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1% ================================================================
2% Chapter Ñ 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 \ref{Griffies_al_MWR01, LeclairMadec2009}.
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 $2$.
170Coriolis and momentum advection terms are evaluated using a leapfrog
171scheme, $i.e.$ the velocity appearing in these expressions is centred in
172time (\textit{now} velocity).
173At the lateral boundaries either free slip, no slip or partial slip boundary
174conditions are applied following Chap.\ref{LBC}.
175
176% -------------------------------------------------------------------------------------------------------------
177%        Vorticity term
178% -------------------------------------------------------------------------------------------------------------
179\subsection   [Vorticity term (\textit{dynvor}) ]
180         {Vorticity term (\mdl{dynvor})}
181\label{DYN_vor}
182%------------------------------------------nam_dynvor----------------------------------------------------
183\namdisplay{namdyn_vor} 
184%-------------------------------------------------------------------------------------------------------------
185
186Four discretisations of the vorticity term (\textit{ln\_dynvor\_xxx}=true) are available:
187conserving potential enstrophy of horizontally non-divergent flow (ENS scheme) ;
188conserving horizontal kinetic energy (ENE scheme) ; conserving potential enstrophy for
189the relative vorticity term and horizontal kinetic energy for the planetary vorticity
190term (MIX scheme) ; or conserving both the potential enstrophy of horizontally non-divergent
191flow and horizontal kinetic energy (EEN scheme) (see  Appendix~\ref{Apdx_C_vor_zad}). In the
192case of ENS, ENE or MIX schemes the land sea mask may be slightly modified to ensure the
193consistency of vorticity term with analytical equations (\textit{ln\_dynvor\_con}=true).
194The vorticity terms are all computed in dedicated routines that can be found in
195the \mdl{dynvor} module.
196
197%-------------------------------------------------------------
198%                 enstrophy conserving scheme
199%-------------------------------------------------------------
200\subsubsection{Enstrophy conserving scheme (\np{ln\_dynvor\_ens}=true)}
201\label{DYN_vor_ens}
202
203In the enstrophy conserving case (ENS scheme), the discrete formulation of the
204vorticity term provides a global conservation of the enstrophy
205($ [ (\zeta +f ) / e_{3f} ]^2 $ in $s$-coordinates) for a horizontally non-divergent
206flow ($i.e.$ $\chi$=$0$), but does not conserve the total kinetic energy. It is given by:
207\begin{equation} \label{Eq_dynvor_ens}
208\left\{ 
209\begin{aligned}
210{+\frac{1}{e_{1u} } } & {\overline {\left( { \frac{\zeta +f}{e_{3f} }} \right)} }^{\,i} 
211                                & {\overline{\overline {\left( {e_{1v}\,e_{3v}\;v} \right)}} }^{\,i, j+1/2}    \\
212{- \frac{1}{e_{2v} } } & {\overline {\left( {\frac{\zeta +f}{e_{3f} }} \right)} }^{\,j} 
213                                & {\overline{\overline {\left( {e_{2u}\,e_{3u}\;u} \right)}} }^{\,i+1/2, j} 
214\end{aligned} 
215 \right.
216\end{equation} 
217
218%-------------------------------------------------------------
219%                 energy conserving scheme
220%-------------------------------------------------------------
221\subsubsection{Energy conserving scheme (\np{ln\_dynvor\_ene}=true)}
222\label{DYN_vor_ene}
223
224The kinetic energy conserving scheme (ENE scheme) conserves the global
225kinetic energy but not the global enstrophy. It is given by:
226\begin{equation} \label{Eq_dynvor_ene}
227\left\{   \begin{aligned}
228{+\frac{1}{e_{1u}}\; {\overline {\left( {\frac{\zeta +f}{e_{3f} }} \right)
229                            \;  \overline {\left( {e_{1v}\,e_{3v}\;v} \right)} ^{\,i+1/2}} }^{\,j} }    \\
230{- \frac{1}{e_{2v}}\; {\overline {\left( {\frac{\zeta +f}{e_{3f} }} \right)
231                            \;  \overline {\left( {e_{2u}\,e_{3u}\;u} \right)} ^{\,j+1/2}} }^{\,i} }
232\end{aligned}    \right.
233\end{equation} 
234
235%-------------------------------------------------------------
236%                 mix energy/enstrophy conserving scheme
237%-------------------------------------------------------------
238\subsubsection{Mixed energy/enstrophy conserving scheme (\np{ln\_dynvor\_mix}=true) }
239\label{DYN_vor_mix}
240
241For the mixed energy/enstrophy conserving scheme (MIX scheme), a mixture of the
242two previous schemes is used. It consists of the ENS scheme (\ref{Eq_dynvor_ens})
243for the relative vorticity term, and of the ENE scheme (\ref{Eq_dynvor_ene}) applied
244to the planetary vorticity term.
245\begin{equation} \label{Eq_dynvor_mix}
246\left\{ {     \begin{aligned}
247 {+\frac{1}{e_{1u} }\; {\overline {\left( {\frac{\zeta }{e_{3f} }} \right)} }^{\,i} 
248 \; {\overline{\overline {\left( {e_{1v}\,e_{3v}\;v} \right)}} }^{\,i,j+1/2} -\frac{1}{e_{1u} }
249 \; {\overline {\left( {\frac{f}{e_{3f} }} \right)
250 \;\overline {\left( {e_{1v}\,e_{3v}\;v} \right)} ^{\,i+1/2}} }^{\,j} } \\
251{-\frac{1}{e_{2v} }\; {\overline {\left( {\frac{\zeta }{e_{3f} }} \right)} }^j
252 \; {\overline{\overline {\left( {e_{2u}\,e_{3u}\;u} \right)}} }^{\,i+1/2,j} +\frac{1}{e_{2v} }
253 \; {\overline {\left( {\frac{f}{e_{3f} }} \right)
254 \;\overline {\left( {e_{2u}\,e_{3u}\;u} \right)} ^{\,j+1/2}} }^{\,i} } \hfill
255\end{aligned}     } \right.
256\end{equation} 
257
258%-------------------------------------------------------------
259%                 energy and enstrophy conserving scheme
260%-------------------------------------------------------------
261\subsubsection{Energy and enstrophy conserving scheme (\np{ln\_dynvor\_een}=true) }
262\label{DYN_vor_een}
263
264In both the ENS and ENE schemes, it is apparent that the combination of $i$ and $j$ 
265averages of the velocity allows for the presence of grid point oscillation structures
266that will be invisible to the operator. These structures are \textit{computational modes} 
267that will be at least partly damped by the momentum diffusion operator ($i.e.$ the
268subgrid-scale advection), but not by the resolved advection term. The ENS and ENE schemes
269therefore do not contribute to dump any grid point noise in the horizontal velocity field.
270Such noise would result in more noise in the vertical velocity field, an undesirable feature.
271This is a well-known characteristic of $C$-grid discretization where $u$ and $v$ are located
272at different grid points, a price worth paying to avoid a double averaging in the pressure
273gradient term as in the $B$-grid.
274\gmcomment{ To circumvent this, Adcroft (ADD REF HERE)
275Nevertheless, this technique strongly distort the phase and group velocity of Rossby waves....}
276
277A very nice solution to the problem of double averaging was proposed by \citet{Arakawa_Hsu_MWR90}.
278The idea is to get rid of the double averaging by considering triad combinations of vorticity.
279It is noteworthy that this solution is conceptually quite similar to the one proposed by
280\citep{Griffies_al_JPO98} for the discretization of the iso-neutral diffusion operator (see App.\ref{Apdx_C}).
281
282The \citet{Arakawa_Hsu_MWR90} vorticity advection scheme for a single layer is modified
283for spherical coordinates as described by \citet{Arakawa_Lamb_MWR81} to obtain the EEN scheme.
