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4\chapter{Ocean Dynamics (DYN)}
9Using the representation described in \autoref{chap:DOM},
10several semi-discrete space forms of the dynamical equations are available depending on
11the vertical coordinate used and on the conservation properties of the vorticity term.
12In all the equations presented here, the masking has been omitted for simplicity.
13One must be aware that all the quantities are masked fields and
14that each time an average or difference operator is used, the resulting field is multiplied by a mask.
16The prognostic ocean dynamics equation can be summarized as follows:
18  \text{NXT} = \dbinom  {\text{VOR} + \text{KEG} + \text {ZAD} }
19  {\text{COR} + \text{ADV}                       }
20  + \text{HPG} + \text{SPG} + \text{LDF} + \text{ZDF}
22NXT stands for next, referring to the time-stepping.
23The first group of terms on the rhs of this equation corresponds to the Coriolis and advection terms that
24are decomposed into either a vorticity part (VOR), a kinetic energy part (KEG) and
25a vertical advection part (ZAD) in the vector invariant formulation,
26or a Coriolis and advection part (COR+ADV) in the flux formulation.
27The terms following these are the pressure gradient contributions
28(HPG, Hydrostatic Pressure Gradient, and SPG, Surface Pressure Gradient);
29and contributions from lateral diffusion (LDF) and vertical diffusion (ZDF),
30which are added to the rhs in the \mdl{dynldf} and \mdl{dynzdf} modules.
31The vertical diffusion term includes the surface and bottom stresses.
32The external forcings and parameterisations require complex inputs
33(surface wind stress calculation using bulk formulae, estimation of mixing coefficients)
34that are carried out in modules SBC, LDF and ZDF and are described in
35\autoref{chap:SBC}, \autoref{chap:LDF} and \autoref{chap:ZDF}, respectively.
37In the present chapter we also describe the diagnostic equations used to compute the horizontal divergence,
38curl of the velocities (\emph{divcur} module) and the vertical velocity (\emph{wzvmod} module).
40The different options available to the user are managed by namelist variables.
41For term \textit{ttt} in the momentum equations, the logical namelist variables are \textit{ln\_dynttt\_xxx},
42where \textit{xxx} is a 3 or 4 letter acronym corresponding to each optional scheme.
43%If a CPP key is used for this term its name is \key{ttt}.
44The corresponding code can be found in the \textit{dynttt\_xxx} module in the DYN directory,
45and it is usually computed in the \textit{dyn\_ttt\_xxx} subroutine.
47The user has the option of extracting and outputting each tendency term from the 3D momentum equations
48(\texttt{trddyn?} defined), as described in \autoref{chap:MISC}.
49Furthermore, the tendency terms associated with the 2D barotropic vorticity balance (when \texttt{trdvor?} is defined)
50can be derived from the 3D terms.
52\gmcomment{STEVEN: not quite sure I've got the sense of the last sentence. does
53MISC correspond to "extracting tendency terms" or "vorticity balance"?}
55%% =================================================================================================
56\section{Sea surface height and diagnostic variables ($\eta$, $\zeta$, $\chi$, $w$)}
59%% =================================================================================================
60\subsection[Horizontal divergence and relative vorticity (\textit{divcur.F90})]{Horizontal divergence and relative vorticity (\protect\mdl{divcur})}
63The vorticity is defined at an $f$-point (\ie\ corner point) as follows:
65  \label{eq:DYN_divcur_cur}
66  \zeta =\frac{1}{e_{1f}\,e_{2f} }\left( {\;\delta_{i+1/2} \left[ {e_{2v}\;v} \right]
67      -\delta_{j+1/2} \left[ {e_{1u}\;u} \right]\;} \right)
70The horizontal divergence is defined at a $T$-point.
71It is given by:
73  % \label{eq:DYN_divcur_div}
74  \chi =\frac{1}{e_{1t}\,e_{2t}\,e_{3t} }
75  \left( {\delta_i \left[ {e_{2u}\,e_{3u}\,u} \right]
76      +\delta_j \left[ {e_{1v}\,e_{3v}\,v} \right]} \right)
79Note that although the vorticity has the same discrete expression in $z$- and $s$-coordinates,
80its physical meaning is not identical.
81$\zeta$ is a pseudo vorticity along $s$-surfaces
82(only pseudo because $(u,v)$ are still defined along geopotential surfaces,
83but are not necessarily defined at the same depth).
85The vorticity and divergence at the \textit{before} step are used in the computation of
86the horizontal diffusion of momentum.
87Note that because they have been calculated prior to the Asselin filtering of the \textit{before} velocities,
88the \textit{before} vorticity and divergence arrays must be included in the restart file to
89ensure perfect restartability.
90The vorticity and divergence at the \textit{now} time step are used for the computation of
91the nonlinear advection and of the vertical velocity respectively.
93%% =================================================================================================
94\subsection[Horizontal divergence and relative vorticity (\textit{sshwzv.F90})]{Horizontal divergence and relative vorticity (\protect\mdl{sshwzv})}
97The sea surface height is given by:
99  \label{eq:DYN_spg_ssh}
100  \begin{aligned}
101    \frac{\partial \eta }{\partial t}
102    &\equiv    \frac{1}{e_{1t} e_{2t} }\sum\limits_k { \left\{  \delta_i \left[ {e_{2u}\,e_{3u}\;u} \right]
103        +\delta_j \left[ {e_{1v}\,e_{3v}\;v} \right]  \right\} }
104    -    \frac{\textit{emp}}{\rho_w }   \\
105    &\equiv    \sum\limits_k {\chi \ e_{3t}}  -  \frac{\textit{emp}}{\rho_w }
106  \end{aligned}
108where \textit{emp} is the surface freshwater budget (evaporation minus precipitation),
109expressed in Kg/m$^2$/s (which is equal to mm/s),
110and $\rho_w$=1,035~Kg/m$^3$ is the reference density of sea water (Boussinesq approximation).
111If river runoff is expressed as a surface freshwater flux (see \autoref{chap:SBC}) then
112\textit{emp} can be written as the evaporation minus precipitation, minus the river runoff.
113The sea-surface height is evaluated using exactly the same time stepping scheme as
114the tracer equation \autoref{eq:TRA_nxt}:
115a leapfrog scheme in combination with an Asselin time filter,
116\ie\ the velocity appearing in \autoref{eq:DYN_spg_ssh} is centred in time (\textit{now} velocity).
117This is of paramount importance.
118Replacing $T$ by the number $1$ in the tracer equation and summing over the water column must lead to
119the sea surface height equation otherwise tracer content will not be conserved
120\citep{griffies.pacanowski.ea_MWR01, leclair.madec_OM09}.
122The vertical velocity is computed by an upward integration of the horizontal divergence starting at the bottom,
123taking into account the change of the thickness of the levels:
125  \label{eq:DYN_wzv}
126  \left\{
127    \begin{aligned}
128      &\left. w \right|_{k_b-1/2} \quad= 0    \qquad \text{where } k_b \text{ is the level just above the sea floor }   \\
129      &\left. w \right|_{k+1/2}     = \left. w \right|_{k-1/2}  +  \left. e_{3t} \right|_{k}\;  \left. \chi \right|_k
130      - \frac{1} {2 \rdt} \left\left. e_{3t}^{t+1}\right|_{k} - \left. e_{3t}^{t-1}\right|_{k}\right)
131    \end{aligned}
132  \right.
135In the case of a non-linear free surface (\texttt{vvl?}), the top vertical velocity is $-\textit{emp}/\rho_w$,
136as changes in the divergence of the barotropic transport are absorbed into the change of the level thicknesses,
137re-orientated downward.
138\gmcomment{not sure of this...  to be modified with the change in emp setting}
139In the case of a linear free surface, the time derivative in \autoref{eq:DYN_wzv} disappears.
140The upper boundary condition applies at a fixed level $z=0$.
141The top vertical velocity is thus equal to the divergence of the barotropic transport
142(\ie\ the first term in the right-hand-side of \autoref{eq:DYN_spg_ssh}).
144Note also that whereas the vertical velocity has the same discrete expression in $z$- and $s$-coordinates,
145its physical meaning is not the same:
146in the second case, $w$ is the velocity normal to the $s$-surfaces.
147Note also that the $k$-axis is re-orientated downwards in the \fortran\ code compared to
148the indexing used in the semi-discrete equations such as \autoref{eq:DYN_wzv}
149(see \autoref{subsec:DOM_Num_Index_vertical}).
152%% =================================================================================================
153\section{Coriolis and advection: vector invariant form}
157  \nlst{namdyn_adv}
158  \caption{\forcode{&namdyn_adv}}
159  \label{lst:namdyn_adv}
162The vector invariant form of the momentum equations is the one most often used in
163applications of the \NEMO\ ocean model.
164The flux form option (see next section) has been present since version $2$.
165Options are defined through the \nam{dyn_adv}{dyn\_adv} namelist variables Coriolis and
166momentum advection terms are evaluated using a leapfrog scheme,
167\ie\ the velocity appearing in these expressions is centred in time (\textit{now} velocity).
168At the lateral boundaries either free slip, no slip or partial slip boundary conditions are applied following
171%% =================================================================================================
172\subsection[Vorticity term (\textit{dynvor.F90})]{Vorticity term (\protect\mdl{dynvor})}
176  \nlst{namdyn_vor}
177  \caption{\forcode{&namdyn_vor}}
178  \label{lst:namdyn_vor}
181Options are defined through the \nam{dyn_vor}{dyn\_vor} namelist variables.
182Four discretisations of the vorticity term (\texttt{ln\_dynvor\_xxx}\forcode{=.true.}) are available:
183conserving potential enstrophy of horizontally non-divergent flow (ENS scheme);
184conserving horizontal kinetic energy (ENE scheme);
185conserving potential enstrophy for the relative vorticity term and
186horizontal kinetic energy for the planetary vorticity term (MIX scheme);
187or conserving both the potential enstrophy of horizontally non-divergent flow and horizontal kinetic energy
188(EEN scheme) (see \autoref{subsec:INVARIANTS_vorEEN}).
189In the case of ENS, ENE or MIX schemes the land sea mask may be slightly modified to ensure the consistency of
190vorticity term with analytical equations (\np[=.true.]{ln_dynvor_con}{ln\_dynvor\_con}).
191The vorticity terms are all computed in dedicated routines that can be found in the \mdl{dynvor} module.
193%                 enstrophy conserving scheme
194%% =================================================================================================
195\subsubsection[Enstrophy conserving scheme (\forcode{ln_dynvor_ens})]{Enstrophy conserving scheme (\protect\np{ln_dynvor_ens}{ln\_dynvor\_ens})}
198In the enstrophy conserving case (ENS scheme),
199the discrete formulation of the vorticity term provides a global conservation of the enstrophy
200($ [ (\zeta +f ) / e_{3f} ]^2 $ in $s$-coordinates) for a horizontally non-divergent flow (\ie\ $\chi$=$0$),
201but does not conserve the total kinetic energy.
202It is given by:
204  \label{eq:DYN_vor_ens}
205  \left\{
206    \begin{aligned}
207      {+\frac{1}{e_{1u} } } & {\overline {\left( { \frac{\zeta +f}{e_{3f} }} \right)} }^{\,i}
208      & {\overline{\overline {\left( {e_{1v}\,e_{3v}\;v} \right)}} }^{\,i, j+1/2}    \\
209      {- \frac{1}{e_{2v} } } & {\overline {\left( {\frac{\zeta +f}{e_{3f} }} \right)} }^{\,j}
210      & {\overline{\overline {\left( {e_{2u}\,e_{3u}\;u} \right)}} }^{\,i+1/2, j}
211    \end{aligned}
212  \right.
215%                 energy conserving scheme
216%% =================================================================================================
217\subsubsection[Energy conserving scheme (\forcode{ln_dynvor_ene})]{Energy conserving scheme (\protect\np{ln_dynvor_ene}{ln\_dynvor\_ene})}
220The kinetic energy conserving scheme (ENE scheme) conserves the global kinetic energy but not the global enstrophy.
