS scheme) ;
187conserving horizontal kinetic energy (ENE scheme) ; conserving potential enstrophy for
188the relative vorticity term and horizontal kinetic energy for the planetary vorticity
189term (MIX scheme) ; or conserving both the potential enstrophy of horizontally non-divergent
190flow and horizontal kinetic energy (EEN scheme) (see \autoref{subsec:C_vorEEN}). In the
191case of ENS, ENE or MIX schemes the land sea mask may be slightly modified to ensure the
192consistency of vorticity term with analytical equations (\np{ln\_dynvor\_con}\forcode{ = .true.}).
193The vorticity terms are all computed in dedicated routines that can be found in
194the \mdl{dynvor} module.
195
196%-------------------------------------------------------------
197%                 enstrophy conserving scheme
198%-------------------------------------------------------------
199\subsubsection{Enstrophy conserving scheme (\protect\np{ln\_dynvor\_ens}\forcode{ = .true.})}
200\label{subsec:DYN_vor_ens}
201
202In the enstrophy conserving case (ENS scheme), the discrete formulation of the
203vorticity term provides a global conservation of the enstrophy
204($ [ (\zeta +f ) / e_{3f} ]^2 $ in $s$-coordinates) for a horizontally non-divergent
205flow ($i.e.$ $\chi$=$0$), but does not conserve the total kinetic energy. It is given by:
206\begin{equation} \label{eq:dynvor_ens}
207\left\{ 
208\begin{aligned}
209{+\frac{1}{e_{1u} } } & {\overline {\left( { \frac{\zeta +f}{e_{3f} }} \right)} }^{\,i} 
210                                & {\overline{\overline {\left( {e_{1v}\,e_{3v}\;v} \right)}} }^{\,i, j+1/2}    \\
211{- \frac{1}{e_{2v} } } & {\overline {\left( {\frac{\zeta +f}{e_{3f} }} \right)} }^{\,j} 
212                                & {\overline{\overline {\left( {e_{2u}\,e_{3u}\;u} \right)}} }^{\,i+1/2, j} 
213\end{aligned} 
214 \right.
215\end{equation} 
216
217%-------------------------------------------------------------
218%                 energy conserving scheme
219%-------------------------------------------------------------
220\subsubsection{Energy conserving scheme (\protect\np{ln\_dynvor\_ene}\forcode{ = .true.})}
221\label{subsec:DYN_vor_ene}
222
223The kinetic energy conserving scheme (ENE scheme) conserves the global
224kinetic energy but not the global enstrophy. It is given by:
225\begin{equation} \label{eq:dynvor_ene}
226\left\{   \begin{aligned}
227{+\frac{1}{e_{1u}}\; {\overline {\left( {\frac{\zeta +f}{e_{3f} }} \right)
228                            \;  \overline {\left( {e_{1v}\,e_{3v}\;v} \right)} ^{\,i+1/2}} }^{\,j} }    \\
229{- \frac{1}{e_{2v}}\; {\overline {\left( {\frac{\zeta +f}{e_{3f} }} \right)
230                            \;  \overline {\left( {e_{2u}\,e_{3u}\;u} \right)} ^{\,j+1/2}} }^{\,i} }
231\end{aligned}    \right.
232\end{equation} 
233
234%-------------------------------------------------------------
235%                 mix energy/enstrophy conserving scheme
236%-------------------------------------------------------------
237\subsubsection{Mixed energy/enstrophy conserving scheme (\protect\np{ln\_dynvor\_mix}\forcode{ = .true.}) }
238\label{subsec:DYN_vor_mix}
239
240For the mixed energy/enstrophy conserving scheme (MIX scheme), a mixture of the
241two previous schemes is used. It consists of the ENS scheme (\autoref{eq:dynvor_ens})
242for the relative vorticity term, and of the ENE scheme (\autoref{eq:dynvor_ene}) applied
243to the planetary vorticity term.
244\begin{equation} \label{eq:dynvor_mix}
245\left\{ {     \begin{aligned}
246 {+\frac{1}{e_{1u} }\; {\overline {\left( {\frac{\zeta }{e_{3f} }} \right)} }^{\,i} 
247 \; {\overline{\overline {\left( {e_{1v}\,e_{3v}\;v} \right)}} }^{\,i,j+1/2} -\frac{1}{e_{1u} }
248 \; {\overline {\left( {\frac{f}{e_{3f} }} \right)
249 \;\overline {\left( {e_{1v}\,e_{3v}\;v} \right)} ^{\,i+1/2}} }^{\,j} } \\
250{-\frac{1}{e_{2v} }\; {\overline {\left( {\frac{\zeta }{e_{3f} }} \right)} }^j
251 \; {\overline{\overline {\left( {e_{2u}\,e_{3u}\;u} \right)}} }^{\,i+1/2,j} +\frac{1}{e_{2v} }
252 \; {\overline {\left( {\frac{f}{e_{3f} }} \right)
253 \;\overline {\left( {e_{2u}\,e_{3u}\;u} \right)} ^{\,j+1/2}} }^{\,i} } \hfill
254\end{aligned}     } \right.
255\end{equation} 
256
257%-------------------------------------------------------------
258%                 energy and enstrophy conserving scheme
259%-------------------------------------------------------------
260\subsubsection{Energy and enstrophy conserving scheme (\protect\np{ln\_dynvor\_een}\forcode{ = .true.}) }
261\label{subsec:DYN_vor_een}
262
263In both the ENS and ENE schemes, it is apparent that the combination of $i$ and $j$ 
264averages of the velocity allows for the presence of grid point oscillation structures
265that will be invisible to the operator. These structures are \textit{computational modes} 
266that will be at least partly damped by the momentum diffusion operator ($i.e.$ the
267subgrid-scale advection), but not by the resolved advection term. The ENS and ENE schemes
268therefore do not contribute to dump any grid point noise in the horizontal velocity field.
269Such noise would result in more noise in the vertical velocity field, an undesirable feature.
270This is a well-known characteristic of $C$-grid discretization where $u$ and $v$ are located
271at different grid points, a price worth paying to avoid a double averaging in the pressure
272gradient term as in the $B$-grid.
273\gmcomment{ To circumvent this, Adcroft (ADD REF HERE)
274Nevertheless, this technique strongly distort the phase and group velocity of Rossby waves....}
275
276A very nice solution to the problem of double averaging was proposed by \citet{Arakawa_Hsu_MWR90}.
277The idea is to get rid of the double averaging by considering triad combinations of vorticity.
278It is noteworthy that this solution is conceptually quite similar to the one proposed by
279\citep{Griffies_al_JPO98} for the discretization of the iso-neutral diffusion operator (see \autoref{apdx:C}).
280
281The \citet{Arakawa_Hsu_MWR90} vorticity advection scheme for a single layer is modified
282for spherical coordinates as described by \citet{Arakawa_Lamb_MWR81} to obtain the EEN scheme.
283First consider the discrete expression of the potential vorticity, $q$, defined at an $f$-point:
284\begin{equation} \label{eq:pot_vor}
285q  = \frac{\zeta +f} {e_{3f} }
286\end{equation}
287where the relative vorticity is defined by (\autoref{eq:divcur_cur}), the Coriolis parameter
288is given by $f=2 \,\Omega \;\sin \varphi _f $ and the layer thickness at $f$-points is:
289\begin{equation} \label{eq:een_e3f}
290e_{3f} = \overline{\overline {e_{3t} }} ^{\,i+1/2,j+1/2}
291\end{equation}
292
293%>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
294\begin{figure}[!ht]    \begin{center}
295\includegraphics[width=0.70\textwidth]{Fig_DYN_een_triad}
296\caption{ \protect\label{fig:DYN_een_triad} 
297Triads used in the energy and enstrophy conserving scheme (een) for
298$u$-component (upper panel) and $v$-component (lower panel).}
299\end{center}   \end{figure}
300%>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
301
302A key point in \autoref{eq:een_e3f} is how the averaging in the \textbf{i}- and \textbf{j}- directions is made.
303It uses the sum of masked t-point vertical scale factor divided either
304by the sum of the four t-point masks (\np{nn\_een\_e3f}\forcode{ = 1}),
305or  just by $4$ (\np{nn\_een\_e3f}\forcode{ = .true.}).
306The latter case preserves the continuity of $e_{3f}$ when one or more of the neighbouring $e_{3t}$ 
307tends to zero and extends by continuity the value of $e_{3f}$ into the land areas.
308This case introduces a sub-grid-scale topography at f-points (with a systematic reduction of $e_{3f}$ 
309when a model level intercept the bathymetry) that tends to reinforce the topostrophy of the flow
310($i.e.$ the tendency of the flow to follow the isobaths) \citep{Penduff_al_OS07}.