284First consider the discrete expression of the potential vorticity, $q$, defined at an $f$-point:
285\begin{equation} \label{Eq_pot_vor}
286q  = \frac{\zeta +f} {e_{3f} }
287\end{equation}
288where the relative vorticity is defined by (\ref{Eq_divcur_cur}), the Coriolis parameter
289is given by $f=2 \,\Omega \;\sin \varphi _f $ and the layer thickness at $f$-points is:
290\begin{equation} \label{Eq_een_e3f}
291e_{3f} = \overline{\overline {e_{3t} }} ^{\,i+1/2,j+1/2}
292\end{equation}
293
294%>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
295\begin{figure}[!ht]    \begin{center}
296\includegraphics[width=0.70\textwidth]{./TexFiles/Figures/Fig_DYN_een_triad.pdf}
297\caption{ \label{Fig_DYN_een_triad} 
298Triads used in the energy and enstrophy conserving scheme (een) for
299$u$-component (upper panel) and $v$-component (lower panel).}
300\end{center}   \end{figure}
301%>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
302
303Note that a key point in \eqref{Eq_een_e3f} is that the averaging in the \textbf{i}- and
304\textbf{j}- directions uses the masked vertical scale factor but is always divided by
305$4$, not by the sum of the masks at the four $T$-points. This preserves the continuity of
306$e_{3f}$ when one or more of the neighbouring $e_{3t}$ tends to zero and
307extends by continuity the value of $e_{3f}$ into the land areas. This feature is essential for
308the $z$-coordinate with partial steps.
309
310Next, the vorticity triads, $ {^i_j}\mathbb{Q}^{i_p}_{j_p}$ can be defined at a $T$-point as
311the following triad combinations of the neighbouring potential vorticities defined at f-points
312(Fig.~\ref{Fig_DYN_een_triad}):
313\begin{equation} \label{Q_triads}
314_i^j \mathbb{Q}^{i_p}_{j_p}
315= \frac{1}{12} \ \left(   q^{i-i_p}_{j+j_p} + q^{i+j_p}_{j+i_p} + q^{i+i_p}_{j-j_p}  \right)
316\end{equation}
317where the indices $i_p$ and $k_p$ take the values: $i_p = -1/2$ or $1/2$ and $j_p = -1/2$ or $1/2$.
318
319Finally, the vorticity terms are represented as:
320\begin{equation} \label{Eq_dynvor_een}
321\left\{ {
322\begin{aligned}
323 +q\,e_3 \, v  &\equiv +\frac{1}{e_{1u} }   \sum_{\substack{i_p,\,k_p}} 
324                         {^{i+1/2-i_p}_j}  \mathbb{Q}^{i_p}_{j_p}  \left( e_{1v}\,e_{3v} \;\right)^{i+1/2-i_p}_{j+j_p}   \\
325 - q\,e_3 \, u     &\equiv -\frac{1}{e_{2v} }    \sum_{\substack{i_p,\,k_p}} 
326                         {^i_{j+1/2-j_p}}  \mathbb{Q}^{i_p}_{j_p}  \left( e_{2u}\,e_{3u} \;\right)^{i+i_p}_{j+1/2-j_p}   \\
327\end{aligned} 
328} \right.
329\end{equation} 
330
331This EEN scheme in fact combines the conservation properties of the ENS and ENE schemes.
332It conserves both total energy and potential enstrophy in the limit of horizontally
333nondivergent flow ($i.e.$ $\chi$=$0$) (see  Appendix~\ref{Apdx_C_vor_zad}).
334Applied to a realistic ocean configuration, it has been shown that it leads to a significant
335reduction of the noise in the vertical velocity field \citep{Le_Sommer_al_OM09}.
336Furthermore, used in combination with a partial steps representation of bottom topography,
337it improves the interaction between current and topography, leading to a larger
338topostrophy of the flow  \citep{Barnier_al_OD06, Penduff_al_OS07}.
339
340%--------------------------------------------------------------------------------------------------------------
341%           Kinetic Energy Gradient term
342%--------------------------------------------------------------------------------------------------------------
343\subsection   [Kinetic Energy Gradient term (\textit{dynkeg})]
344         {Kinetic Energy Gradient term (\mdl{dynkeg})}
345\label{DYN_keg}
346
347As demonstrated in Appendix~\ref{Apdx_C}, there is a single discrete formulation
348of the kinetic energy gradient term that, together with the formulation chosen for
349the vertical advection (see below), conserves the total kinetic energy:
350\begin{equation} \label{Eq_dynkeg}
351\left\{ \begin{aligned}
352 -\frac{1}{2 \; e_{1u} }  & \ \delta _{i+1/2} \left[ {\overline {u^2}^{\,i} + \overline{v^2}^{\,j}} \right]   \\
353 -\frac{1}{2 \; e_{2v} }  & \ \delta _{j+1/2} \left[ {\overline {u^2}^{\,i} + \overline{v^2}^{\,j}} \right]   
354\end{aligned} \right.
355\end{equation} 
356
357%--------------------------------------------------------------------------------------------------------------
358%           Vertical advection term
359%--------------------------------------------------------------------------------------------------------------
360\subsection   [Vertical advection term (\textit{dynzad}) ]
361         {Vertical advection term (\mdl{dynzad}) }
362\label{DYN_zad}
363
364The discrete formulation of the vertical advection, together with the formulation
365chosen for the gradient of kinetic energy (KE) term, conserves the total kinetic
366energy. Indeed, the change of KE due to the vertical advection is exactly
367balanced by the change of KE due to the gradient of KE (see Appendix~\ref{Apdx_C}).
368\begin{equation} \label{Eq_dynzad}
369\left\{     \begin{aligned}
370-\frac{1} {e_{1u}\,e_{2u}\,e_{3u}} &\ \overline{\ \overline{ e_{1t}\,e_{2t}\;w } ^{\,i+1/2}  \;\delta _{k+1/2} \left[ u \right]\  }^{\,k}  \\
371-\frac{1} {e_{1v}\,e_{2v}\,e_{3v}}  &\ \overline{\ \overline{ e_{1t}\,e_{2t}\;w } ^{\,j+1/2}  \;\delta _{k+1/2} \left[ u \right]\  }^{\,k} 
372\end{aligned}         \right.
373\end{equation} 
374
375% ================================================================
376% Coriolis and Advection : flux form
377% ================================================================
378\section{Coriolis and Advection: flux form}
379\label{DYN_adv_cor_flux}
380%------------------------------------------nam_dynadv----------------------------------------------------
381\namdisplay{namdyn_adv} 
382%-------------------------------------------------------------------------------------------------------------
383
384In the flux form (as in the vector invariant form), the Coriolis and momentum
385advection terms are evaluated using a leapfrog scheme, $i.e.$ the velocity
386appearing in their expressions is centred in time (\textit{now} velocity). At the
387lateral boundaries either free slip, no slip or partial slip boundary conditions
388are applied following Chap.\ref{LBC}.
389
390
391%--------------------------------------------------------------------------------------------------------------
392%           Coriolis plus curvature metric terms
393%--------------------------------------------------------------------------------------------------------------
394\subsection   [Coriolis plus curvature metric terms (\textit{dynvor}) ]
395         {Coriolis plus curvature metric terms (\mdl{dynvor}) }
396\label{DYN_cor_flux}
397
398In flux form, the vorticity term reduces to a Coriolis term in which the Coriolis
399parameter has been modified to account for the "metric" term. This altered
400Coriolis parameter is thus discretised at $f$-points. It is given by:
401\begin{multline} \label{Eq_dyncor_metric}
402f+\frac{1}{e_1 e_2 }\left( {v\frac{\partial e_2 }{\partial i}  -  u\frac{\partial e_1 }{\partial j}} \right\\
403   \equiv   f + \frac{1}{e_{1f} e_{2f} } \left( { \ \overline v ^{i+1/2}\delta _{i+1/2} \left[ {e_{2u} } \right] 
404                                                                 -  \overline u ^{j+1/2}\delta _{j+1/2} \left[ {e_{1u} } \right]  }  \ \right)
405\end{multline} 
406
407Any of the (\ref{Eq_dynvor_ens}), (\ref{Eq_dynvor_ene}) and (\ref{Eq_dynvor_een})
408schemes can be used to compute the product of the Coriolis parameter and the
409vorticity. However, the energy-conserving scheme  (\ref{Eq_dynvor_een}) has
410exclusively been used to date. This term is evaluated using a leapfrog scheme,
411$i.e.$ the velocity is centred in time (\textit{now} velocity).