221It is given by:
223  \label{eq:DYN_vor_ene}
224  \left\{
225    \begin{aligned}
226      {+\frac{1}{e_{1u}}\; {\overline {\left( {\frac{\zeta +f}{e_{3f} }} \right)
227            \;  \overline {\left( {e_{1v}\,e_{3v}\;v} \right)} ^{\,i+1/2}} }^{\,j} }    \\
228      {- \frac{1}{e_{2v}}\; {\overline {\left( {\frac{\zeta +f}{e_{3f} }} \right)
229            \;  \overline {\left( {e_{2u}\,e_{3u}\;u} \right)} ^{\,j+1/2}} }^{\,i} }
230    \end{aligned}
231  \right.
234%                 mix energy/enstrophy conserving scheme
235%% =================================================================================================
236\subsubsection[Mixed energy/enstrophy conserving scheme (\forcode{ln_dynvor_mix})]{Mixed energy/enstrophy conserving scheme (\protect\np{ln_dynvor_mix}{ln\_dynvor\_mix})}
239For the mixed energy/enstrophy conserving scheme (MIX scheme), a mixture of the two previous schemes is used.
240It consists of the ENS scheme (\autoref{eq:DYN_vor_ens}) for the relative vorticity term,
241and of the ENE scheme (\autoref{eq:DYN_vor_ene}) applied to the planetary vorticity term.
243  % \label{eq:DYN_vor_mix}
244  \left\{ {
245      \begin{aligned}
246        {+\frac{1}{e_{1u} }\; {\overline {\left( {\frac{\zeta }{e_{3f} }} \right)} }^{\,i}
247          \; {\overline{\overline {\left( {e_{1v}\,e_{3v}\;v} \right)}} }^{\,i,j+1/2} -\frac{1}{e_{1u} }
248          \; {\overline {\left( {\frac{f}{e_{3f} }} \right)
249              \;\overline {\left( {e_{1v}\,e_{3v}\;v} \right)} ^{\,i+1/2}} }^{\,j} } \\
250        {-\frac{1}{e_{2v} }\; {\overline {\left( {\frac{\zeta }{e_{3f} }} \right)} }^j
251          \; {\overline{\overline {\left( {e_{2u}\,e_{3u}\;u} \right)}} }^{\,i+1/2,j} +\frac{1}{e_{2v} }
252          \; {\overline {\left( {\frac{f}{e_{3f} }} \right)
253              \;\overline {\left( {e_{2u}\,e_{3u}\;u} \right)} ^{\,j+1/2}} }^{\,i} } \hfill
254      \end{aligned}
255    } \right.
258%                 energy and enstrophy conserving scheme
259%% =================================================================================================
260\subsubsection[Energy and enstrophy conserving scheme (\forcode{ln_dynvor_een})]{Energy and enstrophy conserving scheme (\protect\np{ln_dynvor_een}{ln\_dynvor\_een})}
263In both the ENS and ENE schemes,
264it is apparent that the combination of $i$ and $j$ averages of the velocity allows for
265the presence of grid point oscillation structures that will be invisible to the operator.
266These structures are \textit{computational modes} that will be at least partly damped by
267the momentum diffusion operator (\ie\ the subgrid-scale advection), but not by the resolved advection term.
268The ENS and ENE schemes therefore do not contribute to dump any grid point noise in the horizontal velocity field.
269Such noise would result in more noise in the vertical velocity field, an undesirable feature.
270This is a well-known characteristic of $C$-grid discretization where
271$u$ and $v$ are located at different grid points,
272a price worth paying to avoid a double averaging in the pressure gradient term as in the $B$-grid.
273\gmcomment{ To circumvent this, Adcroft (ADD REF HERE)
274Nevertheless, this technique strongly distort the phase and group velocity of Rossby waves....}
276A very nice solution to the problem of double averaging was proposed by \citet{arakawa.hsu_MWR90}.
277The idea is to get rid of the double averaging by considering triad combinations of vorticity.
278It is noteworthy that this solution is conceptually quite similar to the one proposed by
279\citep{griffies.gnanadesikan.ea_JPO98} for the discretization of the iso-neutral diffusion operator (see \autoref{apdx:INVARIANTS}).
281The \citet{arakawa.hsu_MWR90} vorticity advection scheme for a single layer is modified
282for spherical coordinates as described by \citet{arakawa.lamb_MWR81} to obtain the EEN scheme.
283First consider the discrete expression of the potential vorticity, $q$, defined at an $f$-point:
285  % \label{eq:DYN_pot_vor}
286  q  = \frac{\zeta +f} {e_{3f} }
288where the relative vorticity is defined by (\autoref{eq:DYN_divcur_cur}),
289the Coriolis parameter is given by $f=2 \,\Omega \;\sin \varphi _f $ and the layer thickness at $f$-points is:
291  \label{eq:DYN_een_e3f}
292  e_{3f} = \overline{\overline {e_{3t} }} ^{\,i+1/2,j+1/2}
296  \centering
297  \includegraphics[width=0.66\textwidth]{Fig_DYN_een_triad}
298  \caption[Triads used in the energy and enstrophy conserving scheme (EEN)]{
299    Triads used in the energy and enstrophy conserving scheme (EEN) for
300    $u$-component (upper panel) and $v$-component (lower panel).}
301  \label{fig:DYN_een_triad}
303% >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
305A key point in \autoref{eq:DYN_een_e3f} is how the averaging in the \textbf{i}- and \textbf{j}- directions is made.
306It uses the sum of masked t-point vertical scale factor divided either by the sum of the four t-point masks
307(\np[=1]{nn_een_e3f}{nn\_een\_e3f}), or just by $4$ (\np[=.true.]{nn_een_e3f}{nn\_een\_e3f}).
308The latter case preserves the continuity of $e_{3f}$ 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.
310This case introduces a sub-grid-scale topography at f-points
311(with a systematic reduction of $e_{3f}$ when a model level intercept the bathymetry)
312that tends to reinforce the topostrophy of the flow
313(\ie\ the tendency of the flow to follow the isobaths) \citep{penduff.le-sommer.ea_OS07}.
315Next, the vorticity triads, $ {^i_j}\mathbb{Q}^{i_p}_{j_p}$ can be defined at a $T$-point as
316the following triad combinations of the neighbouring potential vorticities defined at f-points
319  \label{eq:DYN_Q_triads}
320  _i^j \mathbb{Q}^{i_p}_{j_p}
321  = \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)
323where the indices $i_p$ and $k_p$ take the values: $i_p = -1/2$ or $1/2$ and $j_p = -1/2$ or $1/2$.
325Finally, the vorticity terms are represented as:
327  \label{eq:DYN_vor_een}
328  \left\{ {
329      \begin{aligned}
330        +q\,e_3 \, v    &\equiv +\frac{1}{e_{1u} }   \sum_{\substack{i_p,\,k_p}}
331        {^{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}   \\
332        - q\,e_3 \, u     &\equiv -\frac{1}{e_{2v} }    \sum_{\substack{i_p,\,k_p}}
333        {^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}   \\
334      \end{aligned}
335    } \right.
338This EEN scheme in fact combines the conservation properties of the ENS and ENE schemes.
339It conserves both total energy and potential enstrophy in the limit of horizontally nondivergent flow
340(\ie\ $\chi$=$0$) (see \autoref{subsec:INVARIANTS_vorEEN}).
341Applied to a realistic ocean configuration, it has been shown that it leads to a significant reduction of
342the noise in the vertical velocity field \citep{le-sommer.penduff.ea_OM09}.
343Furthermore, used in combination with a partial steps representation of bottom topography,
344it improves the interaction between current and topography,
345leading to a larger topostrophy of the flow \citep{barnier.madec.ea_OD06, penduff.le-sommer.ea_OS07}.
347%% =================================================================================================
348\subsection[Kinetic energy gradient term (\textit{dynkeg.F90})]{Kinetic energy gradient term (\protect\mdl{dynkeg})}
351As demonstrated in \autoref{apdx:INVARIANTS},
352there is a single discrete formulation of the kinetic energy gradient term that,
353together with the formulation chosen for the vertical advection (see below),
354conserves the total kinetic energy:
356  % \label{eq:DYN_keg}
357  \left\{
358    \begin{aligned}
359      -\frac{1}{2 \; e_{1u} }  & \ \delta_{i+1/2} \left[ {\overline {u^2}^{\,i} + \overline{v^2}^{\,j}} \right]   \\
360      -\frac{1}{2 \; e_{2v} }  & \ \delta_{j+1/2} \left[ {\overline {u^2}^{\,i} + \overline{v^2}^{\,j}} \right]
361    \end{aligned}
362  \right.
365%% =================================================================================================
366\subsection[Vertical advection term (\textit{dynzad.F90})]{Vertical advection term (\protect\mdl{dynzad})}
369The discrete formulation of the vertical advection, t
370ogether with the formulation chosen for the gradient of kinetic energy (KE) term,
371conserves the total kinetic energy.
372Indeed, the change of KE due to the vertical advection is exactly balanced by
373the change of KE due to the gradient of KE (see \autoref{apdx:INVARIANTS}).
375  % \label{eq:DYN_zad}
376  \left\{
377    \begin{aligned}
378      -\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}  \\
379      -\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}
380    \end{aligned}
381  \right.
383When \np[=.true.]{ln_dynzad_zts}{ln\_dynzad\_zts},
384a split-explicit time stepping with 5 sub-timesteps is used on the vertical advection term.
385This option can be useful when the value of the timestep is limited by vertical advection \citep{lemarie.debreu.ea_OM15}.
386Note that in this case,
387a similar split-explicit time stepping should be used on vertical advection of tracer to ensure a better stability,
388an option which is only available with a TVD scheme (see \np{ln_traadv_tvd_zts}{ln\_traadv\_tvd\_zts} in \autoref{subsec:TRA_adv_tvd}).
391%% =================================================================================================
392\section{Coriolis and advection: flux form}
395Options are defined through the \nam{dyn_adv}{dyn\_adv} namelist variables.
396In the flux form (as in the vector invariant form),
397the Coriolis and momentum advection terms are evaluated using a leapfrog scheme,
398\ie\ the velocity appearing in their expressions is centred in time (\textit{now} velocity).
399At the lateral boundaries either free slip,
400no slip or partial slip boundary conditions are applied following \autoref{chap:LBC}.
402%% =================================================================================================
403\subsection[Coriolis plus curvature metric terms (\textit{dynvor.F90})]{Coriolis plus curvature metric terms (\protect\mdl{dynvor})}
406In flux form, the vorticity term reduces to a Coriolis term in which the Coriolis parameter has been modified to account for the "metric" term.
407This altered Coriolis parameter is thus discretised at $f$-points.
408It is given by:
410  % \label{eq:DYN_cor_metric}
411  f+\frac{1}{e_1 e_2 }\left( {v\frac{\partial e_2 }{\partial i}  -  u\frac{\partial e_1 }{\partial j}} \right\\
412  \equiv   f + \frac{1}{e_{1f} e_{2f} } \left( { \ \overline v ^{i+1/2}\delta_{i+1/2} \left[ {e_{2u} } \right]
413      -  \overline u ^{j+1/2}\delta_{j+1/2} \left[ {e_{1u} } \right]  }  \ \right)
416Any of the (\autoref{eq:DYN_vor_ens}), (\autoref{eq:DYN_vor_ene}) and (\autoref{eq:DYN_vor_een}) schemes can be used to
417compute the product of the Coriolis parameter and the vorticity.
418However, the energy-conserving scheme (\autoref{eq:DYN_vor_een}) has exclusively been used to date.
419This term is evaluated using a leapfrog scheme, \ie\ the velocity is centred in time (\textit{now} velocity).