311
312Next, the vorticity triads, $ {^i_j}\mathbb{Q}^{i_p}_{j_p}$ can be defined at a $T$-point as
313the following triad combinations of the neighbouring potential vorticities defined at f-points
314(\autoref{fig:DYN_een_triad}):
315\begin{equation} \label{eq:Q_triads}
316_i^j \mathbb{Q}^{i_p}_{j_p}
317= \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)
318\end{equation}
319where the indices $i_p$ and $k_p$ take the values: $i_p = -1/2$ or $1/2$ and $j_p = -1/2$ or $1/2$.
320
321Finally, the vorticity terms are represented as:
322\begin{equation} \label{eq:dynvor_een}
323\left\{ {
324\begin{aligned}
325 +q\,e_3 \, v  &\equiv +\frac{1}{e_{1u} }   \sum_{\substack{i_p,\,k_p}} 
326                         {^{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}   \\
327 - q\,e_3 \, u     &\equiv -\frac{1}{e_{2v} }    \sum_{\substack{i_p,\,k_p}} 
328                         {^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}   \\
329\end{aligned} 
330} \right.
331\end{equation} 
332
333This EEN scheme in fact combines the conservation properties of the ENS and ENE schemes.
334It conserves both total energy and potential enstrophy in the limit of horizontally
335nondivergent flow ($i.e.$ $\chi$=$0$) (see \autoref{subsec:C_vorEEN}).
336Applied to a realistic ocean configuration, it has been shown that it leads to a significant
337reduction of the noise in the vertical velocity field \citep{Le_Sommer_al_OM09}.
338Furthermore, used in combination with a partial steps representation of bottom topography,
339it improves the interaction between current and topography, leading to a larger
340topostrophy of the flow  \citep{Barnier_al_OD06, Penduff_al_OS07}.
341
342%--------------------------------------------------------------------------------------------------------------
343%           Kinetic Energy Gradient term
344%--------------------------------------------------------------------------------------------------------------
345\subsection{Kinetic energy gradient term (\protect\mdl{dynkeg})}
346\label{subsec:DYN_keg}
347
348As demonstrated in \autoref{apdx:C}, there is a single discrete formulation
349of the kinetic energy gradient term that, together with the formulation chosen for
350the vertical advection (see below), conserves the total kinetic energy:
351\begin{equation} \label{eq:dynkeg}
352\left\{ \begin{aligned}
353 -\frac{1}{2 \; e_{1u} }  & \ \delta _{i+1/2} \left[ {\overline {u^2}^{\,i} + \overline{v^2}^{\,j}} \right]   \\
354 -\frac{1}{2 \; e_{2v} }  & \ \delta _{j+1/2} \left[ {\overline {u^2}^{\,i} + \overline{v^2}^{\,j}} \right]   
355\end{aligned} \right.
356\end{equation} 
357
358%--------------------------------------------------------------------------------------------------------------
359%           Vertical advection term
360%--------------------------------------------------------------------------------------------------------------
361\subsection{Vertical advection term (\protect\mdl{dynzad}) }
362\label{subsec:DYN_zad}
363
364The discrete formulation of the vertical advection, together with the formulation
365chosen for the gradient of kinetic energy (KE) term, conserves the total kinetic
366energy. Indeed, the change of KE due to the vertical advection is exactly
367balanced by the change of KE due to the gradient of KE (see \autoref{apdx:C}).
368\begin{equation} \label{eq:dynzad}
369\left\{     \begin{aligned}
370-\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}  \\
371-\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} 
372\end{aligned}         \right.
373\end{equation} 
374When \np{ln\_dynzad\_zts}\forcode{ = .true.}, a split-explicit time stepping with 5 sub-timesteps is used
375on the vertical advection term.
376This option can be useful when the value of the timestep is limited by vertical advection \citep{Lemarie_OM2015}.
377Note that in this case, a similar split-explicit time stepping should be used on
378vertical advection of tracer to ensure a better stability,
379an option which is only available with a TVD scheme (see \np{ln\_traadv\_tvd\_zts} in \autoref{subsec:TRA_adv_tvd}).
380
381
382% ================================================================
383% Coriolis and Advection : flux form
384% ================================================================
385\section{Coriolis and advection: flux form}
386\label{sec:DYN_adv_cor_flux}
387%------------------------------------------nam_dynadv----------------------------------------------------
388\forfile{../namelists/namdyn_adv} 
389%-------------------------------------------------------------------------------------------------------------
390
391Options are defined through the \ngn{namdyn\_adv} namelist variables.
392In the flux form (as in the vector invariant form), the Coriolis and momentum
393advection terms are evaluated using a leapfrog scheme, $i.e.$ the velocity
394appearing in their expressions is centred in time (\textit{now} velocity). At the
395lateral boundaries either free slip, no slip or partial slip boundary conditions
396are applied following \autoref{chap:LBC}.
397
398
399%--------------------------------------------------------------------------------------------------------------
400%           Coriolis plus curvature metric terms
401%--------------------------------------------------------------------------------------------------------------
402\subsection{Coriolis plus curvature metric terms (\protect\mdl{dynvor}) }
403\label{subsec:DYN_cor_flux}
404
405In flux form, the vorticity term reduces to a Coriolis term in which the Coriolis
406parameter has been modified to account for the "metric" term. This altered
407Coriolis parameter is thus discretised at $f$-points. It is given by:
408\begin{multline} \label{eq:dyncor_metric}
409f+\frac{1}{e_1 e_2 }\left( {v\frac{\partial e_2 }{\partial i}  -  u\frac{\partial e_1 }{\partial j}} \right\\
410   \equiv   f + \frac{1}{e_{1f} e_{2f} } \left( { \ \overline v ^{i+1/2}\delta _{i+1/2} \left[ {e_{2u} } \right] 
411                                                                 -  \overline u ^{j+1/2}\delta _{j+1/2} \left[ {e_{1u} } \right]  }  \ \right)
412\end{multline} 
413
414Any of the (\autoref{eq:dynvor_ens}), (\autoref{eq:dynvor_ene}) and (\autoref{eq:dynvor_een})
415schemes can be used to compute the product of the Coriolis parameter and the
416vorticity. However, the energy-conserving scheme  (\autoref{eq:dynvor_een}) has
417exclusively been used to date. This term is evaluated using a leapfrog scheme,
418$i.e.$ the velocity is centred in time (\textit{now} velocity).
419
420%--------------------------------------------------------------------------------------------------------------
421%           Flux form Advection term
422%--------------------------------------------------------------------------------------------------------------
423\subsection{Flux form advection term (\protect\mdl{dynadv}) }
424\label{subsec:DYN_adv_flux}
425
426The discrete expression of the advection term is given by :
427\begin{equation} \label{eq:dynadv}
428\left\{ 
429\begin{aligned}
430\frac{1}{e_{1u}\,e_{2u}\,e_{3u}} 
431\left(      \delta _{i+1/2} \left[ \overline{e_{2u}\,e_{3u}\;u }^{i       }  \ u_t      \right]   
432          + \delta _{j       } \left[ \overline{e_{1u}\,e_{3u}\;v }^{i+1/2}  \ u_f      \right] \right\ \;   \\
433\left.   + \delta _{k      } \left[ \overline{e_{1w}\,e_{2w}\;w}^{i+1/2}  \ u_{uw} \right] \right)   \\
434\\
435\frac{1}{e_{1v}\,e_{2v}\,e_{3v}} 
436\left(     \delta _{i       } \left[ \overline{e_{2u}\,e_{3u }\;u }^{j+1/2} \ v_f       \right] 
437         + \delta _{j+1/2} \left[ \overline{e_{1u}\,e_{3u }\;v }^{i       } \ v_t       \right] \right\ \, \, \\
438\left.  + \delta _{k      } \left[ \overline{e_{1w}\,e_{2w}\;w}^{j+1/2} \ v_{vw}  \right] \right) \\
439\end{aligned}
440\right.
441\end{equation}
442
443Two advection schemes are available: a $2^{nd}$ order centered finite
444difference scheme, CEN2, or a $3^{rd}$ order upstream biased scheme, UBS.
445The latter is described in \citet{Shchepetkin_McWilliams_OM05}. The schemes are
446selected using the namelist logicals \np{ln\_dynadv\_cen2} and \np{ln\_dynadv\_ubs}.
447In flux form, the schemes differ by the choice of a space and time interpolation to
448define the value of $u$ and $v$ at the centre of each face of $u$- and $v$-cells,
449$i.e.$ at the $T$-, $f$-, and $uw$-points for $u$ and at the $f$-, $T$- and
450$vw$-points for $v$.