412
413%--------------------------------------------------------------------------------------------------------------
414%           Flux form Advection term
415%--------------------------------------------------------------------------------------------------------------
416\subsection   [Flux form Advection term (\textit{dynadv}) ]
417         {Flux form Advection term (\mdl{dynadv}) }
418\label{DYN_adv_flux}
419
420The discrete expression of the advection term is given by :
421\begin{equation} \label{Eq_dynadv}
422\left\{ 
423\begin{aligned}
424\frac{1}{e_{1u}\,e_{2u}\,e_{3u}} 
425\left(      \delta _{i+1/2} \left[ \overline{e_{2u}\,e_{3u}\;u }^{i       }  \ u_t      \right]   
426          + \delta _{j       } \left[ \overline{e_{1u}\,e_{3u}\;v }^{i+1/2}  \ u_f      \right] \right\ \;   \\
427\left.   + \delta _{k      } \left[ \overline{e_{1w}\,e_{2w}\;w}^{i+1/2}  \ u_{uw} \right] \right)   \\
428\\
429\frac{1}{e_{1v}\,e_{2v}\,e_{3v}} 
430\left(     \delta _{i       } \left[ \overline{e_{2u}\,e_{3u }\;u }^{j+1/2} \ v_f       \right] 
431         + \delta _{j+1/2} \left[ \overline{e_{1u}\,e_{3u }\;v }^{i       } \ v_t       \right] \right\ \, \, \\
432\left.  + \delta _{k      } \left[ \overline{e_{1w}\,e_{2w}\;w}^{j+1/2} \ v_{vw}  \right] \right) \\
433\end{aligned}
434\right.
435\end{equation}
436
437Two advection schemes are available: a $2^{nd}$ order centered finite
438difference scheme, CEN2, or a $3^{rd}$ order upstream biased scheme, UBS.
439The latter is described in \citet{Shchepetkin_McWilliams_OM05}. The schemes are
440selected using the namelist logicals \np{ln\_dynadv\_cen2} and \np{ln\_dynadv\_ubs}.
441In flux form, the schemes differ by the choice of a space and time interpolation to
442define the value of $u$ and $v$ at the centre of each face of $u$- and $v$-cells,
443$i.e.$ at the $T$-, $f$-, and $uw$-points for $u$ and at the $f$-, $T$- and
444$vw$-points for $v$.
445
446%-------------------------------------------------------------
447%                 2nd order centred scheme
448%-------------------------------------------------------------
449\subsubsection{$2^{nd}$ order centred scheme (cen2) (\np{ln\_dynadv\_cen2}=true)}
450\label{DYN_adv_cen2}
451
452In the centered $2^{nd}$ order formulation, the velocity is evaluated as the
453mean of the two neighbouring points :
454\begin{equation} \label{Eq_dynadv_cen2}
455\left\{     \begin{aligned}
456 u_T^{cen2} &=\overline u^{i }       \quad &  u_F^{cen2} &=\overline u^{j+1/2}  \quad &  u_{uw}^{cen2} &=\overline u^{k+1/2}   \\
457 v_F^{cen2} &=\overline v ^{i+1/2} \quad & v_F^{cen2} &=\overline v^j      \quad &  v_{vw}^{cen2} &=\overline v ^{k+1/2}  \\
458\end{aligned}      \right.
459\end{equation} 
460
461The scheme is non diffusive (i.e. conserves the kinetic energy) but dispersive
462($i.e.$ it may create false extrema). It is therefore notoriously noisy and must be
463used in conjunction with an explicit diffusion operator to produce a sensible solution.
464The associated time-stepping is performed using a leapfrog scheme in conjunction
465with an Asselin time-filter, so $u$ and $v$ are the \emph{now} velocities.
466
467%-------------------------------------------------------------
468%                 UBS scheme
469%-------------------------------------------------------------
470\subsubsection{Upstream Biased Scheme (UBS) (\np{ln\_dynadv\_ubs}=true)}
471\label{DYN_adv_ubs}
472
473The UBS advection scheme is an upstream biased third order scheme based on
474an upstream-biased parabolic interpolation. For example, the evaluation of
475$u_T^{ubs} $ is done as follows:
476\begin{equation} \label{Eq_dynadv_ubs}
477u_T^{ubs} =\overline u ^i-\;\frac{1}{6}   \begin{cases}
478      u"_{i-1/2}&    \text{if $\ \overline{e_{2u}\,e_{3u} \ u}^i  \geqslant 0$ }    \\
479      u"_{i+1/2}&    \text{if $\ \overline{e_{2u}\,e_{3u} \ u}^i  < 0$ }
480\end{cases}
481\end{equation}
482where $u"_{i+1/2} =\delta _{i+1/2} \left[ {\delta _i \left[ u \right]} \right]$. This results
483in a dissipatively dominant ($i.e.$ hyper-diffusive) truncation error \citep{Shchepetkin_McWilliams_OM05}.
484The overall performance of the advection scheme is similar to that reported in
485\citet{Farrow1995}. It is a relatively good compromise between accuracy and
486smoothness. It is not a \emph{positive} scheme, meaning that false extrema are
487permitted. But the amplitudes of the false extrema are significantly reduced over
488those in the centred second order method. As the scheme already includes
489a diffusion component, it can be used without explicit  lateral diffusion on momentum
490($i.e.$ \np{ln\_dynldf\_lap}=\np{ln\_dynldf\_bilap}=false), and it is recommended to do so.
491
492The UBS scheme is not used in all directions. In the vertical, the centred $2^{nd}$ 
493order evaluation of the advection is preferred, $i.e.$ $u_{uw}^{ubs}$ and
494$u_{vw}^{ubs}$ in \eqref{Eq_dynadv_cen2} are used. UBS is diffusive and is
495associated with vertical mixing of momentum. \gmcomment{ gm  pursue the
496sentence:Since vertical mixing of momentum is a source term of the TKE equation...  }
497
498For stability reasons,  the first term in (\ref{Eq_dynadv_ubs}), which corresponds
499to a second order centred scheme, is evaluated using the \textit{now} velocity
500(centred in time), while the second term, which is the diffusion part of the scheme,
501is evaluated using the \textit{before} velocity (forward in time). This is discussed
502by \citet{Webb_al_JAOT98} in the context of the Quick advection scheme.
503
504Note that the UBS and QUICK (Quadratic Upstream Interpolation for Convective Kinematics)
505schemes only differ by one coefficient. Replacing $1/6$ by $1/8$ in
506(\ref{Eq_dynadv_ubs}) leads to the QUICK advection scheme \citep{Webb_al_JAOT98}.
507This option is not available through a namelist parameter, since the $1/6$ coefficient
508is hard coded. Nevertheless it is quite easy to make the substitution in the
509\mdl{dynadv\_ubs} module and obtain a QUICK scheme.
510
511Note also that in the current version of \mdl{dynadv\_ubs}, there is also the
512possibility of using a $4^{th}$ order evaluation of the advective velocity as in
513ROMS. This is an error and should be suppressed soon.
514%%%
515\gmcomment{action :  this have to be done}
516%%%
517
518% ================================================================
519%           Hydrostatic pressure gradient term
520% ================================================================
521\section  [Hydrostatic pressure gradient (\textit{dynhpg})]
522      {Hydrostatic pressure gradient (\mdl{dynhpg})}
523\label{DYN_hpg}
524%------------------------------------------nam_dynhpg---------------------------------------------------
525\namdisplay{namdyn_hpg} 
526%-------------------------------------------------------------------------------------------------------------
527
528The key distinction between the different algorithms used for the hydrostatic
529pressure gradient is the vertical coordinate used, since HPG is a \emph{horizontal} 
530pressure gradient, $i.e.$ computed along geopotential surfaces. As a result, any
531tilt of the surface of the computational levels will require a specific treatment to
532compute the hydrostatic pressure gradient.
533
534The hydrostatic pressure gradient term is evaluated either using a leapfrog scheme,
535$i.e.$ the density appearing in its expression is centred in time (\emph{now} $\rho$), or
536a semi-implcit scheme. At the lateral boundaries either free slip, no slip or partial slip
537boundary conditions are applied.
