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1% ================================================================
2% Chapter � Ocean Dynamics (DYN)
3% ================================================================
4\chapter{Ocean Dynamics (DYN)}
5\label{DYN}
6\minitoc
7
8% add a figure for  dynvor ens, ene latices
9
10%\vspace{2.cm}
11$\ $\newline      %force an empty line
12
13Using the representation described in Chapter \ref{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 Chapters \ref{SBC}, \ref{LDF} and \ref{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 \textbf{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
53Chap.\ref{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{DYN_divcur_wzv}
67
68%--------------------------------------------------------------------------------------------------------------
69%           Horizontal divergence and relative vorticity
70%--------------------------------------------------------------------------------------------------------------
71\subsection   [Horizontal divergence and relative vorticity (\textit{divcur})]
72         {Horizontal divergence and relative vorticity (\mdl{divcur})}
73\label{DYN_divcur}
74
75The vorticity is defined at an $f$-point ($i.e.$ corner point) as follows:
76\begin{equation} \label{Eq_divcur_cur}
77\zeta =\frac{1}{e_{1f}\,e_{2f} }\left( {\;\delta _{i+1/2} \left[ {e_{2v}\;v} \right]
78                          -\delta _{j+1/2} \left[ {e_{1u}\;u} \right]\;} \right)
79\end{equation} 
80
81The horizontal divergence is defined at a $T$-point. It is given by:
82\begin{equation} \label{Eq_divcur_div}
83\chi =\frac{1}{e_{1t}\,e_{2t}\,e_{3t} }
84      \left( {\delta _i \left[ {e_{2u}\,e_{3u}\,u} \right]
85             +\delta _j \left[ {e_{1v}\,e_{3v}\,v} \right]} \right)
86\end{equation} 
87
88Note that although the vorticity has the same discrete expression in $z$-
89and $s$-coordinates, its physical meaning is not identical. $\zeta$ is a pseudo
90vorticity along $s$-surfaces (only pseudo because $(u,v)$ are still defined along
91geopotential surfaces, but are not necessarily defined at the same depth).
92
93The vorticity and divergence at the \textit{before} step are used in the computation
94of the horizontal diffusion of momentum. Note that because they have been
95calculated prior to the Asselin filtering of the \textit{before} velocities, the
96\textit{before} vorticity and divergence arrays must be included in the restart file
97to ensure perfect restartability. The vorticity and divergence at the \textit{now} 
98time step are used for the computation of the nonlinear advection and of the
99vertical velocity respectively.
100
101%--------------------------------------------------------------------------------------------------------------
102%           Sea Surface Height evolution
103%--------------------------------------------------------------------------------------------------------------
104\subsection   [Sea surface height evolution and vertical velocity (\textit{sshwzv})]
105         {Horizontal divergence and relative vorticity (\mdl{sshwzv})}
106\label{DYN_sshwzv}
107
108The sea surface height is given by :
109\begin{equation} \label{Eq_dynspg_ssh}
110\begin{aligned}
111\frac{\partial \eta }{\partial t}
112&\equiv    \frac{1}{e_{1t} e_{2t} }\sum\limits_k { \left\{  \delta _i \left[ {e_{2u}\,e_{3u}\;u} \right]
113                                                                                  +\delta _j \left[ {e_{1v}\,e_{3v}\;v} \right]  \right\} } 
114           -    \frac{\textit{emp}}{\rho _w }   \\
115&\equiv    \sum\limits_k {\chi \ e_{3t}}  -  \frac{\textit{emp}}{\rho _w }
116\end{aligned}
117\end{equation}
118where \textit{emp} is the surface freshwater budget (evaporation minus precipitation),
119expressed in Kg/m$^2$/s (which is equal to mm/s), and $\rho _w$=1,035~Kg/m$^3$ 
120is the reference density of sea water (Boussinesq approximation). If river runoff is
121expressed as a surface freshwater flux (see \S\ref{SBC}) then \textit{emp} can be
122written as the evaporation minus precipitation, minus the river runoff.
123The sea-surface height is evaluated using exactly the same time stepping scheme
124as the tracer equation \eqref{Eq_tra_nxt}:
125a leapfrog scheme in combination with an Asselin time filter, $i.e.$ the velocity appearing
126in \eqref{Eq_dynspg_ssh} is centred in time (\textit{now} velocity).
127This is of paramount importance. Replacing $T$ by the number $1$ in the tracer equation and summing
128over the water column must lead to the sea surface height equation otherwise tracer content
129will not be conserved \citep{Griffies_al_MWR01, Leclair_Madec_OM09}.
130
131The vertical velocity is computed by an upward integration of the horizontal
132divergence starting at the bottom, taking into account the change of the thickness of the levels :
133\begin{equation} \label{Eq_wzv}
134\left\{   \begin{aligned}
135&\left. w \right|_{k_b-1/2} \quad= 0    \qquad \text{where } k_b \text{ is the level just above the sea floor }   \\
136&\left. w \right|_{k+1/2}     = \left. w \right|_{k-1/2}  +  \left. e_{3t} \right|_{k}\;  \left. \chi \right|_
137                                         - \frac{1} {2 \rdt} \left\left. e_{3t}^{t+1}\right|_{k} - \left. e_{3t}^{t-1}\right|_{k}\right)
138\end{aligned}   \right.
139\end{equation}
140
141In the case of a non-linear free surface (\key{vvl}), the top vertical velocity is $-\textit{emp}/\rho_w$,
142as changes in the divergence of the barotropic transport are absorbed into the change
143of the level thicknesses, re-orientated downward.
144\gmcomment{not sure of this...  to be modified with the change in emp setting}
145In the case of a linear free surface, the time derivative in \eqref{Eq_wzv} disappears.
146The upper boundary condition applies at a fixed level $z=0$. The top vertical velocity
147is thus equal to the divergence of the barotropic transport ($i.e.$ the first term in the
148right-hand-side of \eqref{Eq_dynspg_ssh}).
149
150Note also that whereas the vertical velocity has the same discrete
151expression in $z$- and $s$-coordinates, its physical meaning is not the same:
152in the second case, $w$ is the velocity normal to the $s$-surfaces.
153Note also that the $k$-axis is re-orientated downwards in the \textsc{fortran} code compared
154to the indexing used in the semi-discrete equations such as \eqref{Eq_wzv} 
155(see  \S\ref{DOM_Num_Index_vertical}).
156
157
158% ================================================================
159% Coriolis and Advection terms: vector invariant form
160% ================================================================
161\section{Coriolis and Advection: vector invariant form}
162\label{DYN_adv_cor_vect}
163%-----------------------------------------nam_dynadv----------------------------------------------------
164\namdisplay{namdyn_adv} 
165%-------------------------------------------------------------------------------------------------------------
166
167The vector invariant form of the momentum equations is the one most
168often used in applications of the \NEMO ocean model. The flux form option
169(see next section) has been present since version $2$. Options are defined
170through the \ngn{namdyn\_adv} namelist variables
171Coriolis and momentum advection terms are evaluated using a leapfrog
172scheme, $i.e.$ the velocity appearing in these expressions is centred in
173time (\textit{now} velocity).
174At the lateral boundaries either free slip, no slip or partial slip boundary
175conditions are applied following Chap.\ref{LBC}.
176
177% -------------------------------------------------------------------------------------------------------------
178%        Vorticity term
179% -------------------------------------------------------------------------------------------------------------
180\subsection   [Vorticity term (\textit{dynvor}) ]
181         {Vorticity term (\mdl{dynvor})}
182\label{DYN_vor}
183%------------------------------------------nam_dynvor----------------------------------------------------
184\namdisplay{namdyn_vor} 
185%-------------------------------------------------------------------------------------------------------------
186
187Options are defined through the \ngn{namdyn\_vor} namelist variables.
188Four discretisations of the vorticity term (\textit{ln\_dynvor\_xxx}=true) are available:
189conserving potential enstrophy of horizontally non-divergent flow (ENS scheme) ;
190conserving horizontal kinetic energy (ENE scheme) ; conserving potential enstrophy for
191the relative vorticity term and horizontal kinetic energy for the planetary vorticity
192term (MIX scheme) ; or conserving both the potential enstrophy of horizontally non-divergent
193flow and horizontal kinetic energy (EEN scheme) (see  Appendix~\ref{Apdx_C_vorEEN}). In the
194case of ENS, ENE or MIX schemes the land sea mask may be slightly modified to ensure the
195consistency of vorticity term with analytical equations (\textit{ln\_dynvor\_con}=true).
196The vorticity terms are all computed in dedicated routines that can be found in
197the \mdl{dynvor} module.