421%% =================================================================================================
422\subsection[Flux form advection term (\textit{dynadv.F90})]{Flux form advection term (\protect\mdl{dynadv})}
425The discrete expression of the advection term is given by:
427  % \label{eq:DYN_adv}
428  \left\{
429    \begin{aligned}
430      \frac{1}{e_{1u}\,e_{2u}\,e_{3u}}
431      \left(      \delta_{i+1/2} \left[ \overline{e_{2u}\,e_{3u}\;u }^{i       }  \ u_t      \right]
432        + \delta_{j       } \left[ \overline{e_{1u}\,e_{3u}\;v }^{i+1/2\ u_f      \right] \right\ \;   \\
433      \left.   + \delta_{k      } \left[ \overline{e_{1w}\,e_{2w}\;w}^{i+1/2\ u_{uw} \right] \right)   \\
434      \\
435      \frac{1}{e_{1v}\,e_{2v}\,e_{3v}}
436      \left(     \delta_{i       } \left[ \overline{e_{2u}\,e_{3u }\;u }^{j+1/2} \ v_f       \right]
437        + \delta_{j+1/2} \left[ \overline{e_{1u}\,e_{3u }\;v }^{i       } \ v_t       \right] \right\ \, \, \\
438      \left+ \delta_{k      } \left[ \overline{e_{1w}\,e_{2w}\;w}^{j+1/2} \ v_{vw}  \right] \right) \\
439    \end{aligned}
440  \right.
443Two advection schemes are available:
444a $2^{nd}$ order centered finite difference scheme, CEN2,
445or a $3^{rd}$ order upstream biased scheme, UBS.
446The latter is described in \citet{shchepetkin.mcwilliams_OM05}.
447The schemes are selected using the namelist logicals \np{ln_dynadv_cen2}{ln\_dynadv\_cen2} and \np{ln_dynadv_ubs}{ln\_dynadv\_ubs}.
448In flux form, the schemes differ by the choice of a space and time interpolation to define the value of
449$u$ and $v$ at the centre of each face of $u$- and $v$-cells, \ie\ at the $T$-, $f$-,
450and $uw$-points for $u$ and at the $f$-, $T$- and $vw$-points for $v$.
452%                 2nd order centred scheme
453%% =================================================================================================
454\subsubsection[CEN2: $2^{nd}$ order centred scheme (\forcode{ln_dynadv_cen2})]{CEN2: $2^{nd}$ order centred scheme (\protect\np{ln_dynadv_cen2}{ln\_dynadv\_cen2})}
457In the centered $2^{nd}$ order formulation, the velocity is evaluated as the mean of the two neighbouring points:
459  \label{eq:DYN_adv_cen2}
460  \left\{
461    \begin{aligned}
462      u_T^{cen2} &=\overline u^{i }       \quad &  u_F^{cen2} &=\overline u^{j+1/2}  \quad &  u_{uw}^{cen2} &=\overline u^{k+1/2}   \\
463      v_F^{cen2} &=\overline v ^{i+1/2} \quad & v_F^{cen2} &=\overline v^j    \quad &  v_{vw}^{cen2} &=\overline v ^{k+1/2}  \\
464    \end{aligned}
465  \right.
468The scheme is non diffusive (\ie\ conserves the kinetic energy) but dispersive (\ie\ it may create false extrema).
469It is therefore notoriously noisy and must be used in conjunction with an explicit diffusion operator to
470produce a sensible solution.
471The associated time-stepping is performed using a leapfrog scheme in conjunction with an Asselin time-filter,
472so $u$ and $v$ are the \emph{now} velocities.
474%                 UBS scheme
475%% =================================================================================================
476\subsubsection[UBS: Upstream Biased Scheme (\forcode{ln_dynadv_ubs})]{UBS: Upstream Biased Scheme (\protect\np{ln_dynadv_ubs}{ln\_dynadv\_ubs})}
479The UBS advection scheme is an upstream biased third order scheme based on
480an upstream-biased parabolic interpolation.
481For example, the evaluation of $u_T^{ubs} $ is done as follows:
483  \label{eq:DYN_adv_ubs}
484  u_T^{ubs} =\overline u ^i-\;\frac{1}{6}
485  \begin{cases}
486    u"_{i-1/2}&   \text{if $\ \overline{e_{2u}\,e_{3u} \ u}^i  \geqslant 0$ }    \\
487    u"_{i+1/2}&   \text{if $\ \overline{e_{2u}\,e_{3u} \ u}^i  < 0$ }
488  \end{cases}
490where $u"_{i+1/2} =\delta_{i+1/2} \left[ {\delta_i \left[ u \right]} \right]$.
491This results in a dissipatively dominant (\ie\ hyper-diffusive) truncation error
493The overall performance of the advection scheme is similar to that reported in \citet{farrow.stevens_JPO95}.
494It is a relatively good compromise between accuracy and smoothness.
495It is not a \emph{positive} scheme, meaning that false extrema are permitted.
496But the amplitudes of the false extrema are significantly reduced over those in the centred second order method.
497As the scheme already includes a diffusion component, it can be used without explicit lateral diffusion on momentum
498(\ie\ \np[=]{ln_dynldf_lap}{ln\_dynldf\_lap}\np[=.false.]{ln_dynldf_bilap}{ln\_dynldf\_bilap}),
499and it is recommended to do so.
501The UBS scheme is not used in all directions.
502In the vertical, the centred $2^{nd}$ order evaluation of the advection is preferred, \ie\ $u_{uw}^{ubs}$ and
503$u_{vw}^{ubs}$ in \autoref{eq:DYN_adv_cen2} are used.
504UBS is diffusive and is associated with vertical mixing of momentum. \gmcomment{ gm  pursue the
505sentence:Since vertical mixing of momentum is a source term of the TKE equation...  }
507For stability reasons, the first term in (\autoref{eq:DYN_adv_ubs}),
508which corresponds to a second order centred scheme, is evaluated using the \textit{now} velocity (centred in time),
509while the second term, which is the diffusion part of the scheme,
510is evaluated using the \textit{before} velocity (forward in time).
511This is discussed by \citet{} in the context of the Quick advection scheme.
513Note that the UBS and QUICK (Quadratic Upstream Interpolation for Convective Kinematics) schemes only differ by
514one coefficient.
515Replacing $1/6$ by $1/8$ in (\autoref{eq:DYN_adv_ubs}) leads to the QUICK advection scheme \citep{}.
516This option is not available through a namelist parameter, since the $1/6$ coefficient is hard coded.
517Nevertheless it is quite easy to make the substitution in the \mdl{dynadv\_ubs} module and obtain a QUICK scheme.
519Note also that in the current version of \mdl{dynadv\_ubs},
520there is also the possibility of using a $4^{th}$ order evaluation of the advective velocity as in ROMS.
521This is an error and should be suppressed soon.
523\gmcomment{action :  this have to be done}
526%% =================================================================================================
527\section[Hydrostatic pressure gradient (\textit{dynhpg.F90})]{Hydrostatic pressure gradient (\protect\mdl{dynhpg})}
531  \nlst{namdyn_hpg}
532  \caption{\forcode{&namdyn_hpg}}
533  \label{lst:namdyn_hpg}
536Options are defined through the \nam{dyn_hpg}{dyn\_hpg} namelist variables.
537The key distinction between the different algorithms used for
538the hydrostatic pressure gradient is the vertical coordinate used,
539since HPG is a \emph{horizontal} pressure gradient, \ie\ computed along geopotential surfaces.
540As a result, any tilt of the surface of the computational levels will require a specific treatment to
541compute the hydrostatic pressure gradient.
543The hydrostatic pressure gradient term is evaluated either using a leapfrog scheme,
544\ie\ the density appearing in its expression is centred in time (\emph{now} $\rho$),
545or a semi-implcit scheme.
546At the lateral boundaries either free slip, no slip or partial slip boundary conditions are applied.
548%% =================================================================================================
549\subsection[Full step $Z$-coordinate (\forcode{ln_dynhpg_zco})]{Full step $Z$-coordinate (\protect\np{ln_dynhpg_zco}{ln\_dynhpg\_zco})}
552The hydrostatic pressure can be obtained by integrating the hydrostatic equation vertically from the surface.
553However, the pressure is large at great depth while its horizontal gradient is several orders of magnitude smaller.
554This may lead to large truncation errors in the pressure gradient terms.
555Thus, the two horizontal components of the hydrostatic pressure gradient are computed directly as follows:
557for $k=km$ (surface layer, $jk=1$ in the code)
559  \label{eq:DYN_hpg_zco_surf}
560  \left\{
561    \begin{aligned}
562      \left. \delta_{i+1/2} \left[  p^h          \right] \right|_{k=km}
563      &= \frac{1}{2} g \   \left. \delta_{i+1/2} \left[  e_{3w} \ \rho \right] \right|_{k=km}   \\
564      \left. \delta_{j+1/2} \left[  p^h          \right] \right|_{k=km}
565      &= \frac{1}{2} g \   \left. \delta_{j+1/2} \left[  e_{3w} \ \rho \right] \right|_{k=km}   \\
566    \end{aligned}
567  \right.
570for $1<k<km$ (interior layer)
572  \label{eq:DYN_hpg_zco}
573  \left\{
574    \begin{aligned}
575      \left. \delta_{i+1/2} \left[  p^h          \right] \right|_{k}
576      &=             \left. \delta_{i+1/2} \left[  p^h          \right] \right|_{k-1}
577      +    \frac{1}{2}\;g\;   \left. \delta_{i+1/2} \left[  e_{3w} \ \overline {\rho}^{k+1/2} \right] \right|_{k}   \\
578      \left. \delta_{j+1/2} \left[  p^h          \right] \right|_{k}
579      &=                \left. \delta_{j+1/2} \left[  p^h          \right] \right|_{k-1}
580      +    \frac{1}{2}\;g\;   \left. \delta_{j+1/2} \left[  e_{3w} \ \overline {\rho}^{k+1/2} \right] \right|_{k}   \\
581    \end{aligned}
582  \right.
585Note that the $1/2$ factor in (\autoref{eq:DYN_hpg_zco_surf}) is adequate because of the definition of $e_{3w}$ as
586the vertical derivative of the scale factor at the surface level ($z=0$).
587Note also that in case of variable volume level (\texttt{vvl?} defined),
588the surface pressure gradient is included in \autoref{eq:DYN_hpg_zco_surf} and
589\autoref{eq:DYN_hpg_zco} through the space and time variations of the vertical scale factor $e_{3w}$.
591%% =================================================================================================
592\subsection[Partial step $Z$-coordinate (\forcode{ln_dynhpg_zps})]{Partial step $Z$-coordinate (\protect\np{ln_dynhpg_zps}{ln\_dynhpg\_zps})}
595With partial bottom cells, tracers in horizontally adjacent cells generally live at different depths.
596Before taking horizontal gradients between these tracer points,
597a linear interpolation is used to approximate the deeper tracer as if
598it actually lived at the depth of the shallower tracer point.
600Apart from this modification,
601the horizontal hydrostatic pressure gradient evaluated in the $z$-coordinate with partial step is exactly as in
602the pure $z$-coordinate case.
603As explained in detail in section \autoref{sec:TRA_zpshde},
604the nonlinearity of pressure effects in the equation of state is such that
605it is better to interpolate temperature and salinity vertically before computing the density.
606Horizontal gradients of temperature and salinity are needed for the TRA modules,
607which is the reason why the horizontal gradients of density at the deepest model level are computed in
608module \mdl{zpsdhe} located in the TRA directory and described in \autoref{sec:TRA_zpshde}.
610%% =================================================================================================
611\subsection{$S$- and $Z$-$S$-coordinates}
614Pressure gradient formulations in an $s$-coordinate have been the subject of a vast number of papers
615(\eg, \citet{song_MWR98, shchepetkin.mcwilliams_OM05}).