538
539%--------------------------------------------------------------------------------------------------------------
540%           z-coordinate with full step
541%--------------------------------------------------------------------------------------------------------------
542\subsection   [$z$-coordinate with full step (\np{ln\_dynhpg\_zco}) ]
543         {$z$-coordinate with full step (\np{ln\_dynhpg\_zco}=true)}
544\label{DYN_hpg_zco}
545
546The hydrostatic pressure can be obtained by integrating the hydrostatic equation
547vertically from the surface. However, the pressure is large at great depth while its
548horizontal gradient is several orders of magnitude smaller. This may lead to large
549truncation errors in the pressure gradient terms. Thus, the two horizontal components
550of the hydrostatic pressure gradient are computed directly as follows:
551
552for $k=km$ (surface layer, $jk=1$ in the code)
553\begin{equation} \label{Eq_dynhpg_zco_surf}
554\left\{ \begin{aligned}
555               \left. \delta _{i+1/2} \left[  p^h         \right] \right|_{k=km} 
556&= \frac{1}{2} g \   \left. \delta _{i+1/2} \left[  e_{3w} \ \rho \right] \right|_{k=km}   \\
557                  \left. \delta _{j+1/2} \left[  p^h            \right] \right|_{k=km} 
558&= \frac{1}{2} g \   \left. \delta _{j+1/2} \left[  e_{3w} \ \rho \right] \right|_{k=km}   \\
559\end{aligned} \right.
560\end{equation} 
561
562for $1<k<km$ (interior layer)
563\begin{equation} \label{Eq_dynhpg_zco}
564\left\{ \begin{aligned}
565               \left. \delta _{i+1/2} \left[  p^h         \right] \right|_{k} 
566&=             \left. \delta _{i+1/2} \left[  p^h         \right] \right|_{k-1} 
567+    \frac{1}{2}\;g\;   \left. \delta _{i+1/2} \left[  e_{3w} \ \overline {\rho}^{k+1/2} \right] \right|_{k}   \\
568                  \left. \delta _{j+1/2} \left[  p^h            \right] \right|_{k} 
569&=                \left. \delta _{j+1/2} \left[  p^h            \right] \right|_{k-1} 
570+    \frac{1}{2}\;g\;   \left. \delta _{j+1/2} \left[  e_{3w} \ \overline {\rho}^{k+1/2} \right] \right|_{k}   \\
571\end{aligned} \right.
572\end{equation} 
573
574Note that the $1/2$ factor in (\ref{Eq_dynhpg_zco_surf}) is adequate because of
575the definition of $e_{3w}$ as the vertical derivative of the scale factor at the surface
576level ($z=0$). Note also that in case of variable volume level (\key{vvl} defined), the
577surface pressure gradient is included in \eqref{Eq_dynhpg_zco_surf} and \eqref{Eq_dynhpg_zco} 
578through the space and time variations of the vertical scale factor $e_{3w}$.
579
580%--------------------------------------------------------------------------------------------------------------
581%           z-coordinate with partial step
582%--------------------------------------------------------------------------------------------------------------
583\subsection   [$z$-coordinate with partial step (\np{ln\_dynhpg\_zps})]
584         {$z$-coordinate with partial step (\np{ln\_dynhpg\_zps}=true)}
585\label{DYN_hpg_zps}
586
587With partial bottom cells, tracers in horizontally adjacent cells generally live at
588different depths. Before taking horizontal gradients between these tracer points,
589a linear interpolation is used to approximate the deeper tracer as if it actually lived
590at the depth of the shallower tracer point.
591
592Apart from this modification, the horizontal hydrostatic pressure gradient evaluated
593in the $z$-coordinate with partial step is exactly as in the pure $z$-coordinate case.
594As explained in detail in section \S\ref{TRA_zpshde}, the nonlinearity of pressure
595effects in the equation of state is such that it is better to interpolate temperature and
596salinity vertically before computing the density. Horizontal gradients of temperature
597and salinity are needed for the TRA modules, which is the reason why the horizontal
598gradients of density at the deepest model level are computed in module \mdl{zpsdhe} 
599located in the TRA directory and described in \S\ref{TRA_zpshde}.
600
601%--------------------------------------------------------------------------------------------------------------
602%           s- and s-z-coordinates
603%--------------------------------------------------------------------------------------------------------------
604\subsection{$s$- and $z$-$s$-coordinates}
605\label{DYN_hpg_sco}
606
607Pressure gradient formulations in an $s$-coordinate have been the subject of a vast
608number of papers ($e.g.$, \citet{Song1998, Shchepetkin_McWilliams_OM05}).
609A number of different pressure gradient options are coded but the ROMS-like, density Jacobian with
610cubic polynomial method is currently disabled whilst known bugs are under investigation.
611
612$\bullet$ Traditional coding (see for example \citet{Madec_al_JPO96}: (\np{ln\_dynhpg\_sco}=true)
613\begin{equation} \label{Eq_dynhpg_sco}
614\left\{ \begin{aligned}
615 - \frac{1}                   {\rho_o \, e_{1u}} \;   \delta _{i+1/2} \left[  p^h  \right] 
616+ \frac{g\; \overline {\rho}^{i+1/2}}  {\rho_o \, e_{1u}} \;   \delta _{i+1/2} \left[  z_t   \right]    \\
617 - \frac{1}                   {\rho_o \, e_{2v}} \;   \delta _{j+1/2} \left[  p^h  \right] 
618+ \frac{g\; \overline {\rho}^{j+1/2}}  {\rho_o \, e_{2v}} \;   \delta _{j+1/2} \left[  z_t   \right]    \\
619\end{aligned} \right.
620\end{equation} 
621
622Where the first term is the pressure gradient along coordinates, computed as in
623\eqref{Eq_dynhpg_zco_surf} - \eqref{Eq_dynhpg_zco}, and $z_T$ is the depth of
624the $T$-point evaluated from the sum of the vertical scale factors at the $w$-point
625($e_{3w}$).
626
627$\bullet$ Pressure Jacobian scheme (prj) (a research paper in preparation) (\np{ln\_dynhpg\_prj}=true)
628
629$\bullet$ Density Jacobian with cubic polynomial scheme (DJC) \citep{Shchepetkin_McWilliams_OM05} 
630(\np{ln\_dynhpg\_djc}=true) (currently disabled; under development)
631
632Note that expression \eqref{Eq_dynhpg_sco} is commonly used when the variable volume formulation is
633activated (\key{vvl}) because in that case, even with a flat bottom, the coordinate surfaces are not
634horizontal but follow the free surface \citep{Levier2007}. The pressure jacobian scheme
635(\np{ln\_dynhpg\_prj}=true) is available as an improved option to \np{ln\_dynhpg\_sco}=true when
636\key{vvl} is active.  The pressure Jacobian scheme uses a constrained cubic spline to reconstruct
637the density profile across the water column. This method maintains the monotonicity between the
638density nodes  The pressure can be calculated by analytical integration of the density profile and a
639pressure Jacobian method is used to solve the horizontal pressure gradient. This method can provide
640a more accurate calculation of the horizontal pressure gradient than the standard scheme.
641
642%--------------------------------------------------------------------------------------------------------------
643%           Time-scheme
644%--------------------------------------------------------------------------------------------------------------
645\subsection   [Time-scheme (\np{ln\_dynhpg\_imp}) ]
646         {Time-scheme (\np{ln\_dynhpg\_imp}= true/false)}
647\label{DYN_hpg_imp}
648
649The default time differencing scheme used for the horizontal pressure gradient is
650a leapfrog scheme and therefore the density used in all discrete expressions given
651above is the  \textit{now} density, computed from the \textit{now} temperature and
652salinity. In some specific cases (usually high resolution simulations over an ocean
653domain which includes weakly stratified regions) the physical phenomenon that
654controls the time-step is internal gravity waves (IGWs). A semi-implicit scheme for
655doubling the stability limit associated with IGWs can be used \citep{Brown_Campana_MWR78,
656Maltrud1998}. It involves the evaluation of the hydrostatic pressure gradient as an
657average over the three time levels $t-\rdt$, $t$, and $t+\rdt$ ($i.e.$ 
658\textit{before}\textit{now} and  \textit{after} time-steps), rather than at the central
659time level $t$ only, as in the standard leapfrog scheme.