451
452%-------------------------------------------------------------
453%                 2nd order centred scheme
454%-------------------------------------------------------------
455\subsubsection{CEN2: $2^{nd}$ order centred scheme (\protect\np{ln\_dynadv\_cen2}\forcode{ = .true.})}
456\label{subsec:DYN_adv_cen2}
457
458In the centered $2^{nd}$ order formulation, the velocity is evaluated as the
459mean of the two neighbouring points :
460\begin{equation} \label{eq:dynadv_cen2}
461\left\{     \begin{aligned}
462 u_T^{cen2} &=\overline u^{i }       \quad &  u_F^{cen2} &=\overline u^{j+1/2}  \quad &  u_{uw}^{cen2} &=\overline u^{k+1/2}   \\
463 v_F^{cen2} &=\overline v ^{i+1/2} \quad & v_F^{cen2} &=\overline v^j      \quad &  v_{vw}^{cen2} &=\overline v ^{k+1/2}  \\
464\end{aligned}      \right.
465\end{equation} 
466
467The scheme is non diffusive (i.e. conserves the kinetic energy) but dispersive
468($i.e.$ it may create false extrema). It is therefore notoriously noisy and must be
469used in conjunction with an explicit diffusion operator to produce a sensible solution.
470The associated time-stepping is performed using a leapfrog scheme in conjunction
471with an Asselin time-filter, so $u$ and $v$ are the \emph{now} velocities.
472
473%-------------------------------------------------------------
474%                 UBS scheme
475%-------------------------------------------------------------
476\subsubsection{UBS: Upstream Biased Scheme (\protect\np{ln\_dynadv\_ubs}\forcode{ = .true.})}
477\label{subsec:DYN_adv_ubs}
478
479The UBS advection scheme is an upstream biased third order scheme based on
480an upstream-biased parabolic interpolation. For example, the evaluation of
481$u_T^{ubs} $ is done as follows:
482\begin{equation} \label{eq:dynadv_ubs}
483u_T^{ubs} =\overline u ^i-\;\frac{1}{6}   \begin{cases}
484      u"_{i-1/2}&    \text{if $\ \overline{e_{2u}\,e_{3u} \ u}^i  \geqslant 0$ }    \\
485      u"_{i+1/2}&    \text{if $\ \overline{e_{2u}\,e_{3u} \ u}^i  < 0$ }
486\end{cases}
487\end{equation}
488where $u"_{i+1/2} =\delta _{i+1/2} \left[ {\delta _i \left[ u \right]} \right]$. This results
489in a dissipatively dominant ($i.e.$ hyper-diffusive) truncation error \citep{Shchepetkin_McWilliams_OM05}.
490The overall performance of the advection scheme is similar to that reported in
491\citet{Farrow1995}. It is a relatively good compromise between accuracy and
492smoothness. It is not a \emph{positive} scheme, meaning that false extrema are
493permitted. But the amplitudes of the false extrema are significantly reduced over
494those in the centred second order method. As the scheme already includes
495a diffusion component, it can be used without explicit lateral diffusion on momentum
496($i.e.$ \np{ln\_dynldf\_lap}\forcode{ = }\np{ln\_dynldf\_bilap}\forcode{ = .false.}), and it is recommended to do so.
497
498The UBS scheme is not used in all directions. In the vertical, the centred $2^{nd}$ 
499order evaluation of the advection is preferred, $i.e.$ $u_{uw}^{ubs}$ and
500$u_{vw}^{ubs}$ in \autoref{eq:dynadv_cen2} are used. UBS is diffusive and is
501associated with vertical mixing of momentum. \gmcomment{ gm  pursue the
502sentence:Since vertical mixing of momentum is a source term of the TKE equation...  }
503
504For stability reasons, the first term in (\autoref{eq:dynadv_ubs}), which corresponds
505to a second order centred scheme, is evaluated using the \textit{now} velocity
506(centred in time), while the second term, which is the diffusion part of the scheme,
507is evaluated using the \textit{before} velocity (forward in time). This is discussed
508by \citet{Webb_al_JAOT98} in the context of the Quick advection scheme.
509
510Note that the UBS and QUICK (Quadratic Upstream Interpolation for Convective Kinematics)
511schemes only differ by one coefficient. Replacing $1/6$ by $1/8$ in
512(\autoref{eq:dynadv_ubs}) leads to the QUICK advection scheme \citep{Webb_al_JAOT98}.
513This option is not available through a namelist parameter, since the $1/6$ coefficient
514is hard coded. Nevertheless it is quite easy to make the substitution in the
515\mdl{dynadv\_ubs} module and obtain a QUICK scheme.
516
517Note also that in the current version of \mdl{dynadv\_ubs}, there is also the
518possibility of using a $4^{th}$ order evaluation of the advective velocity as in
519ROMS. This is an error and should be suppressed soon.
520%%%
521\gmcomment{action :  this have to be done}
522%%%
523
524% ================================================================
525%           Hydrostatic pressure gradient term
526% ================================================================
527\section{Hydrostatic pressure gradient (\protect\mdl{dynhpg})}
528\label{sec:DYN_hpg}
529%------------------------------------------nam_dynhpg---------------------------------------------------
530\forfile{../namelists/namdyn_hpg} 
531%-------------------------------------------------------------------------------------------------------------
532
533Options are defined through the \ngn{namdyn\_hpg} namelist variables.
534The key distinction between the different algorithms used for the hydrostatic
535pressure gradient is the vertical coordinate used, since HPG is a \emph{horizontal} 
536pressure gradient, $i.e.$ computed along geopotential surfaces. As a result, any
537tilt of the surface of the computational levels will require a specific treatment to
538compute the hydrostatic pressure gradient.
539
540The hydrostatic pressure gradient term is evaluated either using a leapfrog scheme,
541$i.e.$ the density appearing in its expression is centred in time (\emph{now} $\rho$), or
542a semi-implcit scheme. At the lateral boundaries either free slip, no slip or partial slip
543boundary conditions are applied.
544
545%--------------------------------------------------------------------------------------------------------------
546%           z-coordinate with full step
547%--------------------------------------------------------------------------------------------------------------
548\subsection{Full step $Z$-coordinate (\protect\np{ln\_dynhpg\_zco}\forcode{ = .true.})}
549\label{subsec:DYN_hpg_zco}
550
551The hydrostatic pressure can be obtained by integrating the hydrostatic equation
552vertically from the surface. However, the pressure is large at great depth while its
553horizontal gradient is several orders of magnitude smaller. This may lead to large
554truncation errors in the pressure gradient terms. Thus, the two horizontal components
555of the hydrostatic pressure gradient are computed directly as follows:
556
557for $k=km$ (surface layer, $jk=1$ in the code)
558\begin{equation} \label{eq:dynhpg_zco_surf}
559\left\{ \begin{aligned}
560               \left. \delta _{i+1/2} \left[  p^h         \right] \right|_{k=km} 
561&= \frac{1}{2} g \   \left. \delta _{i+1/2} \left[  e_{3w} \ \rho \right] \right|_{k=km}   \\
562                  \left. \delta _{j+1/2} \left[  p^h            \right] \right|_{k=km} 
563&= \frac{1}{2} g \   \left. \delta _{j+1/2} \left[  e_{3w} \ \rho \right] \right|_{k=km}   \\
564\end{aligned} \right.
565\end{equation} 
566
567for $1<k<km$ (interior layer)
568\begin{equation} \label{eq:dynhpg_zco}
569\left\{ \begin{aligned}
570               \left. \delta _{i+1/2} \left[  p^h         \right] \right|_{k} 
571&=             \left. \delta _{i+1/2} \left[  p^h         \right] \right|_{k-1} 
572+    \frac{1}{2}\;g\;   \left. \delta _{i+1/2} \left[  e_{3w} \ \overline {\rho}^{k+1/2} \right] \right|_{k}   \\
573                  \left. \delta _{j+1/2} \left[  p^h            \right] \right|_{k} 
574&=                \left. \delta _{j+1/2} \left[  p^h            \right] \right|_{k-1} 
575+    \frac{1}{2}\;g\;   \left. \delta _{j+1/2} \left[  e_{3w} \ \overline {\rho}^{k+1/2} \right] \right|_{k}   \\
576\end{aligned} \right.
577\end{equation} 
578
579Note that the $1/2$ factor in (\autoref{eq:dynhpg_zco_surf}) is adequate because of
580the definition of $e_{3w}$ as the vertical derivative of the scale factor at the surface
581level ($z=0$). Note also that in case of variable volume level (\key{vvl} defined), the
582surface pressure gradient is included in \autoref{eq:dynhpg_zco_surf} and \autoref{eq:dynhpg_zco} 
583through the space and time variations of the vertical scale factor $e_{3w}$.
584
585%--------------------------------------------------------------------------------------------------------------
586%           z-coordinate with partial step
587%--------------------------------------------------------------------------------------------------------------
588\subsection{Partial step $Z$-coordinate (\protect\np{ln\_dynhpg\_zps}\forcode{ = .true.})}
589\label{subsec:DYN_hpg_zps}
590
591With partial bottom cells, tracers in horizontally adjacent cells generally live at
592different depths. Before taking horizontal gradients between these tracer points,
593a linear interpolation is used to approximate the deeper tracer as if it actually lived
594at the depth of the shallower tracer point.
