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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
765The split-explicit free surface formulation used in \NEMO (\key{dynspg\_ts} defined),
766also called the time-splitting formulation, follows the one
767proposed by \citet{Griffies_Bk04}. The general idea is to solve the free surface
768equation and the associated barotropic velocity equations with a smaller time
769step than $\rdt$, the time step used for the three dimensional prognostic
770variables (Fig.~\ref{Fig_DYN_dynspg_ts}).
771The size of the small time step, $\rdt_e$ (the external mode or barotropic time step)
772 is provided through the \np{nn\_baro} namelist parameter as:
773$\rdt_e = \rdt / nn\_baro$.
774 
775
776%>   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >
777\begin{figure}[!t]    \begin{center}
778\includegraphics[width=0.90\textwidth]{./TexFiles/Figures/Fig_DYN_dynspg_ts.pdf}
779\caption{  \label{Fig_DYN_dynspg_ts}
780Schematic of the split-explicit time stepping scheme for the external
781and internal modes. Time increases to the right.
782Internal mode time steps (which are also the model time steps) are denoted
783by $t-\rdt$, $t, t+\rdt$, and $t+2\rdt$.
784The curved line represents a leap-frog time step, and the smaller time
785steps $N \rdt_e=\frac{3}{2}\rdt$ are denoted by the zig-zag line.
786The vertically integrated forcing \textbf{M}(t) computed at the model time step $t$ 
787represents the interaction between the external and internal motions.
788While keeping \textbf{M} and freshwater forcing field fixed, a leap-frog
789integration carries the external mode variables (surface height and vertically
790integrated velocity) from $t$ to $t+\frac{3}{2} \rdt$ using N external time
791steps of length $\rdt_e$. Time averaging the external fields over the
792$\frac{2}{3}N+1$ time steps (endpoints included) centers the vertically integrated
793velocity and the sea surface height at the model timestep $t+\rdt$.
794These averaged values are used to update \textbf{M}(t) with both the surface
795pressure gradient and the Coriolis force, therefore providing the $t+\rdt$
796velocity.  The model time stepping scheme can then be achieved by a baroclinic
797leap-frog time step that carries the surface height from $t-\rdt$ to $t+\rdt$}
798\end{center}    \end{figure}
799%>   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >
800
801The split-explicit formulation has a damping effect on external gravity waves,
802which is weaker damping than that for the filtered free surface but still significant, as
803shown by \citet{Levier2007} in the case of an analytical barotropic Kelvin wave.
804
805%>>>>>===============
806\gmcomment{               %%% copy from griffies Book
807
808\textbf{title: Time stepping the barotropic system }
809
810Assume knowledge of the full velocity and tracer fields at baroclinic time $\tau$. Hence,
811we can update the surface height and vertically integrated velocity with a leap-frog
812scheme using the small barotropic time step $\rdt$. We have
813
814\begin{equation} \label{DYN_spg_ts_eta}
815\eta^{(b)}(\tau,t_{n+1}) - \eta^{(b)}(\tau,t_{n+1}) (\tau,t_{n-1})
816   = 2 \rdt \left[-\nabla \cdot \textbf{U}^{(b)}(\tau,t_n) + \text{EMP}_w(\tau) \right] 
817\end{equation}
818\begin{multline} \label{DYN_spg_ts_u}
819\textbf{U}^{(b)}(\tau,t_{n+1}) - \textbf{U}^{(b)}(\tau,t_{n-1}\\
820   = 2\rdt \left[ - f \textbf{k} \times \textbf{U}^{(b)}(\tau,t_{n})
821   - H(\tau) \nabla p_s^{(b)}(\tau,t_{n}) +\textbf{M}(\tau) \right]
822\end{multline}
823\
824
825In these equations, araised (b) denotes values of surface height and vertically integrated velocity updated with the barotropic time steps. The $\tau$ time label on $\eta^{(b)}$ 
826and $U^{(b)}$ 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
827that sets the barotropic time steps via
828\begin{equation} \label{DYN_spg_ts_t}
829t_n=\tau+n\rdt   
