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1% ================================================================
2% Chapter ——— Ocean Dynamics (DYN)
3% ================================================================
4\chapter{Ocean Dynamics (DYN)}
5\label{DYN}
6\minitoc
7
8%\vspace{2.cm}
9$\ $\newline      %force an empty line
10
11Using the representation described in Chapter \ref{DOM}, several semi-discrete
12space forms of the dynamical equations are available depending on the vertical
13coordinate used and on the conservation properties of the vorticity term. In all
14the equations presented here, the masking has been omitted for simplicity.
15One must be aware that all the quantities are masked fields and that each time an
16average or difference operator is used, the resulting field is multiplied by a mask.
17
18The prognostic ocean dynamics equation can be summarized as follows:
19\begin{equation*}
20\text{NXT} = \dbinom {\text{VOR} + \text{KEG} + \text {ZAD} }
21                  {\text{COR} + \text{ADV}                       }
22         + \text{HPG} + \text{SPG} + \text{LDF} + \text{ZDF}
23\end{equation*}
24NXT stands for next, referring to the time-stepping. The first group of terms on
25the rhs of this equation corresponds to the Coriolis and advection
26terms that are decomposed into either a vorticity part (VOR), a kinetic energy part (KEG)
27and a vertical advection part (ZAD) in the vector invariant formulation, or a Coriolis
28and advection part (COR+ADV) in the flux formulation. The terms following these
29are the pressure gradient contributions (HPG, Hydrostatic Pressure Gradient,
30and SPG, Surface Pressure Gradient); and contributions from lateral diffusion
31(LDF) and vertical diffusion (ZDF), which are added to the rhs in the \mdl{dynldf} 
32and \mdl{dynzdf} modules. The vertical diffusion term includes the surface and
33bottom stresses. The external forcings and parameterisations require complex
34inputs (surface wind stress calculation using bulk formulae, estimation of mixing
35coefficients) that are carried out in modules SBC, LDF and ZDF and are described
36in Chapters \ref{SBC}, \ref{LDF} and \ref{ZDF}, respectively.
37
38In the present chapter we also describe the diagnostic equations used to compute
39the horizontal divergence, curl of the velocities (\emph{divcur} module) and
40the vertical velocity (\emph{wzvmod} module).
41
42The different options available to the user are managed by namelist variables.
43For term \textit{ttt} in the momentum equations, the logical namelist variables are \textit{ln\_dynttt\_xxx},
44where \textit{xxx} is a 3 or 4 letter acronym corresponding to each optional scheme.
45If a CPP key is used for this term its name is \textbf{key\_ttt}. The corresponding
46code can be found in the \textit{dynttt\_xxx} module in the DYN directory, and it is
47usually computed in the \textit{dyn\_ttt\_xxx} subroutine.
48
49The user has the option of extracting and outputting each tendency term from the
503D momentum equations (\key{trddyn} defined), as described in
51Chap.\ref{MISC}.  Furthermore, the tendency terms associated with the 2D
52barotropic vorticity balance (when \key{trdvor} is defined) can be derived from the
533D 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$\ $\newline    % force a new ligne
59
60% ================================================================
61% Sea Surface Height evolution & Diagnostics variables
62% ================================================================
63\section{Sea surface height and diagnostic variables ($\eta$, $\zeta$, $\chi$, $w$)}
64\label{DYN_divcur_wzv}
65
66%--------------------------------------------------------------------------------------------------------------
67%           Horizontal divergence and relative vorticity
68%--------------------------------------------------------------------------------------------------------------
69\subsection   [Horizontal divergence and relative vorticity (\textit{divcur})]
70         {Horizontal divergence and relative vorticity (\mdl{divcur})}
71\label{DYN_divcur}
72
73The vorticity is defined at an $f$-point ($i.e.$ corner point) as follows:
74\begin{equation} \label{Eq_divcur_cur}
75\zeta =\frac{1}{e_{1f}\,e_{2f} }\left( {\;\delta _{i+1/2} \left[ {e_{2v}\;v} \right]
76                          -\delta _{j+1/2} \left[ {e_{1u}\;u} \right]\;} \right)
77\end{equation} 
78
79The horizontal divergence is defined at a $T$-point. It is given by:
80\begin{equation} \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\end{equation} 
85
86Note that although the vorticity has the same discrete expression in $z$-
87and $s$-coordinates, its physical meaning is not identical. $\zeta$ is a pseudo
88vorticity along $s$-surfaces (only pseudo because $(u,v)$ are still defined along
89geopotential surfaces, but are not necessarily defined at the same depth).
90
91The vorticity and divergence at the \textit{before} step are used in the computation
92of the horizontal diffusion of momentum. Note that because they have been
93calculated prior to the Asselin filtering of the \textit{before} velocities, the
94\textit{before} vorticity and divergence arrays must be included in the restart file
95to ensure perfect restartability. The vorticity and divergence at the \textit{now} 
96time step are used for the computation of the nonlinear advection and of the
97vertical velocity respectively.
98
99%--------------------------------------------------------------------------------------------------------------
100%           Sea Surface Height evolution
101%--------------------------------------------------------------------------------------------------------------
102\subsection   [Sea surface height evolution and vertical velocity (\textit{sshwzv})]
103         {Horizontal divergence and relative vorticity (\mdl{sshwzv})}
104\label{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 \S\ref{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 \eqref{Eq_tra_nxt}:
123a leapfrog scheme in combination with an Asselin time filter, $i.e.$ the velocity appearing
124in \eqref{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 \eqref{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 \eqref{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 \eqref{Eq_wzv} 
153(see  \S\ref{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{DYN_adv_cor_vect}
161%-----------------------------------------nam_dynadv----------------------------------------------------
162\namdisplay{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 Chap.\ref{LBC}.
174
175% -------------------------------------------------------------------------------------------------------------
176%        Vorticity term
177% -------------------------------------------------------------------------------------------------------------
178\subsection   [Vorticity term (\textit{dynvor}) ]
179         {Vorticity term (\mdl{dynvor})}
180\label{DYN_vor}
181%------------------------------------------nam_dynvor----------------------------------------------------
182\namdisplay{namdyn_vor} 
183%-------------------------------------------------------------------------------------------------------------
184
185Options are defined through the \ngn{namdyn\_vor} namelist variables.
186Four discretisations of the vorticity term (\textit{ln\_dynvor\_xxx}=true) are available:
187conserving potential enstrophy of horizontally non-divergent flow (ENS scheme) ;
188conserving horizontal kinetic energy (ENE scheme) ; conserving potential enstrophy for
189the relative vorticity term and horizontal kinetic energy for the planetary vorticity
190term (MIX scheme) ; or conserving both the potential enstrophy of horizontally non-divergent
191flow and horizontal kinetic energy (EEN scheme) (see  Appendix~\ref{Apdx_C_vorEEN}). In the
192case of ENS, ENE or MIX schemes the land sea mask may be slightly modified to ensure the
193consistency of vorticity term with analytical equations (\textit{ln\_dynvor\_con}=true).
194The vorticity terms are all computed in dedicated routines that can be found in
195the \mdl{dynvor} module.
196
197%-------------------------------------------------------------
198%                 enstrophy conserving scheme
199%-------------------------------------------------------------
200\subsubsection{Enstrophy conserving scheme (\np{ln\_dynvor\_ens}=true)}
201\label{DYN_vor_ens}
202
203In the enstrophy conserving case (ENS scheme), the discrete formulation of the
204vorticity term provides a global conservation of the enstrophy
205($ [ (\zeta +f ) / e_{3f} ]^2 $ in $s$-coordinates) for a horizontally non-divergent
206flow ($i.e.$ $\chi$=$0$), but does not conserve the total kinetic energy. It is given by:
207\begin{equation} \label{Eq_dynvor_ens}
208\left\{ 
209\begin{aligned}
210{+\frac{1}{e_{1u} } } & {\overline {\left( { \frac{\zeta +f}{e_{3f} }} \right)} }^{\,i} 
211                                & {\overline{\overline {\left( {e_{1v}\,e_{3v}\;v} \right)}} }^{\,i, j+1/2}    \\
212{- \frac{1}{e_{2v} } } & {\overline {\left( {\frac{\zeta +f}{e_{3f} }} \right)} }^{\,j} 
213                                & {\overline{\overline {\left( {e_{2u}\,e_{3u}\;u} \right)}} }^{\,i+1/2, j} 
214\end{aligned} 
215 \right.
216\end{equation} 
217
218%-------------------------------------------------------------
219%                 energy conserving scheme
220%-------------------------------------------------------------
221\subsubsection{Energy conserving scheme (\np{ln\_dynvor\_ene}=true)}
222\label{DYN_vor_ene}
223
224The kinetic energy conserving scheme (ENE scheme) conserves the global
225kinetic energy but not the global enstrophy. It is given by:
226\begin{equation} \label{Eq_dynvor_ene}
227\left\{   \begin{aligned}
228{+\frac{1}{e_{1u}}\; {\overline {\left( {\frac{\zeta +f}{e_{3f} }} \right)
229                            \;  \overline {\left( {e_{1v}\,e_{3v}\;v} \right)} ^{\,i+1/2}} }^{\,j} }    \\
230{- \frac{1}{e_{2v}}\; {\overline {\left( {\frac{\zeta +f}{e_{3f} }} \right)
231                            \;  \overline {\left( {e_{2u}\,e_{3u}\;u} \right)} ^{\,j+1/2}} }^{\,i} }
232\end{aligned}    \right.