595
596Apart from this modification, the horizontal hydrostatic pressure gradient evaluated
597in the $z$-coordinate with partial step is exactly as in the pure $z$-coordinate case.
598As explained in detail in section \autoref{sec:TRA_zpshde}, the nonlinearity of pressure
599effects in the equation of state is such that it is better to interpolate temperature and
600salinity vertically before computing the density. Horizontal gradients of temperature
601and salinity are needed for the TRA modules, which is the reason why the horizontal
602gradients of density at the deepest model level are computed in module \mdl{zpsdhe} 
603located in the TRA directory and described in \autoref{sec:TRA_zpshde}.
604
605%--------------------------------------------------------------------------------------------------------------
606%           s- and s-z-coordinates
607%--------------------------------------------------------------------------------------------------------------
608\subsection{$S$- and $Z$-$S$-coordinates}
609\label{subsec:DYN_hpg_sco}
610
611Pressure gradient formulations in an $s$-coordinate have been the subject of a vast
612number of papers ($e.g.$, \citet{Song1998, Shchepetkin_McWilliams_OM05}).
613A number of different pressure gradient options are coded but the ROMS-like, density Jacobian with
614cubic polynomial method is currently disabled whilst known bugs are under investigation.
615
616$\bullet$ Traditional coding (see for example \citet{Madec_al_JPO96}: (\np{ln\_dynhpg\_sco}\forcode{ = .true.})
617\begin{equation} \label{eq:dynhpg_sco}
618\left\{ \begin{aligned}
619 - \frac{1}                   {\rho_o \, e_{1u}} \;   \delta _{i+1/2} \left[  p^h  \right] 
620+ \frac{g\; \overline {\rho}^{i+1/2}}  {\rho_o \, e_{1u}} \;   \delta _{i+1/2} \left[  z_t   \right]    \\
621 - \frac{1}                   {\rho_o \, e_{2v}} \;   \delta _{j+1/2} \left[  p^h  \right] 
622+ \frac{g\; \overline {\rho}^{j+1/2}}  {\rho_o \, e_{2v}} \;   \delta _{j+1/2} \left[  z_t   \right]    \\
623\end{aligned} \right.
624\end{equation} 
625
626Where the first term is the pressure gradient along coordinates, computed as in
627\autoref{eq:dynhpg_zco_surf} - \autoref{eq:dynhpg_zco}, and $z_T$ is the depth of
628the $T$-point evaluated from the sum of the vertical scale factors at the $w$-point
629($e_{3w}$).
630 
631$\bullet$ Traditional coding with adaptation for ice shelf cavities (\np{ln\_dynhpg\_isf}\forcode{ = .true.}).
632This scheme need the activation of ice shelf cavities (\np{ln\_isfcav}\forcode{ = .true.}).
633
634$\bullet$ Pressure Jacobian scheme (prj) (a research paper in preparation) (\np{ln\_dynhpg\_prj}\forcode{ = .true.})
635
636$\bullet$ Density Jacobian with cubic polynomial scheme (DJC) \citep{Shchepetkin_McWilliams_OM05} 
637(\np{ln\_dynhpg\_djc}\forcode{ = .true.}) (currently disabled; under development)
638
639Note that expression \autoref{eq:dynhpg_sco} is commonly used when the variable volume formulation is
640activated (\key{vvl}) because in that case, even with a flat bottom, the coordinate surfaces are not
641horizontal but follow the free surface \citep{Levier2007}. The pressure jacobian scheme
642(\np{ln\_dynhpg\_prj}\forcode{ = .true.}) is available as an improved option to \np{ln\_dynhpg\_sco}\forcode{ = .true.} when
643\key{vvl} is active.  The pressure Jacobian scheme uses a constrained cubic spline to reconstruct
644the density profile across the water column. This method maintains the monotonicity between the
645density nodes  The pressure can be calculated by analytical integration of the density profile and a
646pressure Jacobian method is used to solve the horizontal pressure gradient. This method can provide
647a more accurate calculation of the horizontal pressure gradient than the standard scheme.
648
649\subsection{Ice shelf cavity}
650\label{subsec:DYN_hpg_isf}
651Beneath an ice shelf, the total pressure gradient is the sum of the pressure gradient due to the ice shelf load and
652 the pressure gradient due to the ocean load. If cavity opened (\np{ln\_isfcav}\forcode{ = .true.}) these 2 terms can be
653 calculated by setting \np{ln\_dynhpg\_isf}\forcode{ = .true.}. No other scheme are working with the ice shelf.\\
654
655$\bullet$ The main hypothesis to compute the ice shelf load is that the ice shelf is in an isostatic equilibrium.
656 The top pressure is computed integrating from surface to the base of the ice shelf a reference density profile
657(prescribed as density of a water at 34.4 PSU and -1.9\degC) and corresponds to the water replaced by the ice shelf.
658This top pressure is constant over time. A detailed description of this method is described in \citet{Losch2008}.\\
659
660$\bullet$ The ocean load is computed using the expression \autoref{eq:dynhpg_sco} described in \autoref{subsec:DYN_hpg_sco}.
661
662%--------------------------------------------------------------------------------------------------------------
663%           Time-scheme
664%--------------------------------------------------------------------------------------------------------------
665\subsection{Time-scheme (\protect\np{ln\_dynhpg\_imp}\forcode{ = .true./.false.})}
666\label{subsec:DYN_hpg_imp}
667
668The default time differencing scheme used for the horizontal pressure gradient is
669a leapfrog scheme and therefore the density used in all discrete expressions given
670above is the  \textit{now} density, computed from the \textit{now} temperature and
671salinity. In some specific cases (usually high resolution simulations over an ocean
672domain which includes weakly stratified regions) the physical phenomenon that
673controls the time-step is internal gravity waves (IGWs). A semi-implicit scheme for
674doubling the stability limit associated with IGWs can be used \citep{Brown_Campana_MWR78,
675Maltrud1998}. It involves the evaluation of the hydrostatic pressure gradient as an
676average over the three time levels $t-\rdt$, $t$, and $t+\rdt$ ($i.e.$ 
677\textit{before}\textit{now} and  \textit{after} time-steps), rather than at the central
678time level $t$ only, as in the standard leapfrog scheme.
679
680$\bullet$ leapfrog scheme (\np{ln\_dynhpg\_imp}\forcode{ = .true.}):
681
682\begin{equation} \label{eq:dynhpg_lf}
683\frac{u^{t+\rdt}-u^{t-\rdt}}{2\rdt} = \;\cdots \;
684   -\frac{1}{\rho _o \,e_{1u} }\delta _{i+1/2} \left[ {p_h^t } \right]
685\end{equation}
686
687$\bullet$ semi-implicit scheme (\np{ln\_dynhpg\_imp}\forcode{ = .true.}):
688\begin{equation} \label{eq:dynhpg_imp}
689\frac{u^{t+\rdt}-u^{t-\rdt}}{2\rdt} = \;\cdots \;
690   -\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]
691\end{equation}
692
693The semi-implicit time scheme \autoref{eq:dynhpg_imp} is made possible without
694significant additional computation since the density can be updated to time level
695$t+\rdt$ before computing the horizontal hydrostatic pressure gradient. It can
696be easily shown that the stability limit associated with the hydrostatic pressure
697gradient doubles using \autoref{eq:dynhpg_imp} compared to that using the
698standard leapfrog scheme \autoref{eq:dynhpg_lf}. Note that \autoref{eq:dynhpg_imp} 
699is equivalent to applying a time filter to the pressure gradient to eliminate high
700frequency IGWs. Obviously, when using \autoref{eq:dynhpg_imp}, the doubling of
701the time-step is achievable only if no other factors control the time-step, such as
702the stability limits associated with advection or diffusion.
703
704In practice, the semi-implicit scheme is used when \np{ln\_dynhpg\_imp}\forcode{ = .true.}.
705In this case, we choose to apply the time filter to temperature and salinity used in
706the equation of state, instead of applying it to the hydrostatic pressure or to the
707density, so that no additional storage array has to be defined. The density used to
708compute the hydrostatic pressure gradient (whatever the formulation) is evaluated
709as follows:
710\begin{equation} \label{eq:rho_flt}
711   \rho^t = \rho( \widetilde{T},\widetilde {S},z_t)
712 \quad     \text{with}  \quad 
713   \widetilde{X} = 1 / 4 \left(  X^{t+\rdt} +2 \,X^t + X^{t-\rdt}  \right)
714\end{equation}
715
716Note that in the semi-implicit case, it is necessary to save the filtered density, an
717extra three-dimensional field, in the restart file to restart the model with exact
718reproducibility. This option is controlled by  \np{nn\_dynhpg\_rst}, a namelist parameter.