830\end{equation}
831with $n$ an integer. The density scaled surface pressure is evaluated via
832\begin{equation} \label{DYN_spg_ts_ps}
833p_s^{(b)}(\tau,t_{n}) = \begin{cases}
834   g \;\eta_s^{(b)}(\tau,t_{n}) \;\rho(\tau)_{k=1}) / \rho_&      \text{non-linear case} \\
835   g \;\eta_s^{(b)}(\tau,t_{n}&      \text{linear case} 
836   \end{cases}
837\end{equation}
838To get started, we assume the following initial conditions
839\begin{equation} \label{DYN_spg_ts_eta}
840\begin{split}
841\eta^{(b)}(\tau,t_{n=0}) &= \overline{\eta^{(b)}(\tau)}
842\\
843\eta^{(b)}(\tau,t_{n=1}) &= \eta^{(b)}(\tau,t_{n=0}) + \rdt \ \text{RHS}_{n=0} 
844\end{split}
845\end{equation}
846with
847\begin{equation} \label{DYN_spg_ts_etaF}
848 \overline{\eta^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N \eta^{(b)}(\tau-\rdt,t_{n})
849\end{equation}
850the time averaged surface height taken from the previous barotropic cycle. Likewise,
851\begin{equation} \label{DYN_spg_ts_u}
852\textbf{U}^{(b)}(\tau,t_{n=0}) = \overline{\textbf{U}^{(b)}(\tau)}   \\
853\\
854\textbf{U}(\tau,t_{n=1}) = \textbf{U}^{(b)}(\tau,t_{n=0}) + \rdt \ \text{RHS}_{n=0}   
855\end{equation}
856with
857\begin{equation} \label{DYN_spg_ts_u}
858 \overline{\textbf{U}^{(b)}(\tau)} 
859   = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau-\rdt,t_{n})
860\end{equation}
861the time averaged vertically integrated transport. Notably, there is no Robert-Asselin time filter used in the barotropic portion of the integration.
862
863Upon 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 $\tau + \rdt \tau$ 
864\begin{equation} \label{DYN_spg_ts_u}
865\textbf{U}(\tau+\rdt) = \overline{\textbf{U}^{(b)}(\tau+\rdt)} 
866   = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau,t_{n})
867\end{equation}
868The surface height on the new baroclinic time step is then determined via a baroclinic leap-frog using the following form
869
870\begin{equation} \label{DYN_spg_ts_ssh}
871\eta(\tau+\Delta) - \eta^{F}(\tau-\Delta) = 2\rdt \ \left[ - \nabla \cdot \textbf{U}(\tau) + \text{EMP}_w \right] 
872\end{equation}
873
874 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).
875 
876In general, some form of time filter is needed to maintain integrity of the surface
877height field due to the leap-frog splitting mode in equation \ref{DYN_spg_ts_ssh}. We
878have tried various forms of such filtering, with the following method discussed in
879\cite{Griffies_al_MWR01} chosen due to its stability and reasonably good maintenance of
880tracer conservation properties (see Section ??)
881
882\begin{equation} \label{DYN_spg_ts_sshf}
883\eta^{F}(\tau-\Delta) =  \overline{\eta^{(b)}(\tau)} 
884\end{equation}
885Another approach tried was
886
887\begin{equation} \label{DYN_spg_ts_sshf2}
888\eta^{F}(\tau-\Delta) = \eta(\tau)
889   + (\alpha/2) \left[\overline{\eta^{(b)}}(\tau+\rdt)
890                + \overline{\eta^{(b)}}(\tau-\rdt) -2 \;\eta(\tau) \right]
891\end{equation}
892
893which is useful since it isolates all the time filtering aspects into the term multiplied
894by $\alpha$. This isolation allows for an easy check that tracer conservation is exact when
895eliminating 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.
896
897}            %%end gm comment (copy of griffies book)
898
899%>>>>>===============
900
901
902%--------------------------------------------------------------------------------------------------------------
903% Filtered free surface formulation
904%--------------------------------------------------------------------------------------------------------------
905\subsection{Filtered free surface (\key{dynspg\_flt})}
906\label{DYN_spg_fltp}
907
908The filtered formulation follows the \citet{Roullet_Madec_JGR00} implementation.
909The extra term introduced in the equations (see \S\ref{PE_free_surface}) is solved implicitly.
910The elliptic solvers available in the code are documented in \S\ref{MISC}.