233\end{equation} 
234
235%-------------------------------------------------------------
236%                 mix energy/enstrophy conserving scheme
237%-------------------------------------------------------------
238\subsubsection{Mixed energy/enstrophy conserving scheme (\np{ln\_dynvor\_mix}=true) }
239\label{DYN_vor_mix}
240
241For the mixed energy/enstrophy conserving scheme (MIX scheme), a mixture of the
242two previous schemes is used. It consists of the ENS scheme (\ref{Eq_dynvor_ens})
243for the relative vorticity term, and of the ENE scheme (\ref{Eq_dynvor_ene}) applied
244to the planetary vorticity term.
245\begin{equation} \label{Eq_dynvor_mix}
246\left\{ {     \begin{aligned}
247 {+\frac{1}{e_{1u} }\; {\overline {\left( {\frac{\zeta }{e_{3f} }} \right)} }^{\,i} 
248 \; {\overline{\overline {\left( {e_{1v}\,e_{3v}\;v} \right)}} }^{\,i,j+1/2} -\frac{1}{e_{1u} }
249 \; {\overline {\left( {\frac{f}{e_{3f} }} \right)
250 \;\overline {\left( {e_{1v}\,e_{3v}\;v} \right)} ^{\,i+1/2}} }^{\,j} } \\
251{-\frac{1}{e_{2v} }\; {\overline {\left( {\frac{\zeta }{e_{3f} }} \right)} }^j
252 \; {\overline{\overline {\left( {e_{2u}\,e_{3u}\;u} \right)}} }^{\,i+1/2,j} +\frac{1}{e_{2v} }
253 \; {\overline {\left( {\frac{f}{e_{3f} }} \right)
254 \;\overline {\left( {e_{2u}\,e_{3u}\;u} \right)} ^{\,j+1/2}} }^{\,i} } \hfill
255\end{aligned}     } \right.
256\end{equation} 
257
258%-------------------------------------------------------------
259%                 energy and enstrophy conserving scheme
260%-------------------------------------------------------------
261\subsubsection{Energy and enstrophy conserving scheme (\np{ln\_dynvor\_een}=true) }
262\label{DYN_vor_een}
263
264In both the ENS and ENE schemes, it is apparent that the combination of $i$ and $j$ 
265averages of the velocity allows for the presence of grid point oscillation structures
266that will be invisible to the operator. These structures are \textit{computational modes} 
267that will be at least partly damped by the momentum diffusion operator ($i.e.$ the
268subgrid-scale advection), but not by the resolved advection term. The ENS and ENE schemes
269therefore do not contribute to dump any grid point noise in the horizontal velocity field.
270Such noise would result in more noise in the vertical velocity field, an undesirable feature.
271This is a well-known characteristic of $C$-grid discretization where $u$ and $v$ are located
272at different grid points, a price worth paying to avoid a double averaging in the pressure
273gradient term as in the $B$-grid.
274\gmcomment{ To circumvent this, Adcroft (ADD REF HERE)
275Nevertheless, this technique strongly distort the phase and group velocity of Rossby waves....}
276
277A very nice solution to the problem of double averaging was proposed by \citet{Arakawa_Hsu_MWR90}.
278The idea is to get rid of the double averaging by considering triad combinations of vorticity.
279It is noteworthy that this solution is conceptually quite similar to the one proposed by
280\citep{Griffies_al_JPO98} for the discretization of the iso-neutral diffusion operator (see App.\ref{Apdx_C}).
281
282The \citet{Arakawa_Hsu_MWR90} vorticity advection scheme for a single layer is modified
283for spherical coordinates as described by \citet{Arakawa_Lamb_MWR81} to obtain the EEN scheme.
284First consider the discrete expression of the potential vorticity, $q$, defined at an $f$-point:
285\begin{equation} \label{Eq_pot_vor}
286q  = \frac{\zeta +f} {e_{3f} }
287\end{equation}
288where the relative vorticity is defined by (\ref{Eq_divcur_cur}), the Coriolis parameter
289is given by $f=2 \,\Omega \;\sin \varphi _f $ and the layer thickness at $f$-points is:
290\begin{equation} \label{Eq_een_e3f}
291e_{3f} = \overline{\overline {e_{3t} }} ^{\,i+1/2,j+1/2}
292\end{equation}
293
294%>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
295\begin{figure}[!ht]    \begin{center}
296\includegraphics[width=0.70\textwidth]{./TexFiles/Figures/Fig_DYN_een_triad.pdf}
297\caption{ \label{Fig_DYN_een_triad} 
298Triads used in the energy and enstrophy conserving scheme (een) for
299$u$-component (upper panel) and $v$-component (lower panel).}
300\end{center}   \end{figure}
301%>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
302
303Note that a key point in \eqref{Eq_een_e3f} is that the averaging in the \textbf{i}- and
304\textbf{j}- directions uses the masked vertical scale factor but is always divided by
305$4$, not by the sum of the masks at the four $T$-points. This preserves the continuity of
306$e_{3f}$ when one or more of the neighbouring $e_{3t}$ tends to zero and
307extends by continuity the value of $e_{3f}$ into the land areas. This feature is essential for
308the $z$-coordinate with partial steps.
309
310Next, the vorticity triads, $ {^i_j}\mathbb{Q}^{i_p}_{j_p}$ can be defined at a $T$-point as
311the following triad combinations of the neighbouring potential vorticities defined at f-points
312(Fig.~\ref{Fig_DYN_een_triad}):
313\begin{equation} \label{Q_triads}
314_i^j \mathbb{Q}^{i_p}_{j_p}
315= \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)
316\end{equation}
317where the indices $i_p$ and $k_p$ take the values: $i_p = -1/2$ or $1/2$ and $j_p = -1/2$ or $1/2$.
318
319Finally, the vorticity terms are represented as:
320\begin{equation} \label{Eq_dynvor_een}
321\left\{ {
322\begin{aligned}
323 +q\,e_3 \, v  &\equiv +\frac{1}{e_{1u} }   \sum_{\substack{i_p,\,k_p}} 
324                         {^{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}   \\
325 - q\,e_3 \, u     &\equiv -\frac{1}{e_{2v} }    \sum_{\substack{i_p,\,k_p}} 
326                         {^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}   \\
327\end{aligned} 
328} \right.
329\end{equation} 
330
331This EEN scheme in fact combines the conservation properties of the ENS and ENE schemes.
332It conserves both total energy and potential enstrophy in the limit of horizontally
333nondivergent flow ($i.e.$ $\chi$=$0$) (see  Appendix~\ref{Apdx_C_vorEEN}).
334Applied to a realistic ocean configuration, it has been shown that it leads to a significant
335reduction of the noise in the vertical velocity field \citep{Le_Sommer_al_OM09}.
336Furthermore, used in combination with a partial steps representation of bottom topography,
337it improves the interaction between current and topography, leading to a larger
338topostrophy of the flow  \citep{Barnier_al_OD06, Penduff_al_OS07}.
339
340%--------------------------------------------------------------------------------------------------------------
341%           Kinetic Energy Gradient term
342%--------------------------------------------------------------------------------------------------------------
343\subsection   [Kinetic Energy Gradient term (\textit{dynkeg})]
344         {Kinetic Energy Gradient term (\mdl{dynkeg})}
345\label{DYN_keg}
346
347As demonstrated in Appendix~\ref{Apdx_C}, there is a single discrete formulation
348of the kinetic energy gradient term that, together with the formulation chosen for
349the vertical advection (see below), conserves the total kinetic energy:
350\begin{equation} \label{Eq_dynkeg}
351\left\{ \begin{aligned}
352 -\frac{1}{2 \; e_{1u} }  & \ \delta _{i+1/2} \left[ {\overline {u^2}^{\,i} + \overline{v^2}^{\,j}} \right]   \\
353 -\frac{1}{2 \; e_{2v} }  & \ \delta _{j+1/2} \left[ {\overline {u^2}^{\,i} + \overline{v^2}^{\,j}} \right]   
354\end{aligned} \right.
355\end{equation} 
356
357%--------------------------------------------------------------------------------------------------------------
358%           Vertical advection term
359%--------------------------------------------------------------------------------------------------------------
360\subsection   [Vertical advection term (\textit{dynzad}) ]
361         {Vertical advection term (\mdl{dynzad}) }
362\label{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 Appendix~\ref{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} 
374
375% ================================================================
376% Coriolis and Advection : flux form
377% ================================================================
378\section{Coriolis and Advection: flux form}
379\label{DYN_adv_cor_flux}
380%------------------------------------------nam_dynadv----------------------------------------------------
381\namdisplay{namdyn_adv} 
382%-------------------------------------------------------------------------------------------------------------
383
384Options are defined through the \ngn{namdyn\_adv} namelist variables.
385In the flux form (as in the vector invariant form), the Coriolis and momentum
386advection terms are evaluated using a leapfrog scheme, $i.e.$ the velocity
387appearing in their expressions is centred in time (\textit{now} velocity). At the
388lateral boundaries either free slip, no slip or partial slip boundary conditions
389are applied following Chap.\ref{LBC}.