616A number of different pressure gradient options are coded but the ROMS-like,
617density Jacobian with cubic polynomial method is currently disabled whilst known bugs are under investigation.
619$\bullet$ Traditional coding (see for example \citet{madec.delecluse.ea_JPO96}: (\np[=.true.]{ln_dynhpg_sco}{ln\_dynhpg\_sco})
621  \label{eq:DYN_hpg_sco}
622  \left\{
623    \begin{aligned}
624      - \frac{1}                 {\rho_o \, e_{1u}} \;   \delta_{i+1/2} \left[  p^h  \right]
625      + \frac{g\; \overline {\rho}^{i+1/2}}  {\rho_o \, e_{1u}} \;   \delta_{i+1/2} \left[  z_t   \right]    \\
626      - \frac{1}                 {\rho_o \, e_{2v}} \;   \delta_{j+1/2} \left[  p^h  \right]
627      + \frac{g\; \overline {\rho}^{j+1/2}}  {\rho_o \, e_{2v}} \;   \delta_{j+1/2} \left[  z_t   \right]    \\
628    \end{aligned}
629  \right.
632Where the first term is the pressure gradient along coordinates,
633computed as in \autoref{eq:DYN_hpg_zco_surf} - \autoref{eq:DYN_hpg_zco},
634and $z_T$ is the depth of the $T$-point evaluated from the sum of the vertical scale factors at the $w$-point
637$\bullet$ Traditional coding with adaptation for ice shelf cavities (\np[=.true.]{ln_dynhpg_isf}{ln\_dynhpg\_isf}).
638This scheme need the activation of ice shelf cavities (\np[=.true.]{ln_isfcav}{ln\_isfcav}).
640$\bullet$ Pressure Jacobian scheme (prj) (a research paper in preparation) (\np[=.true.]{ln_dynhpg_prj}{ln\_dynhpg\_prj})
642$\bullet$ Density Jacobian with cubic polynomial scheme (DJC) \citep{shchepetkin.mcwilliams_OM05}
643(\np[=.true.]{ln_dynhpg_djc}{ln\_dynhpg\_djc}) (currently disabled; under development)
645Note that expression \autoref{eq:DYN_hpg_sco} is commonly used when the variable volume formulation is activated
646(\texttt{vvl?}) because in that case, even with a flat bottom,
647the coordinate surfaces are not horizontal but follow the free surface \citep{levier.treguier.ea_rpt07}.
648The pressure jacobian scheme (\np[=.true.]{ln_dynhpg_prj}{ln\_dynhpg\_prj}) is available as
649an improved option to \np[=.true.]{ln_dynhpg_sco}{ln\_dynhpg\_sco} when \texttt{vvl?} is active.
650The pressure Jacobian scheme uses a constrained cubic spline to
651reconstruct the density profile across the water column.
652This method maintains the monotonicity between the density nodes.
653The pressure can be calculated by analytical integration of the density profile and
654a pressure Jacobian method is used to solve the horizontal pressure gradient.
655This method can provide a more accurate calculation of the horizontal pressure gradient than the standard scheme.
657%% =================================================================================================
658\subsection{Ice shelf cavity}
661Beneath an ice shelf, the total pressure gradient is the sum of the pressure gradient due to the ice shelf load and
662the pressure gradient due to the ocean load (\np[=.true.]{ln_dynhpg_isf}{ln\_dynhpg\_isf}).\\
664The main hypothesis to compute the ice shelf load is that the ice shelf is in an isostatic equilibrium.
665The top pressure is computed integrating from surface to the base of the ice shelf a reference density profile
666(prescribed as density of a water at 34.4 PSU and -1.9\deg{C}) and
667corresponds to the water replaced by the ice shelf.
668This top pressure is constant over time.
669A detailed description of this method is described in \citet{losch_JGR08}.\\
671The pressure gradient due to ocean load is computed using the expression \autoref{eq:DYN_hpg_sco} described in
674%% =================================================================================================
675\subsection[Time-scheme (\forcode{ln_dynhpg_imp})]{Time-scheme (\protect\np{ln_dynhpg_imp}{ln\_dynhpg\_imp})}
678The default time differencing scheme used for the horizontal pressure gradient is a leapfrog scheme and
679therefore the density used in all discrete expressions given above is the  \textit{now} density,
680computed from the \textit{now} temperature and salinity.
681In some specific cases
682(usually high resolution simulations over an ocean domain which includes weakly stratified regions)
683the physical phenomenon that controls the time-step is internal gravity waves (IGWs).
684A semi-implicit scheme for doubling the stability limit associated with IGWs can be used
685\citep{brown.campana_MWR78, maltrud.smith.ea_JGR98}.
686It involves the evaluation of the hydrostatic pressure gradient as
687an average over the three time levels $t-\rdt$, $t$, and $t+\rdt$
688(\ie\ \textit{before}, \textit{now} and  \textit{after} time-steps),
689rather than at the central time level $t$ only, as in the standard leapfrog scheme.
691$\bullet$ leapfrog scheme (\np[=.true.]{ln_dynhpg_imp}{ln\_dynhpg\_imp}):
694  \label{eq:DYN_hpg_lf}
695  \frac{u^{t+\rdt}-u^{t-\rdt}}{2\rdt} = \;\cdots \;
696  -\frac{1}{\rho_o \,e_{1u} }\delta_{i+1/2} \left[ {p_h^t } \right]
699$\bullet$ semi-implicit scheme (\np[=.true.]{ln_dynhpg_imp}{ln\_dynhpg\_imp}):
701  \label{eq:DYN_hpg_imp}
702  \frac{u^{t+\rdt}-u^{t-\rdt}}{2\rdt} = \;\cdots \;
703  -\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]
706The semi-implicit time scheme \autoref{eq:DYN_hpg_imp} is made possible without
707significant additional computation since the density can be updated to time level $t+\rdt$ before
708computing the horizontal hydrostatic pressure gradient.
709It can be easily shown that the stability limit associated with the hydrostatic pressure gradient doubles using
710\autoref{eq:DYN_hpg_imp} compared to that using the standard leapfrog scheme \autoref{eq:DYN_hpg_lf}.
711Note that \autoref{eq:DYN_hpg_imp} is equivalent to applying a time filter to the pressure gradient to
712eliminate high frequency IGWs.
713Obviously, when using \autoref{eq:DYN_hpg_imp},
714the doubling of the time-step is achievable only if no other factors control the time-step,
715such as the stability limits associated with advection or diffusion.
717In practice, the semi-implicit scheme is used when \np[=.true.]{ln_dynhpg_imp}{ln\_dynhpg\_imp}.
718In this case, we choose to apply the time filter to temperature and salinity used in the equation of state,
719instead of applying it to the hydrostatic pressure or to the density,
720so that no additional storage array has to be defined.
721The density used to compute the hydrostatic pressure gradient (whatever the formulation) is evaluated as follows:
723  % \label{eq:DYN_rho_flt}
724  \rho^t = \rho( \widetilde{T},\widetilde {S},z_t)
725  \quad    \text{with}  \quad
726  \widetilde{X} = 1 / 4 \left(  X^{t+\rdt} +2 \,X^t + X^{t-\rdt\right)
729Note that in the semi-implicit case, it is necessary to save the filtered density,
730an extra three-dimensional field, in the restart file to restart the model with exact reproducibility.
731This option is controlled by  \np{nn_dynhpg_rst}{nn\_dynhpg\_rst}, a namelist parameter.
733%% =================================================================================================
734\section[Surface pressure gradient (\textit{dynspg.F90})]{Surface pressure gradient (\protect\mdl{dynspg})}
738  \nlst{namdyn_spg}
739  \caption{\forcode{&namdyn_spg}}
740  \label{lst:namdyn_spg}
743Options are defined through the \nam{dyn_spg}{dyn\_spg} namelist variables.
744The surface pressure gradient term is related to the representation of the free surface (\autoref{sec:MB_hor_pg}).
745The main distinction is between the fixed volume case (linear free surface) and
746the variable volume case (nonlinear free surface, \texttt{vvl?} is defined).
747In the linear free surface case (\autoref{subsec:MB_free_surface})
748the vertical scale factors $e_{3}$ are fixed in time,
749while they are time-dependent in the nonlinear case (\autoref{subsec:MB_free_surface}).
750With both linear and nonlinear free surface, external gravity waves are allowed in the equations,
751which imposes a very small time step when an explicit time stepping is used.
752Two methods are proposed to allow a longer time step for the three-dimensional equations:
753the filtered free surface, which is a modification of the continuous equations (see \autoref{eq:MB_flt?}),
754and the split-explicit free surface described below.
755The extra term introduced in the filtered method is calculated implicitly,
756so that the update of the next velocities is done in module \mdl{dynspg\_flt} and not in \mdl{dynnxt}.
758The form of the surface pressure gradient term depends on how the user wants to
759handle the fast external gravity waves that are a solution of the analytical equation (\autoref{sec:MB_hor_pg}).
760Three formulations are available, all controlled by a CPP key (ln\_dynspg\_xxx):
761an explicit formulation which requires a small time step;
762a filtered free surface formulation which allows a larger time step by
763adding a filtering term into the momentum equation;
764and a split-explicit free surface formulation, described below, which also allows a larger time step.
766The extra term introduced in the filtered method is calculated implicitly, so that a solver is used to compute it.
767As a consequence the update of the $next$ velocities is done in module \mdl{dynspg\_flt} and not in \mdl{dynnxt}.
769%% =================================================================================================
770\subsection[Explicit free surface (\forcode{ln_dynspg_exp})]{Explicit free surface (\protect\np{ln_dynspg_exp}{ln\_dynspg\_exp})}
773In the explicit free surface formulation (\np{ln_dynspg_exp}{ln\_dynspg\_exp} set to true),
774the model time step is chosen to be small enough to resolve the external gravity waves
775(typically a few tens of seconds).
776The surface pressure gradient, evaluated using a leap-frog scheme (\ie\ centered in time),
777is thus simply given by :
779  \label{eq:DYN_spg_exp}
780  \left\{
781    \begin{aligned}
782      - \frac{1}{e_{1u}\,\rho_o} \; \delta_{i+1/2} \left[  \,\rho \,\eta\,  \right]    \\
783      - \frac{1}{e_{2v}\,\rho_o} \; \delta_{j+1/2} \left[  \,\rho \,\eta\,  \right]
784    \end{aligned}
785  \right.
788Note that in the non-linear free surface case (\ie\ \texttt{vvl?} defined),
789the surface pressure gradient is already included in the momentum tendency through
790the level thickness variation allowed in the computation of the hydrostatic pressure gradient.
791Thus, nothing is done in the \mdl{dynspg\_exp} module.
793%% =================================================================================================
794\subsection[Split-explicit free surface (\forcode{ln_dynspg_ts})]{Split-explicit free surface (\protect\np{ln_dynspg_ts}{ln\_dynspg\_ts})}
799The split-explicit free surface formulation used in \NEMO\ (\np{ln_dynspg_ts}{ln\_dynspg\_ts} set to true),
800also called the time-splitting formulation, follows the one proposed by \citet{shchepetkin.mcwilliams_OM05}.
801The general idea is to solve the free surface equation and the associated barotropic velocity equations with
802a smaller time step than $\rdt$, the time step used for the three dimensional prognostic variables
804The size of the small time step, $\rdt_e$ (the external mode or barotropic time step) is provided through
805the \np{nn_baro}{nn\_baro} namelist parameter as: $\rdt_e = \rdt / nn\_baro$.
806This parameter can be optionally defined automatically (\np[=.true.]{ln_bt_nn_auto}{ln\_bt\_nn\_auto}) considering that
807the stability of the barotropic system is essentially controled by external waves propagation.
808Maximum Courant number is in that case time independent, and easily computed online from the input bathymetry.
809Therefore, $\rdt_e$ is adjusted so that the Maximum allowed Courant number is smaller than \np{rn_bt_cmax}{rn\_bt\_cmax}.