660
661$\bullet$ leapfrog scheme (\np{ln\_dynhpg\_imp}=true):
662
663\begin{equation} \label{Eq_dynhpg_lf}
664\frac{u^{t+\rdt}-u^{t-\rdt}}{2\rdt} = \;\cdots \;
665   -\frac{1}{\rho _o \,e_{1u} }\delta _{i+1/2} \left[ {p_h^t } \right]
666\end{equation}
667
668$\bullet$ semi-implicit scheme (\np{ln\_dynhpg\_imp}=true):
669\begin{equation} \label{Eq_dynhpg_imp}
670\frac{u^{t+\rdt}-u^{t-\rdt}}{2\rdt} = \;\cdots \;
671   -\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]
672\end{equation}
673
674The semi-implicit time scheme \eqref{Eq_dynhpg_imp} is made possible without
675significant additional computation since the density can be updated to time level
676$t+\rdt$ before computing the horizontal hydrostatic pressure gradient. It can
677be easily shown that the stability limit associated with the hydrostatic pressure
678gradient doubles using \eqref{Eq_dynhpg_imp} compared to that using the
679standard leapfrog scheme \eqref{Eq_dynhpg_lf}. Note that \eqref{Eq_dynhpg_imp} 
680is equivalent to applying a time filter to the pressure gradient to eliminate high
681frequency IGWs. Obviously, when using \eqref{Eq_dynhpg_imp}, the doubling of
682the time-step is achievable only if no other factors control the time-step, such as
683the stability limits associated with advection or diffusion.
684
685In practice, the semi-implicit scheme is used when \np{ln\_dynhpg\_imp}=true.
686In this case, we choose to apply the time filter to temperature and salinity used in
687the equation of state, instead of applying it to the hydrostatic pressure or to the
688density, so that no additional storage array has to be defined. The density used to
689compute the hydrostatic pressure gradient (whatever the formulation) is evaluated
690as follows:
691\begin{equation} \label{Eq_rho_flt}
692   \rho^t = \rho( \widetilde{T},\widetilde {S},z_t)
693 \quad     \text{with}  \quad 
694   \widetilde{X} = 1 / 4 \left(  X^{t+\rdt} +2 \,X^t + X^{t-\rdt}  \right)
695\end{equation}
696
697Note that in the semi-implicit case, it is necessary to save the filtered density, an
698extra three-dimensional field, in the restart file to restart the model with exact
699reproducibility. This option is controlled by  \np{nn\_dynhpg\_rst}, a namelist parameter.
700
701% ================================================================
702% Surface Pressure Gradient
703% ================================================================
704\section  [Surface pressure gradient (\textit{dynspg}) ]
705      {Surface pressure gradient (\mdl{dynspg})}
706\label{DYN_spg}
707%-----------------------------------------nam_dynspg----------------------------------------------------
708\namdisplay{namdyn_spg} 
709%------------------------------------------------------------------------------------------------------------
710
711$\ $\newline      %force an empty line
712
713%%%
714The 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}.
715
716%%%
717
718
719The form of the surface pressure gradient term depends on how the user wants to handle
720the fast external gravity waves that are a solution of the analytical equation (\S\ref{PE_hor_pg}).
721Three formulations are available, all controlled by a CPP key (ln\_dynspg\_xxx):
722an explicit formulation which requires a small time step ;
723a filtered free surface formulation which allows a larger time step by adding a filtering
724term into the momentum equation ;
725and a split-explicit free surface formulation, described below, which also allows a larger time step.
726
727The extra term introduced in the filtered method is calculated
728implicitly, so that a solver is used to compute it. As a consequence the update of the $next$ 
729velocities is done in module \mdl{dynspg\_flt} and not in \mdl{dynnxt}.
730
731
732
733%--------------------------------------------------------------------------------------------------------------
734% Explicit free surface formulation
735%--------------------------------------------------------------------------------------------------------------
736\subsection{Explicit free surface (\key{dynspg\_exp})}
737\label{DYN_spg_exp}
738
739In the explicit free surface formulation (\key{dynspg\_exp} defined), the model time step
740is chosen to be small enough to resolve the external gravity waves (typically a few tens of seconds).
741The surface pressure gradient, evaluated using a leap-frog scheme ($i.e.$ centered in time),
742is thus simply given by :
743\begin{equation} \label{Eq_dynspg_exp}
744\left\{ \begin{aligned}
745 - \frac{1}{e_{1u}\,\rho_o} \;   \delta _{i+1/2} \left[  \,\rho \,\eta\,  \right]   \\
746 - \frac{1}{e_{2v}\,\rho_o} \;   \delta _{j+1/2} \left[  \,\rho \,\eta\,  \right] 
747\end{aligned} \right.
748\end{equation} 
749
750Note that in the non-linear free surface case ($i.e.$ \key{vvl} defined), the surface pressure
751gradient is already included in the momentum tendency  through the level thickness variation
752allowed in the computation of the hydrostatic pressure gradient. Thus, nothing is done in the \mdl{dynspg\_exp} module.
753
754%--------------------------------------------------------------------------------------------------------------
755% Split-explict free surface formulation
756%--------------------------------------------------------------------------------------------------------------
757\subsection{Split-Explicit free surface (\key{dynspg\_ts})}
758\label{DYN_spg_ts}
759
760The split-explicit free surface formulation used in \NEMO (\key{dynspg\_ts} defined),
761also called the time-splitting formulation, follows the one
762proposed by \citet{Griffies_Bk04}. The general idea is to solve the free surface
763equation and the associated barotropic velocity equations with a smaller time
764step than $\rdt$, the time step used for the three dimensional prognostic
765variables (Fig.~\ref{Fig_DYN_dynspg_ts}).
766The size of the small time step, $\rdt_e$ (the external mode or barotropic time step)
767 is provided through the \np{nn\_baro} namelist parameter as:
768$\rdt_e = \rdt / nn\_baro$.
769 
770
771%>   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >
772\begin{figure}[!t]    \begin{center}
773\includegraphics[width=0.90\textwidth]{./TexFiles/Figures/Fig_DYN_dynspg_ts.pdf}
774\caption{  \label{Fig_DYN_dynspg_ts}
775Schematic of the split-explicit time stepping scheme for the external
776and internal modes. Time increases to the right.
777Internal mode time steps (which are also the model time steps) are denoted
778by $t-\rdt$, $t, t+\rdt$, and $t+2\rdt$.
779The curved line represents a leap-frog time step, and the smaller time
780steps $N \rdt_e=\frac{3}{2}\rdt$ are denoted by the zig-zag line.
781The vertically integrated forcing \textbf{M}(t) computed at the model time step $t$ 
782represents the interaction between the external and internal motions.
783While keeping \textbf{M} and freshwater forcing field fixed, a leap-frog
784integration carries the external mode variables (surface height and vertically
785integrated velocity) from $t$ to $t+\frac{3}{2} \rdt$ using N external time
786steps of length $\rdt_e$. Time averaging the external fields over the
787$\frac{2}{3}N+1$ time steps (endpoints included) centers the vertically integrated
788velocity and the sea surface height at the model timestep $t+\rdt$.
789These averaged values are used to update \textbf{M}(t) with both the surface
790pressure gradient and the Coriolis force, therefore providing the $t+\rdt$
791velocity.  The model time stepping scheme can then be achieved by a baroclinic
792leap-frog time step that carries the surface height from $t-\rdt$ to $t+\rdt$}
793\end{center}    \end{figure}
794%>   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >
795
796The split-explicit formulation has a damping effect on external gravity waves,
797which is weaker damping than that for the filtered free surface but still significant, as
798shown by \citet{Levier2007} in the case of an analytical barotropic Kelvin wave.