390
391
392%--------------------------------------------------------------------------------------------------------------
393%           Coriolis plus curvature metric terms
394%--------------------------------------------------------------------------------------------------------------
395\subsection   [Coriolis plus curvature metric terms (\textit{dynvor}) ]
396         {Coriolis plus curvature metric terms (\mdl{dynvor}) }
397\label{DYN_cor_flux}
398
399In flux form, the vorticity term reduces to a Coriolis term in which the Coriolis
400parameter has been modified to account for the "metric" term. This altered
401Coriolis parameter is thus discretised at $f$-points. It is given by:
402\begin{multline} \label{Eq_dyncor_metric}
403f+\frac{1}{e_1 e_2 }\left( {v\frac{\partial e_2 }{\partial i}  -  u\frac{\partial e_1 }{\partial j}} \right\\
404   \equiv   f + \frac{1}{e_{1f} e_{2f} } \left( { \ \overline v ^{i+1/2}\delta _{i+1/2} \left[ {e_{2u} } \right] 
405                                                                 -  \overline u ^{j+1/2}\delta _{j+1/2} \left[ {e_{1u} } \right]  }  \ \right)
406\end{multline} 
407
408Any of the (\ref{Eq_dynvor_ens}), (\ref{Eq_dynvor_ene}) and (\ref{Eq_dynvor_een})
409schemes can be used to compute the product of the Coriolis parameter and the
410vorticity. However, the energy-conserving scheme  (\ref{Eq_dynvor_een}) has
411exclusively been used to date. This term is evaluated using a leapfrog scheme,
412$i.e.$ the velocity is centred in time (\textit{now} velocity).
413
414%--------------------------------------------------------------------------------------------------------------
415%           Flux form Advection term
416%--------------------------------------------------------------------------------------------------------------
417\subsection   [Flux form Advection term (\textit{dynadv}) ]
418         {Flux form Advection term (\mdl{dynadv}) }
419\label{DYN_adv_flux}
420
421The discrete expression of the advection term is given by :
422\begin{equation} \label{Eq_dynadv}
423\left\{ 
424\begin{aligned}
425\frac{1}{e_{1u}\,e_{2u}\,e_{3u}} 
426\left(      \delta _{i+1/2} \left[ \overline{e_{2u}\,e_{3u}\;u }^{i       }  \ u_t      \right]   
427          + \delta _{j       } \left[ \overline{e_{1u}\,e_{3u}\;v }^{i+1/2}  \ u_f      \right] \right\ \;   \\
428\left.   + \delta _{k      } \left[ \overline{e_{1w}\,e_{2w}\;w}^{i+1/2}  \ u_{uw} \right] \right)   \\
429\\
430\frac{1}{e_{1v}\,e_{2v}\,e_{3v}} 
431\left(     \delta _{i       } \left[ \overline{e_{2u}\,e_{3u }\;u }^{j+1/2} \ v_f       \right] 
432         + \delta _{j+1/2} \left[ \overline{e_{1u}\,e_{3u }\;v }^{i       } \ v_t       \right] \right\ \, \, \\
433\left.  + \delta _{k      } \left[ \overline{e_{1w}\,e_{2w}\;w}^{j+1/2} \ v_{vw}  \right] \right) \\
434\end{aligned}
435\right.
436\end{equation}
437
438Two advection schemes are available: a $2^{nd}$ order centered finite
439difference scheme, CEN2, or a $3^{rd}$ order upstream biased scheme, UBS.
440The latter is described in \citet{Shchepetkin_McWilliams_OM05}. The schemes are
441selected using the namelist logicals \np{ln\_dynadv\_cen2} and \np{ln\_dynadv\_ubs}.
442In flux form, the schemes differ by the choice of a space and time interpolation to
443define the value of $u$ and $v$ at the centre of each face of $u$- and $v$-cells,
444$i.e.$ at the $T$-, $f$-, and $uw$-points for $u$ and at the $f$-, $T$- and
445$vw$-points for $v$.
446
447%-------------------------------------------------------------
448%                 2nd order centred scheme
449%-------------------------------------------------------------
450\subsubsection{$2^{nd}$ order centred scheme (cen2) (\np{ln\_dynadv\_cen2}=true)}
451\label{DYN_adv_cen2}
452
453In the centered $2^{nd}$ order formulation, the velocity is evaluated as the
454mean of the two neighbouring points :
455\begin{equation} \label{Eq_dynadv_cen2}
456\left\{     \begin{aligned}
457 u_T^{cen2} &=\overline u^{i }       \quad &  u_F^{cen2} &=\overline u^{j+1/2}  \quad &  u_{uw}^{cen2} &=\overline u^{k+1/2}   \\
458 v_F^{cen2} &=\overline v ^{i+1/2} \quad & v_F^{cen2} &=\overline v^j      \quad &  v_{vw}^{cen2} &=\overline v ^{k+1/2}  \\
459\end{aligned}      \right.
460\end{equation} 
461
462The scheme is non diffusive (i.e. conserves the kinetic energy) but dispersive
463($i.e.$ it may create false extrema). It is therefore notoriously noisy and must be
464used in conjunction with an explicit diffusion operator to produce a sensible solution.
465The associated time-stepping is performed using a leapfrog scheme in conjunction
466with an Asselin time-filter, so $u$ and $v$ are the \emph{now} velocities.
467
468%-------------------------------------------------------------
469%                 UBS scheme
470%-------------------------------------------------------------
471\subsubsection{Upstream Biased Scheme (UBS) (\np{ln\_dynadv\_ubs}=true)}
472\label{DYN_adv_ubs}
473
474The UBS advection scheme is an upstream biased third order scheme based on
475an upstream-biased parabolic interpolation. For example, the evaluation of
476$u_T^{ubs} $ is done as follows:
477\begin{equation} \label{Eq_dynadv_ubs}
478u_T^{ubs} =\overline u ^i-\;\frac{1}{6}   \begin{cases}
479      u"_{i-1/2}&    \text{if $\ \overline{e_{2u}\,e_{3u} \ u}^i  \geqslant 0$ }    \\
480      u"_{i+1/2}&    \text{if $\ \overline{e_{2u}\,e_{3u} \ u}^i  < 0$ }
481\end{cases}
482\end{equation}
483where $u"_{i+1/2} =\delta _{i+1/2} \left[ {\delta _i \left[ u \right]} \right]$. This results
484in a dissipatively dominant ($i.e.$ hyper-diffusive) truncation error \citep{Shchepetkin_McWilliams_OM05}.
485The overall performance of the advection scheme is similar to that reported in
486\citet{Farrow1995}. It is a relatively good compromise between accuracy and
487smoothness. It is not a \emph{positive} scheme, meaning that false extrema are
488permitted. But the amplitudes of the false extrema are significantly reduced over
489those in the centred second order method. As the scheme already includes
490a diffusion component, it can be used without explicit  lateral diffusion on momentum
491($i.e.$ \np{ln\_dynldf\_lap}=\np{ln\_dynldf\_bilap}=false), and it is recommended to do so.
492
493The UBS scheme is not used in all directions. In the vertical, the centred $2^{nd}$ 
494order evaluation of the advection is preferred, $i.e.$ $u_{uw}^{ubs}$ and
495$u_{vw}^{ubs}$ in \eqref{Eq_dynadv_cen2} are used. UBS is diffusive and is
496associated with vertical mixing of momentum. \gmcomment{ gm  pursue the
497sentence:Since vertical mixing of momentum is a source term of the TKE equation...  }
498
499For stability reasons,  the first term in (\ref{Eq_dynadv_ubs}), which corresponds
500to a second order centred scheme, is evaluated using the \textit{now} velocity
501(centred in time), while the second term, which is the diffusion part of the scheme,
502is evaluated using the \textit{before} velocity (forward in time). This is discussed
503by \citet{Webb_al_JAOT98} in the context of the Quick advection scheme.
504
505Note that the UBS and QUICK (Quadratic Upstream Interpolation for Convective Kinematics)
506schemes only differ by one coefficient. Replacing $1/6$ by $1/8$ in
507(\ref{Eq_dynadv_ubs}) leads to the QUICK advection scheme \citep{Webb_al_JAOT98}.
508This option is not available through a namelist parameter, since the $1/6$ coefficient
509is hard coded. Nevertheless it is quite easy to make the substitution in the
510\mdl{dynadv\_ubs} module and obtain a QUICK scheme.
511
512Note also that in the current version of \mdl{dynadv\_ubs}, there is also the
513possibility of using a $4^{th}$ order evaluation of the advective velocity as in
514ROMS. This is an error and should be suppressed soon.
515%%%
516\gmcomment{action :  this have to be done}
517%%%
518
519% ================================================================
520%           Hydrostatic pressure gradient term
521% ================================================================
522\section  [Hydrostatic pressure gradient (\textit{dynhpg})]
523      {Hydrostatic pressure gradient (\mdl{dynhpg})}
524\label{DYN_hpg}
525%------------------------------------------nam_dynhpg---------------------------------------------------
526\namdisplay{namdyn_hpg} 
527%-------------------------------------------------------------------------------------------------------------
528
529Options are defined through the \ngn{namdyn\_hpg} namelist variables.
530The key distinction between the different algorithms used for the hydrostatic
531pressure gradient is the vertical coordinate used, since HPG is a \emph{horizontal} 
532pressure gradient, $i.e.$ computed along geopotential surfaces. As a result, any
533tilt of the surface of the computational levels will require a specific treatment to
534compute the hydrostatic pressure gradient.
535
536The hydrostatic pressure gradient term is evaluated either using a leapfrog scheme,
537$i.e.$ the density appearing in its expression is centred in time (\emph{now} $\rho$), or
538a semi-implcit scheme. At the lateral boundaries either free slip, no slip or partial slip
539boundary conditions are applied.