719
720% ================================================================
721% Surface Pressure Gradient
722% ================================================================
723\section{Surface pressure gradient (\protect\mdl{dynspg})}
724\label{sec:DYN_spg}
725%-----------------------------------------nam_dynspg----------------------------------------------------
726\forfile{../namelists/namdyn_spg} 
727%------------------------------------------------------------------------------------------------------------
728
729$\ $\newline      %force an empty line
730
731Options are defined through the \ngn{namdyn\_spg} namelist variables.
732The surface pressure gradient term is related to the representation of the free surface (\autoref{sec:PE_hor_pg}).
733The main distinction is between the fixed volume case (linear free surface) and the variable volume case
734(nonlinear free surface, \key{vvl} is defined). In the linear free surface case (\autoref{subsec:PE_free_surface})
735the vertical scale factors $e_{3}$ are fixed in time, while they are time-dependent in the nonlinear case
736(\autoref{subsec:PE_free_surface}).
737With both linear and nonlinear free surface, external gravity waves are allowed in the equations,
738which imposes a very small time step when an explicit time stepping is used.
739Two methods are proposed to allow a longer time step for the three-dimensional equations:
740the filtered free surface, which is a modification of the continuous equations (see \autoref{eq:PE_flt}),
741and the split-explicit free surface described below.
742The extra term introduced in the filtered method is calculated implicitly,
743so that the update of the next velocities is done in module \mdl{dynspg\_flt} and not in \mdl{dynnxt}.
744
745
746The form of the surface pressure gradient term depends on how the user wants to handle
747the fast external gravity waves that are a solution of the analytical equation (\autoref{sec:PE_hor_pg}).
748Three formulations are available, all controlled by a CPP key (ln\_dynspg\_xxx):
749an explicit formulation which requires a small time step ;
750a filtered free surface formulation which allows a larger time step by adding a filtering
751term into the momentum equation ;
752and a split-explicit free surface formulation, described below, which also allows a larger time step.
753
754The extra term introduced in the filtered method is calculated
755implicitly, so that a solver is used to compute it. As a consequence the update of the $next$ 
756velocities is done in module \mdl{dynspg\_flt} and not in \mdl{dynnxt}.
757
758
759%--------------------------------------------------------------------------------------------------------------
760% Explicit free surface formulation
761%--------------------------------------------------------------------------------------------------------------
762\subsection{Explicit free surface (\protect\key{dynspg\_exp})}
763\label{subsec:DYN_spg_exp}
764
765In the explicit free surface formulation (\key{dynspg\_exp} defined), the model time step
766is chosen to be small enough to resolve the external gravity waves (typically a few tens of seconds).
767The surface pressure gradient, evaluated using a leap-frog scheme ($i.e.$ centered in time),
768is thus simply given by :
769\begin{equation} \label{eq:dynspg_exp}
770\left\{ \begin{aligned}
771 - \frac{1}{e_{1u}\,\rho_o} \;   \delta _{i+1/2} \left[  \,\rho \,\eta\,  \right]   \\
772 - \frac{1}{e_{2v}\,\rho_o} \;   \delta _{j+1/2} \left[  \,\rho \,\eta\,  \right] 
773\end{aligned} \right.
774\end{equation} 
775
776Note that in the non-linear free surface case ($i.e.$ \key{vvl} defined), the surface pressure
777gradient is already included in the momentum tendency  through the level thickness variation
778allowed in the computation of the hydrostatic pressure gradient. Thus, nothing is done in the \mdl{dynspg\_exp} module.
779
780%--------------------------------------------------------------------------------------------------------------
781% Split-explict free surface formulation
782%--------------------------------------------------------------------------------------------------------------
783\subsection{Split-explicit free surface (\protect\key{dynspg\_ts})}
784\label{subsec:DYN_spg_ts}
785%------------------------------------------namsplit-----------------------------------------------------------
786%\forfile{../namelists/namsplit}
787%-------------------------------------------------------------------------------------------------------------
788
789The split-explicit free surface formulation used in \NEMO (\key{dynspg\_ts} defined),
790also called the time-splitting formulation, follows the one
791proposed by \citet{Shchepetkin_McWilliams_OM05}. The general idea is to solve the free surface
792equation and the associated barotropic velocity equations with a smaller time
793step than $\rdt$, the time step used for the three dimensional prognostic
794variables (\autoref{fig:DYN_dynspg_ts}).
795The size of the small time step, $\rdt_e$ (the external mode or barotropic time step)
796 is provided through the \np{nn\_baro} namelist parameter as:
797$\rdt_e = \rdt / nn\_baro$. This parameter can be optionally defined automatically (\np{ln\_bt\_nn\_auto}\forcode{ = .true.})
798considering that the stability of the barotropic system is essentially controled by external waves propagation.
799Maximum Courant number is in that case time independent, and easily computed online from the input bathymetry.
800Therefore, $\rdt_e$ is adjusted so that the Maximum allowed Courant number is smaller than \np{rn\_bt\_cmax}.
801
802%%%
803The barotropic mode solves the following equations:
804\begin{subequations} \label{eq:BT}
805  \begin{equation}     \label{eq:BT_dyn}
806\frac{\partial {\rm \overline{{\bf U}}_h} }{\partial t}=
807 -f\;{\rm {\bf k}}\times {\rm \overline{{\bf U}}_h} 
808-g\nabla _h \eta -\frac{c_b^{\textbf U}}{H+\eta} \rm {\overline{{\bf U}}_h} + \rm {\overline{\bf G}}
809  \end{equation}
810
811  \begin{equation} \label{eq:BT_ssh}
812\frac{\partial \eta }{\partial t}=-\nabla \cdot \left[ {\left( {H+\eta } \right) \; {\rm{\bf \overline{U}}}_h \,} \right]+P-E
813  \end{equation}
814\end{subequations}
815where $\rm {\overline{\bf G}}$ is a forcing term held constant, containing coupling term between modes, surface atmospheric forcing as well as slowly varying barotropic terms not explicitly computed to gain efficiency. The third term on the right hand side of \autoref{eq:BT_dyn} represents the bottom stress (see section \autoref{sec:ZDF_bfr}), explicitly accounted for at each barotropic iteration. Temporal discretization of the system above follows a three-time step Generalized Forward Backward algorithm detailed in \citet{Shchepetkin_McWilliams_OM05}. AB3-AM4 coefficients used in \NEMO follow the second-order accurate, "multi-purpose" stability compromise as defined in \citet{Shchepetkin_McWilliams_Bk08} (see their figure 12, lower left).
816
817%>   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >
818\begin{figure}[!t]    \begin{center}
819\includegraphics[width=0.7\textwidth]{Fig_DYN_dynspg_ts}
820\caption{  \protect\label{fig:DYN_dynspg_ts}
821Schematic of the split-explicit time stepping scheme for the external
822and internal modes. Time increases to the right. In this particular exemple,
823a boxcar averaging window over $nn\_baro$ barotropic time steps is used ($nn\_bt\_flt=1$) and $nn\_baro=5$.
824Internal mode time steps (which are also the model time steps) are denoted
825by $t-\rdt$, $t$ and $t+\rdt$. Variables with $k$ superscript refer to instantaneous barotropic variables,
826$< >$ and $<< >>$ operator refer to time filtered variables using respectively primary (red vertical bars) and secondary weights (blue vertical bars).
827The former are used to obtain time filtered quantities at $t+\rdt$ while the latter are used to obtain time averaged
828transports to advect tracers.
829a) Forward time integration: \protect\np{ln\_bt\_fw}\forcode{ = .true.}\protect\np{ln\_bt\_av}\forcode{ = .true.}.
830b) Centred time integration: \protect\np{ln\_bt\_fw}\forcode{ = .false.}, \protect\np{ln\_bt\_av}\forcode{ = .true.}.
831c) Forward time integration with no time filtering (POM-like scheme): \protect\np{ln\_bt\_fw}\forcode{ = .true.}, \protect\np{ln\_bt\_av}\forcode{ = .false.}. }
832\end{center}    \end{figure}
833%>   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >
834
835In the default case (\np{ln\_bt\_fw}\forcode{ = .true.}), the external mode is integrated
836between \textit{now} and  \textit{after} baroclinic time-steps (\autoref{fig:DYN_dynspg_ts}a). To avoid aliasing of fast barotropic motions into three dimensional equations, time filtering is eventually applied on barotropic
837quantities (\np{ln\_bt\_av}\forcode{ = .true.}). In that case, the integration is extended slightly beyond  \textit{after} time step to provide time filtered quantities.
838These are used for the subsequent initialization of the barotropic mode in the following baroclinic step.
839Since external mode equations written at baroclinic time steps finally follow a forward time stepping scheme,
840asselin filtering is not applied to barotropic quantities. \\
841Alternatively, one can choose to integrate barotropic equations starting
842from \textit{before} time step (\np{ln\_bt\_fw}\forcode{ = .false.}). Although more computationaly expensive ( \np{nn\_baro} additional iterations are indeed necessary), the baroclinic to barotropic forcing term given at \textit{now} time step
843become centred in the middle of the integration window. It can easily be shown that this property
844removes part of splitting errors between modes, which increases the overall numerical robustness.