799
800%>>>>>===============
801\gmcomment{               %%% copy from griffies Book
802
803\textbf{title: Time stepping the barotropic system }
804
805Assume knowledge of the full velocity and tracer fields at baroclinic time $\tau$. Hence,
806we can update the surface height and vertically integrated velocity with a leap-frog
807scheme using the small barotropic time step $\rdt$. We have
808
809\begin{equation} \label{DYN_spg_ts_eta}
810\eta^{(b)}(\tau,t_{n+1}) - \eta^{(b)}(\tau,t_{n+1}) (\tau,t_{n-1})
811   = 2 \rdt \left[-\nabla \cdot \textbf{U}^{(b)}(\tau,t_n) + \text{EMP}_w(\tau) \right] 
812\end{equation}
813\begin{multline} \label{DYN_spg_ts_u}
814\textbf{U}^{(b)}(\tau,t_{n+1}) - \textbf{U}^{(b)}(\tau,t_{n-1}\\
815   = 2\rdt \left[ - f \textbf{k} \times \textbf{U}^{(b)}(\tau,t_{n})
816   - H(\tau) \nabla p_s^{(b)}(\tau,t_{n}) +\textbf{M}(\tau) \right]
817\end{multline}
818\
819
820In these equations, araised (b) denotes values of surface height and vertically integrated velocity updated with the barotropic time steps. The $\tau$ time label on $\eta^{(b)}$ 
821and $U^{(b)}$ 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
822that sets the barotropic time steps via
823\begin{equation} \label{DYN_spg_ts_t}
824t_n=\tau+n\rdt   
825\end{equation}
826with $n$ an integer. The density scaled surface pressure is evaluated via
827\begin{equation} \label{DYN_spg_ts_ps}
828p_s^{(b)}(\tau,t_{n}) = \begin{cases}
829   g \;\eta_s^{(b)}(\tau,t_{n}) \;\rho(\tau)_{k=1}) / \rho_&      \text{non-linear case} \\
830   g \;\eta_s^{(b)}(\tau,t_{n}&      \text{linear case} 
831   \end{cases}
832\end{equation}
833To get started, we assume the following initial conditions
834\begin{equation} \label{DYN_spg_ts_eta}
835\begin{split}
836\eta^{(b)}(\tau,t_{n=0}) &= \overline{\eta^{(b)}(\tau)}
837\\
838\eta^{(b)}(\tau,t_{n=1}) &= \eta^{(b)}(\tau,t_{n=0}) + \rdt \ \text{RHS}_{n=0} 
839\end{split}
840\end{equation}
841with
842\begin{equation} \label{DYN_spg_ts_etaF}
843 \overline{\eta^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N \eta^{(b)}(\tau-\rdt,t_{n})
844\end{equation}
845the time averaged surface height taken from the previous barotropic cycle. Likewise,
846\begin{equation} \label{DYN_spg_ts_u}
847\textbf{U}^{(b)}(\tau,t_{n=0}) = \overline{\textbf{U}^{(b)}(\tau)}   \\
848\\
849\textbf{U}(\tau,t_{n=1}) = \textbf{U}^{(b)}(\tau,t_{n=0}) + \rdt \ \text{RHS}_{n=0}   
850\end{equation}
851with
852\begin{equation} \label{DYN_spg_ts_u}
853 \overline{\textbf{U}^{(b)}(\tau)} 
854   = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau-\rdt,t_{n})
855\end{equation}
856the time averaged vertically integrated transport. Notably, there is no Robert-Asselin time filter used in the barotropic portion of the integration.
857
858Upon 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 $\tau + \rdt \tau$ 
859\begin{equation} \label{DYN_spg_ts_u}
860\textbf{U}(\tau+\rdt) = \overline{\textbf{U}^{(b)}(\tau+\rdt)} 
861   = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau,t_{n})
862\end{equation}
863The surface height on the new baroclinic time step is then determined via a baroclinic leap-frog using the following form
864
865\begin{equation} \label{DYN_spg_ts_ssh}
866\eta(\tau+\Delta) - \eta^{F}(\tau-\Delta) = 2\rdt \ \left[ - \nabla \cdot \textbf{U}(\tau) + \text{EMP}_w \right] 
867\end{equation}
868
869 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).
870 
871In general, some form of time filter is needed to maintain integrity of the surface
872height field due to the leap-frog splitting mode in equation \ref{DYN_spg_ts_ssh}. We
873have tried various forms of such filtering, with the following method discussed in
874\cite{Griffies_al_MWR01} chosen due to its stability and reasonably good maintenance of
875tracer conservation properties (see Section ??)
876
877\begin{equation} \label{DYN_spg_ts_sshf}
878\eta^{F}(\tau-\Delta) =  \overline{\eta^{(b)}(\tau)} 
879\end{equation}
880Another approach tried was
881
882\begin{equation} \label{DYN_spg_ts_sshf2}
883\eta^{F}(\tau-\Delta) = \eta(\tau)
884   + (\alpha/2) \left[\overline{\eta^{(b)}}(\tau+\rdt)
885                + \overline{\eta^{(b)}}(\tau-\rdt) -2 \;\eta(\tau) \right]
886\end{equation}
887
888which is useful since it isolates all the time filtering aspects into the term multiplied
889by $\alpha$. This isolation allows for an easy check that tracer conservation is exact when
890eliminating 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.
891
892}            %%end gm comment (copy of griffies book)
893
894%>>>>>===============
895
896
897%--------------------------------------------------------------------------------------------------------------
898% Filtered free surface formulation
899%--------------------------------------------------------------------------------------------------------------
900\subsection{Filtered free surface (\key{dynspg\_flt})}
901\label{DYN_spg_fltp}
902
903The filtered formulation follows the \citet{Roullet_Madec_JGR00} implementation.
904The extra term introduced in the equations (see \S\ref{PE_free_surface}) is solved implicitly.
905The elliptic solvers available in the code are documented in \S\ref{MISC}.
906
907%% gm %%======>>>>   given here the discrete eqs provided to the solver
908\gmcomment{               %%% copy from chap-model basics
909\begin{equation} \label{Eq_spg_flt}
910\frac{\partial {\rm {\bf U}}_h }{\partial t}= {\rm {\bf M}}
911- g \nabla \left( \tilde{\rho} \ \eta \right)
912- g \ T_c \nabla \left( \widetilde{\rho} \ \partial_t \eta \right)
913\end{equation}
914where $T_c$, is a parameter with dimensions of time which characterizes the force,
915$\widetilde{\rho} = \rho / \rho_o$ is the dimensionless density, and $\rm {\bf M}$ 
916represents the collected contributions of the Coriolis, hydrostatic pressure gradient,
917non-linear and viscous terms in \eqref{Eq_PE_dyn}.
918}   %end gmcomment
919
920Note that in the linear free surface formulation (\key{vvl} not defined), the ocean depth
921is time-independent and so is the matrix to be inverted. It is computed once and for all and applies to all ocean time steps.
922
923% ================================================================
924% Lateral diffusion term
925% ================================================================
926\section  [Lateral diffusion term (\textit{dynldf})]
927      {Lateral diffusion term (\mdl{dynldf})}
928\label{DYN_ldf}
929%------------------------------------------nam_dynldf----------------------------------------------------
930\namdisplay{namdyn_ldf} 
931%-------------------------------------------------------------------------------------------------------------
932
933The options available for lateral diffusion are to use either laplacian
934(rotated or not) or biharmonic operators. The coefficients may be constant
935or spatially variable; the description of the coefficients is found in the chapter
936on lateral physics (Chap.\ref{LDF}). The lateral diffusion of momentum is
937evaluated using a forward scheme, $i.e.$ the velocity appearing in its expression
938is the \textit{before} velocity in time, except for the pure vertical component
939that appears when a tensor of rotation is used. This latter term is solved
940implicitly together with the vertical diffusion term (see \S\ref{DOM_nxt})
941
942At the lateral boundaries either free slip, no slip or partial slip boundary
943conditions are applied according to the user's choice (see Chap.\ref{LBC}).
944
945% ================================================================
946\subsection   [Iso-level laplacian operator (\np{ln\_dynldf\_lap}) ]
947         {Iso-level laplacian operator (\np{ln\_dynldf\_lap}=true)}
948\label{DYN_ldf_lap}
949
950For lateral iso-level diffusion, the discrete operator is:
951\begin{equation} \label{Eq_dynldf_lap}
952\left\{ \begin{aligned}
953 D_u^{l{\rm {\bf U}}} =\frac{1}{e_{1u} }\delta _{i+1/2} \left[ {A_T^{lm} 
954\;\chi } \right]-\frac{1}{e_{2u} {\kern 1pt}e_{3u} }\delta _j \left[
955{A_f^{lm} \;e_{3f} \zeta } \right] \\ 
956\\
957 D_v^{l{\rm {\bf U}}} =\frac{1}{e_{2v} }\delta _{j+1/2} \left[ {A_T^{lm} 
958\;\chi } \right]+\frac{1}{e_{1v} {\kern 1pt}e_{3v} }\delta _i \left[
959{A_f^{lm} \;e_{3f} \zeta } \right] \\ 
960\end{aligned} \right.
961\end{equation} 
962
963As explained in \S\ref{PE_ldf}, this formulation (as the gradient of a divergence
964and curl of the vorticity) preserves symmetry and ensures a complete
965separation between the vorticity and divergence parts of the momentum diffusion.