540
541%--------------------------------------------------------------------------------------------------------------
542%           z-coordinate with full step
543%--------------------------------------------------------------------------------------------------------------
544\subsection   [$z$-coordinate with full step (\np{ln\_dynhpg\_zco}) ]
545         {$z$-coordinate with full step (\np{ln\_dynhpg\_zco}=true)}
546\label{DYN_hpg_zco}
547
548The hydrostatic pressure can be obtained by integrating the hydrostatic equation
549vertically from the surface. However, the pressure is large at great depth while its
550horizontal gradient is several orders of magnitude smaller. This may lead to large
551truncation errors in the pressure gradient terms. Thus, the two horizontal components
552of the hydrostatic pressure gradient are computed directly as follows:
553
554for $k=km$ (surface layer, $jk=1$ in the code)
555\begin{equation} \label{Eq_dynhpg_zco_surf}
556\left\{ \begin{aligned}
557               \left. \delta _{i+1/2} \left[  p^h         \right] \right|_{k=km} 
558&= \frac{1}{2} g \   \left. \delta _{i+1/2} \left[  e_{3w} \ \rho \right] \right|_{k=km}   \\
559                  \left. \delta _{j+1/2} \left[  p^h            \right] \right|_{k=km} 
560&= \frac{1}{2} g \   \left. \delta _{j+1/2} \left[  e_{3w} \ \rho \right] \right|_{k=km}   \\
561\end{aligned} \right.
562\end{equation} 
563
564for $1<k<km$ (interior layer)
565\begin{equation} \label{Eq_dynhpg_zco}
566\left\{ \begin{aligned}
567               \left. \delta _{i+1/2} \left[  p^h         \right] \right|_{k} 
568&=             \left. \delta _{i+1/2} \left[  p^h         \right] \right|_{k-1} 
569+    \frac{1}{2}\;g\;   \left. \delta _{i+1/2} \left[  e_{3w} \ \overline {\rho}^{k+1/2} \right] \right|_{k}   \\
570                  \left. \delta _{j+1/2} \left[  p^h            \right] \right|_{k} 
571&=                \left. \delta _{j+1/2} \left[  p^h            \right] \right|_{k-1} 
572+    \frac{1}{2}\;g\;   \left. \delta _{j+1/2} \left[  e_{3w} \ \overline {\rho}^{k+1/2} \right] \right|_{k}   \\
573\end{aligned} \right.
574\end{equation} 
575
576Note that the $1/2$ factor in (\ref{Eq_dynhpg_zco_surf}) is adequate because of
577the definition of $e_{3w}$ as the vertical derivative of the scale factor at the surface
578level ($z=0$). Note also that in case of variable volume level (\key{vvl} defined), the
579surface pressure gradient is included in \eqref{Eq_dynhpg_zco_surf} and \eqref{Eq_dynhpg_zco} 
580through the space and time variations of the vertical scale factor $e_{3w}$.
581
582%--------------------------------------------------------------------------------------------------------------
583%           z-coordinate with partial step
584%--------------------------------------------------------------------------------------------------------------
585\subsection   [$z$-coordinate with partial step (\np{ln\_dynhpg\_zps})]
586         {$z$-coordinate with partial step (\np{ln\_dynhpg\_zps}=true)}
587\label{DYN_hpg_zps}
588
589With partial bottom cells, tracers in horizontally adjacent cells generally live at
590different depths. Before taking horizontal gradients between these tracer points,
591a linear interpolation is used to approximate the deeper tracer as if it actually lived
592at the depth of the shallower tracer point.
593
594Apart from this modification, the horizontal hydrostatic pressure gradient evaluated
595in the $z$-coordinate with partial step is exactly as in the pure $z$-coordinate case.
596As explained in detail in section \S\ref{TRA_zpshde}, the nonlinearity of pressure
597effects in the equation of state is such that it is better to interpolate temperature and
598salinity vertically before computing the density. Horizontal gradients of temperature
599and salinity are needed for the TRA modules, which is the reason why the horizontal
600gradients of density at the deepest model level are computed in module \mdl{zpsdhe} 
601located in the TRA directory and described in \S\ref{TRA_zpshde}.
602
603%--------------------------------------------------------------------------------------------------------------
604%           s- and s-z-coordinates
605%--------------------------------------------------------------------------------------------------------------
606\subsection{$s$- and $z$-$s$-coordinates}
607\label{DYN_hpg_sco}
608
609Pressure gradient formulations in an $s$-coordinate have been the subject of a vast
610number of papers ($e.g.$, \citet{Song1998, Shchepetkin_McWilliams_OM05}).
611A number of different pressure gradient options are coded but the ROMS-like, density Jacobian with
612cubic polynomial method is currently disabled whilst known bugs are under investigation.
613
614$\bullet$ Traditional coding (see for example \citet{Madec_al_JPO96}: (\np{ln\_dynhpg\_sco}=true)
615\begin{equation} \label{Eq_dynhpg_sco}
616\left\{ \begin{aligned}
617 - \frac{1}                   {\rho_o \, e_{1u}} \;   \delta _{i+1/2} \left[  p^h  \right] 
618+ \frac{g\; \overline {\rho}^{i+1/2}}  {\rho_o \, e_{1u}} \;   \delta _{i+1/2} \left[  z_t   \right]    \\
619 - \frac{1}                   {\rho_o \, e_{2v}} \;   \delta _{j+1/2} \left[  p^h  \right] 
620+ \frac{g\; \overline {\rho}^{j+1/2}}  {\rho_o \, e_{2v}} \;   \delta _{j+1/2} \left[  z_t   \right]    \\
621\end{aligned} \right.
622\end{equation} 
623
624Where the first term is the pressure gradient along coordinates, computed as in
625\eqref{Eq_dynhpg_zco_surf} - \eqref{Eq_dynhpg_zco}, and $z_T$ is the depth of
626the $T$-point evaluated from the sum of the vertical scale factors at the $w$-point
627($e_{3w}$).
628 
629$\bullet$ Traditional coding with adaptation for ice shelf cavities (\np{ln\_dynhpg\_isf}=true).
630This scheme need the activation of ice shelf cavities (\np{ln\_isfcav}=true).
631
632$\bullet$ Pressure Jacobian scheme (prj) (a research paper in preparation) (\np{ln\_dynhpg\_prj}=true)
633
634$\bullet$ Density Jacobian with cubic polynomial scheme (DJC) \citep{Shchepetkin_McWilliams_OM05} 
635(\np{ln\_dynhpg\_djc}=true) (currently disabled; under development)
636
637Note that expression \eqref{Eq_dynhpg_sco} is commonly used when the variable volume formulation is
638activated (\key{vvl}) because in that case, even with a flat bottom, the coordinate surfaces are not
639horizontal but follow the free surface \citep{Levier2007}. The pressure jacobian scheme
640(\np{ln\_dynhpg\_prj}=true) is available as an improved option to \np{ln\_dynhpg\_sco}=true when
641\key{vvl} is active.  The pressure Jacobian scheme uses a constrained cubic spline to reconstruct
642the density profile across the water column. This method maintains the monotonicity between the
643density nodes  The pressure can be calculated by analytical integration of the density profile and a
644pressure Jacobian method is used to solve the horizontal pressure gradient. This method can provide
645a more accurate calculation of the horizontal pressure gradient than the standard scheme.
646
647%--------------------------------------------------------------------------------------------------------------
648%           Time-scheme
649%--------------------------------------------------------------------------------------------------------------
650\subsection   [Time-scheme (\np{ln\_dynhpg\_imp}) ]
651         {Time-scheme (\np{ln\_dynhpg\_imp}= true/false)}
652\label{DYN_hpg_imp}
653
654The default time differencing scheme used for the horizontal pressure gradient is
655a leapfrog scheme and therefore the density used in all discrete expressions given
656above is the  \textit{now} density, computed from the \textit{now} temperature and
657salinity. In some specific cases (usually high resolution simulations over an ocean
658domain which includes weakly stratified regions) the physical phenomenon that
659controls the time-step is internal gravity waves (IGWs). A semi-implicit scheme for
660doubling the stability limit associated with IGWs can be used \citep{Brown_Campana_MWR78,
661Maltrud1998}. It involves the evaluation of the hydrostatic pressure gradient as an
662average over the three time levels $t-\rdt$, $t$, and $t+\rdt$ ($i.e.$ 
663\textit{before}\textit{now} and  \textit{after} time-steps), rather than at the central
664time level $t$ only, as in the standard leapfrog scheme.
665
666$\bullet$ leapfrog scheme (\np{ln\_dynhpg\_imp}=true):
667
668\begin{equation} \label{Eq_dynhpg_lf}
669\frac{u^{t+\rdt}-u^{t-\rdt}}{2\rdt} = \;\cdots \;
670   -\frac{1}{\rho _o \,e_{1u} }\delta _{i+1/2} \left[ {p_h^t } \right]
671\end{equation}
672
673$\bullet$ semi-implicit scheme (\np{ln\_dynhpg\_imp}=true):
674\begin{equation} \label{Eq_dynhpg_imp}
675\frac{u^{t+\rdt}-u^{t-\rdt}}{2\rdt} = \;\cdots \;
676   -\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]
677\end{equation}
678
679The semi-implicit time scheme \eqref{Eq_dynhpg_imp} is made possible without
680significant additional computation since the density can be updated to time level
681$t+\rdt$ before computing the horizontal hydrostatic pressure gradient. It can
682be easily shown that the stability limit associated with the hydrostatic pressure
683gradient doubles using \eqref{Eq_dynhpg_imp} compared to that using the
684standard leapfrog scheme \eqref{Eq_dynhpg_lf}. Note that \eqref{Eq_dynhpg_imp} 
685is equivalent to applying a time filter to the pressure gradient to eliminate high
686frequency IGWs. Obviously, when using \eqref{Eq_dynhpg_imp}, the doubling of
687the time-step is achievable only if no other factors control the time-step, such as
688the stability limits associated with advection or diffusion.