812The barotropic mode solves the following equations:
813% \begin{subequations}
814%  \label{eq:DYN_BT}
816  \label{eq:DYN_BT_dyn}
817  \frac{\partial {\mathrm \overline{{\mathbf U}}_h} }{\partial t}=
818  -f\;{\mathrm {\mathbf k}}\times {\mathrm \overline{{\mathbf U}}_h}
819  -g\nabla _h \eta -\frac{c_b^{\textbf U}}{H+\eta} \mathrm {\overline{{\mathbf U}}_h} + \mathrm {\overline{\mathbf G}}
822  % \label{eq:DYN_BT_ssh}
823  \frac{\partial \eta }{\partial t}=-\nabla \cdot \left[ {\left( {H+\eta } \right) \; {\mathrm{\mathbf \overline{U}}}_h \,} \right]+P-E
825% \end{subequations}
826where $\mathrm {\overline{\mathbf G}}$ is a forcing term held constant, containing coupling term between modes,
827surface atmospheric forcing as well as slowly varying barotropic terms not explicitly computed to gain efficiency.
828The third term on the right hand side of \autoref{eq:DYN_BT_dyn} represents the bottom stress
829(see section \autoref{sec:ZDF_drg}), explicitly accounted for at each barotropic iteration.
830Temporal discretization of the system above follows a three-time step Generalized Forward Backward algorithm
831detailed in \citet{shchepetkin.mcwilliams_OM05}.
832AB3-AM4 coefficients used in \NEMO\ follow the second-order accurate,
833"multi-purpose" stability compromise as defined in \citet{shchepetkin.mcwilliams_ibk09}
834(see their figure 12, lower left).
836%>   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >
838  \centering
839  \includegraphics[width=0.66\textwidth]{Fig_DYN_dynspg_ts}
840  \caption[Split-explicit time stepping scheme for the external and internal modes]{
841    Schematic of the split-explicit time stepping scheme for the external and internal modes.
842    Time increases to the right.
843    In this particular exemple,
844    a boxcar averaging window over \np{nn_baro}{nn\_baro} barotropic time steps is used
845    (\np[=1]{nn_bt_flt}{nn\_bt\_flt}) and \np[=5]{nn_baro}{nn\_baro}.
846    Internal mode time steps (which are also the model time steps) are denoted by
847    $t-\rdt$, $t$ and $t+\rdt$.
848    Variables with $k$ superscript refer to instantaneous barotropic variables,
849    $< >$ and $<< >>$ operator refer to time filtered variables using respectively primary
850    (red vertical bars) and secondary weights (blue vertical bars).
851    The former are used to obtain time filtered quantities at $t+\rdt$ while
852    the latter are used to obtain time averaged transports to advect tracers.
853    a) Forward time integration:
854    \protect\np[=.true.]{ln_bt_fw}{ln\_bt\_fw}\protect\np[=.true.]{ln_bt_av}{ln\_bt\_av}.
855    b) Centred time integration:
856    \protect\np[=.false.]{ln_bt_fw}{ln\_bt\_fw}, \protect\np[=.true.]{ln_bt_av}{ln\_bt\_av}.
857    c) Forward time integration with no time filtering (POM-like scheme):
858    \protect\np[=.true.]{ln_bt_fw}{ln\_bt\_fw}\protect\np[=.false.]{ln_bt_av}{ln\_bt\_av}.}
859  \label{fig:DYN_spg_ts}
861%>   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >
863In the default case (\np[=.true.]{ln_bt_fw}{ln\_bt\_fw}),
864the external mode is integrated between \textit{now} and \textit{after} baroclinic time-steps
866To avoid aliasing of fast barotropic motions into three dimensional equations,
867time filtering is eventually applied on barotropic quantities (\np[=.true.]{ln_bt_av}{ln\_bt\_av}).
868In that case, the integration is extended slightly beyond \textit{after} time step to
869provide time filtered quantities.
870These are used for the subsequent initialization of the barotropic mode in the following baroclinic step.
871Since external mode equations written at baroclinic time steps finally follow a forward time stepping scheme,
872asselin filtering is not applied to barotropic quantities.\\
873Alternatively, one can choose to integrate barotropic equations starting from \textit{before} time step
875Although more computationaly expensive ( \np{nn_baro}{nn\_baro} additional iterations are indeed necessary),
876the baroclinic to barotropic forcing term given at \textit{now} time step become centred in
877the middle of the integration window.
878It can easily be shown that this property removes part of splitting errors between modes,
879which increases the overall numerical robustness.
880%references to Patrick Marsaleix' work here. Also work done by SHOM group.
884As far as tracer conservation is concerned,
885barotropic velocities used to advect tracers must also be updated at \textit{now} time step.
886This implies to change the traditional order of computations in \NEMO:
887most of momentum trends (including the barotropic mode calculation) updated first, tracers' after.
888This \textit{de facto} makes semi-implicit hydrostatic pressure gradient
889(see section \autoref{subsec:DYN_hpg_imp})
890and time splitting not compatible.
891Advective barotropic velocities are obtained by using a secondary set of filtering weights,
892uniquely defined from the filter coefficients used for the time averaging (\citet{shchepetkin.mcwilliams_OM05}).
893Consistency between the time averaged continuity equation and the time stepping of tracers is here the key to
894obtain exact conservation.
898One can eventually choose to feedback instantaneous values by not using any time filter
900In that case, external mode equations are continuous in time,
901\ie\ they are not re-initialized when starting a new sub-stepping sequence.
902This is the method used so far in the POM model, the stability being maintained by
903refreshing at (almost) each barotropic time step advection and horizontal diffusion terms.
904Since the latter terms have not been added in \NEMO\ for computational efficiency,
905removing time filtering is not recommended except for debugging purposes.
906This may be used for instance to appreciate the damping effect of the standard formulation on
907external gravity waves in idealized or weakly non-linear cases.
908Although the damping is lower than for the filtered free surface,
909it is still significant as shown by \citet{levier.treguier.ea_rpt07} in the case of an analytical barotropic Kelvin wave.
912\gmcomment{               %%% copy from griffies Book
914\textbf{title: Time stepping the barotropic system }
916Assume knowledge of the full velocity and tracer fields at baroclinic time $\tau$.
917Hence, we can update the surface height and vertically integrated velocity with a leap-frog scheme using
918the small barotropic time step $\rdt$.
919We have
922  % \label{eq:DYN_spg_ts_eta}
923  \eta^{(b)}(\tau,t_{n+1}) - \eta^{(b)}(\tau,t_{n+1}) (\tau,t_{n-1})
924  = 2 \rdt \left[-\nabla \cdot \textbf{U}^{(b)}(\tau,t_n) + \text{EMP}_w(\tau) \right]
927  % \label{eq:DYN_spg_ts_u}
928  \textbf{U}^{(b)}(\tau,t_{n+1}) - \textbf{U}^{(b)}(\tau,t_{n-1}\\
929  = 2\rdt \left[ - f \textbf{k} \times \textbf{U}^{(b)}(\tau,t_{n})
930    - H(\tau) \nabla p_s^{(b)}(\tau,t_{n}) +\textbf{M}(\tau) \right]
934In these equations, araised (b) denotes values of surface height and vertically integrated velocity updated with
935the barotropic time steps.
936The $\tau$ time label on $\eta^{(b)}$ and $U^{(b)}$ denotes the baroclinic time at which
937the vertically integrated forcing $\textbf{M}(\tau)$
938(note that this forcing includes the surface freshwater forcing),
939the tracer fields, the freshwater flux $\text{EMP}_w(\tau)$,
940and total depth of the ocean $H(\tau)$ are held for the duration of the barotropic time stepping over
941a single cycle.
942This is also the time that sets the barotropic time steps via
944  % \label{eq:DYN_spg_ts_t}
945  t_n=\tau+n\rdt
947with $n$ an integer.
948The density scaled surface pressure is evaluated via
950  % \label{eq:DYN_spg_ts_ps}
951  p_s^{(b)}(\tau,t_{n}) =
952  \begin{cases}
953    g \;\eta_s^{(b)}(\tau,t_{n}) \;\rho(\tau)_{k=1}) / \rho_o  &      \text{non-linear case} \\
954    g \;\eta_s^{(b)}(\tau,t_{n})  &      \text{linear case}
955  \end{cases}
957To get started, we assume the following initial conditions
959  % \label{eq:DYN_spg_ts_eta}
960  \begin{split}
961    \eta^{(b)}(\tau,t_{n=0}) &= \overline{\eta^{(b)}(\tau)}    \\
962    \eta^{(b)}(\tau,t_{n=1}) &= \eta^{(b)}(\tau,t_{n=0}) + \rdt \ \text{RHS}_{n=0}
963  \end{split}
967  % \label{eq:DYN_spg_ts_etaF}
968  \overline{\eta^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N \eta^{(b)}(\tau-\rdt,t_{n})
970the time averaged surface height taken from the previous barotropic cycle.
973  % \label{eq:DYN_spg_ts_u}
974  \textbf{U}^{(b)}(\tau,t_{n=0}) = \overline{\textbf{U}^{(b)}(\tau)} \\ \\
975  \textbf{U}(\tau,t_{n=1}) = \textbf{U}^{(b)}(\tau,t_{n=0}) + \rdt \ \text{RHS}_{n=0}
979  % \label{eq:DYN_spg_ts_u}
980  \overline{\textbf{U}^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau-\rdt,t_{n})
982the time averaged vertically integrated transport.
983Notably, there is no Robert-Asselin time filter used in the barotropic portion of the integration.
985Upon reaching $t_{n=N} = \tau + 2\rdt \tau$ ,
986the vertically integrated velocity is time averaged to produce the updated vertically integrated velocity at
987baroclinic time $\tau + \rdt \tau$
989  % \label{eq:DYN_spg_ts_u}
990  \textbf{U}(\tau+\rdt) = \overline{\textbf{U}^{(b)}(\tau+\rdt)} = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau,t_{n})
992The surface height on the new baroclinic time step is then determined via a baroclinic leap-frog using
993the following form
996  \label{eq:DYN_spg_ts_ssh}
997  \eta(\tau+\Delta) - \eta^{F}(\tau-\Delta) = 2\rdt \ \left[ - \nabla \cdot \textbf{U}(\tau) + \text{EMP}_w \right]
1000The use of this "big-leap-frog" scheme for the surface height ensures compatibility between
1001the mass/volume budgets and the tracer budgets.
1002More discussion of this point is provided in Chapter 10 (see in particular Section 10.2).
1004In general, some form of time filter is needed to maintain integrity of the surface height field due to
1005the leap-frog splitting mode in equation \autoref{eq:DYN_spg_ts_ssh}.
1006We have tried various forms of such filtering,
1007with the following method discussed in \cite{griffies.pacanowski.ea_MWR01} chosen due to
1008its stability and reasonably good maintenance of tracer conservation properties (see ??).
1011  \label{eq:DYN_spg_ts_sshf}
1012  \eta^{F}(\tau-\Delta) =  \overline{\eta^{(b)}(\tau)}
1014Another approach tried was
1017  % \label{eq:DYN_spg_ts_sshf2}
1018  \eta^{F}(\tau-\Delta) = \eta(\tau)
1019  + (\alpha/2) \left[\overline{\eta^{(b)}}(\tau+\rdt)
1020    + \overline{\eta^{(b)}}(\tau-\rdt) -2 \;\eta(\tau) \right]
1023which is useful since it isolates all the time filtering aspects into the term multiplied by $\alpha$.
1024This isolation allows for an easy check that tracer conservation is exact when
1025eliminating tracer and surface height time filtering (see ?? for more complete discussion).
1026However, in the general case with a non-zero $\alpha$,
1027the filter \autoref{eq:DYN_spg_ts_sshf} was found to be more conservative, and so is recommended.