911
912%% gm %%======>>>>   given here the discrete eqs provided to the solver
913\gmcomment{               %%% copy from chap-model basics
914\begin{equation} \label{Eq_spg_flt}
915\frac{\partial {\rm {\bf U}}_h }{\partial t}= {\rm {\bf M}}
916- g \nabla \left( \tilde{\rho} \ \eta \right)
917- g \ T_c \nabla \left( \widetilde{\rho} \ \partial_t \eta \right)
918\end{equation}
919where $T_c$, is a parameter with dimensions of time which characterizes the force,
920$\widetilde{\rho} = \rho / \rho_o$ is the dimensionless density, and $\rm {\bf M}$ 
921represents the collected contributions of the Coriolis, hydrostatic pressure gradient,
922non-linear and viscous terms in \eqref{Eq_PE_dyn}.
923}   %end gmcomment
924
925Note that in the linear free surface formulation (\key{vvl} not defined), the ocean depth
926is time-independent and so is the matrix to be inverted. It is computed once and for all and applies to all ocean time steps.
927
928% ================================================================
929% Lateral diffusion term
930% ================================================================
931\section  [Lateral diffusion term (\textit{dynldf})]
932      {Lateral diffusion term (\mdl{dynldf})}
933\label{DYN_ldf}
934%------------------------------------------nam_dynldf----------------------------------------------------
935\namdisplay{namdyn_ldf} 
936%-------------------------------------------------------------------------------------------------------------
937
938Options are defined through the \ngn{namdyn\_ldf} namelist variables.
939The options available for lateral diffusion are to use either laplacian
940(rotated or not) or biharmonic operators. The coefficients may be constant
941or spatially variable; the description of the coefficients is found in the chapter
942on lateral physics (Chap.\ref{LDF}). The lateral diffusion of momentum is
943evaluated using a forward scheme, $i.e.$ the velocity appearing in its expression
944is the \textit{before} velocity in time, except for the pure vertical component
945that appears when a tensor of rotation is used. This latter term is solved
946implicitly together with the vertical diffusion term (see \S\ref{STP})
947
948At the lateral boundaries either free slip, no slip or partial slip boundary
949conditions are applied according to the user's choice (see Chap.\ref{LBC}).
950
951% ================================================================
952\subsection   [Iso-level laplacian operator (\np{ln\_dynldf\_lap}) ]
953         {Iso-level laplacian operator (\np{ln\_dynldf\_lap}=true)}
954\label{DYN_ldf_lap}
955
956For lateral iso-level diffusion, the discrete operator is:
957\begin{equation} \label{Eq_dynldf_lap}
958\left\{ \begin{aligned}
959 D_u^{l{\rm {\bf U}}} =\frac{1}{e_{1u} }\delta _{i+1/2} \left[ {A_T^{lm} 
960\;\chi } \right]-\frac{1}{e_{2u} {\kern 1pt}e_{3u} }\delta _j \left[
961{A_f^{lm} \;e_{3f} \zeta } \right] \\ 
962\\
963 D_v^{l{\rm {\bf U}}} =\frac{1}{e_{2v} }\delta _{j+1/2} \left[ {A_T^{lm} 
964\;\chi } \right]+\frac{1}{e_{1v} {\kern 1pt}e_{3v} }\delta _i \left[
965{A_f^{lm} \;e_{3f} \zeta } \right] \\ 
966\end{aligned} \right.
967\end{equation} 
968
969As explained in \S\ref{PE_ldf}, this formulation (as the gradient of a divergence
970and curl of the vorticity) preserves symmetry and ensures a complete
971separation between the vorticity and divergence parts of the momentum diffusion.