198
199%-------------------------------------------------------------
200%                 enstrophy conserving scheme
201%-------------------------------------------------------------
202\subsubsection{Enstrophy conserving scheme (\np{ln\_dynvor\_ens}=true)}
203\label{DYN_vor_ens}
204
205In the enstrophy conserving case (ENS scheme), the discrete formulation of the
206vorticity term provides a global conservation of the enstrophy
207($ [ (\zeta +f ) / e_{3f} ]^2 $ in $s$-coordinates) for a horizontally non-divergent
208flow ($i.e.$ $\chi$=$0$), but does not conserve the total kinetic energy. It is given by:
209\begin{equation} \label{Eq_dynvor_ens}
210\left\{ 
211\begin{aligned}
212{+\frac{1}{e_{1u} } } & {\overline {\left( { \frac{\zeta +f}{e_{3f} }} \right)} }^{\,i} 
213                                & {\overline{\overline {\left( {e_{1v}\,e_{3v}\;v} \right)}} }^{\,i, j+1/2}    \\
214{- \frac{1}{e_{2v} } } & {\overline {\left( {\frac{\zeta +f}{e_{3f} }} \right)} }^{\,j} 
215                                & {\overline{\overline {\left( {e_{2u}\,e_{3u}\;u} \right)}} }^{\,i+1/2, j} 
216\end{aligned} 
217 \right.
218\end{equation} 
219
220%-------------------------------------------------------------
221%                 energy conserving scheme
222%-------------------------------------------------------------
223\subsubsection{Energy conserving scheme (\np{ln\_dynvor\_ene}=true)}
224\label{DYN_vor_ene}
225
226The kinetic energy conserving scheme (ENE scheme) conserves the global
227kinetic energy but not the global enstrophy. It is given by:
228\begin{equation} \label{Eq_dynvor_ene}
229\left\{   \begin{aligned}
230{+\frac{1}{e_{1u}}\; {\overline {\left( {\frac{\zeta +f}{e_{3f} }} \right)
231                            \;  \overline {\left( {e_{1v}\,e_{3v}\;v} \right)} ^{\,i+1/2}} }^{\,j} }    \\
232{- \frac{1}{e_{2v}}\; {\overline {\left( {\frac{\zeta +f}{e_{3f} }} \right)
233                            \;  \overline {\left( {e_{2u}\,e_{3u}\;u} \right)} ^{\,j+1/2}} }^{\,i} }
234\end{aligned}    \right.
235\end{equation} 
236
237%-------------------------------------------------------------
238%                 mix energy/enstrophy conserving scheme
239%-------------------------------------------------------------
240\subsubsection{Mixed energy/enstrophy conserving scheme (\np{ln\_dynvor\_mix}=true) }
241\label{DYN_vor_mix}
242
243For the mixed energy/enstrophy conserving scheme (MIX scheme), a mixture of the
244two previous schemes is used. It consists of the ENS scheme (\ref{Eq_dynvor_ens})
245for the relative vorticity term, and of the ENE scheme (\ref{Eq_dynvor_ene}) applied
246to the planetary vorticity term.
247\begin{equation} \label{Eq_dynvor_mix}
248\left\{ {     \begin{aligned}
249 {+\frac{1}{e_{1u} }\; {\overline {\left( {\frac{\zeta }{e_{3f} }} \right)} }^{\,i} 
250 \; {\overline{\overline {\left( {e_{1v}\,e_{3v}\;v} \right)}} }^{\,i,j+1/2} -\frac{1}{e_{1u} }
251 \; {\overline {\left( {\frac{f}{e_{3f} }} \right)
252 \;\overline {\left( {e_{1v}\,e_{3v}\;v} \right)} ^{\,i+1/2}} }^{\,j} } \\
253{-\frac{1}{e_{2v} }\; {\overline {\left( {\frac{\zeta }{e_{3f} }} \right)} }^j
254 \; {\overline{\overline {\left( {e_{2u}\,e_{3u}\;u} \right)}} }^{\,i+1/2,j} +\frac{1}{e_{2v} }
255 \; {\overline {\left( {\frac{f}{e_{3f} }} \right)
256 \;\overline {\left( {e_{2u}\,e_{3u}\;u} \right)} ^{\,j+1/2}} }^{\,i} } \hfill
257\end{aligned}     } \right.
258\end{equation} 
259
260%-------------------------------------------------------------
261%                 energy and enstrophy conserving scheme
262%-------------------------------------------------------------
263\subsubsection{Energy and enstrophy conserving scheme (\np{ln\_dynvor\_een}=true) }
264\label{DYN_vor_een}
265
266In both the ENS and ENE schemes, it is apparent that the combination of $i$ and $j$ 
267averages of the velocity allows for the presence of grid point oscillation structures
268that will be invisible to the operator. These structures are \textit{computational modes} 
269that will be at least partly damped by the momentum diffusion operator ($i.e.$ the
270subgrid-scale advection), but not by the resolved advection term. The ENS and ENE schemes
271therefore do not contribute to dump any grid point noise in the horizontal velocity field.
272Such noise would result in more noise in the vertical velocity field, an undesirable feature.
273This is a well-known characteristic of $C$-grid discretization where $u$ and $v$ are located
274at different grid points, a price worth paying to avoid a double averaging in the pressure
275gradient term as in the $B$-grid.
276\gmcomment{ To circumvent this, Adcroft (ADD REF HERE)
277Nevertheless, this technique strongly distort the phase and group velocity of Rossby waves....}
278
279A very nice solution to the problem of double averaging was proposed by \citet{Arakawa_Hsu_MWR90}.
280The idea is to get rid of the double averaging by considering triad combinations of vorticity.
281It is noteworthy that this solution is conceptually quite similar to the one proposed by
282\citep{Griffies_al_JPO98} for the discretization of the iso-neutral diffusion operator (see App.\ref{Apdx_C}).
283
284The \citet{Arakawa_Hsu_MWR90} vorticity advection scheme for a single layer is modified
285for spherical coordinates as described by \citet{Arakawa_Lamb_MWR81} to obtain the EEN scheme.
286First consider the discrete expression of the potential vorticity, $q$, defined at an $f$-point:
287\begin{equation} \label{Eq_pot_vor}
288q  = \frac{\zeta +f} {e_{3f} }
289\end{equation}
290where the relative vorticity is defined by (\ref{Eq_divcur_cur}), the Coriolis parameter
291is given by $f=2 \,\Omega \;\sin \varphi _f $ and the layer thickness at $f$-points is:
292\begin{equation} \label{Eq_een_e3f}
293e_{3f} = \overline{\overline {e_{3t} }} ^{\,i+1/2,j+1/2}
294\end{equation}
295
296%>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
297\begin{figure}[!ht]    \begin{center}
298\includegraphics[width=0.70\textwidth]{./TexFiles/Figures/Fig_DYN_een_triad.pdf}
299\caption{ \label{Fig_DYN_een_triad} 
300Triads used in the energy and enstrophy conserving scheme (een) for
301$u$-component (upper panel) and $v$-component (lower panel).}
302\end{center}   \end{figure}
303%>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
304
305Note that a key point in \eqref{Eq_een_e3f} is that the averaging in the \textbf{i}- and
306\textbf{j}- directions uses the masked vertical scale factor but is always divided by
307$4$, not by the sum of the masks at the four $T$-points. This preserves the continuity of
308$e_{3f}$ when one or more of the neighbouring $e_{3t}$ tends to zero and
309extends by continuity the value of $e_{3f}$ into the land areas. This feature is essential for
310the $z$-coordinate with partial steps.
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(Fig.~\ref{Fig_DYN_een_triad}):
315\begin{equation} \label{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  Appendix~\ref{Apdx_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 (\textit{dynkeg})]
346         {Kinetic Energy Gradient term (\mdl{dynkeg})}
347\label{DYN_keg}
348
349As demonstrated in Appendix~\ref{Apdx_C}, there is a single discrete formulation
350of the kinetic energy gradient term that, together with the formulation chosen for
351the vertical advection (see below), conserves the total kinetic energy:
352\begin{equation} \label{Eq_dynkeg}
353\left\{ \begin{aligned}
354 -\frac{1}{2 \; e_{1u} }  & \ \delta _{i+1/2} \left[ {\overline {u^2}^{\,i} + \overline{v^2}^{\,j}} \right]   \\
355 -\frac{1}{2 \; e_{2v} }  & \ \delta _{j+1/2} \left[ {\overline {u^2}^{\,i} + \overline{v^2}^{\,j}} \right]   
356\end{aligned} \right.
357\end{equation} 
358
359%--------------------------------------------------------------------------------------------------------------
360%           Vertical advection term
361%--------------------------------------------------------------------------------------------------------------
362\subsection   [Vertical advection term (\textit{dynzad}) ]
363         {Vertical advection term (\mdl{dynzad}) }
364\label{DYN_zad}
365
366The discrete formulation of the vertical advection, together with the formulation
367chosen for the gradient of kinetic energy (KE) term, conserves the total kinetic
368energy. Indeed, the change of KE due to the vertical advection is exactly
369balanced by the change of KE due to the gradient of KE (see Appendix~\ref{Apdx_C}).
370\begin{equation} \label{Eq_dynzad}
371\left\{     \begin{aligned}
372-\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}  \\
373-\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} 
374\end{aligned}         \right.