689
690In practice, the semi-implicit scheme is used when \np{ln\_dynhpg\_imp}=true.
691In this case, we choose to apply the time filter to temperature and salinity used in
692the equation of state, instead of applying it to the hydrostatic pressure or to the
693density, so that no additional storage array has to be defined. The density used to
694compute the hydrostatic pressure gradient (whatever the formulation) is evaluated
695as follows:
696\begin{equation} \label{Eq_rho_flt}
697   \rho^t = \rho( \widetilde{T},\widetilde {S},z_t)
698 \quad     \text{with}  \quad 
699   \widetilde{X} = 1 / 4 \left(  X^{t+\rdt} +2 \,X^t + X^{t-\rdt}  \right)
700\end{equation}
701
702Note that in the semi-implicit case, it is necessary to save the filtered density, an
703extra three-dimensional field, in the restart file to restart the model with exact
704reproducibility. This option is controlled by  \np{nn\_dynhpg\_rst}, a namelist parameter.
705
706% ================================================================
707% Surface Pressure Gradient
708% ================================================================
709\section  [Surface pressure gradient (\textit{dynspg}) ]
710      {Surface pressure gradient (\mdl{dynspg})}
711\label{DYN_spg}
712%-----------------------------------------nam_dynspg----------------------------------------------------
713\namdisplay{namdyn_spg} 
714%------------------------------------------------------------------------------------------------------------
715
716$\ $\newline      %force an empty line
717
718%%%
719Options are defined through the \ngn{namdyn\_spg} namelist variables.
720The surface pressure gradient term is related to the representation of the free surface (\S\ref{PE_hor_pg}). The main distinction is between the fixed volume case (linear free surface) and the variable volume case (nonlinear free surface, \key{vvl} is defined). In the linear free surface case (\S\ref{PE_free_surface}) the vertical scale factors $e_{3}$ are fixed in time, while they are time-dependent in the nonlinear case (\S\ref{PE_free_surface}). With both linear and nonlinear free surface, external gravity waves are allowed in the equations, which imposes a very small time step when an explicit time stepping is used. Two methods are proposed to allow a longer time step for the three-dimensional equations: the filtered free surface, which is a modification of the continuous equations (see \eqref{Eq_PE_flt}), and the split-explicit free surface described below. The extra term introduced in the filtered method is calculated implicitly, so that the update of the next velocities is done in module \mdl{dynspg\_flt} and not in \mdl{dynnxt}.
721
722%%%
723
724
725The form of the surface pressure gradient term depends on how the user wants to handle
726the fast external gravity waves that are a solution of the analytical equation (\S\ref{PE_hor_pg}).
727Three formulations are available, all controlled by a CPP key (ln\_dynspg\_xxx):
728an explicit formulation which requires a small time step ;
729a filtered free surface formulation which allows a larger time step by adding a filtering
730term into the momentum equation ;
731and a split-explicit free surface formulation, described below, which also allows a larger time step.
732
733The extra term introduced in the filtered method is calculated
734implicitly, so that a solver is used to compute it. As a consequence the update of the $next$ 
735velocities is done in module \mdl{dynspg\_flt} and not in \mdl{dynnxt}.
736
737
738
739%--------------------------------------------------------------------------------------------------------------
740% Explicit free surface formulation
741%--------------------------------------------------------------------------------------------------------------
742\subsection{Explicit free surface (\key{dynspg\_exp})}
743\label{DYN_spg_exp}
744
745In the explicit free surface formulation (\key{dynspg\_exp} defined), the model time step
746is chosen to be small enough to resolve the external gravity waves (typically a few tens of seconds).
747The surface pressure gradient, evaluated using a leap-frog scheme ($i.e.$ centered in time),
748is thus simply given by :
749\begin{equation} \label{Eq_dynspg_exp}
750\left\{ \begin{aligned}
751 - \frac{1}{e_{1u}\,\rho_o} \;   \delta _{i+1/2} \left[  \,\rho \,\eta\,  \right]   \\
752 - \frac{1}{e_{2v}\,\rho_o} \;   \delta _{j+1/2} \left[  \,\rho \,\eta\,  \right] 
753\end{aligned} \right.
754\end{equation} 
755
756Note that in the non-linear free surface case ($i.e.$ \key{vvl} defined), the surface pressure
757gradient is already included in the momentum tendency  through the level thickness variation
758allowed in the computation of the hydrostatic pressure gradient. Thus, nothing is done in the \mdl{dynspg\_exp} module.
759
760%--------------------------------------------------------------------------------------------------------------
761% Split-explict free surface formulation
762%--------------------------------------------------------------------------------------------------------------
763\subsection{Split-Explicit free surface (\key{dynspg\_ts})}
764\label{DYN_spg_ts}
765%------------------------------------------namsplit-----------------------------------------------------------
766\namdisplay{namsplit} 
767%-------------------------------------------------------------------------------------------------------------
768
769The split-explicit free surface formulation used in \NEMO (\key{dynspg\_ts} defined),
770also called the time-splitting formulation, follows the one
771proposed by \citet{Shchepetkin_McWilliams_OM05}. The general idea is to solve the free surface
772equation and the associated barotropic velocity equations with a smaller time
773step than $\rdt$, the time step used for the three dimensional prognostic
774variables (Fig.~\ref{Fig_DYN_dynspg_ts}).
775The size of the small time step, $\rdt_e$ (the external mode or barotropic time step)
776 is provided through the \np{nn\_baro} namelist parameter as:
777$\rdt_e = \rdt / nn\_baro$. This parameter can be optionally defined automatically (\np{ln\_bt\_nn\_auto}=true)
778considering that the stability of the barotropic system is essentially controled by external waves propagation.
779Maximum allowed Courant number is in that case time independent, and easily computed online from the input bathymetry.
780
781%%%
782The barotropic mode solves the following equations:
783\begin{subequations} \label{Eq_BT}
784  \begin{equation}     \label{Eq_BT_dyn}
785\frac{\partial {\rm \overline{{\bf U}}_h} }{\partial t}=
786 -f\;{\rm {\bf k}}\times {\rm \overline{{\bf U}}_h} 
787-g\nabla _h \eta -\frac{c_b^{\textbf U}}{H+\eta} \rm {\overline{{\bf U}}_h} + \rm {\overline{\bf G}}
788  \end{equation}
789
790  \begin{equation} \label{Eq_BT_ssh}
791\frac{\partial \eta }{\partial t}=-\nabla \cdot \left[ {\left( {H+\eta } \right) \; {\rm{\bf \overline{U}}}_h \,} \right]+P-E
792  \end{equation}
793\end{subequations}
794where $\rm {\overline{\bf G}}$ 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 \eqref{Eq_BT_dyn} represents the bottom stress (see section \S\ref{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).
795
796%>   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >
797\begin{figure}[!t]    \begin{center}
798\includegraphics[width=0.7\textwidth]{./TexFiles/Figures/Fig_DYN_dynspg_ts.pdf}
799\caption{  \label{Fig_DYN_dynspg_ts}
800Schematic of the split-explicit time stepping scheme for the external
801and internal modes. Time increases to the right. In this particular exemple,
802a boxcar averaging window over $nn\_baro$ barotropic time steps is used ($nn\_bt\_filt=1$) and $nn\_baro=5$.
803Internal mode time steps (which are also the model time steps) are denoted
804by $t-\rdt$, $t$ and $t+\rdt$. Variables with $k$ superscript refer to instantaneous barotropic variables,
805$< >$ and $<< >>$ operator refer to time filtered variables using respectively primary (red vertical bars) and secondary weights (blue vertical bars).
806The former are used to obtain time filtered quantities at $t+\rdt$ while the latter are used to obtain time averaged
807transports to advect tracers.
808a) Forward time integration: \np{ln\_bt\_fw}=true, \np{ln\_bt\_ave}=true.
809b) Centred time integration: \np{ln\_bt\_fw}=false, \np{ln\_bt\_ave}=true.
810c) Forward time integration with no time filtering (POM-like scheme): \np{ln\_bt\_fw}=true, \np{ln\_bt\_ave}=false. }
811\end{center}    \end{figure}
812%>   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >
813
814In the default case (\np{ln\_bt\_fw}=true), the external mode is integrated
815between \textit{now} and  \textit{after} baroclinic time-steps (Fig.~\ref{Fig_DYN_dynspg_ts}a). To avoid aliasing of fast barotropic motions into three dimensional equations, time filtering is eventually applied on barotropic
816quantities (\np{ln\_bt\_ave}=true). In that case, the integration is extended slightly beyond  \textit{after} time step to provide time filtered quantities.
817These are used for the subsequent initialization of the barotropic mode in the following baroclinic step.