972
973%--------------------------------------------------------------------------------------------------------------
974%           Rotated laplacian operator
975%--------------------------------------------------------------------------------------------------------------
976\subsection   [Rotated laplacian operator (\np{ln\_dynldf\_iso}) ]
977         {Rotated laplacian operator (\np{ln\_dynldf\_iso}=true)}
978\label{DYN_ldf_iso}
979
980A rotation of the lateral momentum diffusion operator is needed in several cases:
981for iso-neutral diffusion in the $z$-coordinate (\np{ln\_dynldf\_iso}=true) and for
982either iso-neutral (\np{ln\_dynldf\_iso}=true) or geopotential
983(\np{ln\_dynldf\_hor}=true) diffusion in the $s$-coordinate. In the partial step
984case, coordinates are horizontal except at the deepest level and no
985rotation is performed when \np{ln\_dynldf\_hor}=true. The diffusion operator
986is defined simply as the divergence of down gradient momentum fluxes on each
987momentum component. It must be emphasized that this formulation ignores
988constraints on the stress tensor such as symmetry. The resulting discrete
989representation is:
990\begin{equation} \label{Eq_dyn_ldf_iso}
991\begin{split}
992 D_u^{l\textbf{U}} &= \frac{1}{e_{1u} \, e_{2u} \, e_{3u} } \\
993&  \left\{\quad  {\delta _{i+1/2} \left[ {A_T^{lm}  \left(
994    {\frac{e_{2t} \; e_{3t} }{e_{1t} } \,\delta _{i}[u]
995   -e_{2t} \; r_{1t} \,\overline{\overline {\delta _{k+1/2}[u]}}^{\,i,\,k}}
996 \right)} \right]}   \right.
997\\ 
998& \qquad +\ \delta_j \left[ {A_f^{lm} \left( {\frac{e_{1f}\,e_{3f} }{e_{2f} 
999}\,\delta _{j+1/2} [u] - e_{1f}\, r_{2f} 
1000\,\overline{\overline {\delta _{k+1/2} [u]}} ^{\,j+1/2,\,k}} 
1001\right)} \right]
1002\\ 
1003&\qquad +\ \delta_k \left[ {A_{uw}^{lm} \left( {-e_{2u} \, r_{1uw} \,\overline{\overline 
1004{\delta_{i+1/2} [u]}}^{\,i+1/2,\,k+1/2} } 
1005\right.} \right.
1006\\ 
1007&  \ \qquad \qquad \qquad \quad\
1008- e_{1u} \, r_{2uw} \,\overline{\overline {\delta_{j+1/2} [u]}} ^{\,j,\,k+1/2}
1009\\ 
1010& \left. {\left. { \ \qquad \qquad \qquad \ \ \ \left. {\
1011+\frac{e_{1u}\, e_{2u} }{e_{3uw} }\,\left( {r_{1uw}^2+r_{2uw}^2} 
1012\right)\,\delta_{k+1/2} [u]} \right)} \right]\;\;\;} \right\} 
1013\\
1014\\
1015 D_v^{l\textbf{V}} &= \frac{1}{e_{1v} \, e_{2v} \, e_{3v} }    \\
1016&  \left\{\quad  {\delta _{i+1/2} \left[ {A_f^{lm}  \left(
1017    {\frac{e_{2f} \; e_{3f} }{e_{1f} } \,\delta _{i+1/2}[v]
1018   -e_{2f} \; r_{1f} \,\overline{\overline {\delta _{k+1/2}[v]}}^{\,i+1/2,\,k}}
1019 \right)} \right]}   \right.
1020\\ 
1021& \qquad +\ \delta_j \left[ {A_T^{lm} \left( {\frac{e_{1t}\,e_{3t} }{e_{2t} 
1022}\,\delta _{j} [v] - e_{1t}\, r_{2t} 
1023\,\overline{\overline {\delta _{k+1/2} [v]}} ^{\,j,\,k}} 
1024\right)} \right]
1025\\ 
1026& \qquad +\ \delta_k \left[ {A_{vw}^{lm} \left( {-e_{2v} \, r_{1vw} \,\overline{\overline 
1027{\delta_{i+1/2} [v]}}^{\,i+1/2,\,k+1/2} }\right.} \right.
1028\\
1029&  \ \qquad \qquad \qquad \quad\
1030- e_{1v} \, r_{2vw} \,\overline{\overline {\delta_{j+1/2} [v]}} ^{\,j+1/2,\,k+1/2}
1031\\ 
1032& \left. {\left. { \ \qquad \qquad \qquad \ \ \ \left. {\
1033+\frac{e_{1v}\, e_{2v} }{e_{3vw} }\,\left( {r_{1vw}^2+r_{2vw}^2} 
1034\right)\,\delta_{k+1/2} [v]} \right)} \right]\;\;\;} \right\} 
1035 \end{split}
1036\end{equation}
1037where $r_1$ and $r_2$ are the slopes between the surface along which the
1038diffusion operator acts and the surface of computation ($z$- or $s$-surfaces).