966
967%--------------------------------------------------------------------------------------------------------------
968%           Rotated laplacian operator
969%--------------------------------------------------------------------------------------------------------------
970\subsection   [Rotated laplacian operator (\np{ln\_dynldf\_iso}) ]
971         {Rotated laplacian operator (\np{ln\_dynldf\_iso}=true)}
972\label{DYN_ldf_iso}
973
974A rotation of the lateral momentum diffusion operator is needed in several cases:
975for iso-neutral diffusion in the $z$-coordinate (\np{ln\_dynldf\_iso}=true) and for
976either iso-neutral (\np{ln\_dynldf\_iso}=true) or geopotential
977(\np{ln\_dynldf\_hor}=true) diffusion in the $s$-coordinate. In the partial step
978case, coordinates are horizontal except at the deepest level and no
979rotation is performed when \np{ln\_dynldf\_hor}=true. The diffusion operator
980is defined simply as the divergence of down gradient momentum fluxes on each
981momentum component. It must be emphasized that this formulation ignores
982constraints on the stress tensor such as symmetry. The resulting discrete
983representation is:
984\begin{equation} \label{Eq_dyn_ldf_iso}
985\begin{split}
986 D_u^{l\textbf{U}} &= \frac{1}{e_{1u} \, e_{2u} \, e_{3u} } \\
987&  \left\{\quad  {\delta _{i+1/2} \left[ {A_T^{lm}  \left(
988    {\frac{e_{2t} \; e_{3t} }{e_{1t} } \,\delta _{i}[u]
989   -e_{2t} \; r_{1t} \,\overline{\overline {\delta _{k+1/2}[u]}}^{\,i,\,k}}
990 \right)} \right]}   \right.
991\\ 
992& \qquad +\ \delta_j \left[ {A_f^{lm} \left( {\frac{e_{1f}\,e_{3f} }{e_{2f} 
993}\,\delta _{j+1/2} [u] - e_{1f}\, r_{2f} 
994\,\overline{\overline {\delta _{k+1/2} [u]}} ^{\,j+1/2,\,k}} 
995\right)} \right]
996\\ 
997&\qquad +\ \delta_k \left[ {A_{uw}^{lm} \left( {-e_{2u} \, r_{1uw} \,\overline{\overline 
998{\delta_{i+1/2} [u]}}^{\,i+1/2,\,k+1/2} } 
999\right.} \right.
1000\\ 
1001&  \ \qquad \qquad \qquad \quad\
1002- e_{1u} \, r_{2uw} \,\overline{\overline {\delta_{j+1/2} [u]}} ^{\,j,\,k+1/2}
1003\\ 
1004& \left. {\left. { \ \qquad \qquad \qquad \ \ \ \left. {\
1005+\frac{e_{1u}\, e_{2u} }{e_{3uw} }\,\left( {r_{1uw}^2+r_{2uw}^2} 
1006\right)\,\delta_{k+1/2} [u]} \right)} \right]\;\;\;} \right\} 
1007\\
1008\\
1009 D_v^{l\textbf{V}} &= \frac{1}{e_{1v} \, e_{2v} \, e_{3v} }    \\
1010&  \left\{\quad  {\delta _{i+1/2} \left[ {A_f^{lm}  \left(
1011    {\frac{e_{2f} \; e_{3f} }{e_{1f} } \,\delta _{i+1/2}[v]
1012   -e_{2f} \; r_{1f} \,\overline{\overline {\delta _{k+1/2}[v]}}^{\,i+1/2,\,k}}
1013 \right)} \right]}   \right.
1014\\ 
1015& \qquad +\ \delta_j \left[ {A_T^{lm} \left( {\frac{e_{1t}\,e_{3t} }{e_{2t} 
1016}\,\delta _{j} [v] - e_{1t}\, r_{2t} 
1017\,\overline{\overline {\delta _{k+1/2} [v]}} ^{\,j,\,k}} 
1018\right)} \right]
1019\\ 
1020& \qquad +\ \delta_k \left[ {A_{vw}^{lm} \left( {-e_{2v} \, r_{1vw} \,\overline{\overline 
1021{\delta_{i+1/2} [v]}}^{\,i+1/2,\,k+1/2} }\right.} \right.
1022\\
1023&  \ \qquad \qquad \qquad \quad\
1024- e_{1v} \, r_{2vw} \,\overline{\overline {\delta_{j+1/2} [v]}} ^{\,j+1/2,\,k+1/2}
1025\\ 
1026& \left. {\left. { \ \qquad \qquad \qquad \ \ \ \left. {\
1027+\frac{e_{1v}\, e_{2v} }{e_{3vw} }\,\left( {r_{1vw}^2+r_{2vw}^2} 
1028\right)\,\delta_{k+1/2} [v]} \right)} \right]\;\;\;} \right\} 
1029 \end{split}
1030\end{equation}
1031where $r_1$ and $r_2$ are the slopes between the surface along which the
1032diffusion operator acts and the surface of computation ($z$- or $s$-surfaces).
1033The way these slopes are evaluated is given in the lateral physics chapter
1034(Chap.\ref{LDF}).
1035
1036%--------------------------------------------------------------------------------------------------------------
1037%           Iso-level bilaplacian operator
1038%--------------------------------------------------------------------------------------------------------------
1039\subsection   [Iso-level bilaplacian operator (\np{ln\_dynldf\_bilap})]
1040         {Iso-level bilaplacian operator (\np{ln\_dynldf\_bilap}=true)}
1041\label{DYN_ldf_bilap}
1042
1043The lateral fourth order operator formulation on momentum is obtained by
1044applying \eqref{Eq_dynldf_lap} twice. It requires an additional assumption on
1045boundary conditions: the first derivative term normal to the coast depends on
1046the free or no-slip lateral boundary conditions chosen, while the third
1047derivative terms normal to the coast are set to zero (see Chap.\ref{LBC}).
1048%%%
1049\gmcomment{add a remark on the the change in the position of the coefficient}
1050%%%
1051
1052% ================================================================
1053%           Vertical diffusion term
1054% ================================================================
1055\section  [Vertical diffusion term (\mdl{dynzdf})]
1056      {Vertical diffusion term (\mdl{dynzdf})}
1057\label{DYN_zdf}
1058%----------------------------------------------namzdf------------------------------------------------------
1059\namdisplay{namzdf} 
1060%-------------------------------------------------------------------------------------------------------------
1061
1062The large vertical diffusion coefficient found in the surface mixed layer together
1063with high vertical resolution implies that in the case of explicit time stepping there
1064would be too restrictive a constraint on the time step. Two time stepping schemes
1065can be used for the vertical diffusion term : $(a)$ a forward time differencing
1066scheme (\np{ln\_zdfexp}=true) using a time splitting technique
1067(\np{nn\_zdfexp} $>$ 1) or $(b)$ a backward (or implicit) time differencing scheme
1068(\np{ln\_zdfexp}=false) (see \S\ref{DOM_nxt}). Note that namelist variables
1069\np{ln\_zdfexp} and \np{nn\_zdfexp} apply to both tracers and dynamics.
1070
1071The formulation of the vertical subgrid scale physics is the same whatever
1072the vertical coordinate is. The vertical diffusion operators given by
1073\eqref{Eq_PE_zdf} take the following semi-discrete space form:
1074\begin{equation} \label{Eq_dynzdf}
1075\left\{   \begin{aligned}
1076D_u^{vm} &\equiv \frac{1}{e_{3u}} \ \delta _k \left[ \frac{A_{uw}^{vm} }{e_{3uw} }
1077                              \ \delta _{k+1/2} [\,u\,]         \right]     \\
1078\\
1079D_v^{vm} &\equiv \frac{1}{e_{3v}} \ \delta _k \left[ \frac{A_{vw}^{vm} }{e_{3vw} }
1080                              \ \delta _{k+1/2} [\,v\,]         \right]
1081\end{aligned}   \right.
1082\end{equation} 
1083where $A_{uw}^{vm} $ and $A_{vw}^{vm} $ are the vertical eddy viscosity and
1084diffusivity coefficients. The way these coefficients are evaluated
1085depends on the vertical physics used (see \S\ref{ZDF}).