845%references to Patrick Marsaleix' work here. Also work done by SHOM group.
846
847%%%
848
849As far as tracer conservation is concerned, barotropic velocities used to advect tracers must also be updated
850at \textit{now} time step. This implies to change the traditional order of computations in \NEMO: most of momentum 
851trends (including the barotropic mode calculation) updated first, tracers' after. This \textit{de facto} makes semi-implicit hydrostatic
852pressure gradient (see section \autoref{subsec:DYN_hpg_imp}) and time splitting not compatible.
853Advective barotropic velocities are obtained by using a secondary set of filtering weights, uniquely defined from the filter
854coefficients used for the time averaging (\citet{Shchepetkin_McWilliams_OM05}). Consistency between the time averaged continuity equation and the time stepping of tracers is here the key to obtain exact conservation.
855
856%%%
857
858One can eventually choose to feedback instantaneous values by not using any time filter (\np{ln\_bt\_av}\forcode{ = .false.}).
859In that case, external mode equations are continuous in time, ie they are not re-initialized when starting a new
860sub-stepping sequence. This is the method used so far in the POM model, the stability being maintained by refreshing at (almost)
861each barotropic time step advection and horizontal diffusion terms. Since the latter terms have not been added in \NEMO for
862computational efficiency, removing time filtering is not recommended except for debugging purposes.
863This may be used for instance to appreciate the damping effect of the standard formulation on external gravity waves in idealized or weakly non-linear cases. Although the damping is lower than for the filtered free surface, it is still significant as shown by \citet{Levier2007} in the case of an analytical barotropic Kelvin wave.
864
865%>>>>>===============
866\gmcomment{               %%% copy from griffies Book
867
868\textbf{title: Time stepping the barotropic system }
869
870Assume knowledge of the full velocity and tracer fields at baroclinic time $\tau$. Hence,
871we can update the surface height and vertically integrated velocity with a leap-frog
872scheme using the small barotropic time step $\rdt$. We have
873
874\begin{equation} \label{eq:DYN_spg_ts_eta}
875\eta^{(b)}(\tau,t_{n+1}) - \eta^{(b)}(\tau,t_{n+1}) (\tau,t_{n-1})
876   = 2 \rdt \left[-\nabla \cdot \textbf{U}^{(b)}(\tau,t_n) + \text{EMP}_w(\tau) \right] 
877\end{equation}
878\begin{multline} \label{eq:DYN_spg_ts_u}
879\textbf{U}^{(b)}(\tau,t_{n+1}) - \textbf{U}^{(b)}(\tau,t_{n-1}\\
880   = 2\rdt \left[ - f \textbf{k} \times \textbf{U}^{(b)}(\tau,t_{n})
881   - H(\tau) \nabla p_s^{(b)}(\tau,t_{n}) +\textbf{M}(\tau) \right]
882\end{multline}
883\
884
885In these equations, araised (b) denotes values of surface height and vertically integrated velocity updated with the barotropic time steps. The $\tau$ time label on $\eta^{(b)}$ 
886and $U^{(b)}$ denotes the baroclinic time at which the vertically integrated forcing $\textbf{M}(\tau)$ (note that this forcing includes the surface freshwater forcing), the tracer fields, the freshwater flux $\text{EMP}_w(\tau)$, and total depth of the ocean $H(\tau)$ are held for the duration of the barotropic time stepping over a single cycle. This is also the time
887that sets the barotropic time steps via
888\begin{equation} \label{eq:DYN_spg_ts_t}
889t_n=\tau+n\rdt   
890\end{equation}
891with $n$ an integer. The density scaled surface pressure is evaluated via
892\begin{equation} \label{eq:DYN_spg_ts_ps}
893p_s^{(b)}(\tau,t_{n}) = \begin{cases}
894   g \;\eta_s^{(b)}(\tau,t_{n}) \;\rho(\tau)_{k=1}) / \rho_&      \text{non-linear case} \\
895   g \;\eta_s^{(b)}(\tau,t_{n}&      \text{linear case} 
896   \end{cases}
897\end{equation}
898To get started, we assume the following initial conditions
899\begin{equation} \label{eq:DYN_spg_ts_eta}
900\begin{split}
901\eta^{(b)}(\tau,t_{n=0}) &= \overline{\eta^{(b)}(\tau)}
902\\
903\eta^{(b)}(\tau,t_{n=1}) &= \eta^{(b)}(\tau,t_{n=0}) + \rdt \ \text{RHS}_{n=0} 
904\end{split}
905\end{equation}
906with
907\begin{equation} \label{eq:DYN_spg_ts_etaF}
908 \overline{\eta^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N \eta^{(b)}(\tau-\rdt,t_{n})
909\end{equation}
910the time averaged surface height taken from the previous barotropic cycle. Likewise,
911\begin{equation} \label{eq:DYN_spg_ts_u}
912\textbf{U}^{(b)}(\tau,t_{n=0}) = \overline{\textbf{U}^{(b)}(\tau)}   \\
913\\
914\textbf{U}(\tau,t_{n=1}) = \textbf{U}^{(b)}(\tau,t_{n=0}) + \rdt \ \text{RHS}_{n=0}   
915\end{equation}
916with
917\begin{equation} \label{eq:DYN_spg_ts_u}
918 \overline{\textbf{U}^{(b)}(\tau)} 
919   = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau-\rdt,t_{n})
920\end{equation}
921the time averaged vertically integrated transport. Notably, there is no Robert-Asselin time filter used in the barotropic portion of the integration.
922
923Upon reaching $t_{n=N} = \tau + 2\rdt \tau$ , the vertically integrated velocity is time averaged to produce the updated vertically integrated velocity at baroclinic time $\tau + \rdt \tau$ 
924\begin{equation} \label{eq:DYN_spg_ts_u}
925\textbf{U}(\tau+\rdt) = \overline{\textbf{U}^{(b)}(\tau+\rdt)} 
926   = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau,t_{n})
927\end{equation}
928The surface height on the new baroclinic time step is then determined via a baroclinic leap-frog using the following form
929
930\begin{equation} \label{eq:DYN_spg_ts_ssh}
931\eta(\tau+\Delta) - \eta^{F}(\tau-\Delta) = 2\rdt \ \left[ - \nabla \cdot \textbf{U}(\tau) + \text{EMP}_w \right] 
932\end{equation}
933
934 The use of this "big-leap-frog" scheme for the surface height ensures compatibility between the mass/volume budgets and the tracer budgets. More discussion of this point is provided in Chapter 10 (see in particular Section 10.2).
935 
936In general, some form of time filter is needed to maintain integrity of the surface
937height field due to the leap-frog splitting mode in equation \autoref{eq:DYN_spg_ts_ssh}. We
938have tried various forms of such filtering, with the following method discussed in
939\cite{Griffies_al_MWR01} chosen due to its stability and reasonably good maintenance of
940tracer conservation properties (see ??)
941
942\begin{equation} \label{eq:DYN_spg_ts_sshf}
943\eta^{F}(\tau-\Delta) =  \overline{\eta^{(b)}(\tau)} 
944\end{equation}
945Another approach tried was
946
947\begin{equation} \label{eq:DYN_spg_ts_sshf2}
948\eta^{F}(\tau-\Delta) = \eta(\tau)
949   + (\alpha/2) \left[\overline{\eta^{(b)}}(\tau+\rdt)
950                + \overline{\eta^{(b)}}(\tau-\rdt) -2 \;\eta(\tau) \right]
951\end{equation}
952
953which is useful since it isolates all the time filtering aspects into the term multiplied
954by $\alpha$. This isolation allows for an easy check that tracer conservation is exact when
955eliminating tracer and surface height time filtering (see ?? for more complete discussion). However, in the general case with a non-zero $\alpha$, the filter \autoref{eq:DYN_spg_ts_sshf} was found to be more conservative, and so is recommended.
956
957}            %%end gm comment (copy of griffies book)
958
959%>>>>>===============
960
961
962%--------------------------------------------------------------------------------------------------------------
963% Filtered free surface formulation
964%--------------------------------------------------------------------------------------------------------------
965\subsection{Filtered free surface (\protect\key{dynspg\_flt})}
966\label{subsec:DYN_spg_fltp}
967
968The filtered formulation follows the \citet{Roullet_Madec_JGR00} implementation.
969The extra term introduced in the equations (see \autoref{subsec:PE_free_surface}) is solved implicitly.
970The elliptic solvers available in the code are documented in \autoref{chap:MISC}.