1039The way these slopes are evaluated is given in the lateral physics chapter
1040(Chap.\ref{LDF}).
1041
1042%--------------------------------------------------------------------------------------------------------------
1043%           Iso-level bilaplacian operator
1044%--------------------------------------------------------------------------------------------------------------
1045\subsection   [Iso-level bilaplacian operator (\np{ln\_dynldf\_bilap})]
1046         {Iso-level bilaplacian operator (\np{ln\_dynldf\_bilap}=true)}
1047\label{DYN_ldf_bilap}
1048
1049The lateral fourth order operator formulation on momentum is obtained by
1050applying \eqref{Eq_dynldf_lap} twice. It requires an additional assumption on
1051boundary conditions: the first derivative term normal to the coast depends on
1052the free or no-slip lateral boundary conditions chosen, while the third
1053derivative terms normal to the coast are set to zero (see Chap.\ref{LBC}).
1054%%%
1055\gmcomment{add a remark on the the change in the position of the coefficient}
1056%%%
1057
1058% ================================================================
1059%           Vertical diffusion term
1060% ================================================================
1061\section  [Vertical diffusion term (\mdl{dynzdf})]
1062      {Vertical diffusion term (\mdl{dynzdf})}
1063\label{DYN_zdf}
1064%----------------------------------------------namzdf------------------------------------------------------
1065\namdisplay{namzdf} 
1066%-------------------------------------------------------------------------------------------------------------
1067
1068Options are defined through the \ngn{namzdf} namelist variables.
1069The large vertical diffusion coefficient found in the surface mixed layer together
1070with high vertical resolution implies that in the case of explicit time stepping there
1071would be too restrictive a constraint on the time step. Two time stepping schemes
1072can be used for the vertical diffusion term : $(a)$ a forward time differencing
1073scheme (\np{ln\_zdfexp}=true) using a time splitting technique
1074(\np{nn\_zdfexp} $>$ 1) or $(b)$ a backward (or implicit) time differencing scheme
1075(\np{ln\_zdfexp}=false) (see \S\ref{STP}). Note that namelist variables
1076\np{ln\_zdfexp} and \np{nn\_zdfexp} apply to both tracers and dynamics.
1077
1078The formulation of the vertical subgrid scale physics is the same whatever
1079the vertical coordinate is. The vertical diffusion operators given by
1080\eqref{Eq_PE_zdf} take the following semi-discrete space form:
1081\begin{equation} \label{Eq_dynzdf}
1082\left\{   \begin{aligned}
1083D_u^{vm} &\equiv \frac{1}{e_{3u}} \ \delta _k \left[ \frac{A_{uw}^{vm} }{e_{3uw} }
1084                              \ \delta _{k+1/2} [\,u\,]         \right]     \\
1085\\
1086D_v^{vm} &\equiv \frac{1}{e_{3v}} \ \delta _k \left[ \frac{A_{vw}^{vm} }{e_{3vw} }
1087                              \ \delta _{k+1/2} [\,v\,]         \right]
1088\end{aligned}   \right.
1089\end{equation} 
1090where $A_{uw}^{vm} $ and $A_{vw}^{vm} $ are the vertical eddy viscosity and
1091diffusivity coefficients. The way these coefficients are evaluated
1092depends on the vertical physics used (see \S\ref{ZDF}).