1029}            %%end gm comment (copy of griffies book)
1033%% =================================================================================================
1034\subsection{Filtered free surface (\forcode{dynspg_flt?})}
1037The filtered formulation follows the \citet{roullet.madec_JGR00} implementation.
1038The extra term introduced in the equations (see \autoref{subsec:MB_free_surface}) is solved implicitly.
1039The elliptic solvers available in the code are documented in \autoref{chap:MISC}.
1041%% gm %%======>>>>   given here the discrete eqs provided to the solver
1042\gmcomment{               %%% copy from chap-model basics
1043  \[
1044    % \label{eq:DYN_spg_flt}
1045    \frac{\partial {\mathrm {\mathbf U}}_h }{\partial t}= {\mathrm {\mathbf M}}
1046    - g \nabla \left( \tilde{\rho} \ \eta \right)
1047    - g \ T_c \nabla \left( \widetilde{\rho} \ \partial_t \eta \right)
1048  \]
1049  where $T_c$, is a parameter with dimensions of time which characterizes the force,
1050  $\widetilde{\rho} = \rho / \rho_o$ is the dimensionless density,
1051  and $\mathrm {\mathbf M}$ represents the collected contributions of the Coriolis, hydrostatic pressure gradient,
1052  non-linear and viscous terms in \autoref{eq:MB_dyn}.
1053}   %end gmcomment
1055Note that in the linear free surface formulation (\texttt{vvl?} not defined),
1056the ocean depth is time-independent and so is the matrix to be inverted.
1057It is computed once and for all and applies to all ocean time steps.
1059%% =================================================================================================
1060\section[Lateral diffusion term and operators (\textit{dynldf.F90})]{Lateral diffusion term and operators (\protect\mdl{dynldf})}
1064  \nlst{namdyn_ldf}
1065  \caption{\forcode{&namdyn_ldf}}
1066  \label{lst:namdyn_ldf}
1069Options are defined through the \nam{dyn_ldf}{dyn\_ldf} namelist variables.
1070The options available for lateral diffusion are to use either laplacian (rotated or not) or biharmonic operators.
1071The coefficients may be constant or spatially variable;
1072the description of the coefficients is found in the chapter on lateral physics (\autoref{chap:LDF}).
1073The lateral diffusion of momentum is evaluated using a forward scheme,
1074\ie\ the velocity appearing in its expression is the \textit{before} velocity in time,
1075except for the pure vertical component that appears when a tensor of rotation is used.
1076This latter term is solved implicitly together with the vertical diffusion term (see \autoref{chap:TD}).
1078At the lateral boundaries either free slip,
1079no slip or partial slip boundary conditions are applied according to the user's choice (see \autoref{chap:LBC}).
1082  Hyperviscous operators are frequently used in the simulation of turbulent flows to
1083  control the dissipation of unresolved small scale features.
1084  Their primary role is to provide strong dissipation at the smallest scale supported by
1085  the grid while minimizing the impact on the larger scale features.
1086  Hyperviscous operators are thus designed to be more scale selective than the traditional,
1087  physically motivated Laplace operator.
1088  In finite difference methods,
1089  the biharmonic operator is frequently the method of choice to achieve this scale selective dissipation since
1090  its damping time (\ie\ its spin down time) scale like $\lambda^{-4}$ for disturbances of wavelength $\lambda$
1091  (so that short waves damped more rapidelly than long ones),
1092  whereas the Laplace operator damping time scales only like $\lambda^{-2}$.
1095%% =================================================================================================
1096\subsection[Iso-level laplacian (\forcode{ln_dynldf_lap})]{Iso-level laplacian operator (\protect\np{ln_dynldf_lap}{ln\_dynldf\_lap})}
1099For lateral iso-level diffusion, the discrete operator is:
1101  \label{eq:DYN_ldf_lap}
1102  \left\{
1103    \begin{aligned}
1104      D_u^{l{\mathrm {\mathbf U}}} =\frac{1}{e_{1u} }\delta_{i+1/2} \left[ {A_T^{lm}
1105          \;\chi } \right]-\frac{1}{e_{2u} {\kern 1pt}e_{3u} }\delta_j \left[
1106        {A_f^{lm} \;e_{3f} \zeta } \right] \\ \\
1107      D_v^{l{\mathrm {\mathbf U}}} =\frac{1}{e_{2v} }\delta_{j+1/2} \left[ {A_T^{lm}
1108          \;\chi } \right]+\frac{1}{e_{1v} {\kern 1pt}e_{3v} }\delta_i \left[
1109        {A_f^{lm} \;e_{3f} \zeta } \right]
1110    \end{aligned}
1111  \right.
1114As explained in \autoref{subsec:MB_ldf},
1115this formulation (as the gradient of a divergence and curl of the vorticity) preserves symmetry and
1116ensures a complete separation between the vorticity and divergence parts of the momentum diffusion.
1118%% =================================================================================================
1119\subsection[Rotated laplacian (\forcode{ln_dynldf_iso})]{Rotated laplacian operator (\protect\np{ln_dynldf_iso}{ln\_dynldf\_iso})}
1122A rotation of the lateral momentum diffusion operator is needed in several cases:
1123for iso-neutral diffusion in the $z$-coordinate (\np[=.true.]{ln_dynldf_iso}{ln\_dynldf\_iso}) and
1124for either iso-neutral (\np[=.true.]{ln_dynldf_iso}{ln\_dynldf\_iso}) or
1125geopotential (\np[=.true.]{ln_dynldf_hor}{ln\_dynldf\_hor}) diffusion in the $s$-coordinate.
1126In the partial step case, coordinates are horizontal except at the deepest level and
1127no rotation is performed when \np[=.true.]{ln_dynldf_hor}{ln\_dynldf\_hor}.
1128The diffusion operator is defined simply as the divergence of down gradient momentum fluxes on
1129each momentum component.
1130It must be emphasized that this formulation ignores constraints on the stress tensor such as symmetry.
1131The resulting discrete representation is:
1133  \label{eq:DYN_ldf_iso}
1134  \begin{split}
1135    D_u^{l\textbf{U}} &= \frac{1}{e_{1u} \, e_{2u} \, e_{3u} } \\
1136    &  \left\{\quad  {\delta_{i+1/2} \left[ {A_T^{lm}  \left(
1137              {\frac{e_{2t} \; e_{3t} }{e_{1t} } \,\delta_{i}[u]
1138                -e_{2t} \; r_{1t} \,\overline{\overline {\delta_{k+1/2}[u]}}^{\,i,\,k}}
1139            \right)} \right]}    \right. \\
1140    & \qquad +\ \delta_j \left[ {A_f^{lm} \left( {\frac{e_{1f}\,e_{3f} }{e_{2f}
1141            }\,\delta_{j+1/2} [u] - e_{1f}\, r_{2f}
1142            \,\overline{\overline {\delta_{k+1/2} [u]}} ^{\,j+1/2,\,k}}
1143        \right)} \right] \\
1144    &\qquad +\ \delta_k \left[ {A_{uw}^{lm} \left( {-e_{2u} \, r_{1uw} \,\overline{\overline
1145              {\delta_{i+1/2} [u]}}^{\,i+1/2,\,k+1/2} }
1146        \right.} \right. \\
1147    &  \ \qquad \qquad \qquad \quad\
1148    - e_{1u} \, r_{2uw} \,\overline{\overline {\delta_{j+1/2} [u]}} ^{\,j,\,k+1/2} \\
1149    & \left. {\left. { \ \qquad \qquad \qquad \ \ \ \left. {\
1150                +\frac{e_{1u}\, e_{2u} }{e_{3uw} }\,\left( {r_{1uw}^2+r_{2uw}^2}
1151                \right)\,\delta_{k+1/2} [u]} \right)} \right]\;\;\;} \right\} \\ \\
1152    D_v^{l\textbf{V}} &= \frac{1}{e_{1v} \, e_{2v} \, e_{3v} } \\
1153    &  \left\{\quad  {\delta_{i+1/2} \left[ {A_f^{lm}  \left(
1154              {\frac{e_{2f} \; e_{3f} }{e_{1f} } \,\delta_{i+1/2}[v]
1155                -e_{2f} \; r_{1f} \,\overline{\overline {\delta_{k+1/2}[v]}}^{\,i+1/2,\,k}}
1156            \right)} \right]}    \right. \\
1157    & \qquad +\ \delta_j \left[ {A_T^{lm} \left( {\frac{e_{1t}\,e_{3t} }{e_{2t}
1158            }\,\delta_{j} [v] - e_{1t}\, r_{2t}
1159            \,\overline{\overline {\delta_{k+1/2} [v]}} ^{\,j,\,k}}
1160        \right)} \right] \\
1161    & \qquad +\ \delta_k \left[ {A_{vw}^{lm} \left( {-e_{2v} \, r_{1vw} \,\overline{\overline
1162              {\delta_{i+1/2} [v]}}^{\,i+1/2,\,k+1/2} }\right.} \right. \\
1163    &  \ \qquad \qquad \qquad \quad\
1164    - e_{1v} \, r_{2vw} \,\overline{\overline {\delta_{j+1/2} [v]}} ^{\,j+1/2,\,k+1/2} \\
1165    & \left. {\left. { \ \qquad \qquad \qquad \ \ \ \left. {\
1166                +\frac{e_{1v}\, e_{2v} }{e_{3vw} }\,\left( {r_{1vw}^2+r_{2vw}^2}
1167                \right)\,\delta_{k+1/2} [v]} \right)} \right]\;\;\;} \right\}
1168  \end{split}
1170where $r_1$ and $r_2$ are the slopes between the surface along which the diffusion operator acts and
1171the surface of computation ($z$- or $s$-surfaces).
1172The way these slopes are evaluated is given in the lateral physics chapter (\autoref{chap:LDF}).
1174%% =================================================================================================
1175\subsection[Iso-level bilaplacian (\forcode{ln_dynldf_bilap})]{Iso-level bilaplacian operator (\protect\np{ln_dynldf_bilap}{ln\_dynldf\_bilap})}
1178The lateral fourth order operator formulation on momentum is obtained by applying \autoref{eq:DYN_ldf_lap} twice.
1179It requires an additional assumption on boundary conditions:
1180the first derivative term normal to the coast depends on the free or no-slip lateral boundary conditions chosen,
1181while the third derivative terms normal to the coast are set to zero (see \autoref{chap:LBC}).
1183\gmcomment{add a remark on the the change in the position of the coefficient}
1186%% =================================================================================================
1187\section[Vertical diffusion term (\textit{dynzdf.F90})]{Vertical diffusion term (\protect\mdl{dynzdf})}
1190Options are defined through the \nam{zdf}{zdf} namelist variables.
1191The large vertical diffusion coefficient found in the surface mixed layer together with high vertical resolution implies that in the case of explicit time stepping there would be too restrictive a constraint on the time step.
1192Two time stepping schemes can be used for the vertical diffusion term:
1193$(a)$ a forward time differencing scheme
1194(\np[=.true.]{ln_zdfexp}{ln\_zdfexp}) using a time splitting technique (\np{nn_zdfexp}{nn\_zdfexp} $>$ 1) or
1195$(b)$ a backward (or implicit) time differencing scheme (\np[=.false.]{ln_zdfexp}{ln\_zdfexp})
1196(see \autoref{chap:TD}).
1197Note that namelist variables \np{ln_zdfexp}{ln\_zdfexp} and \np{nn_zdfexp}{nn\_zdfexp} apply to both tracers and dynamics.
1199The formulation of the vertical subgrid scale physics is the same whatever the vertical coordinate is.