818Since external mode equations written at baroclinic time steps finally follow a forward time stepping scheme,
819asselin filtering is not applied to barotropic quantities. \\
820Alternatively, one can choose to integrate barotropic equations starting
821from \textit{before} time step (\np{ln\_bt\_fw}=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
822become centred in the middle of the integration window. It can easily be shown that this property
823removes part of splitting errors between modes, which increases the overall numerical robustness.
824%references to Patrick Marsaleix' work here. Also work done by SHOM group.
825
826%%%
827
828As far as tracer conservation is concerned, barotropic velocities used to advect tracers must also be updated
829at \textit{now} time step. This implies to change the traditional order of computations in \NEMO: most of momentum 
830trends (including the barotropic mode calculation) updated first, tracers' after. This \textit{de facto} makes semi-implicit hydrostatic
831pressure gradient (see section \S\ref{DYN_hpg_imp}) and time splitting not compatible.
832Advective barotropic velocities are obtained by using a secondary set of filtering weights, uniquely defined from the filter
833coefficients 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.
834
835%%%
836
837One can eventually choose to feedback instantaneous values by not using any time filter (\np{ln\_bt\_ave}=false).
838In that case, external mode equations are continuous in time, ie they are not re-initialized when starting a new
839sub-stepping sequence. This is the method used so far in the POM model, the stability being maintained by refreshing at (almost)
840each barotropic time step advection and horizontal diffusion terms. Since the latter terms have not been added in \NEMO for
841computational efficiency, removing time filtering is not recommended except for debugging purposes.
842This 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.
843
844%>>>>>===============
845\gmcomment{               %%% copy from griffies Book
846
847\textbf{title: Time stepping the barotropic system }
848
849Assume knowledge of the full velocity and tracer fields at baroclinic time $\tau$. Hence,
850we can update the surface height and vertically integrated velocity with a leap-frog
851scheme using the small barotropic time step $\rdt$. We have
852
853\begin{equation} \label{DYN_spg_ts_eta}
854\eta^{(b)}(\tau,t_{n+1}) - \eta^{(b)}(\tau,t_{n+1}) (\tau,t_{n-1})
855   = 2 \rdt \left[-\nabla \cdot \textbf{U}^{(b)}(\tau,t_n) + \text{EMP}_w(\tau) \right] 
856\end{equation}
857\begin{multline} \label{DYN_spg_ts_u}
858\textbf{U}^{(b)}(\tau,t_{n+1}) - \textbf{U}^{(b)}(\tau,t_{n-1}\\
859   = 2\rdt \left[ - f \textbf{k} \times \textbf{U}^{(b)}(\tau,t_{n})
860   - H(\tau) \nabla p_s^{(b)}(\tau,t_{n}) +\textbf{M}(\tau) \right]
861\end{multline}
862\
863
864In these equations, araised (b) denotes values of surface height and vertically integrated velocity updated with the barotropic time steps. The $\tau$ time label on $\eta^{(b)}$ 
865and $U^{(b)}$ 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
866that sets the barotropic time steps via
867\begin{equation} \label{DYN_spg_ts_t}
868t_n=\tau+n\rdt   
869\end{equation}
870with $n$ an integer. The density scaled surface pressure is evaluated via
871\begin{equation} \label{DYN_spg_ts_ps}
872p_s^{(b)}(\tau,t_{n}) = \begin{cases}
873   g \;\eta_s^{(b)}(\tau,t_{n}) \;\rho(\tau)_{k=1}) / \rho_&      \text{non-linear case} \\
874   g \;\eta_s^{(b)}(\tau,t_{n}&      \text{linear case} 
875   \end{cases}
876\end{equation}
877To get started, we assume the following initial conditions
878\begin{equation} \label{DYN_spg_ts_eta}
879\begin{split}
880\eta^{(b)}(\tau,t_{n=0}) &= \overline{\eta^{(b)}(\tau)}
881\\
882\eta^{(b)}(\tau,t_{n=1}) &= \eta^{(b)}(\tau,t_{n=0}) + \rdt \ \text{RHS}_{n=0} 
883\end{split}
884\end{equation}
885with
886\begin{equation} \label{DYN_spg_ts_etaF}
887 \overline{\eta^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N \eta^{(b)}(\tau-\rdt,t_{n})
888\end{equation}
889the time averaged surface height taken from the previous barotropic cycle. Likewise,
890\begin{equation} \label{DYN_spg_ts_u}
891\textbf{U}^{(b)}(\tau,t_{n=0}) = \overline{\textbf{U}^{(b)}(\tau)}   \\
892\\
893\textbf{U}(\tau,t_{n=1}) = \textbf{U}^{(b)}(\tau,t_{n=0}) + \rdt \ \text{RHS}_{n=0}   
894\end{equation}
895with
896\begin{equation} \label{DYN_spg_ts_u}
897 \overline{\textbf{U}^{(b)}(\tau)} 
898   = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau-\rdt,t_{n})
899\end{equation}
900the time averaged vertically integrated transport. Notably, there is no Robert-Asselin time filter used in the barotropic portion of the integration.
901
902Upon 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 $\tau + \rdt \tau$ 
903\begin{equation} \label{DYN_spg_ts_u}
904\textbf{U}(\tau+\rdt) = \overline{\textbf{U}^{(b)}(\tau+\rdt)} 
905   = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau,t_{n})
906\end{equation}
907The surface height on the new baroclinic time step is then determined via a baroclinic leap-frog using the following form
908
909\begin{equation} \label{DYN_spg_ts_ssh}
910\eta(\tau+\Delta) - \eta^{F}(\tau-\Delta) = 2\rdt \ \left[ - \nabla \cdot \textbf{U}(\tau) + \text{EMP}_w \right] 
911\end{equation}
912
913 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).
914 
915In general, some form of time filter is needed to maintain integrity of the surface
916height field due to the leap-frog splitting mode in equation \ref{DYN_spg_ts_ssh}. We
917have tried various forms of such filtering, with the following method discussed in
918\cite{Griffies_al_MWR01} chosen due to its stability and reasonably good maintenance of
919tracer conservation properties (see Section ??)
920
921\begin{equation} \label{DYN_spg_ts_sshf}
922\eta^{F}(\tau-\Delta) =  \overline{\eta^{(b)}(\tau)} 
923\end{equation}
924Another approach tried was
925
926\begin{equation} \label{DYN_spg_ts_sshf2}
927\eta^{F}(\tau-\Delta) = \eta(\tau)
928   + (\alpha/2) \left[\overline{\eta^{(b)}}(\tau+\rdt)
929                + \overline{\eta^{(b)}}(\tau-\rdt) -2 \;\eta(\tau) \right]
930\end{equation}
931
932which is useful since it isolates all the time filtering aspects into the term multiplied
933by $\alpha$. This isolation allows for an easy check that tracer conservation is exact when
934eliminating tracer and surface height time filtering (see Section ?? for more complete discussion). However, in the general case with a non-zero $\alpha$, the filter \ref{DYN_spg_ts_sshf} was found to be more conservative, and so is recommended.
935
936}            %%end gm comment (copy of griffies book)
937
938%>>>>>===============
939
940
941%--------------------------------------------------------------------------------------------------------------
942% Filtered free surface formulation
943%--------------------------------------------------------------------------------------------------------------
944\subsection{Filtered free surface (\key{dynspg\_flt})}
945\label{DYN_spg_fltp}
946
947The filtered formulation follows the \citet{Roullet_Madec_JGR00} implementation.
948The extra term introduced in the equations (see \S\ref{PE_free_surface}) is solved implicitly.
949The elliptic solvers available in the code are documented in \S\ref{MISC}.
950
951%% gm %%======>>>>   given here the discrete eqs provided to the solver
952\gmcomment{               %%% copy from chap-model basics
953\begin{equation} \label{Eq_spg_flt}
954\frac{\partial {\rm {\bf U}}_h }{\partial t}= {\rm {\bf M}}
955- g \nabla \left( \tilde{\rho} \ \eta \right)
956- g \ T_c \nabla \left( \widetilde{\rho} \ \partial_t \eta \right)
957\end{equation}
958where $T_c$, is a parameter with dimensions of time which characterizes the force,
959$\widetilde{\rho} = \rho / \rho_o$ is the dimensionless density, and $\rm {\bf M}$ 
960represents the collected contributions of the Coriolis, hydrostatic pressure gradient,
961non-linear and viscous terms in \eqref{Eq_PE_dyn}.
962}   %end gmcomment
963
964Note that in the linear free surface formulation (\key{vvl} not defined), the ocean depth
965is time-independent and so is the matrix to be inverted. It is computed once and for all and applies to all ocean time steps.
966
967% ================================================================
968% Lateral diffusion term
969% ================================================================
970\section  [Lateral diffusion term (\textit{dynldf})]
971      {Lateral diffusion term (\mdl{dynldf})}
972\label{DYN_ldf}
973%------------------------------------------nam_dynldf----------------------------------------------------
974\namdisplay{namdyn_ldf} 
975%-------------------------------------------------------------------------------------------------------------
976
977Options are defined through the \ngn{namdyn\_ldf} namelist variables.
978The options available for lateral diffusion are to use either laplacian
979(rotated or not) or biharmonic operators. The coefficients may be constant
980or spatially variable; the description of the coefficients is found in the chapter
981on lateral physics (Chap.\ref{LDF}). The lateral diffusion of momentum is
982evaluated using a forward scheme, $i.e.$ the velocity appearing in its expression
983is the \textit{before} velocity in time, except for the pure vertical component
984that appears when a tensor of rotation is used. This latter term is solved
985implicitly together with the vertical diffusion term (see \S\ref{STP})
986
987At the lateral boundaries either free slip, no slip or partial slip boundary
988conditions are applied according to the user's choice (see Chap.\ref{LBC}).