1093
1094The surface boundary condition on momentum is the stress exerted by
1095the wind. At the surface, the momentum fluxes are prescribed as the boundary
1096condition on the vertical turbulent momentum fluxes,
1097\begin{equation} \label{Eq_dynzdf_sbc}
1098\left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{z=1}
1099    = \frac{1}{\rho _o} \binom{\tau _u}{\tau _v }
1100\end{equation}
1101where $\left( \tau _u ,\tau _v \right)$ are the two components of the wind stress
1102vector in the (\textbf{i},\textbf{j}) coordinate system. The high mixing coefficients
1103in the surface mixed layer ensure that the surface wind stress is distributed in
1104the vertical over the mixed layer depth. If the vertical mixing coefficient
1105is small (when no mixed layer scheme is used) the surface stress enters only
1106the top model level, as a body force. The surface wind stress is calculated
1107in the surface module routines (SBC, see Chap.\ref{SBC})
1108
1109The turbulent flux of momentum at the bottom of the ocean is specified through
1110a bottom friction parameterisation (see \S\ref{ZDF_bfr})
1111
1112% ================================================================
1113% External Forcing
1114% ================================================================
1115\section{External Forcings}
1116\label{DYN_forcing}
1117
1118Besides the surface and bottom stresses (see the above section) which are
1119introduced as boundary conditions on the vertical mixing, two other forcings
1120enter the dynamical equations.
1121
1122One is the effect of atmospheric pressure on the ocean dynamics.
1123Another forcing term is the tidal potential.
1124Both of which will be introduced into the reference version soon.
1125
1126\gmcomment{atmospheric pressure is there!!!!    include its description }
1127
1128% ================================================================
1129% Time evolution term
1130% ================================================================
1131\section  [Time evolution term (\textit{dynnxt})]
1132      {Time evolution term (\mdl{dynnxt})}
1133\label{DYN_nxt}
1134
1135%----------------------------------------------namdom----------------------------------------------------
1136\namdisplay{namdom} 
1137%-------------------------------------------------------------------------------------------------------------
1138
1139Options are defined through the \ngn{namdom} namelist variables.
1140The general framework for dynamics time stepping is a leap-frog scheme,
1141$i.e.$ a three level centred time scheme associated with an Asselin time filter
1142(cf. Chap.\ref{STP}). The scheme is applied to the velocity, except when using
1143the flux form of momentum advection (cf. \S\ref{DYN_adv_cor_flux}) in the variable
1144volume case (\key{vvl} defined), where it has to be applied to the thickness
1145weighted velocity (see \S\ref{Apdx_A_momentum}
1146
1147$\bullet$ vector invariant form or linear free surface (\np{ln\_dynhpg\_vec}=true ; \key{vvl} not defined):
1148\begin{equation} \label{Eq_dynnxt_vec}
1149\left\{   \begin{aligned}
1150&u^{t+\rdt} = u_f^{t-\rdt} + 2\rdt  \ \text{RHS}_u^t     \\
1151&u_f^t \;\quad = u^t+\gamma \,\left[ {u_f^{t-\rdt} -2u^t+u^{t+\rdt}} \right]
1152\end{aligned}   \right.
1153\end{equation} 
1154
1155$\bullet$ flux form and nonlinear free surface (\np{ln\_dynhpg\_vec}=false ; \key{vvl} defined):
1156\begin{equation} \label{Eq_dynnxt_flux}
1157\left\{   \begin{aligned}
1158&\left(e_{3u}\,u\right)^{t+\rdt} = \left(e_{3u}\,u\right)_f^{t-\rdt} + 2\rdt \; e_{3u} \;\text{RHS}_u^t     \\
1159&\left(e_{3u}\,u\right)_f^t \;\quad = \left(e_{3u}\,u\right)^t
1160  +\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]
1161\end{aligned}   \right.
1162\end{equation} 
1163where RHS is the right hand side of the momentum equation, the subscript $f$ 
1164denotes filtered values and $\gamma$ is the Asselin coefficient. $\gamma$ is
1165initialized as \np{nn\_atfp} (namelist parameter). Its default value is \np{nn\_atfp} = $10^{-3}$.
1166In both cases, the modified Asselin filter is not applied since perfect conservation
1167is not an issue for the momentum equations.
1168
1169Note that with the filtered free surface, the update of the \textit{after} velocities
1170is done in the \mdl{dynsp\_flt} module, and only array swapping
1171and Asselin filtering is done in \mdl{dynnxt}.