375\end{equation} 
376
377% ================================================================
378% Coriolis and Advection : flux form
379% ================================================================
380\section{Coriolis and Advection: flux form}
381\label{DYN_adv_cor_flux}
382%------------------------------------------nam_dynadv----------------------------------------------------
383\namdisplay{namdyn_adv} 
384%-------------------------------------------------------------------------------------------------------------
385
386Options are defined through the \ngn{namdyn\_adv} namelist variables.
387In the flux form (as in the vector invariant form), the Coriolis and momentum
388advection terms are evaluated using a leapfrog scheme, $i.e.$ the velocity
389appearing in their expressions is centred in time (\textit{now} velocity). At the
390lateral boundaries either free slip, no slip or partial slip boundary conditions
391are applied following Chap.\ref{LBC}.
392
393
394%--------------------------------------------------------------------------------------------------------------
395%           Coriolis plus curvature metric terms
396%--------------------------------------------------------------------------------------------------------------
397\subsection   [Coriolis plus curvature metric terms (\textit{dynvor}) ]
398         {Coriolis plus curvature metric terms (\mdl{dynvor}) }
399\label{DYN_cor_flux}
400
401In flux form, the vorticity term reduces to a Coriolis term in which the Coriolis
402parameter has been modified to account for the "metric" term. This altered
403Coriolis parameter is thus discretised at $f$-points. It is given by:
404\begin{multline} \label{Eq_dyncor_metric}
405f+\frac{1}{e_1 e_2 }\left( {v\frac{\partial e_2 }{\partial i}  -  u\frac{\partial e_1 }{\partial j}} \right\\
406   \equiv   f + \frac{1}{e_{1f} e_{2f} } \left( { \ \overline v ^{i+1/2}\delta _{i+1/2} \left[ {e_{2u} } \right] 
407                                                                 -  \overline u ^{j+1/2}\delta _{j+1/2} \left[ {e_{1u} } \right]  }  \ \right)
408\end{multline} 
409
410Any of the (\ref{Eq_dynvor_ens}), (\ref{Eq_dynvor_ene}) and (\ref{Eq_dynvor_een})
411schemes can be used to compute the product of the Coriolis parameter and the
412vorticity. However, the energy-conserving scheme  (\ref{Eq_dynvor_een}) has
413exclusively been used to date. This term is evaluated using a leapfrog scheme,
414$i.e.$ the velocity is centred in time (\textit{now} velocity).
415
416%--------------------------------------------------------------------------------------------------------------
417%           Flux form Advection term
418%--------------------------------------------------------------------------------------------------------------
419\subsection   [Flux form Advection term (\textit{dynadv}) ]
420         {Flux form Advection term (\mdl{dynadv}) }
421\label{DYN_adv_flux}
422
423The discrete expression of the advection term is given by :
424\begin{equation} \label{Eq_dynadv}
425\left\{ 
426\begin{aligned}
427\frac{1}{e_{1u}\,e_{2u}\,e_{3u}} 
428\left(      \delta _{i+1/2} \left[ \overline{e_{2u}\,e_{3u}\;u }^{i       }  \ u_t      \right]   
429          + \delta _{j       } \left[ \overline{e_{1u}\,e_{3u}\;v }^{i+1/2}  \ u_f      \right] \right\ \;   \\
430\left.   + \delta _{k      } \left[ \overline{e_{1w}\,e_{2w}\;w}^{i+1/2}  \ u_{uw} \right] \right)   \\
431\\
432\frac{1}{e_{1v}\,e_{2v}\,e_{3v}} 
433\left(     \delta _{i       } \left[ \overline{e_{2u}\,e_{3u }\;u }^{j+1/2} \ v_f       \right] 
434         + \delta _{j+1/2} \left[ \overline{e_{1u}\,e_{3u }\;v }^{i       } \ v_t       \right] \right\ \, \, \\
435\left.  + \delta _{k      } \left[ \overline{e_{1w}\,e_{2w}\;w}^{j+1/2} \ v_{vw}  \right] \right) \\
436\end{aligned}
437\right.
438\end{equation}
439
440Two advection schemes are available: a $2^{nd}$ order centered finite
441difference scheme, CEN2, or a $3^{rd}$ order upstream biased scheme, UBS.
442The latter is described in \citet{Shchepetkin_McWilliams_OM05}. The schemes are
443selected using the namelist logicals \np{ln\_dynadv\_cen2} and \np{ln\_dynadv\_ubs}.
444In flux form, the schemes differ by the choice of a space and time interpolation to
445define the value of $u$ and $v$ at the centre of each face of $u$- and $v$-cells,
446$i.e.$ at the $T$-, $f$-, and $uw$-points for $u$ and at the $f$-, $T$- and
447$vw$-points for $v$.
448
449%-------------------------------------------------------------
450%                 2nd order centred scheme
451%-------------------------------------------------------------
452\subsubsection{$2^{nd}$ order centred scheme (cen2) (\np{ln\_dynadv\_cen2}=true)}
453\label{DYN_adv_cen2}
454
455In the centered $2^{nd}$ order formulation, the velocity is evaluated as the
456mean of the two neighbouring points :
457\begin{equation} \label{Eq_dynadv_cen2}
458\left\{     \begin{aligned}
459 u_T^{cen2} &=\overline u^{i }       \quad &  u_F^{cen2} &=\overline u^{j+1/2}  \quad &  u_{uw}^{cen2} &=\overline u^{k+1/2}   \\
460 v_F^{cen2} &=\overline v ^{i+1/2} \quad & v_F^{cen2} &=\overline v^j      \quad &  v_{vw}^{cen2} &=\overline v ^{k+1/2}  \\
461\end{aligned}      \right.
462\end{equation} 
463
464The scheme is non diffusive (i.e. conserves the kinetic energy) but dispersive
465($i.e.$ it may create false extrema). It is therefore notoriously noisy and must be
466used in conjunction with an explicit diffusion operator to produce a sensible solution.
467The associated time-stepping is performed using a leapfrog scheme in conjunction
468with an Asselin time-filter, so $u$ and $v$ are the \emph{now} velocities.
469
470%-------------------------------------------------------------
471%                 UBS scheme
472%-------------------------------------------------------------
473\subsubsection{Upstream Biased Scheme (UBS) (\np{ln\_dynadv\_ubs}=true)}
474\label{DYN_adv_ubs}
475
476The UBS advection scheme is an upstream biased third order scheme based on
477an upstream-biased parabolic interpolation. For example, the evaluation of
478$u_T^{ubs} $ is done as follows:
479\begin{equation} \label{Eq_dynadv_ubs}
480u_T^{ubs} =\overline u ^i-\;\frac{1}{6}   \begin{cases}
481      u"_{i-1/2}&    \text{if $\ \overline{e_{2u}\,e_{3u} \ u}^i  \geqslant 0$ }    \\
482      u"_{i+1/2}&    \text{if $\ \overline{e_{2u}\,e_{3u} \ u}^i  < 0$ }
483\end{cases}
484\end{equation}
485where $u"_{i+1/2} =\delta _{i+1/2} \left[ {\delta _i \left[ u \right]} \right]$. This results
486in a dissipatively dominant ($i.e.$ hyper-diffusive) truncation error \citep{Shchepetkin_McWilliams_OM05}.
487The overall performance of the advection scheme is similar to that reported in
488\citet{Farrow1995}. It is a relatively good compromise between accuracy and
489smoothness. It is not a \emph{positive} scheme, meaning that false extrema are
490permitted. But the amplitudes of the false extrema are significantly reduced over
491those in the centred second order method. As the scheme already includes
492a diffusion component, it can be used without explicit  lateral diffusion on momentum
493($i.e.$ \np{ln\_dynldf\_lap}=\np{ln\_dynldf\_bilap}=false), and it is recommended to do so.
494
495The UBS scheme is not used in all directions. In the vertical, the centred $2^{nd}$ 
496order evaluation of the advection is preferred, $i.e.$ $u_{uw}^{ubs}$ and
497$u_{vw}^{ubs}$ in \eqref{Eq_dynadv_cen2} are used. UBS is diffusive and is
498associated with vertical mixing of momentum. \gmcomment{ gm  pursue the
499sentence:Since vertical mixing of momentum is a source term of the TKE equation...  }
500
501For stability reasons,  the first term in (\ref{Eq_dynadv_ubs}), which corresponds
502to a second order centred scheme, is evaluated using the \textit{now} velocity
503(centred in time), while the second term, which is the diffusion part of the scheme,
504is evaluated using the \textit{before} velocity (forward in time). This is discussed
505by \citet{Webb_al_JAOT98} in the context of the Quick advection scheme.
506
507Note that the UBS and QUICK (Quadratic Upstream Interpolation for Convective Kinematics)
508schemes only differ by one coefficient. Replacing $1/6$ by $1/8$ in
509(\ref{Eq_dynadv_ubs}) leads to the QUICK advection scheme \citep{Webb_al_JAOT98}.