971
972%% gm %%======>>>>   given here the discrete eqs provided to the solver
973\gmcomment{               %%% copy from chap-model basics
974\begin{equation} \label{eq:spg_flt}
975\frac{\partial {\rm {\bf U}}_h }{\partial t}= {\rm {\bf M}}
976- g \nabla \left( \tilde{\rho} \ \eta \right)
977- g \ T_c \nabla \left( \widetilde{\rho} \ \partial_t \eta \right)
978\end{equation}
979where $T_c$, is a parameter with dimensions of time which characterizes the force,
980$\widetilde{\rho} = \rho / \rho_o$ is the dimensionless density, and $\rm {\bf M}$ 
981represents the collected contributions of the Coriolis, hydrostatic pressure gradient,
982non-linear and viscous terms in \autoref{eq:PE_dyn}.
983}   %end gmcomment
984
985Note that in the linear free surface formulation (\key{vvl} not defined), the ocean depth
986is time-independent and so is the matrix to be inverted. It is computed once and for all and applies to all ocean time steps.
987
988% ================================================================
989% Lateral diffusion term
990% ================================================================
991\section{Lateral diffusion term and operators (\protect\mdl{dynldf})}
992\label{sec:DYN_ldf}
993%------------------------------------------nam_dynldf----------------------------------------------------
994\forfile{../namelists/namdyn_ldf} 
995%-------------------------------------------------------------------------------------------------------------
996
997Options are defined through the \ngn{namdyn\_ldf} namelist variables.
998The options available for lateral diffusion are to use either laplacian
999(rotated or not) or biharmonic operators. The coefficients may be constant
1000or spatially variable; the description of the coefficients is found in the chapter
1001on lateral physics (\autoref{chap:LDF}). The lateral diffusion of momentum is
1002evaluated using a forward scheme, $i.e.$ the velocity appearing in its expression
1003is the \textit{before} velocity in time, except for the pure vertical component
1004that appears when a tensor of rotation is used. This latter term is solved
1005implicitly together with the vertical diffusion term (see \autoref{chap:STP})
1006
1007At the lateral boundaries either free slip, no slip or partial slip boundary
1008conditions are applied according to the user's choice (see \autoref{chap:LBC}).
1009
1010\gmcomment{
1011Hyperviscous operators are frequently used in the simulation of turbulent flows to control
1012the dissipation of unresolved small scale features.
1013Their primary role is to provide strong dissipation at the smallest scale supported by the grid
1014while minimizing the impact on the larger scale features.
1015Hyperviscous operators are thus designed to be more scale selective than the traditional,
1016physically motivated Laplace operator.
1017In finite difference methods, the biharmonic operator is frequently the method of choice to achieve
1018this scale selective dissipation since its damping time ($i.e.$ its spin down time)
1019scale like $\lambda^{-4}$ for disturbances of wavelength $\lambda$ 
1020(so that short waves damped more rapidelly than long ones),
1021whereas the Laplace operator damping time scales only like $\lambda^{-2}$.
1022}
1023
1024% ================================================================
1025\subsection[Iso-level laplacian (\protect\np{ln\_dynldf\_lap}\forcode{ = .true.})]
1026            {Iso-level laplacian operator (\protect\np{ln\_dynldf\_lap}\forcode{ = .true.})}
1027\label{subsec:DYN_ldf_lap}
1028
1029For lateral iso-level diffusion, the discrete operator is:
1030\begin{equation} \label{eq:dynldf_lap}
1031\left\{ \begin{aligned}
1032 D_u^{l{\rm {\bf U}}} =\frac{1}{e_{1u} }\delta _{i+1/2} \left[ {A_T^{lm} 
1033\;\chi } \right]-\frac{1}{e_{2u} {\kern 1pt}e_{3u} }\delta _j \left[
1034{A_f^{lm} \;e_{3f} \zeta } \right] \\ 
1035\\
1036 D_v^{l{\rm {\bf U}}} =\frac{1}{e_{2v} }\delta _{j+1/2} \left[ {A_T^{lm} 
1037\;\chi } \right]+\frac{1}{e_{1v} {\kern 1pt}e_{3v} }\delta _i \left[
1038{A_f^{lm} \;e_{3f} \zeta } \right] \\ 
1039\end{aligned} \right.
1040\end{equation} 
1041
1042As explained in \autoref{subsec:PE_ldf}, this formulation (as the gradient of a divergence
1043and curl of the vorticity) preserves symmetry and ensures a complete
1044separation between the vorticity and divergence parts of the momentum diffusion.
1045
1046%--------------------------------------------------------------------------------------------------------------
1047%           Rotated laplacian operator
1048%--------------------------------------------------------------------------------------------------------------
1049\subsection[Rotated laplacian (\protect\np{ln\_dynldf\_iso}\forcode{ = .true.})]
1050            {Rotated laplacian operator (\protect\np{ln\_dynldf\_iso}\forcode{ = .true.})}
1051\label{subsec:DYN_ldf_iso}
1052
1053A rotation of the lateral momentum diffusion operator is needed in several cases:
1054for iso-neutral diffusion in the $z$-coordinate (\np{ln\_dynldf\_iso}\forcode{ = .true.}) and for
1055either iso-neutral (\np{ln\_dynldf\_iso}\forcode{ = .true.}) or geopotential
1056(\np{ln\_dynldf\_hor}\forcode{ = .true.}) diffusion in the $s$-coordinate. In the partial step
1057case, coordinates are horizontal except at the deepest level and no
1058rotation is performed when \np{ln\_dynldf\_hor}\forcode{ = .true.}. The diffusion operator
1059is defined simply as the divergence of down gradient momentum fluxes on each
1060momentum component. It must be emphasized that this formulation ignores
1061constraints on the stress tensor such as symmetry. The resulting discrete
1062representation is:
1063\begin{equation} \label{eq:dyn_ldf_iso}
1064\begin{split}
1065 D_u^{l\textbf{U}} &= \frac{1}{e_{1u} \, e_{2u} \, e_{3u} } \\
1066&  \left\{\quad  {\delta _{i+1/2} \left[ {A_T^{lm}  \left(
1067    {\frac{e_{2t} \; e_{3t} }{e_{1t} } \,\delta _{i}[u]
1068   -e_{2t} \; r_{1t} \,\overline{\overline {\delta _{k+1/2}[u]}}^{\,i,\,k}}
1069 \right)} \right]}   \right.
1070\\ 
1071& \qquad +\ \delta_j \left[ {A_f^{lm} \left( {\frac{e_{1f}\,e_{3f} }{e_{2f} 
1072}\,\delta _{j+1/2} [u] - e_{1f}\, r_{2f} 
1073\,\overline{\overline {\delta _{k+1/2} [u]}} ^{\,j+1/2,\,k}} 
1074\right)} \right]
1075\\ 
1076&\qquad +\ \delta_k \left[ {A_{uw}^{lm} \left( {-e_{2u} \, r_{1uw} \,\overline{\overline 
1077{\delta_{i+1/2} [u]}}^{\,i+1/2,\,k+1/2} } 
1078\right.} \right.
1079\\ 
1080&  \ \qquad \qquad \qquad \quad\
1081- e_{1u} \, r_{2uw} \,\overline{\overline {\delta_{j+1/2} [u]}} ^{\,j,\,k+1/2}
1082\\ 
1083& \left. {\left. { \ \qquad \qquad \qquad \ \ \ \left. {\
1084+\frac{e_{1u}\, e_{2u} }{e_{3uw} }\,\left( {r_{1uw}^2+r_{2uw}^2} 
1085\right)\,\delta_{k+1/2} [u]} \right)} \right]\;\;\;} \right\} 
1086\\
1087\\
1088 D_v^{l\textbf{V}} &= \frac{1}{e_{1v} \, e_{2v} \, e_{3v} }    \\
1089&  \left\{\quad  {\delta _{i+1/2} \left[ {A_f^{lm}  \left(
1090    {\frac{e_{2f} \; e_{3f} }{e_{1f} } \,\delta _{i+1/2}[v]
1091   -e_{2f} \; r_{1f} \,\overline{\overline {\delta _{k+1/2}[v]}}^{\,i+1/2,\,k}}
1092 \right)} \right]}   \right.
1093\\ 
1094& \qquad +\ \delta_j \left[ {A_T^{lm} \left( {\frac{e_{1t}\,e_{3t} }{e_{2t} 
1095}\,\delta _{j} [v] - e_{1t}\, r_{2t} 
1096\,\overline{\overline {\delta _{k+1/2} [v]}} ^{\,j,\,k}} 
1097\right)} \right]
1098\\ 
1099& \qquad +\ \delta_k \left[ {A_{vw}^{lm} \left( {-e_{2v} \, r_{1vw} \,\overline{\overline 
1100{\delta_{i+1/2} [v]}}^{\,i+1/2,\,k+1/2} }\right.} \right.