1172
1173% ================================================================
1174% Neptune effect
1175% ================================================================
1176\section  [Neptune effect (\textit{dynnept})]
1177                {Neptune effect (\mdl{dynnept})}
1178\label{DYN_nept}
1179
1180The "Neptune effect" (thus named in \citep{HollowayOM86}) is a
1181parameterisation of the potentially large effect of topographic form stress
1182(caused by eddies) in driving the ocean circulation. Originally developed for
1183low-resolution models, in which it was applied via a Laplacian (second-order)
1184diffusion-like term in the momentum equation, it can also be applied in eddy
1185permitting or resolving models, in which a more scale-selective bilaplacian
1186(fourth-order) implementation is preferred. This mechanism has a
1187significant effect on boundary currents (including undercurrents), and the
1188upwelling of deep water near continental shelves.
1189
1190The theoretical basis for the method can be found in
1191\citep{HollowayJPO92}, including the explanation of why form stress is not
1192necessarily a drag force, but may actually drive the flow.
1193\citep{HollowayJPO94} demonstrate the effects of the parameterisation in
1194the GFDL-MOM model, at a horizontal resolution of about 1.8 degrees.
1195\citep{HollowayOM08} demonstrate the biharmonic version of the
1196parameterisation in a global run of the POP model, with an average horizontal
1197grid spacing of about 32km.
1198
1199The NEMO implementation is a simplified form of that supplied by
1200Greg Holloway, the testing of which was described in \citep{HollowayJGR09}.
1201The major simplification is that a time invariant Neptune velocity
1202field is assumed.  This is computed only once, during start-up, and
1203made available to the rest of the code via a module.  Vertical
1204diffusive terms are also ignored, and the model topography itself
1205is used, rather than a separate topographic dataset as in
1206\citep{HollowayOM08}.  This implementation is only in the iso-level
1207formulation, as is the case anyway for the bilaplacian operator.
1208
1209The velocity field is derived from a transport stream function given by:
1210
1211\begin{equation} \label{Eq_dynnept_sf}
1212\psi = -fL^2H
1213\end{equation}
1214
1215where $L$ is a latitude-dependant length scale given by:
1216
1217\begin{equation} \label{Eq_dynnept_ls}
1218L = l_1 + (l_2 -l_1)\left ( {1 + \cos 2\phi \over 2 } \right )
1219\end{equation}
1220
1221where $\phi$ is latitude and $l_1$ and $l_2$ are polar and equatorial length scales respectively.
1222Neptune velocity components, $u^*$, $v^*$ are derived from the stremfunction as:
1223
1224\begin{equation} \label{Eq_dynnept_vel}
1225u^* = -{1\over H} {\partial \psi \over \partial y}\ \ \  ,\ \ \ v^* = {1\over H} {\partial \psi \over \partial x}
1226\end{equation}
1227
1228\smallskip
1229%----------------------------------------------namdom----------------------------------------------------
1230\namdisplay{namdyn_nept}
1231%--------------------------------------------------------------------------------------------------------
1232\smallskip
1233
1234The Neptune effect is enabled when \np{ln\_neptsimp}=true (default=false).
1235\np{ln\_smooth\_neptvel} controls whether a scale-selective smoothing is applied
1236to the Neptune effect flow field (default=false) (this smoothing method is as
1237used by Holloway).  \np{rn\_tslse} and \np{rn\_tslsp} are the equatorial and
1238polar values respectively of the length-scale parameter $L$ used in determining
1239the Neptune stream function \eqref{Eq_dynnept_sf} and \eqref{Eq_dynnept_ls}.
1240Values at intermediate latitudes are given by a cosine fit, mimicking the
1241variation of the deformation radius with latitude.  The default values of 12km
1242and 3km are those given in \citep{HollowayJPO94}, appropriate for a coarse
1243resolution model. The finer resolution study of \citep{HollowayOM08} increased
1244the values of L by a factor of $\sqrt 2$ to 17km and 4.2km, thus doubling the
1245stream function for a given topography.
1246
1247The simple formulation for ($u^*$, $v^*$) can give unacceptably large velocities
1248in shallow water, and \citep{HollowayOM08} add an offset to the depth in the
1249denominator to control this problem. In this implementation we offer instead (at
1250the suggestion of G. Madec) the option of ramping down the Neptune flow field to
1251zero over a finite depth range. The switch \np{ln\_neptramp} activates this
1252option (default=false), in which case velocities at depths greater than
1253\np{rn\_htrmax} are unaltered, but ramp down linearly with depth to zero at a
1254depth of \np{rn\_htrmin} (and shallower).
1255
1256% ================================================================
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