510This option is not available through a namelist parameter, since the $1/6$ coefficient
511is hard coded. Nevertheless it is quite easy to make the substitution in the
512\mdl{dynadv\_ubs} module and obtain a QUICK scheme.
513
514Note also that in the current version of \mdl{dynadv\_ubs}, there is also the
515possibility of using a $4^{th}$ order evaluation of the advective velocity as in
516ROMS. This is an error and should be suppressed soon.
517%%%
518\gmcomment{action :  this have to be done}
519%%%
520
521% ================================================================
522%           Hydrostatic pressure gradient term
523% ================================================================
524\section  [Hydrostatic pressure gradient (\textit{dynhpg})]
525      {Hydrostatic pressure gradient (\mdl{dynhpg})}
526\label{DYN_hpg}
527%------------------------------------------nam_dynhpg---------------------------------------------------
528\namdisplay{namdyn_hpg} 
529%-------------------------------------------------------------------------------------------------------------
530
531Options are defined through the \ngn{namdyn\_hpg} namelist variables.
532The key distinction between the different algorithms used for the hydrostatic
533pressure gradient is the vertical coordinate used, since HPG is a \emph{horizontal} 
534pressure gradient, $i.e.$ computed along geopotential surfaces. As a result, any
535tilt of the surface of the computational levels will require a specific treatment to
536compute the hydrostatic pressure gradient.
537
538The hydrostatic pressure gradient term is evaluated either using a leapfrog scheme,
539$i.e.$ the density appearing in its expression is centred in time (\emph{now} $\rho$), or
540a semi-implcit scheme. At the lateral boundaries either free slip, no slip or partial slip
541boundary conditions are applied.
542
543%--------------------------------------------------------------------------------------------------------------
544%           z-coordinate with full step
545%--------------------------------------------------------------------------------------------------------------
546\subsection   [$z$-coordinate with full step (\np{ln\_dynhpg\_zco}) ]
547         {$z$-coordinate with full step (\np{ln\_dynhpg\_zco}=true)}
548\label{DYN_hpg_zco}
549
550The hydrostatic pressure can be obtained by integrating the hydrostatic equation
551vertically from the surface. However, the pressure is large at great depth while its
552horizontal gradient is several orders of magnitude smaller. This may lead to large
553truncation errors in the pressure gradient terms. Thus, the two horizontal components
554of the hydrostatic pressure gradient are computed directly as follows:
555
556for $k=km$ (surface layer, $jk=1$ in the code)
557\begin{equation} \label{Eq_dynhpg_zco_surf}
558\left\{ \begin{aligned}
559               \left. \delta _{i+1/2} \left[  p^h         \right] \right|_{k=km} 
560&= \frac{1}{2} g \   \left. \delta _{i+1/2} \left[  e_{3w} \ \rho \right] \right|_{k=km}   \\
561                  \left. \delta _{j+1/2} \left[  p^h            \right] \right|_{k=km} 
562&= \frac{1}{2} g \   \left. \delta _{j+1/2} \left[  e_{3w} \ \rho \right] \right|_{k=km}   \\
563\end{aligned} \right.
564\end{equation} 
565
566for $1<k<km$ (interior layer)
567\begin{equation} \label{Eq_dynhpg_zco}
568\left\{ \begin{aligned}
569               \left. \delta _{i+1/2} \left[  p^h         \right] \right|_{k} 
570&=             \left. \delta _{i+1/2} \left[  p^h         \right] \right|_{k-1} 
571+    \frac{1}{2}\;g\;   \left. \delta _{i+1/2} \left[  e_{3w} \ \overline {\rho}^{k+1/2} \right] \right|_{k}   \\
572                  \left. \delta _{j+1/2} \left[  p^h            \right] \right|_{k} 
573&=                \left. \delta _{j+1/2} \left[  p^h            \right] \right|_{k-1} 
574+    \frac{1}{2}\;g\;   \left. \delta _{j+1/2} \left[  e_{3w} \ \overline {\rho}^{k+1/2} \right] \right|_{k}   \\
575\end{aligned} \right.
576\end{equation} 
577
578Note that the $1/2$ factor in (\ref{Eq_dynhpg_zco_surf}) is adequate because of
579the definition of $e_{3w}$ as the vertical derivative of the scale factor at the surface
580level ($z=0$). Note also that in case of variable volume level (\key{vvl} defined), the
581surface pressure gradient is included in \eqref{Eq_dynhpg_zco_surf} and \eqref{Eq_dynhpg_zco} 
582through the space and time variations of the vertical scale factor $e_{3w}$.
583
584%--------------------------------------------------------------------------------------------------------------
585%           z-coordinate with partial step
586%--------------------------------------------------------------------------------------------------------------
587\subsection   [$z$-coordinate with partial step (\np{ln\_dynhpg\_zps})]
588         {$z$-coordinate with partial step (\np{ln\_dynhpg\_zps}=true)}
589\label{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 \S\ref{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 \S\ref{TRA_zpshde}.
604
605%--------------------------------------------------------------------------------------------------------------
606%           s- and s-z-coordinates
607%--------------------------------------------------------------------------------------------------------------
608\subsection{$s$- and $z$-$s$-coordinates}
609\label{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}=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\eqref{Eq_dynhpg_zco_surf} - \eqref{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$\bullet$ Pressure Jacobian scheme (prj) (a research paper in preparation) (\np{ln\_dynhpg\_prj}=true)
632
633$\bullet$ Density Jacobian with cubic polynomial scheme (DJC) \citep{Shchepetkin_McWilliams_OM05} 
634(\np{ln\_dynhpg\_djc}=true) (currently disabled; under development)
635
636Note that expression \eqref{Eq_dynhpg_sco} is commonly used when the variable volume formulation is
637activated (\key{vvl}) because in that case, even with a flat bottom, the coordinate surfaces are not
638horizontal but follow the free surface \citep{Levier2007}. The pressure jacobian scheme
639(\np{ln\_dynhpg\_prj}=true) is available as an improved option to \np{ln\_dynhpg\_sco}=true when
640\key{vvl} is active.  The pressure Jacobian scheme uses a constrained cubic spline to reconstruct
641the density profile across the water column. This method maintains the monotonicity between the
642density nodes  The pressure can be calculated by analytical integration of the density profile and a
643pressure Jacobian method is used to solve the horizontal pressure gradient. This method can provide
644a more accurate calculation of the horizontal pressure gradient than the standard scheme.
645
646%--------------------------------------------------------------------------------------------------------------
647%           Time-scheme
648%--------------------------------------------------------------------------------------------------------------
649\subsection   [Time-scheme (\np{ln\_dynhpg\_imp}) ]
650         {Time-scheme (\np{ln\_dynhpg\_imp}= true/false)}
651\label{DYN_hpg_imp}
652
653The default time differencing scheme used for the horizontal pressure gradient is
654a leapfrog scheme and therefore the density used in all discrete expressions given
655above is the  \textit{now} density, computed from the \textit{now} temperature and
656salinity. In some specific cases (usually high resolution simulations over an ocean
657domain which includes weakly stratified regions) the physical phenomenon that
658controls the time-step is internal gravity waves (IGWs). A semi-implicit scheme for
659doubling the stability limit associated with IGWs can be used \citep{Brown_Campana_MWR78,
660Maltrud1998}. It involves the evaluation of the hydrostatic pressure gradient as an
661average over the three time levels $t-\rdt$, $t$, and $t+\rdt$ ($i.e.$ 
662\textit{before}\textit{now} and  \textit{after} time-steps), rather than at the central
663time level $t$ only, as in the standard leapfrog scheme.
664
665$\bullet$ leapfrog scheme (\np{ln\_dynhpg\_imp}=true):
666
667\begin{equation} \label{Eq_dynhpg_lf}
668\frac{u^{t+\rdt}-u^{t-\rdt}}{2\rdt} = \;\cdots \;
669   -\frac{1}{\rho _o \,e_{1u} }\delta _{i+1/2} \left[ {p_h^t } \right]
670\end{equation}
671
672$\bullet$ semi-implicit scheme (\np{ln\_dynhpg\_imp}=true):
673\begin{equation} \label{Eq_dynhpg_imp}
674\frac{u^{t+\rdt}-u^{t-\rdt}}{2\rdt} = \;\cdots \;
675   -\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]
676\end{equation}
677
678The semi-implicit time scheme \eqref{Eq_dynhpg_imp} is made possible without
679significant additional computation since the density can be updated to time level
680$t+\rdt$ before computing the horizontal hydrostatic pressure gradient. It can
681be easily shown that the stability limit associated with the hydrostatic pressure
682gradient doubles using \eqref{Eq_dynhpg_imp} compared to that using the
683standard leapfrog scheme \eqref{Eq_dynhpg_lf}. Note that \eqref{Eq_dynhpg_imp} 
684is equivalent to applying a time filter to the pressure gradient to eliminate high
685frequency IGWs. Obviously, when using \eqref{Eq_dynhpg_imp}, the doubling of
686the time-step is achievable only if no other factors control the time-step, such as
687the stability limits associated with advection or diffusion.