1200The vertical diffusion operators given by \autoref{eq:MB_zdf} take the following semi-discrete space form:
1202  % \label{eq:DYN_zdf}
1203  \left\{
1204    \begin{aligned}
1205      D_u^{vm} &\equiv \frac{1}{e_{3u}} \ \delta_k \left[ \frac{A_{uw}^{vm} }{e_{3uw} }
1206        \ \delta_{k+1/2} [\,u\,]         \right]     \\
1207      \\
1208      D_v^{vm} &\equiv \frac{1}{e_{3v}} \ \delta_k \left[ \frac{A_{vw}^{vm} }{e_{3vw} }
1209        \ \delta_{k+1/2} [\,v\,]         \right]
1210    \end{aligned}
1211  \right.
1213where $A_{uw}^{vm} $ and $A_{vw}^{vm} $ are the vertical eddy viscosity and diffusivity coefficients.
1214The way these coefficients are evaluated depends on the vertical physics used (see \autoref{chap:ZDF}).
1216The surface boundary condition on momentum is the stress exerted by the wind.
1217At the surface, the momentum fluxes are prescribed as the boundary condition on
1218the vertical turbulent momentum fluxes,
1220  \label{eq:DYN_zdf_sbc}
1221  \left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{z=1}
1222  = \frac{1}{\rho_o} \binom{\tau_u}{\tau_v }
1224where $\left( \tau_u ,\tau_v \right)$ are the two components of the wind stress vector in
1225the (\textbf{i},\textbf{j}) coordinate system.
1226The high mixing coefficients in the surface mixed layer ensure that the surface wind stress is distributed in
1227the vertical over the mixed layer depth.
1228If the vertical mixing coefficient is small (when no mixed layer scheme is used)
1229the surface stress enters only the top model level, as a body force.
1230The surface wind stress is calculated in the surface module routines (SBC, see \autoref{chap:SBC}).
1232The turbulent flux of momentum at the bottom of the ocean is specified through a bottom friction parameterisation
1233(see \autoref{sec:ZDF_drg})
1235%% =================================================================================================
1236\section{External forcings}
1239Besides the surface and bottom stresses (see the above section)
1240which are introduced as boundary conditions on the vertical mixing,
1241three other forcings may enter the dynamical equations by affecting the surface pressure gradient.
1243(1) When \np[=.true.]{ln_apr_dyn}{ln\_apr\_dyn} (see \autoref{sec:SBC_apr}),
1244the atmospheric pressure is taken into account when computing the surface pressure gradient.
1246(2) When \np[=.true.]{ln_tide_pot}{ln\_tide\_pot} and \np[=.true.]{ln_tide}{ln\_tide} (see \autoref{sec:SBC_tide}),
1247the tidal potential is taken into account when computing the surface pressure gradient.
1249(3) When \np[=2]{nn_ice_embd}{nn\_ice\_embd} and LIM or CICE is used
1250(\ie\ when the sea-ice is embedded in the ocean),
1251the snow-ice mass is taken into account when computing the surface pressure gradient.
1254\gmcomment{ missing : the lateral boundary condition !!!   another external forcing
1255 }
1257%% =================================================================================================
1258\section{Wetting and drying }
1261There are two main options for wetting and drying code (wd):
1262(a) an iterative limiter (il) and (b) a directional limiter (dl).
1263The directional limiter is based on the scheme developed by \cite{warner.defne.ea_CG13} for RO
1265which was in turn based on ideas developed for POM by \cite{oey_OM06}. The iterative
1266limiter is a new scheme.  The iterative limiter is activated by setting $\mathrm{ln\_wd\_il} = \mathrm{.true.}$
1267and $\mathrm{ln\_wd\_dl} = \mathrm{.false.}$. The directional limiter is activated
1268by setting $\mathrm{ln\_wd\_dl} = \mathrm{.true.}$ and $\mathrm{ln\_wd\_il} = \mathrm{.false.}$.
1271  \nlst{namwad}
1272  \caption{\forcode{&namwad}}
1273  \label{lst:namwad}
1276The following terminology is used. The depth of the topography (positive downwards)
1277at each $(i,j)$ point is the quantity stored in array $\mathrm{ht\_wd}$ in the \NEMO\ code.
1278The height of the free surface (positive upwards) is denoted by $ \mathrm{ssh}$. Given the sign
1279conventions used, the water depth, $h$, is the height of the free surface plus the depth of the
1280topography (i.e. $\mathrm{ssh} + \mathrm{ht\_wd}$).
1282Both wd schemes take all points in the domain below a land elevation of $\mathrm{rn\_wdld}$ to be
1283covered by water. They require the topography specified with a model
1284configuration to have negative depths at points where the land is higher than the
1285topography's reference sea-level. The vertical grid in \NEMO\ is normally computed relative to an
1286initial state with zero sea surface height elevation.
1287The user can choose to compute the vertical grid and heights in the model relative to
1288a non-zero reference height for the free surface. This choice affects the calculation of the metrics and depths
1289(i.e. the $\mathrm{e3t\_0, ht\_0}$ etc. arrays).
1291Points where the water depth is less than $\mathrm{rn\_wdmin1}$ are interpreted as ``dry''.
1292$\mathrm{rn\_wdmin1}$ is usually chosen to be of order $0.05$m but extreme topographies
1293with very steep slopes require larger values for normal choices of time-step. Surface fluxes
1294are also switched off for dry cells to prevent freezing, boiling etc. of very thin water layers.
1295The fluxes are tappered down using a $\mathrm{tanh}$ weighting function
1296to no flux as the dry limit $\mathrm{rn\_wdmin1}$ is approached. Even wet cells can be very shallow.
1297The depth at which to start tapering is controlled by the user by setting $\mathrm{rn\_wd\_sbcdep}$.
1298The fraction $(<1)$ of sufrace fluxes to use at this depth is set by $\mathrm{rn\_wd\_sbcfra}$.
1300Both versions of the code have been tested in six test cases provided in the WAD\_TEST\_CASES configuration
1301and in ``realistic'' configurations covering parts of the north-west European shelf.
1302All these configurations have used pure sigma coordinates. It is expected that
1303the wetting and drying code will work in domains with more general s-coordinates provided
1304the coordinates are pure sigma in the region where wetting and drying actually occurs.
1306The next sub-section descrbies the directional limiter and the following sub-section the iterative limiter.
1307The final sub-section covers some additional considerations that are relevant to both schemes.
1310%   Iterative limiters
1311%% =================================================================================================
1312\subsection[Directional limiter (\textit{wet\_dry.F90})]{Directional limiter (\mdl{wet\_dry})}
1315The principal idea of the directional limiter is that
1316water should not be allowed to flow out of a dry tracer cell (i.e. one whose water depth is less than \np{rn_wdmin1}{rn\_wdmin1}).
1318All the changes associated with this option are made to the barotropic solver for the non-linear
1319free surface code within dynspg\_ts.
1320On each barotropic sub-step the scheme determines the direction of the flow across each face of all the tracer cells
1321and sets the flux across the face to zero when the flux is from a dry tracer cell. This prevents cells
1322whose depth is rn\_wdmin1 or less from drying out further. The scheme does not force $h$ (the water depth) at tracer cells
1323to be at least the minimum depth and hence is able to conserve mass / volume.
1325The flux across each $u$-face of a tracer cell is multiplied by a factor zuwdmask (an array which depends on ji and jj).
1326If the user sets \np[=.false.]{ln_wd_dl_ramp}{ln\_wd\_dl\_ramp} then zuwdmask is 1 when the
1327flux is from a cell with water depth greater than \np{rn_wdmin1}{rn\_wdmin1} and 0 otherwise. If the user sets
1328\np[=.true.]{ln_wd_dl_ramp}{ln\_wd\_dl\_ramp} the flux across the face is ramped down as the water depth decreases
1329from 2 * \np{rn_wdmin1}{rn\_wdmin1} to \np{rn_wdmin1}{rn\_wdmin1}. The use of this ramp reduced grid-scale noise in idealised test cases.
1331At the point where the flux across a $u$-face is multiplied by zuwdmask , we have chosen
1332also to multiply the corresponding velocity on the ``now'' step at that face by zuwdmask. We could have
1333chosen not to do that and to allow fairly large velocities to occur in these ``dry'' cells.
1334The rationale for setting the velocity to zero is that it is the momentum equations that are being solved
1335and the total momentum of the upstream cell (treating it as a finite volume) should be considered
1336to be its depth times its velocity. This depth is considered to be zero at ``dry'' $u$-points consistent with its
1337treatment in the calculation of the flux of mass across the cell face.
1340\cite{warner.defne.ea_CG13} state that in their scheme the velocity masks at the cell faces for the baroclinic
1341timesteps are set to 0 or 1 depending on whether the average of the masks over the barotropic sub-steps is respectively less than
1342or greater than 0.5. That scheme does not conserve tracers in integrations started from constant tracer
1343fields (tracers independent of $x$, $y$ and $z$). Our scheme conserves constant tracers because
1344the velocities used at the tracer cell faces on the baroclinic timesteps are carefully calculated by dynspg\_ts
1345to equal their mean value during the barotropic steps. If the user sets \np[=.true.]{ln_wd_dl_bc}{ln\_wd\_dl\_bc}, the
1346baroclinic velocities are also multiplied by a suitably weighted average of zuwdmask.
1348%   Iterative limiters
1350%% =================================================================================================
1351\subsection[Iterative limiter (\textit{wet\_dry.F90})]{Iterative limiter (\mdl{wet\_dry})}
1354%% =================================================================================================
1355\subsubsection[Iterative flux limiter (\textit{wet\_dry.F90})]{Iterative flux limiter (\mdl{wet\_dry})}
1358The iterative limiter modifies the fluxes across the faces of cells that are either already ``dry''
1359or may become dry within the next time-step using an iterative method.
1361The flux limiter for the barotropic flow (devised by Hedong Liu) can be understood as follows:
1363The continuity equation for the total water depth in a column
1365  \label{eq:DYN_wd_continuity}
1366  \frac{\partial h}{\partial t} + \mathbf{\nabla.}(h\mathbf{u}) = 0 .
1368can be written in discrete form  as
1371  \label{eq:DYN_wd_continuity_2}
1372  \frac{e_1 e_2}{\Delta t} ( h_{i,j}(t_{n+1}) - h_{i,j}(t_e) )
1373  &= - ( \mathrm{flxu}_{i+1,j} - \mathrm{flxu}_{i,j}  + \mathrm{flxv}_{i,j+1} - \mathrm{flxv}_{i,j} ) \\
1374  &= \mathrm{zzflx}_{i,j} .
1377In the above $h$ is the depth of the water in the column at point $(i,j)$,
1378$\mathrm{flxu}_{i+1,j}$ is the flux out of the ``eastern'' face of the cell and
1379$\mathrm{flxv}_{i,j+1}$ the flux out of the ``northern'' face of the cell; $t_{n+1}$ is
1380the new timestep, $t_e$ is the old timestep (either $t_b$ or $t_n$) and $ \Delta t =
1381t_{n+1} - t_e$; $e_1 e_2$ is the area of the tracer cells centred at $(i,j)$ and
1382$\mathrm{zzflx}$ is the sum of the fluxes through all the faces.
1384The flux limiter splits the flux $\mathrm{zzflx}$ into fluxes that are out of the cell
1385(zzflxp) and fluxes that are into the cell (zzflxn).  Clearly
1388  \label{eq:DYN_wd_zzflx_p_n_1}
1389  \mathrm{zzflx}_{i,j} = \mathrm{zzflxp}_{i,j} + \mathrm{zzflxn}_{i,j} .
1392The flux limiter iteratively adjusts the fluxes $\mathrm{flxu}$ and $\mathrm{flxv}$ until
1393none of the cells will ``dry out''. To be precise the fluxes are limited until none of the
1394cells has water depth less than $\mathrm{rn\_wdmin1}$ on step $n+1$.