989
990% ================================================================
991\subsection   [Iso-level laplacian operator (\np{ln\_dynldf\_lap}) ]
992         {Iso-level laplacian operator (\np{ln\_dynldf\_lap}=true)}
993\label{DYN_ldf_lap}
994
995For lateral iso-level diffusion, the discrete operator is:
996\begin{equation} \label{Eq_dynldf_lap}
997\left\{ \begin{aligned}
998 D_u^{l{\rm {\bf U}}} =\frac{1}{e_{1u} }\delta _{i+1/2} \left[ {A_T^{lm} 
999\;\chi } \right]-\frac{1}{e_{2u} {\kern 1pt}e_{3u} }\delta _j \left[
1000{A_f^{lm} \;e_{3f} \zeta } \right] \\ 
1001\\
1002 D_v^{l{\rm {\bf U}}} =\frac{1}{e_{2v} }\delta _{j+1/2} \left[ {A_T^{lm} 
1003\;\chi } \right]+\frac{1}{e_{1v} {\kern 1pt}e_{3v} }\delta _i \left[
1004{A_f^{lm} \;e_{3f} \zeta } \right] \\ 
1005\end{aligned} \right.
1006\end{equation} 
1007
1008As explained in \S\ref{PE_ldf}, this formulation (as the gradient of a divergence
1009and curl of the vorticity) preserves symmetry and ensures a complete
1010separation between the vorticity and divergence parts of the momentum diffusion.
1011
1012%--------------------------------------------------------------------------------------------------------------
1013%           Rotated laplacian operator
1014%--------------------------------------------------------------------------------------------------------------
1015\subsection   [Rotated laplacian operator (\np{ln\_dynldf\_iso}) ]
1016         {Rotated laplacian operator (\np{ln\_dynldf\_iso}=true)}
1017\label{DYN_ldf_iso}
1018
1019A rotation of the lateral momentum diffusion operator is needed in several cases:
1020for iso-neutral diffusion in the $z$-coordinate (\np{ln\_dynldf\_iso}=true) and for
1021either iso-neutral (\np{ln\_dynldf\_iso}=true) or geopotential
1022(\np{ln\_dynldf\_hor}=true) diffusion in the $s$-coordinate. In the partial step
1023case, coordinates are horizontal except at the deepest level and no
1024rotation is performed when \np{ln\_dynldf\_hor}=true. The diffusion operator
1025is defined simply as the divergence of down gradient momentum fluxes on each
1026momentum component. It must be emphasized that this formulation ignores
1027constraints on the stress tensor such as symmetry. The resulting discrete
1028representation is:
1029\begin{equation} \label{Eq_dyn_ldf_iso}
1030\begin{split}
1031 D_u^{l\textbf{U}} &= \frac{1}{e_{1u} \, e_{2u} \, e_{3u} } \\
1032&  \left\{\quad  {\delta _{i+1/2} \left[ {A_T^{lm}  \left(
1033    {\frac{e_{2t} \; e_{3t} }{e_{1t} } \,\delta _{i}[u]
1034   -e_{2t} \; r_{1t} \,\overline{\overline {\delta _{k+1/2}[u]}}^{\,i,\,k}}
1035 \right)} \right]}   \right.
1036\\ 
1037& \qquad +\ \delta_j \left[ {A_f^{lm} \left( {\frac{e_{1f}\,e_{3f} }{e_{2f} 
1038}\,\delta _{j+1/2} [u] - e_{1f}\, r_{2f} 
1039\,\overline{\overline {\delta _{k+1/2} [u]}} ^{\,j+1/2,\,k}} 
1040\right)} \right]
1041\\ 
1042&\qquad +\ \delta_k \left[ {A_{uw}^{lm} \left( {-e_{2u} \, r_{1uw} \,\overline{\overline 
1043{\delta_{i+1/2} [u]}}^{\,i+1/2,\,k+1/2} } 
1044\right.} \right.
1045\\ 
1046&  \ \qquad \qquad \qquad \quad\
1047- e_{1u} \, r_{2uw} \,\overline{\overline {\delta_{j+1/2} [u]}} ^{\,j,\,k+1/2}
1048\\ 
1049& \left. {\left. { \ \qquad \qquad \qquad \ \ \ \left. {\
1050+\frac{e_{1u}\, e_{2u} }{e_{3uw} }\,\left( {r_{1uw}^2+r_{2uw}^2} 
1051\right)\,\delta_{k+1/2} [u]} \right)} \right]\;\;\;} \right\} 
1052\\
1053\\
1054 D_v^{l\textbf{V}} &= \frac{1}{e_{1v} \, e_{2v} \, e_{3v} }    \\
1055&  \left\{\quad  {\delta _{i+1/2} \left[ {A_f^{lm}  \left(
1056    {\frac{e_{2f} \; e_{3f} }{e_{1f} } \,\delta _{i+1/2}[v]
1057   -e_{2f} \; r_{1f} \,\overline{\overline {\delta _{k+1/2}[v]}}^{\,i+1/2,\,k}}
1058 \right)} \right]}   \right.
1059\\ 
1060& \qquad +\ \delta_j \left[ {A_T^{lm} \left( {\frac{e_{1t}\,e_{3t} }{e_{2t} 
1061}\,\delta _{j} [v] - e_{1t}\, r_{2t} 
1062\,\overline{\overline {\delta _{k+1/2} [v]}} ^{\,j,\,k}} 
1063\right)} \right]
1064\\ 
1065& \qquad +\ \delta_k \left[ {A_{vw}^{lm} \left( {-e_{2v} \, r_{1vw} \,\overline{\overline 
1066{\delta_{i+1/2} [v]}}^{\,i+1/2,\,k+1/2} }\right.} \right.
1067\\
1068&  \ \qquad \qquad \qquad \quad\
1069- e_{1v} \, r_{2vw} \,\overline{\overline {\delta_{j+1/2} [v]}} ^{\,j+1/2,\,k+1/2}
1070\\ 
1071& \left. {\left. { \ \qquad \qquad \qquad \ \ \ \left. {\
1072+\frac{e_{1v}\, e_{2v} }{e_{3vw} }\,\left( {r_{1vw}^2+r_{2vw}^2} 
1073\right)\,\delta_{k+1/2} [v]} \right)} \right]\;\;\;} \right\} 
1074 \end{split}
1075\end{equation}
1076where $r_1$ and $r_2$ are the slopes between the surface along which the
1077diffusion operator acts and the surface of computation ($z$- or $s$-surfaces).
1078The way these slopes are evaluated is given in the lateral physics chapter
1079(Chap.\ref{LDF}).
1080
1081%--------------------------------------------------------------------------------------------------------------
1082%           Iso-level bilaplacian operator
1083%--------------------------------------------------------------------------------------------------------------
1084\subsection   [Iso-level bilaplacian operator (\np{ln\_dynldf\_bilap})]
1085         {Iso-level bilaplacian operator (\np{ln\_dynldf\_bilap}=true)}
1086\label{DYN_ldf_bilap}
1087
1088The lateral fourth order operator formulation on momentum is obtained by
1089applying \eqref{Eq_dynldf_lap} twice. It requires an additional assumption on
1090boundary conditions: the first derivative term normal to the coast depends on
1091the free or no-slip lateral boundary conditions chosen, while the third
1092derivative terms normal to the coast are set to zero (see Chap.\ref{LBC}).
1093%%%
1094\gmcomment{add a remark on the the change in the position of the coefficient}
1095%%%
1096
1097% ================================================================
1098%           Vertical diffusion term
1099% ================================================================
1100\section  [Vertical diffusion term (\mdl{dynzdf})]
1101      {Vertical diffusion term (\mdl{dynzdf})}
1102\label{DYN_zdf}
1103%----------------------------------------------namzdf------------------------------------------------------
1104\namdisplay{namzdf} 
1105%-------------------------------------------------------------------------------------------------------------
1106
1107Options are defined through the \ngn{namzdf} namelist variables.
1108The large vertical diffusion coefficient found in the surface mixed layer together
1109with high vertical resolution implies that in the case of explicit time stepping there
1110would be too restrictive a constraint on the time step. Two time stepping schemes
1111can be used for the vertical diffusion term : $(a)$ a forward time differencing
1112scheme (\np{ln\_zdfexp}=true) using a time splitting technique
1113(\np{nn\_zdfexp} $>$ 1) or $(b)$ a backward (or implicit) time differencing scheme
1114(\np{ln\_zdfexp}=false) (see \S\ref{STP}). Note that namelist variables
1115\np{ln\_zdfexp} and \np{nn\_zdfexp} apply to both tracers and dynamics.
1116
1117The formulation of the vertical subgrid scale physics is the same whatever
1118the vertical coordinate is. The vertical diffusion operators given by
1119\eqref{Eq_PE_zdf} take the following semi-discrete space form:
1120\begin{equation} \label{Eq_dynzdf}
1121\left\{   \begin{aligned}
1122D_u^{vm} &\equiv \frac{1}{e_{3u}} \ \delta _k \left[ \frac{A_{uw}^{vm} }{e_{3uw} }
1123                              \ \delta _{k+1/2} [\,u\,]         \right]     \\
1124\\
1125D_v^{vm} &\equiv \frac{1}{e_{3v}} \ \delta _k \left[ \frac{A_{vw}^{vm} }{e_{3vw} }
1126                              \ \delta _{k+1/2} [\,v\,]         \right]
1127\end{aligned}   \right.