688
689In practice, the semi-implicit scheme is used when \np{ln\_dynhpg\_imp}=true.
690In this case, we choose to apply the time filter to temperature and salinity used in
691the equation of state, instead of applying it to the hydrostatic pressure or to the
692density, so that no additional storage array has to be defined. The density used to
693compute the hydrostatic pressure gradient (whatever the formulation) is evaluated
694as follows:
695\begin{equation} \label{Eq_rho_flt}
696   \rho^t = \rho( \widetilde{T},\widetilde {S},z_t)
697 \quad     \text{with}  \quad 
698   \widetilde{X} = 1 / 4 \left(  X^{t+\rdt} +2 \,X^t + X^{t-\rdt}  \right)
699\end{equation}
700
701Note that in the semi-implicit case, it is necessary to save the filtered density, an
702extra three-dimensional field, in the restart file to restart the model with exact
703reproducibility. This option is controlled by  \np{nn\_dynhpg\_rst}, a namelist parameter.
704
705% ================================================================
706% Surface Pressure Gradient
707% ================================================================
708\section  [Surface pressure gradient (\textit{dynspg}) ]
709      {Surface pressure gradient (\mdl{dynspg})}
710\label{DYN_spg}
711%-----------------------------------------nam_dynspg----------------------------------------------------
712\namdisplay{namdyn_spg} 
713%------------------------------------------------------------------------------------------------------------
714
715$\ $\newline      %force an empty line
716
717%%%
718Options are defined through the \ngn{namdyn\_spg} namelist variables.
719The 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}.
720
721%%%
722
723
724The form of the surface pressure gradient term depends on how the user wants to handle
725the fast external gravity waves that are a solution of the analytical equation (\S\ref{PE_hor_pg}).
726Three formulations are available, all controlled by a CPP key (ln\_dynspg\_xxx):
727an explicit formulation which requires a small time step ;
728a filtered free surface formulation which allows a larger time step by adding a filtering
729term into the momentum equation ;
730and a split-explicit free surface formulation, described below, which also allows a larger time step.
731
732The extra term introduced in the filtered method is calculated
733implicitly, so that a solver is used to compute it. As a consequence the update of the $next$ 
734velocities is done in module \mdl{dynspg\_flt} and not in \mdl{dynnxt}.
735
736
737
738%--------------------------------------------------------------------------------------------------------------
739% Explicit free surface formulation
740%--------------------------------------------------------------------------------------------------------------
741\subsection{Explicit free surface (\key{dynspg\_exp})}
742\label{DYN_spg_exp}
743
744In the explicit free surface formulation (\key{dynspg\_exp} defined), the model time step
745is chosen to be small enough to resolve the external gravity waves (typically a few tens of seconds).
746The surface pressure gradient, evaluated using a leap-frog scheme ($i.e.$ centered in time),
747is thus simply given by :
748\begin{equation} \label{Eq_dynspg_exp}
749\left\{ \begin{aligned}
750 - \frac{1}{e_{1u}\,\rho_o} \;   \delta _{i+1/2} \left[  \,\rho \,\eta\,  \right]   \\
751 - \frac{1}{e_{2v}\,\rho_o} \;   \delta _{j+1/2} \left[  \,\rho \,\eta\,  \right] 
752\end{aligned} \right.
753\end{equation} 
754
755Note that in the non-linear free surface case ($i.e.$ \key{vvl} defined), the surface pressure
756gradient is already included in the momentum tendency  through the level thickness variation
757allowed in the computation of the hydrostatic pressure gradient. Thus, nothing is done in the \mdl{dynspg\_exp} module.
758
759%--------------------------------------------------------------------------------------------------------------
760% Split-explict free surface formulation
761%--------------------------------------------------------------------------------------------------------------
762\subsection{Split-Explicit free surface (\key{dynspg\_ts})}
763\label{DYN_spg_ts}
764%------------------------------------------namsplit-----------------------------------------------------------
765\namdisplay{namsplit} 
766%-------------------------------------------------------------------------------------------------------------
767
768The split-explicit free surface formulation used in \NEMO (\key{dynspg\_ts} defined),
769also called the time-splitting formulation, follows the one
770proposed by \citet{Shchepetkin_McWilliams_OM05}. The general idea is to solve the free surface
771equation and the associated barotropic velocity equations with a smaller time
772step than $\rdt$, the time step used for the three dimensional prognostic
773variables (Fig.~\ref{Fig_DYN_dynspg_ts}).
774The size of the small time step, $\rdt_e$ (the external mode or barotropic time step)
775 is provided through the \np{nn\_baro} namelist parameter as:
776$\rdt_e = \rdt / nn\_baro$. This parameter can be optionally defined automatically (\np{ln\_bt\_nn\_auto}=true)
777considering that the stability of the barotropic system is essentially controled by external waves propagation.
778Maximum allowed Courant number is in that case time independent, and easily computed online from the input bathymetry.
779
780%%%
781The barotropic mode solves the following equations:
782\begin{subequations} \label{Eq_BT}
783  \begin{equation}     \label{Eq_BT_dyn}
784\frac{\partial {\rm \overline{{\bf U}}_h} }{\partial t}=
785 -f\;{\rm {\bf k}}\times {\rm \overline{{\bf U}}_h} 
786-g\nabla _h \eta -\frac{c_b^{\textbf U}}{H+\eta} \rm {\overline{{\bf U}}_h} + \rm {\overline{\bf G}}
787  \end{equation}
788
789  \begin{equation} \label{Eq_BT_ssh}
790\frac{\partial \eta }{\partial t}=-\nabla \cdot \left[ {\left( {H+\eta } \right) \; {\rm{\bf \overline{U}}}_h \,} \right]+P-E
791  \end{equation}
792\end{subequations}
793where $\rm {\overline{\bf G}}$ 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).
794
795%>   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >
796\begin{figure}[!t]    \begin{center}
797\includegraphics[width=0.7\textwidth]{./TexFiles/Figures/Fig_DYN_dynspg_ts.pdf}
798\caption{  \label{Fig_DYN_dynspg_ts}
799Schematic of the split-explicit time stepping scheme for the external
800and internal modes. Time increases to the right. In this particular exemple,
801a boxcar averaging window over $nn\_baro$ barotropic time steps is used ($nn\_bt\_filt=1$) and $nn\_baro=5$.
802Internal mode time steps (which are also the model time steps) are denoted
803by $t-\rdt$, $t$ and $t+\rdt$. Variables with $k$ superscript refer to instantaneous barotropic variables,
804$< >$ and $<< >>$ operator refer to time filtered variables using respectively primary (red vertical bars) and secondary weights (blue vertical bars).
805The former are used to obtain time filtered quantities at $t+\rdt$ while the latter are used to obtain time averaged
806transports to advect tracers.
807a) Forward time integration: \np{ln\_bt\_fw}=true, \np{ln\_bt\_ave}=true.
808b) Centred time integration: \np{ln\_bt\_fw}=false, \np{ln\_bt\_ave}=true.
809c) Forward time integration with no time filtering (POM-like scheme): \np{ln\_bt\_fw}=true, \np{ln\_bt\_ave}=false. }
810\end{center}    \end{figure}
811%>   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >
812
813In the default case (\np{ln\_bt\_fw}=true), the external mode is integrated
814between \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
815quantities (\np{ln\_bt\_ave}=true). In that case, the integration is extended slightly beyond  \textit{after} time step to provide time filtered quantities.
816These are used for the subsequent initialization of the barotropic mode in the following baroclinic step.
817Since external mode equations written at baroclinic time steps finally follow a forward time stepping scheme,
818asselin filtering is not applied to barotropic quantities. \\
819Alternatively, one can choose to integrate barotropic equations starting
820from \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
821become centred in the middle of the integration window. It can easily be shown that this property
822removes part of splitting errors between modes, which increases the overall numerical robustness.
823%references to Patrick Marsaleix' work here. Also work done by SHOM group.
824
825%%%
826
827As far as tracer conservation is concerned, barotropic velocities used to advect tracers must also be updated
828at \textit{now} time step. This implies to change the traditional order of computations in \NEMO: most of momentum 
829trends (including the barotropic mode calculation) updated first, tracers' after. This \textit{de facto} makes semi-implicit hydrostatic
830pressure gradient (see section \S\ref{DYN_hpg_imp}) and time splitting not compatible.
831Advective barotropic velocities are obtained by using a secondary set of filtering weights, uniquely defined from the filter
832coefficients 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.
833
834%%%
835
836One can eventually choose to feedback instantaneous values by not using any time filter (\np{ln\_bt\_ave}=false).
837In that case, external mode equations are continuous in time, ie they are not re-initialized when starting a new
838sub-stepping sequence. This is the method used so far in the POM model, the stability being maintained by refreshing at (almost)
839each barotropic time step advection and horizontal diffusion terms. Since the latter terms have not been added in \NEMO for
840computational efficiency, removing time filtering is not recommended except for debugging purposes.