1396Let the fluxes on the $m$th iteration step be denoted by $\mathrm{flxu}^{(m)}$ and
1397$\mathrm{flxv}^{(m)}$.  Then the adjustment is achieved by seeking a set of coefficients,
1398$\mathrm{zcoef}_{i,j}^{(m)}$ such that:
1401  \label{eq:DYN_wd_continuity_coef}
1402  \begin{split}
1403    \mathrm{zzflxp}^{(m)}_{i,j} =& \mathrm{zcoef}_{i,j}^{(m)} \mathrm{zzflxp}^{(0)}_{i,j} \\
1404    \mathrm{zzflxn}^{(m)}_{i,j} =& \mathrm{zcoef}_{i,j}^{(m)} \mathrm{zzflxn}^{(0)}_{i,j}
1405  \end{split}
1408where the coefficients are $1.0$ generally but can vary between $0.0$ and $1.0$ around
1409cells that would otherwise dry.
1411The iteration is initialised by setting
1414  \label{eq:DYN_wd_zzflx_initial}
1415  \mathrm{zzflxp^{(0)}}_{i,j} = \mathrm{zzflxp}_{i,j} , \quad  \mathrm{zzflxn^{(0)}}_{i,j} = \mathrm{zzflxn}_{i,j} .
1418The fluxes out of cell $(i,j)$ are updated at the $m+1$th iteration if the depth of the
1419cell on timestep $t_e$, namely $h_{i,j}(t_e)$, is less than the total flux out of the cell
1420times the timestep divided by the cell area. Using (\autoref{eq:DYN_wd_continuity_2}) this
1421condition is
1424  \label{eq:DYN_wd_continuity_if}
1425  h_{i,j}(t_e)  - \mathrm{rn\_wdmin1} <  \frac{\Delta t}{e_1 e_2} ( \mathrm{zzflxp}^{(m)}_{i,j} + \mathrm{zzflxn}^{(m)}_{i,j} ) .
1428Rearranging (\autoref{eq:DYN_wd_continuity_if}) we can obtain an expression for the maximum
1429outward flux that can be allowed and still maintain the minimum wet depth:
1432  \label{eq:DYN_wd_max_flux}
1433  \begin{split}
1434    \mathrm{zzflxp}^{(m+1)}_{i,j} = \Big[ (h_{i,j}(t_e) & - \mathrm{rn\_wdmin1} - \mathrm{rn\_wdmin2})  \frac{e_1 e_2}{\Delta t} \phantom{]} \\
1435    \phantom{[} & -  \mathrm{zzflxn}^{(m)}_{i,j} \Big]
1436  \end{split}
1439Note a small tolerance ($\mathrm{rn\_wdmin2}$) has been introduced here {\itshape [Q: Why is
1440this necessary/desirable?]}. Substituting from (\autoref{eq:DYN_wd_continuity_coef}) gives an
1441expression for the coefficient needed to multiply the outward flux at this cell in order
1442to avoid drying.
1445  \label{eq:DYN_wd_continuity_nxtcoef}
1446  \begin{split}
1447    \mathrm{zcoef}^{(m+1)}_{i,j} = \Big[ (h_{i,j}(t_e) & - \mathrm{rn\_wdmin1} - \mathrm{rn\_wdmin2})  \frac{e_1 e_2}{\Delta t} \phantom{]} \\
1448    \phantom{[} & -  \mathrm{zzflxn}^{(m)}_{i,j} \Big] \frac{1}{ \mathrm{zzflxp}^{(0)}_{i,j} }
1449  \end{split}
1452Only the outward flux components are altered but, of course, outward fluxes from one cell
1453are inward fluxes to adjacent cells and the balance in these cells may need subsequent
1454adjustment; hence the iterative nature of this scheme.  Note, for example, that the flux
1455across the ``eastern'' face of the $(i,j)$th cell is only updated at the $m+1$th iteration
1456if that flux at the $m$th iteration is out of the $(i,j)$th cell. If that is the case then
1457the flux across that face is into the $(i+1,j)$ cell and that flux will not be updated by
1458the calculation for the $(i+1,j)$th cell. In this sense the updates to the fluxes across
1459the faces of the cells do not ``compete'' (they do not over-write each other) and one
1460would expect the scheme to converge relatively quickly. The scheme is flux based so
1461conserves mass. It also conserves constant tracers for the same reason that the
1462directional limiter does.
1465%      Surface pressure gradients
1466%% =================================================================================================
1467\subsubsection[Modification of surface pressure gradients (\textit{dynhpg.F90})]{Modification of surface pressure gradients (\mdl{dynhpg})}
1470At ``dry'' points the water depth is usually close to $\mathrm{rn\_wdmin1}$. If the
1471topography is sloping at these points the sea-surface will have a similar slope and there
1472will hence be very large horizontal pressure gradients at these points. The WAD modifies
1473the magnitude but not the sign of the surface pressure gradients (zhpi and zhpj) at such
1474points by mulitplying them by positive factors (zcpx and zcpy respectively) that lie
1475between $0$ and $1$.
1477We describe how the scheme works for the ``eastward'' pressure gradient, zhpi, calculated
1478at the $(i,j)$th $u$-point. The scheme uses the ht\_wd depths and surface heights at the
1479neighbouring $(i+1,j)$ and $(i,j)$ tracer points.  zcpx is calculated using two logicals
1480variables, $\mathrm{ll\_tmp1}$ and $\mathrm{ll\_tmp2}$ which are evaluated for each grid
1481column.  The three possible combinations are illustrated in \autoref{fig:DYN_WAD_dynhpg}.
1485  \centering
1486  \includegraphics[width=0.66\textwidth]{Fig_WAD_dynhpg}
1487  \caption[Combinations controlling the limiting of the horizontal pressure gradient in
1488  wetting and drying regimes]{
1489    Three possible combinations of the logical variables controlling the
1490    limiting of the horizontal pressure gradient in wetting and drying regimes}
1491  \label{fig:DYN_WAD_dynhpg}
1495The first logical, $\mathrm{ll\_tmp1}$, is set to true if and only if the water depth at
1496both neighbouring points is greater than $\mathrm{rn\_wdmin1} + \mathrm{rn\_wdmin2}$ and
1497the minimum height of the sea surface at the two points is greater than the maximum height
1498of the topography at the two points:
1501  \label{eq:DYN_ll_tmp1}
1502  \begin{split}
1503    \mathrm{ll\_tmp1}  = & \mathrm{MIN(sshn(ji,jj), sshn(ji+1,jj))} > \\
1504                     & \quad \mathrm{MAX(-ht\_wd(ji,jj), -ht\_wd(ji+1,jj))\  .and.} \\
1505                     & \mathrm{MAX(sshn(ji,jj) + ht\_wd(ji,jj),} \\
1506                     & \mathrm{\phantom{MAX(}sshn(ji+1,jj) + ht\_wd(ji+1,jj))} >\\
1507                     & \quad\quad\mathrm{rn\_wdmin1 + rn\_wdmin2 }
1508  \end{split}
1511The second logical, $\mathrm{ll\_tmp2}$, is set to true if and only if the maximum height
1512of the sea surface at the two points is greater than the maximum height of the topography
1513at the two points plus $\mathrm{rn\_wdmin1} + \mathrm{rn\_wdmin2}$
1516  \label{eq:DYN_ll_tmp2}
1517  \begin{split}
1518    \mathrm{ ll\_tmp2 } = & \mathrm{( ABS( sshn(ji,jj) - sshn(ji+1,jj) ) > 1.E-12 )\ .AND.}\\
1519    & \mathrm{( MAX(sshn(ji,jj), sshn(ji+1,jj)) > } \\
1520    & \mathrm{\phantom{(} MAX(-ht\_wd(ji,jj), -ht\_wd(ji+1,jj)) + rn\_wdmin1 + rn\_wdmin2}) .
1521  \end{split}
1524If $\mathrm{ll\_tmp1}$ is true then the surface pressure gradient, zhpi at the $(i,j)$
1525point is unmodified. If both logicals are false zhpi is set to zero.
1527If $\mathrm{ll\_tmp1}$ is true and $\mathrm{ll\_tmp2}$ is false then the surface pressure
1528gradient is multiplied through by zcpx which is the absolute value of the difference in
1529the water depths at the two points divided by the difference in the surface heights at the
1530two points. Thus the sign of the sea surface height gradient is retained but the magnitude
1531of the pressure force is determined by the difference in water depths rather than the
1532difference in surface height between the two points. Note that dividing by the difference
1533between the sea surface heights can be problematic if the heights approach parity. An
1534additional condition is applied to $\mathrm{ ll\_tmp2 }$ to ensure it is .false. in such
1537%% =================================================================================================
1538\subsubsection[Additional considerations (\textit{usrdef\_zgr.F90})]{Additional considerations (\mdl{usrdef\_zgr})}
1541In the very shallow water where wetting and drying occurs the parametrisation of
1542bottom drag is clearly very important. In order to promote stability
1543it is sometimes useful to calculate the bottom drag using an implicit time-stepping approach.
1545Suitable specifcation of the surface heat flux in wetting and drying domains in forced and
1546coupled simulations needs further consideration. In order to prevent freezing or boiling
1547in uncoupled integrations the net surface heat fluxes need to be appropriately limited.
1549%      The WAD test cases
1550%% =================================================================================================
1551\subsection[The WAD test cases (\textit{usrdef\_zgr.F90})]{The WAD test cases (\mdl{usrdef\_zgr})}
1554See the WAD tests MY\_DOC documention for details of the WAD test cases.
1558%% =================================================================================================
1559\section[Time evolution term (\textit{dynnxt.F90})]{Time evolution term (\protect\mdl{dynnxt})}
1563Options are defined through the \nam{dom}{dom} namelist variables.
1564The general framework for dynamics time stepping is a leap-frog scheme,
1565\ie\ a three level centred time scheme associated with an Asselin time filter (cf. \autoref{chap:TD}).
1566The scheme is applied to the velocity, except when
1567using the flux form of momentum advection (cf. \autoref{sec:DYN_adv_cor_flux})
1568in the variable volume case (\texttt{vvl?} defined),
1569where it has to be applied to the thickness weighted velocity (see \autoref{sec:SCOORD_momentum})
1571$\bullet$ vector invariant form or linear free surface
1572(\np[=.true.]{ln_dynhpg_vec}{ln\_dynhpg\_vec} ; \texttt{vvl?} not defined):
1574  % \label{eq:DYN_nxt_vec}
1575  \left\{
1576    \begin{aligned}
1577      &u^{t+\rdt} = u_f^{t-\rdt} + 2\rdt  \ \text{RHS}_u^t     \\
1578      &u_f^t \;\quad = u^t+\gamma \,\left[ {u_f^{t-\rdt} -2u^t+u^{t+\rdt}} \right]
1579    \end{aligned}
1580  \right.
1583$\bullet$ flux form and nonlinear free surface
1584(\np[=.false.]{ln_dynhpg_vec}{ln\_dynhpg\_vec} ; \texttt{vvl?} defined):
1586  % \label{eq:DYN_nxt_flux}
1587  \left\{
1588    \begin{aligned}
1589      &\left(e_{3u}\,u\right)^{t+\rdt} = \left(e_{3u}\,u\right)_f^{t-\rdt} + 2\rdt \; e_{3u} \;\text{RHS}_u^t     \\
1590      &\left(e_{3u}\,u\right)_f^t \;\quad = \left(e_{3u}\,u\right)^t
1591      +\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]
1592    \end{aligned}
1593  \right.
1595where RHS is the right hand side of the momentum equation,
1596the subscript $f$ denotes filtered values and $\gamma$ is the Asselin coefficient.
1597$\gamma$ is initialized as \np{nn_atfp}{nn\_atfp} (namelist parameter).
1598Its default value is \np[=10.e-3]{nn_atfp}{nn\_atfp}.
1599In both cases, the modified Asselin filter is not applied since perfect conservation is not an issue for
1600the momentum equations.
1602Note that with the filtered free surface,
1603the update of the \textit{after} velocities is done in the \mdl{dynsp\_flt} module,
1604and only array swapping and Asselin filtering is done in \mdl{dynnxt}.
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