1128\end{equation} 
1129where $A_{uw}^{vm} $ and $A_{vw}^{vm} $ are the vertical eddy viscosity and
1130diffusivity coefficients. The way these coefficients are evaluated
1131depends on the vertical physics used (see \S\ref{ZDF}).
1132
1133The surface boundary condition on momentum is the stress exerted by
1134the wind. At the surface, the momentum fluxes are prescribed as the boundary
1135condition on the vertical turbulent momentum fluxes,
1136\begin{equation} \label{Eq_dynzdf_sbc}
1137\left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{z=1}
1138    = \frac{1}{\rho _o} \binom{\tau _u}{\tau _v }
1139\end{equation}
1140where $\left( \tau _u ,\tau _v \right)$ are the two components of the wind stress
1141vector in the (\textbf{i},\textbf{j}) coordinate system. The high mixing coefficients
1142in the surface mixed layer ensure that the surface wind stress is distributed in
1143the vertical over the mixed layer depth. If the vertical mixing coefficient
1144is small (when no mixed layer scheme is used) the surface stress enters only
1145the top model level, as a body force. The surface wind stress is calculated
1146in the surface module routines (SBC, see Chap.\ref{SBC})
1147
1148The turbulent flux of momentum at the bottom of the ocean is specified through
1149a bottom friction parameterisation (see \S\ref{ZDF_bfr})
1150
1151% ================================================================
1152% External Forcing
1153% ================================================================
1154\section{External Forcings}
1155\label{DYN_forcing}
1156
1157Besides the surface and bottom stresses (see the above section) which are
1158introduced as boundary conditions on the vertical mixing, two other forcings
1159enter the dynamical equations.
1160
1161One is the effect of atmospheric pressure on the ocean dynamics.
1162Another forcing term is the tidal potential.
1163Both of which will be introduced into the reference version soon.
1164
1165\gmcomment{atmospheric pressure is there!!!!    include its description }
1166
1167% ================================================================
1168% Time evolution term
1169% ================================================================
1170\section  [Time evolution term (\textit{dynnxt})]
1171      {Time evolution term (\mdl{dynnxt})}
1172\label{DYN_nxt}
1173
1174%----------------------------------------------namdom----------------------------------------------------
1175\namdisplay{namdom} 
1176%-------------------------------------------------------------------------------------------------------------
1177
1178Options are defined through the \ngn{namdom} namelist variables.
1179The general framework for dynamics time stepping is a leap-frog scheme,
1180$i.e.$ a three level centred time scheme associated with an Asselin time filter
1181(cf. Chap.\ref{STP}). The scheme is applied to the velocity, except when using
1182the flux form of momentum advection (cf. \S\ref{DYN_adv_cor_flux}) in the variable
1183volume case (\key{vvl} defined), where it has to be applied to the thickness
1184weighted velocity (see \S\ref{Apdx_A_momentum}
1185
1186$\bullet$ vector invariant form or linear free surface (\np{ln\_dynhpg\_vec}=true ; \key{vvl} not defined):
1187\begin{equation} \label{Eq_dynnxt_vec}
1188\left\{   \begin{aligned}
1189&u^{t+\rdt} = u_f^{t-\rdt} + 2\rdt  \ \text{RHS}_u^t     \\
1190&u_f^t \;\quad = u^t+\gamma \,\left[ {u_f^{t-\rdt} -2u^t+u^{t+\rdt}} \right]
1191\end{aligned}   \right.
1192\end{equation} 
1193
1194$\bullet$ flux form and nonlinear free surface (\np{ln\_dynhpg\_vec}=false ; \key{vvl} defined):
1195\begin{equation} \label{Eq_dynnxt_flux}
1196\left\{   \begin{aligned}
1197&\left(e_{3u}\,u\right)^{t+\rdt} = \left(e_{3u}\,u\right)_f^{t-\rdt} + 2\rdt \; e_{3u} \;\text{RHS}_u^t     \\
1198&\left(e_{3u}\,u\right)_f^t \;\quad = \left(e_{3u}\,u\right)^t
1199  +\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]
1200\end{aligned}   \right.
1201\end{equation} 
1202where RHS is the right hand side of the momentum equation, the subscript $f$ 
1203denotes filtered values and $\gamma$ is the Asselin coefficient. $\gamma$ is
1204initialized as \np{nn\_atfp} (namelist parameter). Its default value is \np{nn\_atfp} = $10^{-3}$.
1205In both cases, the modified Asselin filter is not applied since perfect conservation
1206is not an issue for the momentum equations.
1207
1208Note that with the filtered free surface, the update of the \textit{after} velocities
1209is done in the \mdl{dynsp\_flt} module, and only array swapping
1210and Asselin filtering is done in \mdl{dynnxt}.
1211
1212% ================================================================
1213% Neptune effect
1214% ================================================================
1215\section  [Neptune effect (\textit{dynnept})]
1216                {Neptune effect (\mdl{dynnept})}
1217\label{DYN_nept}
1218
1219The "Neptune effect" (thus named in \citep{HollowayOM86}) is a
1220parameterisation of the potentially large effect of topographic form stress
1221(caused by eddies) in driving the ocean circulation. Originally developed for
1222low-resolution models, in which it was applied via a Laplacian (second-order)
1223diffusion-like term in the momentum equation, it can also be applied in eddy
1224permitting or resolving models, in which a more scale-selective bilaplacian
1225(fourth-order) implementation is preferred. This mechanism has a
1226significant effect on boundary currents (including undercurrents), and the
1227upwelling of deep water near continental shelves.
1228
1229The theoretical basis for the method can be found in
1230\citep{HollowayJPO92}, including the explanation of why form stress is not
1231necessarily a drag force, but may actually drive the flow.
1232\citep{HollowayJPO94} demonstrate the effects of the parameterisation in
1233the GFDL-MOM model, at a horizontal resolution of about 1.8 degrees.
1234\citep{HollowayOM08} demonstrate the biharmonic version of the
1235parameterisation in a global run of the POP model, with an average horizontal
1236grid spacing of about 32km.
1237
1238The NEMO implementation is a simplified form of that supplied by
1239Greg Holloway, the testing of which was described in \citep{HollowayJGR09}.
1240The major simplification is that a time invariant Neptune velocity
1241field is assumed.  This is computed only once, during start-up, and
1242made available to the rest of the code via a module.  Vertical
1243diffusive terms are also ignored, and the model topography itself
1244is used, rather than a separate topographic dataset as in
1245\citep{HollowayOM08}.  This implementation is only in the iso-level
1246formulation, as is the case anyway for the bilaplacian operator.
1247
1248The velocity field is derived from a transport stream function given by:
1249
1250\begin{equation} \label{Eq_dynnept_sf}
1251\psi = -fL^2H
1252\end{equation}
1253
1254where $L$ is a latitude-dependant length scale given by:
1255
1256\begin{equation} \label{Eq_dynnept_ls}
1257L = l_1 + (l_2 -l_1)\left ( {1 + \cos 2\phi \over 2 } \right )
1258\end{equation}
1259
1260where $\phi$ is latitude and $l_1$ and $l_2$ are polar and equatorial length scales respectively.
1261Neptune velocity components, $u^*$, $v^*$ are derived from the stremfunction as:
1262
1263\begin{equation} \label{Eq_dynnept_vel}
1264u^* = -{1\over H} {\partial \psi \over \partial y}\ \ \  ,\ \ \ v^* = {1\over H} {\partial \psi \over \partial x}
1265\end{equation}
1266
1267\smallskip
1268%----------------------------------------------namdom----------------------------------------------------
1269\namdisplay{namdyn_nept}
1270%--------------------------------------------------------------------------------------------------------
1271\smallskip
1272
1273The Neptune effect is enabled when \np{ln\_neptsimp}=true (default=false).
1274\np{ln\_smooth\_neptvel} controls whether a scale-selective smoothing is applied
1275to the Neptune effect flow field (default=false) (this smoothing method is as
1276used by Holloway).  \np{rn\_tslse} and \np{rn\_tslsp} are the equatorial and
1277polar values respectively of the length-scale parameter $L$ used in determining
1278the Neptune stream function \eqref{Eq_dynnept_sf} and \eqref{Eq_dynnept_ls}.
1279Values at intermediate latitudes are given by a cosine fit, mimicking the
1280variation of the deformation radius with latitude.  The default values of 12km
1281and 3km are those given in \citep{HollowayJPO94}, appropriate for a coarse
1282resolution model. The finer resolution study of \citep{HollowayOM08} increased
1283the values of L by a factor of $\sqrt 2$ to 17km and 4.2km, thus doubling the
1284stream function for a given topography.
1285
1286The simple formulation for ($u^*$, $v^*$) can give unacceptably large velocities
1287in shallow water, and \citep{HollowayOM08} add an offset to the depth in the
1288denominator to control this problem. In this implementation we offer instead (at
1289the suggestion of G. Madec) the option of ramping down the Neptune flow field to
1290zero over a finite depth range. The switch \np{ln\_neptramp} activates this
1291option (default=false), in which case velocities at depths greater than
1292\np{rn\_htrmax} are unaltered, but ramp down linearly with depth to zero at a
1293depth of \np{rn\_htrmin} (and shallower).
1294
1295% ================================================================
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