1086
1087The surface boundary condition on momentum is the stress exerted by
1088the wind. At the surface, the momentum fluxes are prescribed as the boundary
1089condition on the vertical turbulent momentum fluxes,
1090\begin{equation} \label{Eq_dynzdf_sbc}
1091\left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{z=1}
1092    = \frac{1}{\rho _o} \binom{\tau _u}{\tau _v }
1093\end{equation}
1094where $\left( \tau _u ,\tau _v \right)$ are the two components of the wind stress
1095vector in the (\textbf{i},\textbf{j}) coordinate system. The high mixing coefficients
1096in the surface mixed layer ensure that the surface wind stress is distributed in
1097the vertical over the mixed layer depth. If the vertical mixing coefficient
1098is small (when no mixed layer scheme is used) the surface stress enters only
1099the top model level, as a body force. The surface wind stress is calculated
1100in the surface module routines (SBC, see Chap.\ref{SBC})
1101
1102The turbulent flux of momentum at the bottom of the ocean is specified through
1103a bottom friction parameterisation (see \S\ref{ZDF_bfr})
1104
1105% ================================================================
1106% External Forcing
1107% ================================================================
1108\section{External Forcings}
1109\label{DYN_forcing}
1110
1111Besides the surface and bottom stresses (see the above section) which are
1112introduced as boundary conditions on the vertical mixing, two other forcings
1113enter the dynamical equations.
1114
1115One is the effect of atmospheric pressure on the ocean dynamics.
1116Another forcing term is the tidal potential.
1117Both of which will be introduced into the reference version soon.
1118
1119\gmcomment{atmospheric pressure is there!!!!    include its description }
1120
1121% ================================================================
1122% Time evolution term
1123% ================================================================
1124\section  [Time evolution term (\textit{dynnxt})]
1125      {Time evolution term (\mdl{dynnxt})}
1126\label{DYN_nxt}
1127
1128%----------------------------------------------namdom----------------------------------------------------
1129\namdisplay{namdom} 
1130%-------------------------------------------------------------------------------------------------------------
1131
1132The general framework for dynamics time stepping is a leap-frog scheme,
1133$i.e.$ a three level centred time scheme associated with an Asselin time filter
1134(cf. Chap.\ref{STP}). The scheme is applied to the velocity, except when using
1135the flux form of momentum advection (cf. \S\ref{DYN_adv_cor_flux}) in the variable
1136volume case (\key{vvl} defined), where it has to be applied to the thickness
1137weighted velocity (see \S\ref{Apdx_A_momentum}
1138
1139$\bullet$ vector invariant form or linear free surface (\np{ln\_dynhpg\_vec}=true ; \key{vvl} not defined):
1140\begin{equation} \label{Eq_dynnxt_vec}
1141\left\{   \begin{aligned}
1142&u^{t+\rdt} = u_f^{t-\rdt} + 2\rdt  \ \text{RHS}_u^t     \\
1143&u_f^t \;\quad = u^t+\gamma \,\left[ {u_f^{t-\rdt} -2u^t+u^{t+\rdt}} \right]
1144\end{aligned}   \right.
1145\end{equation} 
1146
1147$\bullet$ flux form and nonlinear free surface (\np{ln\_dynhpg\_vec}=false ; \key{vvl} defined):
1148\begin{equation} \label{Eq_dynnxt_flux}
1149\left\{   \begin{aligned}
1150&\left(e_{3u}\,u\right)^{t+\rdt} = \left(e_{3u}\,u\right)_f^{t-\rdt} + 2\rdt \; e_{3u} \;\text{RHS}_u^t     \\
1151&\left(e_{3u}\,u\right)_f^t \;\quad = \left(e_{3u}\,u\right)^t
1152  +\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]
1153\end{aligned}   \right.
1154\end{equation} 
1155where RHS is the right hand side of the momentum equation, the subscript $f$ 
1156denotes filtered values and $\gamma$ is the Asselin coefficient. $\gamma$ is
1157initialized as \np{nn\_atfp} (namelist parameter). Its default value is \np{nn\_atfp} = $10^{-3}$.
1158In both cases, the modified Asselin filter is not applied since perfect conservation
1159is not an issue for the momentum equations.
1160
1161Note that with the filtered free surface, the update of the \textit{after} velocities
1162is done in the \mdl{dynsp\_flt} module, and only array swapping
1163and Asselin filtering is done in \mdl{dynnxt}.
1164
1165% ================================================================
1166% Neptune effect
1167% ================================================================
1168\section  [Neptune effect (\textit{dynnept})]
1169                {Neptune effect (\mdl{dynnept})}
1170\label{DYN_nept}
1171
1172The "Neptune effect" (thus named in \citep{HollowayOM86}) is a
1173parameterisation of the potentially large effect of topographic form stress
1174(caused by eddies) in driving the ocean circulation. Originally developed for
1175low-resolution models, in which it was applied via a Laplacian (second-order)
1176diffusion-like term in the momentum equation, it can also be applied in eddy
1177permitting or resolving models, in which a more scale-selective bilaplacian
1178(fourth-order) implementation is preferred. This mechanism has a
1179significant effect on boundary currents (including undercurrents), and the
1180upwelling of deep water near continental shelves.
1181
1182The theoretical basis for the method can be found in
1183\citep{HollowayJPO92}, including the explanation of why form stress is not
1184necessarily a drag force, but may actually drive the flow.
1185\citep{HollowayJPO94} demonstrate the effects of the parameterisation in
1186the GFDL-MOM model, at a horizontal resolution of about 1.8 degrees.
1187\citep{HollowayOM08} demonstrate the biharmonic version of the
1188parameterisation in a global run of the POP model, with an average horizontal
1189grid spacing of about 32km.
1190
1191The NEMO implementation is a simplified form of that supplied by
1192Greg Holloway, the testing of which was described in \citep{HollowayJGR09}.
1193The major simplification is that a time invariant Neptune velocity
1194field is assumed.  This is computed only once, during start-up, and
1195made available to the rest of the code via a module.  Vertical
1196diffusive terms are also ignored, and the model topography itself
1197is used, rather than a separate topographic dataset as in
1198\citep{HollowayOM08}.  This implementation is only in the iso-level
1199formulation, as is the case anyway for the bilaplacian operator.
1200
1201The velocity field is derived from a transport stream function given by:
1202
1203\begin{equation} \label{Eq_dynnept_sf}
1204\psi = -fL^2H
1205\end{equation}
1206
1207where $L$ is a latitude-dependant length scale given by:
1208
1209\begin{equation} \label{Eq_dynnept_ls}
1210L = l_1 + (l_2 -l_1)\left ( {1 + \cos 2\phi \over 2 } \right )
1211\end{equation}
1212
1213where $\phi$ is latitude and $l_1$ and $l_2$ are polar and equatorial length scales respectively.
1214Neptune velocity components, $u^*$, $v^*$ are derived from the stremfunction as:
1215
1216\begin{equation} \label{Eq_dynnept_vel}
1217u^* = -{1\over H} {\partial \psi \over \partial y}\ \ \  ,\ \ \ v^* = {1\over H} {\partial \psi \over \partial x}
1218\end{equation}
1219
1220\smallskip
1221%----------------------------------------------namdom----------------------------------------------------
1222\namdisplay{namdyn_nept}
1223%--------------------------------------------------------------------------------------------------------
1224\smallskip
1225
1226The Neptune effect is enabled when \np{ln\_neptsimp}=true (default=false).
1227\np{ln\_smooth\_neptvel} controls whether a scale-selective smoothing is applied
1228to the Neptune effect flow field (default=false) (this smoothing method is as
1229used by Holloway).  \np{rn\_tslse} and \np{rn\_tslsp} are the equatorial and
1230polar values respectively of the length-scale parameter $L$ used in determining
1231the Neptune stream function \eqref{Eq_dynnept_sf} and \eqref{Eq_dynnept_ls}.
1232Values at intermediate latitudes are given by a cosine fit, mimicking the
1233variation of the deformation radius with latitude.  The default values of 12km
1234and 3km are those given in \citep{HollowayJPO94}, appropriate for a coarse
1235resolution model. The finer resolution study of \citep{HollowayOM08} increased
1236the values of L by a factor of $\sqrt 2$ to 17km and 4.2km, thus doubling the
1237stream function for a given topography.
1238
1239The simple formulation for ($u^*$, $v^*$) can give unacceptably large velocities
1240in shallow water, and \citep{HollowayOM08} add an offset to the depth in the
1241denominator to control this problem. In this implementation we offer instead (at
1242the suggestion of G. Madec) the option of ramping down the Neptune flow field to
1243zero over a finite depth range. The switch \np{ln\_neptramp} activates this
1244option (default=false), in which case velocities at depths greater than
1245\np{rn\_htrmax} are unaltered, but ramp down linearly with depth to zero at a
1246depth of \np{rn\_htrmin} (and shallower).
1247
1248% ================================================================
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