841This 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.
842
843%>>>>>===============
844\gmcomment{               %%% copy from griffies Book
845
846\textbf{title: Time stepping the barotropic system }
847
848Assume knowledge of the full velocity and tracer fields at baroclinic time $\tau$. Hence,
849we can update the surface height and vertically integrated velocity with a leap-frog
850scheme using the small barotropic time step $\rdt$. We have
851
852\begin{equation} \label{DYN_spg_ts_eta}
853\eta^{(b)}(\tau,t_{n+1}) - \eta^{(b)}(\tau,t_{n+1}) (\tau,t_{n-1})
854   = 2 \rdt \left[-\nabla \cdot \textbf{U}^{(b)}(\tau,t_n) + \text{EMP}_w(\tau) \right] 
855\end{equation}
856\begin{multline} \label{DYN_spg_ts_u}
857\textbf{U}^{(b)}(\tau,t_{n+1}) - \textbf{U}^{(b)}(\tau,t_{n-1}\\
858   = 2\rdt \left[ - f \textbf{k} \times \textbf{U}^{(b)}(\tau,t_{n})
859   - H(\tau) \nabla p_s^{(b)}(\tau,t_{n}) +\textbf{M}(\tau) \right]
860\end{multline}
861\
862
863In these equations, araised (b) denotes values of surface height and vertically integrated velocity updated with the barotropic time steps. The $\tau$ time label on $\eta^{(b)}$ 
864and $U^{(b)}$ 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
865that sets the barotropic time steps via
866\begin{equation} \label{DYN_spg_ts_t}
867t_n=\tau+n\rdt   
868\end{equation}
869with $n$ an integer. The density scaled surface pressure is evaluated via
870\begin{equation} \label{DYN_spg_ts_ps}
871p_s^{(b)}(\tau,t_{n}) = \begin{cases}
872   g \;\eta_s^{(b)}(\tau,t_{n}) \;\rho(\tau)_{k=1}) / \rho_&      \text{non-linear case} \\
873   g \;\eta_s^{(b)}(\tau,t_{n}&      \text{linear case} 
874   \end{cases}
875\end{equation}
876To get started, we assume the following initial conditions
877\begin{equation} \label{DYN_spg_ts_eta}
878\begin{split}
879\eta^{(b)}(\tau,t_{n=0}) &= \overline{\eta^{(b)}(\tau)}
880\\
881\eta^{(b)}(\tau,t_{n=1}) &= \eta^{(b)}(\tau,t_{n=0}) + \rdt \ \text{RHS}_{n=0} 
882\end{split}
883\end{equation}
884with
885\begin{equation} \label{DYN_spg_ts_etaF}
886 \overline{\eta^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N \eta^{(b)}(\tau-\rdt,t_{n})
887\end{equation}
888the time averaged surface height taken from the previous barotropic cycle. Likewise,
889\begin{equation} \label{DYN_spg_ts_u}
890\textbf{U}^{(b)}(\tau,t_{n=0}) = \overline{\textbf{U}^{(b)}(\tau)}   \\
891\\
892\textbf{U}(\tau,t_{n=1}) = \textbf{U}^{(b)}(\tau,t_{n=0}) + \rdt \ \text{RHS}_{n=0}   
893\end{equation}
894with
895\begin{equation} \label{DYN_spg_ts_u}
896 \overline{\textbf{U}^{(b)}(\tau)} 
897   = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau-\rdt,t_{n})
898\end{equation}
899the time averaged vertically integrated transport. Notably, there is no Robert-Asselin time filter used in the barotropic portion of the integration.
900
901Upon 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 $\tau + \rdt \tau$ 
902\begin{equation} \label{DYN_spg_ts_u}
903\textbf{U}(\tau+\rdt) = \overline{\textbf{U}^{(b)}(\tau+\rdt)} 
904   = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau,t_{n})
905\end{equation}
906The surface height on the new baroclinic time step is then determined via a baroclinic leap-frog using the following form
907
908\begin{equation} \label{DYN_spg_ts_ssh}
909\eta(\tau+\Delta) - \eta^{F}(\tau-\Delta) = 2\rdt \ \left[ - \nabla \cdot \textbf{U}(\tau) + \text{EMP}_w \right] 
910\end{equation}
911
912 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).
913 
914In general, some form of time filter is needed to maintain integrity of the surface
915height field due to the leap-frog splitting mode in equation \ref{DYN_spg_ts_ssh}. We
916have tried various forms of such filtering, with the following method discussed in
917\cite{Griffies_al_MWR01} chosen due to its stability and reasonably good maintenance of
918tracer conservation properties (see Section ??)
919
920\begin{equation} \label{DYN_spg_ts_sshf}
921\eta^{F}(\tau-\Delta) =  \overline{\eta^{(b)}(\tau)} 
922\end{equation}
923Another approach tried was
924
925\begin{equation} \label{DYN_spg_ts_sshf2}
926\eta^{F}(\tau-\Delta) = \eta(\tau)
927   + (\alpha/2) \left[\overline{\eta^{(b)}}(\tau+\rdt)
928                + \overline{\eta^{(b)}}(\tau-\rdt) -2 \;\eta(\tau) \right]
929\end{equation}
930
931which is useful since it isolates all the time filtering aspects into the term multiplied
932by $\alpha$. This isolation allows for an easy check that tracer conservation is exact when
933eliminating 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.
934
935}            %%end gm comment (copy of griffies book)
936
937%>>>>>===============
938
939
940%--------------------------------------------------------------------------------------------------------------
941% Filtered free surface formulation
942%--------------------------------------------------------------------------------------------------------------
943\subsection{Filtered free surface (\key{dynspg\_flt})}
944\label{DYN_spg_fltp}
945
946The filtered formulation follows the \citet{Roullet_Madec_JGR00} implementation.
947The extra term introduced in the equations (see \S\ref{PE_free_surface}) is solved implicitly.
948The elliptic solvers available in the code are documented in \S\ref{MISC}.
949
950%% gm %%======>>>>   given here the discrete eqs provided to the solver
951\gmcomment{               %%% copy from chap-model basics
952\begin{equation} \label{Eq_spg_flt}
953\frac{\partial {\rm {\bf U}}_h }{\partial t}= {\rm {\bf M}}
954- g \nabla \left( \tilde{\rho} \ \eta \right)
955- g \ T_c \nabla \left( \widetilde{\rho} \ \partial_t \eta \right)
956\end{equation}
957where $T_c$, is a parameter with dimensions of time which characterizes the force,
958$\widetilde{\rho} = \rho / \rho_o$ is the dimensionless density, and $\rm {\bf M}$ 
959represents the collected contributions of the Coriolis, hydrostatic pressure gradient,
960non-linear and viscous terms in \eqref{Eq_PE_dyn}.
961}   %end gmcomment
962
963Note that in the linear free surface formulation (\key{vvl} not defined), the ocean depth
964is time-independent and so is the matrix to be inverted. It is computed once and for all and applies to all ocean time steps.
965
966% ================================================================
967% Lateral diffusion term
968% ================================================================
969\section  [Lateral diffusion term (\textit{dynldf})]
970      {Lateral diffusion term (\mdl{dynldf})}
971\label{DYN_ldf}
972%------------------------------------------nam_dynldf----------------------------------------------------
973\namdisplay{namdyn_ldf} 
974%-------------------------------------------------------------------------------------------------------------
975
976Options are defined through the \ngn{namdyn\_ldf} namelist variables.
977The options available for lateral diffusion are to use either laplacian
978(rotated or not) or biharmonic operators. The coefficients may be constant
979or spatially variable; the description of the coefficients is found in the chapter
980on lateral physics (Chap.\ref{LDF}). The lateral diffusion of momentum is
981evaluated using a forward scheme, $i.e.$ the velocity appearing in its expression
982is the \textit{before} velocity in time, except for the pure vertical component
983that appears when a tensor of rotation is used. This latter term is solved
984implicitly together with the vertical diffusion term (see \S\ref{STP})
985
986At the lateral boundaries either free slip, no slip or partial slip boundary
987conditions are applied according to the user's choice (see Chap.\ref{LBC}).
988
989% ================================================================
990\subsection   [Iso-level laplacian operator (\np{ln\_dynldf\_lap}) ]
991         {Iso-level laplacian operator (\np{ln\_dynldf\_lap}=true)}
992\label{DYN_ldf_lap}
993
994For lateral iso-level diffusion, the discrete operator is:
995\begin{equation} \label{Eq_dynldf_lap}
996\left\{ \begin{aligned}
997 D_u^{l{\rm {\bf U}}} =\frac{1}{e_{1u} }\delta _{i+1/2} \left[ {A_T^{lm} 
998\;\chi } \right]-\frac{1}{e_{2u} {\kern 1pt}e_{3u} }\delta _j \left[
999{A_f^{lm} \;e_{3f} \zeta } \right] \\ 
1000\\
1001 D_v^{l{\rm {\bf U}}} =\frac{1}{e_{2v} }\delta _{j+1/2} \left[ {A_T^{lm} 
1002\;\chi } \right]+\frac{1}{e_{1v} {\kern 1pt}e_{3v} }\delta _i \left[
1003{A_f^{lm} \;e_{3f} \zeta } \right] \\ 
1004\end{aligned} \right.