1101\\
1102&  \ \qquad \qquad \qquad \quad\
1103- e_{1v} \, r_{2vw} \,\overline{\overline {\delta_{j+1/2} [v]}} ^{\,j+1/2,\,k+1/2}
1104\\ 
1105& \left. {\left. { \ \qquad \qquad \qquad \ \ \ \left. {\
1106+\frac{e_{1v}\, e_{2v} }{e_{3vw} }\,\left( {r_{1vw}^2+r_{2vw}^2} 
1107\right)\,\delta_{k+1/2} [v]} \right)} \right]\;\;\;} \right\} 
1108 \end{split}
1109\end{equation}
1110where $r_1$ and $r_2$ are the slopes between the surface along which the
1111diffusion operator acts and the surface of computation ($z$- or $s$-surfaces).
1112The way these slopes are evaluated is given in the lateral physics chapter
1113(\autoref{chap:LDF}).
1114
1115%--------------------------------------------------------------------------------------------------------------
1116%           Iso-level bilaplacian operator
1117%--------------------------------------------------------------------------------------------------------------
1118\subsection[Iso-level bilaplacian (\protect\np{ln\_dynldf\_bilap}\forcode{ = .true.})]
1119            {Iso-level bilaplacian operator (\protect\np{ln\_dynldf\_bilap}\forcode{ = .true.})}
1120\label{subsec:DYN_ldf_bilap}
1121
1122The lateral fourth order operator formulation on momentum is obtained by
1123applying \autoref{eq:dynldf_lap} twice. It requires an additional assumption on
1124boundary conditions: the first derivative term normal to the coast depends on
1125the free or no-slip lateral boundary conditions chosen, while the third
1126derivative terms normal to the coast are set to zero (see \autoref{chap:LBC}).
1127%%%
1128\gmcomment{add a remark on the the change in the position of the coefficient}
1129%%%
1130
1131% ================================================================
1132%           Vertical diffusion term
1133% ================================================================
1134\section{Vertical diffusion term (\protect\mdl{dynzdf})}
1135\label{sec:DYN_zdf}
1136%----------------------------------------------namzdf------------------------------------------------------
1137\forfile{../namelists/namzdf} 
1138%-------------------------------------------------------------------------------------------------------------
1139
1140Options are defined through the \ngn{namzdf} namelist variables.
1141The large vertical diffusion coefficient found in the surface mixed layer together
1142with high vertical resolution implies that in the case of explicit time stepping there
1143would be too restrictive a constraint on the time step. Two time stepping schemes
1144can be used for the vertical diffusion term : $(a)$ a forward time differencing
1145scheme (\np{ln\_zdfexp}\forcode{ = .true.}) using a time splitting technique
1146(\np{nn\_zdfexp} $>$ 1) or $(b)$ a backward (or implicit) time differencing scheme
1147(\np{ln\_zdfexp}\forcode{ = .false.}) (see \autoref{chap:STP}). Note that namelist variables
1148\np{ln\_zdfexp} and \np{nn\_zdfexp} apply to both tracers and dynamics.
1149
1150The formulation of the vertical subgrid scale physics is the same whatever
1151the vertical coordinate is. The vertical diffusion operators given by
1152\autoref{eq:PE_zdf} take the following semi-discrete space form:
1153\begin{equation} \label{eq:dynzdf}
1154\left\{   \begin{aligned}
1155D_u^{vm} &\equiv \frac{1}{e_{3u}} \ \delta _k \left[ \frac{A_{uw}^{vm} }{e_{3uw} }
1156                              \ \delta _{k+1/2} [\,u\,]         \right]     \\
1157\\
1158D_v^{vm} &\equiv \frac{1}{e_{3v}} \ \delta _k \left[ \frac{A_{vw}^{vm} }{e_{3vw} }
1159                              \ \delta _{k+1/2} [\,v\,]         \right]
1160\end{aligned}   \right.
1161\end{equation} 
1162where $A_{uw}^{vm} $ and $A_{vw}^{vm} $ are the vertical eddy viscosity and
1163diffusivity coefficients. The way these coefficients are evaluated
1164depends on the vertical physics used (see \autoref{chap:ZDF}).
1165
1166The surface boundary condition on momentum is the stress exerted by
1167the wind. At the surface, the momentum fluxes are prescribed as the boundary
1168condition on the vertical turbulent momentum fluxes,
1169\begin{equation} \label{eq:dynzdf_sbc}
1170\left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{z=1}
1171    = \frac{1}{\rho _o} \binom{\tau _u}{\tau _v }
1172\end{equation}
1173where $\left( \tau _u ,\tau _v \right)$ are the two components of the wind stress
1174vector in the (\textbf{i},\textbf{j}) coordinate system. The high mixing coefficients
1175in the surface mixed layer ensure that the surface wind stress is distributed in
1176the vertical over the mixed layer depth. If the vertical mixing coefficient
1177is small (when no mixed layer scheme is used) the surface stress enters only
1178the top model level, as a body force. The surface wind stress is calculated
1179in the surface module routines (SBC, see \autoref{chap:SBC})
1180
1181The turbulent flux of momentum at the bottom of the ocean is specified through
1182a bottom friction parameterisation (see \autoref{sec:ZDF_bfr})
1183
1184% ================================================================
1185% External Forcing
1186% ================================================================
1187\section{External forcings}
1188\label{sec:DYN_forcing}
1189
1190Besides the surface and bottom stresses (see the above section) which are
1191introduced as boundary conditions on the vertical mixing, three other forcings
1192may enter the dynamical equations by affecting the surface pressure gradient.
1193
1194(1) When \np{ln\_apr\_dyn}\forcode{ = .true.} (see \autoref{sec:SBC_apr}), the atmospheric pressure is taken
1195into account when computing the surface pressure gradient.
1196
1197(2) When \np{ln\_tide\_pot}\forcode{ = .true.} and \np{ln\_tide}\forcode{ = .true.} (see \autoref{sec:SBC_tide}),
1198the tidal potential is taken into account when computing the surface pressure gradient.
1199
1200(3) When \np{nn\_ice\_embd}\forcode{ = 2} and LIM or CICE is used ($i.e.$ when the sea-ice is embedded in the ocean),
1201the snow-ice mass is taken into account when computing the surface pressure gradient.
1202
1203
1204\gmcomment{ missing : the lateral boundary condition !!!   another external forcing
1205 }
1206
1207% ================================================================
1208% Time evolution term
1209% ================================================================
1210\section{Time evolution term (\protect\mdl{dynnxt})}
1211\label{sec:DYN_nxt}
1212
1213%----------------------------------------------namdom----------------------------------------------------
1214\forfile{../namelists/namdom} 
1215%-------------------------------------------------------------------------------------------------------------
1216
1217Options are defined through the \ngn{namdom} namelist variables.
1218The general framework for dynamics time stepping is a leap-frog scheme,
1219$i.e.$ a three level centred time scheme associated with an Asselin time filter
1220(cf. \autoref{chap:STP}). The scheme is applied to the velocity, except when using
1221the flux form of momentum advection (cf. \autoref{sec:DYN_adv_cor_flux}) in the variable
1222volume case (\key{vvl} defined), where it has to be applied to the thickness
1223weighted velocity (see \autoref{sec:A_momentum}
1224
1225$\bullet$ vector invariant form or linear free surface (\np{ln\_dynhpg\_vec}\forcode{ = .true.} ; \key{vvl} not defined):
1226\begin{equation} \label{eq:dynnxt_vec}
1227\left\{   \begin{aligned}
1228&u^{t+\rdt} = u_f^{t-\rdt} + 2\rdt  \ \text{RHS}_u^t     \\
1229&u_f^t \;\quad = u^t+\gamma \,\left[ {u_f^{t-\rdt} -2u^t+u^{t+\rdt}} \right]
1230\end{aligned}   \right.
1231\end{equation} 
1232
1233$\bullet$ flux form and nonlinear free surface (\np{ln\_dynhpg\_vec}\forcode{ = .false.} ; \key{vvl} defined):
1234\begin{equation} \label{eq:dynnxt_flux}
1235\left\{   \begin{aligned}
1236&\left(e_{3u}\,u\right)^{t+\rdt} = \left(e_{3u}\,u\right)_f^{t-\rdt} + 2\rdt \; e_{3u} \;\text{RHS}_u^t     \\
1237&\left(e_{3u}\,u\right)_f^t \;\quad = \left(e_{3u}\,u\right)^t
1238  +\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]
1239\end{aligned}   \right.
1240\end{equation} 
1241where RHS is the right hand side of the momentum equation, the subscript $f$ 
1242denotes filtered values and $\gamma$ is the Asselin coefficient. $\gamma$ is
1243initialized as \np{nn\_atfp} (namelist parameter). Its default value is \np{nn\_atfp}\forcode{ = 10.e-3}.
1244In both cases, the modified Asselin filter is not applied since perfect conservation
1245is not an issue for the momentum equations.
1246
1247Note that with the filtered free surface, the update of the \textit{after} velocities
1248is done in the \mdl{dynsp\_flt} module, and only array swapping
1249and Asselin filtering is done in \mdl{dynnxt}.
1250
1251% ================================================================
1252\end{document}
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