1005\end{equation} 
1006
1007As explained in \S\ref{PE_ldf}, this formulation (as the gradient of a divergence
1008and curl of the vorticity) preserves symmetry and ensures a complete
1009separation between the vorticity and divergence parts of the momentum diffusion.
1010
1011%--------------------------------------------------------------------------------------------------------------
1012%           Rotated laplacian operator
1013%--------------------------------------------------------------------------------------------------------------
1014\subsection   [Rotated laplacian operator (\np{ln\_dynldf\_iso}) ]
1015         {Rotated laplacian operator (\np{ln\_dynldf\_iso}=true)}
1016\label{DYN_ldf_iso}
1017
1018A rotation of the lateral momentum diffusion operator is needed in several cases:
1019for iso-neutral diffusion in the $z$-coordinate (\np{ln\_dynldf\_iso}=true) and for
1020either iso-neutral (\np{ln\_dynldf\_iso}=true) or geopotential
1021(\np{ln\_dynldf\_hor}=true) diffusion in the $s$-coordinate. In the partial step
1022case, coordinates are horizontal except at the deepest level and no
1023rotation is performed when \np{ln\_dynldf\_hor}=true. The diffusion operator
1024is defined simply as the divergence of down gradient momentum fluxes on each
1025momentum component. It must be emphasized that this formulation ignores
1026constraints on the stress tensor such as symmetry. The resulting discrete
1027representation is:
1028\begin{equation} \label{Eq_dyn_ldf_iso}
1029\begin{split}
1030 D_u^{l\textbf{U}} &= \frac{1}{e_{1u} \, e_{2u} \, e_{3u} } \\
1031&  \left\{\quad  {\delta _{i+1/2} \left[ {A_T^{lm}  \left(
1032    {\frac{e_{2t} \; e_{3t} }{e_{1t} } \,\delta _{i}[u]
1033   -e_{2t} \; r_{1t} \,\overline{\overline {\delta _{k+1/2}[u]}}^{\,i,\,k}}
1034 \right)} \right]}   \right.
1035\\ 
1036& \qquad +\ \delta_j \left[ {A_f^{lm} \left( {\frac{e_{1f}\,e_{3f} }{e_{2f} 
1037}\,\delta _{j+1/2} [u] - e_{1f}\, r_{2f} 
1038\,\overline{\overline {\delta _{k+1/2} [u]}} ^{\,j+1/2,\,k}} 
1039\right)} \right]
1040\\ 
1041&\qquad +\ \delta_k \left[ {A_{uw}^{lm} \left( {-e_{2u} \, r_{1uw} \,\overline{\overline 
1042{\delta_{i+1/2} [u]}}^{\,i+1/2,\,k+1/2} } 
1043\right.} \right.
1044\\ 
1045&  \ \qquad \qquad \qquad \quad\
1046- e_{1u} \, r_{2uw} \,\overline{\overline {\delta_{j+1/2} [u]}} ^{\,j,\,k+1/2}
1047\\ 
1048& \left. {\left. { \ \qquad \qquad \qquad \ \ \ \left. {\
1049+\frac{e_{1u}\, e_{2u} }{e_{3uw} }\,\left( {r_{1uw}^2+r_{2uw}^2} 
1050\right)\,\delta_{k+1/2} [u]} \right)} \right]\;\;\;} \right\} 
1051\\
1052\\
1053 D_v^{l\textbf{V}} &= \frac{1}{e_{1v} \, e_{2v} \, e_{3v} }    \\
1054&  \left\{\quad  {\delta _{i+1/2} \left[ {A_f^{lm}  \left(
1055    {\frac{e_{2f} \; e_{3f} }{e_{1f} } \,\delta _{i+1/2}[v]
1056   -e_{2f} \; r_{1f} \,\overline{\overline {\delta _{k+1/2}[v]}}^{\,i+1/2,\,k}}
1057 \right)} \right]}   \right.
1058\\ 
1059& \qquad +\ \delta_j \left[ {A_T^{lm} \left( {\frac{e_{1t}\,e_{3t} }{e_{2t} 
1060}\,\delta _{j} [v] - e_{1t}\, r_{2t} 
1061\,\overline{\overline {\delta _{k+1/2} [v]}} ^{\,j,\,k}} 
1062\right)} \right]
1063\\ 
1064& \qquad +\ \delta_k \left[ {A_{vw}^{lm} \left( {-e_{2v} \, r_{1vw} \,\overline{\overline 
1065{\delta_{i+1/2} [v]}}^{\,i+1/2,\,k+1/2} }\right.} \right.
1066\\
1067&  \ \qquad \qquad \qquad \quad\
1068- e_{1v} \, r_{2vw} \,\overline{\overline {\delta_{j+1/2} [v]}} ^{\,j+1/2,\,k+1/2}
1069\\ 
1070& \left. {\left. { \ \qquad \qquad \qquad \ \ \ \left. {\
1071+\frac{e_{1v}\, e_{2v} }{e_{3vw} }\,\left( {r_{1vw}^2+r_{2vw}^2} 
1072\right)\,\delta_{k+1/2} [v]} \right)} \right]\;\;\;} \right\} 
1073 \end{split}
1074\end{equation}
1075where $r_1$ and $r_2$ are the slopes between the surface along which the
1076diffusion operator acts and the surface of computation ($z$- or $s$-surfaces).
1077The way these slopes are evaluated is given in the lateral physics chapter
1078(Chap.\ref{LDF}).
1079
1080%--------------------------------------------------------------------------------------------------------------
1081%           Iso-level bilaplacian operator
1082%--------------------------------------------------------------------------------------------------------------
1083\subsection   [Iso-level bilaplacian operator (\np{ln\_dynldf\_bilap})]
1084         {Iso-level bilaplacian operator (\np{ln\_dynldf\_bilap}=true)}
1085\label{DYN_ldf_bilap}
1086
1087The lateral fourth order operator formulation on momentum is obtained by
1088applying \eqref{Eq_dynldf_lap} twice. It requires an additional assumption on
1089boundary conditions: the first derivative term normal to the coast depends on
1090the free or no-slip lateral boundary conditions chosen, while the third
1091derivative terms normal to the coast are set to zero (see Chap.\ref{LBC}).
1092%%%
1093\gmcomment{add a remark on the the change in the position of the coefficient}
1094%%%
1095
1096% ================================================================
1097%           Vertical diffusion term
1098% ================================================================
1099\section  [Vertical diffusion term (\mdl{dynzdf})]
1100      {Vertical diffusion term (\mdl{dynzdf})}
1101\label{DYN_zdf}
1102%----------------------------------------------namzdf------------------------------------------------------
1103\namdisplay{namzdf} 
1104%-------------------------------------------------------------------------------------------------------------
1105
1106Options are defined through the \ngn{namzdf} namelist variables.
1107The large vertical diffusion coefficient found in the surface mixed layer together
1108with high vertical resolution implies that in the case of explicit time stepping there
1109would be too restrictive a constraint on the time step. Two time stepping schemes
1110can be used for the vertical diffusion term : $(a)$ a forward time differencing
1111scheme (\np{ln\_zdfexp}=true) using a time splitting technique
1112(\np{nn\_zdfexp} $>$ 1) or $(b)$ a backward (or implicit) time differencing scheme
1113(\np{ln\_zdfexp}=false) (see \S\ref{STP}). Note that namelist variables
1114\np{ln\_zdfexp} and \np{nn\_zdfexp} apply to both tracers and dynamics.
1115
1116The formulation of the vertical subgrid scale physics is the same whatever
1117the vertical coordinate is. The vertical diffusion operators given by
1118\eqref{Eq_PE_zdf} take the following semi-discrete space form:
1119\begin{equation} \label{Eq_dynzdf}
1120\left\{   \begin{aligned}
1121D_u^{vm} &\equiv \frac{1}{e_{3u}} \ \delta _k \left[ \frac{A_{uw}^{vm} }{e_{3uw} }
1122                              \ \delta _{k+1/2} [\,u\,]         \right]     \\
1123\\
1124D_v^{vm} &\equiv \frac{1}{e_{3v}} \ \delta _k \left[ \frac{A_{vw}^{vm} }{e_{3vw} }
1125                              \ \delta _{k+1/2} [\,v\,]         \right]
1126\end{aligned}   \right.
1127\end{equation} 
1128where $A_{uw}^{vm} $ and $A_{vw}^{vm} $ are the vertical eddy viscosity and
1129diffusivity coefficients. The way these coefficients are evaluated
1130depends on the vertical physics used (see \S\ref{ZDF}).
1131
1132The surface boundary condition on momentum is the stress exerted by
1133the wind. At the surface, the momentum fluxes are prescribed as the boundary
1134condition on the vertical turbulent momentum fluxes,
1135\begin{equation} \label{Eq_dynzdf_sbc}
1136\left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{z=1}
1137    = \frac{1}{\rho _o} \binom{\tau _u}{\tau _v }
1138\end{equation}
1139where $\left( \tau _u ,\tau _v \right)$ are the two components of the wind stress
1140vector in the (\textbf{i},\textbf{j}) coordinate system. The high mixing coefficients
1141in the surface mixed layer ensure that the surface wind stress is distributed in
1142the vertical over the mixed layer depth. If the vertical mixing coefficient
1143is small (when no mixed layer scheme is used) the surface stress enters only
1144the top model level, as a body force. The surface wind stress is calculated
1145in the surface module routines (SBC, see Chap.\ref{SBC})
1146
1147The turbulent flux of momentum at the bottom of the ocean is specified through
1148a bottom friction parameterisation (see \S\ref{ZDF_bfr})
1149
1150% ================================================================
1151% External Forcing
1152% ================================================================
1153\section{External Forcings}
1154\label{DYN_forcing}
1155
1156Besides the surface and bottom stresses (see the above section) which are
1157introduced as boundary conditions on the vertical mixing, two other forcings
1158enter the dynamical equations.
1159
1160One is the effect of atmospheric pressure on the ocean dynamics.
1161Another forcing term is the tidal potential.
1162Both of which will be introduced into the reference version soon.
1163
1164\gmcomment{atmospheric pressure is there!!!!    include its description }
1165
1166% ================================================================
1167% Time evolution term
1168% ================================================================
1169\section  [Time evolution term (\textit{dynnxt})]
1170      {Time evolution term (\mdl{dynnxt})}
1171\label{DYN_nxt}
1172
1173%----------------------------------------------namdom----------------------------------------------------
1174\namdisplay{namdom} 
1175%-------------------------------------------------------------------------------------------------------------
1176
1177Options are defined through the \ngn{namdom} namelist variables.
1178The general framework for dynamics time stepping is a leap-frog scheme,
1179$i.e.$ a three level centred time scheme associated with an Asselin time filter
1180(cf. Chap.\ref{STP}). The scheme is applied to the velocity, except when using
1181the flux form of momentum advection (cf. \S\ref{DYN_adv_cor_flux}) in the variable
1182volume case (\key{vvl} defined), where it has to be applied to the thickness
1183weighted velocity (see \S\ref{Apdx_A_momentum}
1184
1185$\bullet$ vector invariant form or linear free surface (\np{ln\_dynhpg\_vec}=true ; \key{vvl} not defined):
1186\begin{equation} \label{Eq_dynnxt_vec}
1187\left\{   \begin{aligned}
1188&u^{t+\rdt} = u_f^{t-\rdt} + 2\rdt  \ \text{RHS}_u^t     \\
1189&u_f^t \;\quad = u^t+\gamma \,\left[ {u_f^{t-\rdt} -2u^t+u^{t+\rdt}} \right]
1190\end{aligned}   \right.
1191\end{equation} 
1192
1193$\bullet$ flux form and nonlinear free surface (\np{ln\_dynhpg\_vec}=false ; \key{vvl} defined):
1194\begin{equation} \label{Eq_dynnxt_flux}
1195\left\{   \begin{aligned}
1196&\left(e_{3u}\,u\right)^{t+\rdt} = \left(e_{3u}\,u\right)_f^{t-\rdt} + 2\rdt \; e_{3u} \;\text{RHS}_u^t     \\
1197&\left(e_{3u}\,u\right)_f^t \;\quad = \left(e_{3u}\,u\right)^t
1198  +\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]
1199\end{aligned}   \right.
1200\end{equation} 
1201where RHS is the right hand side of the momentum equation, the subscript $f$ 
1202denotes filtered values and $\gamma$ is the Asselin coefficient. $\gamma$ is
1203initialized as \np{nn\_atfp} (namelist parameter). Its default value is \np{nn\_atfp} = $10^{-3}$.
1204In both cases, the modified Asselin filter is not applied since perfect conservation
1205is not an issue for the momentum equations.
1206
1207Note that with the filtered free surface, the update of the \textit{after} velocities
1208is done in the \mdl{dynsp\_flt} module, and only array swapping
1209and Asselin filtering is done in \mdl{dynnxt}.
1210
1211% ================================================================
1212% Neptune effect
1213% ================================================================
1214\section  [Neptune effect (\textit{dynnept})]
1215                {Neptune effect (\mdl{dynnept})}
1216\label{DYN_nept}
1217
1218The "Neptune effect" (thus named in \citep{HollowayOM86}) is a
1219parameterisation of the potentially large effect of topographic form stress
1220(caused by eddies) in driving the ocean circulation. Originally developed for
1221low-resolution models, in which it was applied via a Laplacian (second-order)
1222diffusion-like term in the momentum equation, it can also be applied in eddy
1223permitting or resolving models, in which a more scale-selective bilaplacian
1224(fourth-order) implementation is preferred. This mechanism has a
1225significant effect on boundary currents (including undercurrents), and the
1226upwelling of deep water near continental shelves.
1227
1228The theoretical basis for the method can be found in
1229\citep{HollowayJPO92}, including the explanation of why form stress is not
1230necessarily a drag force, but may actually drive the flow.
1231\citep{HollowayJPO94} demonstrate the effects of the parameterisation in
1232the GFDL-MOM model, at a horizontal resolution of about 1.8 degrees.
1233\citep{HollowayOM08} demonstrate the biharmonic version of the
1234parameterisation in a global run of the POP model, with an average horizontal
1235grid spacing of about 32km.
1236
1237The NEMO implementation is a simplified form of that supplied by
1238Greg Holloway, the testing of which was described in \citep{HollowayJGR09}.
1239The major simplification is that a time invariant Neptune velocity
1240field is assumed.  This is computed only once, during start-up, and
1241made available to the rest of the code via a module.  Vertical
1242diffusive terms are also ignored, and the model topography itself
1243is used, rather than a separate topographic dataset as in
1244\citep{HollowayOM08}.  This implementation is only in the iso-level
1245formulation, as is the case anyway for the bilaplacian operator.
1246
1247The velocity field is derived from a transport stream function given by:
1248
1249\begin{equation} \label{Eq_dynnept_sf}
1250\psi = -fL^2H
1251\end{equation}
1252
1253where $L$ is a latitude-dependant length scale given by:
1254
1255\begin{equation} \label{Eq_dynnept_ls}
1256L = l_1 + (l_2 -l_1)\left ( {1 + \cos 2\phi \over 2 } \right )
1257\end{equation}
1258
1259where $\phi$ is latitude and $l_1$ and $l_2$ are polar and equatorial length scales respectively.
1260Neptune velocity components, $u^*$, $v^*$ are derived from the stremfunction as:
1261
1262\begin{equation} \label{Eq_dynnept_vel}
1263u^* = -{1\over H} {\partial \psi \over \partial y}\ \ \  ,\ \ \ v^* = {1\over H} {\partial \psi \over \partial x}
1264\end{equation}
1265
1266\smallskip
1267%----------------------------------------------namdom----------------------------------------------------
1268\namdisplay{namdyn_nept}
1269%--------------------------------------------------------------------------------------------------------
1270\smallskip
1271
1272The Neptune effect is enabled when \np{ln\_neptsimp}=true (default=false).
1273\np{ln\_smooth\_neptvel} controls whether a scale-selective smoothing is applied
1274to the Neptune effect flow field (default=false) (this smoothing method is as
1275used by Holloway).  \np{rn\_tslse} and \np{rn\_tslsp} are the equatorial and
1276polar values respectively of the length-scale parameter $L$ used in determining
1277the Neptune stream function \eqref{Eq_dynnept_sf} and \eqref{Eq_dynnept_ls}.
1278Values at intermediate latitudes are given by a cosine fit, mimicking the
1279variation of the deformation radius with latitude.  The default values of 12km
1280and 3km are those given in \citep{HollowayJPO94}, appropriate for a coarse
1281resolution model. The finer resolution study of \citep{HollowayOM08} increased
1282the values of L by a factor of $\sqrt 2$ to 17km and 4.2km, thus doubling the
1283stream function for a given topography.
1284
1285The simple formulation for ($u^*$, $v^*$) can give unacceptably large velocities
1286in shallow water, and \citep{HollowayOM08} add an offset to the depth in the
1287denominator to control this problem. In this implementation we offer instead (at
1288the suggestion of G. Madec) the option of ramping down the Neptune flow field to
1289zero over a finite depth range. The switch \np{ln\_neptramp} activates this
1290option (default=false), in which case velocities at depths greater than
1291\np{rn\_htrmax} are unaltered, but ramp down linearly with depth to zero at a
1292depth of \np{rn\_htrmin} (and shallower).
1293
1294% ================================================================
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