chap_DYN.tex

\documentclass[../main/NEMO_manual]{subfiles}

\begin{document}

\chapter{Ocean Dynamics (DYN)}
\label{chap:DYN}

\chaptertoc

\paragraph{Changes record} ~\\

{\footnotesize
  \begin{tabularx}{\textwidth}{l||X|X}
    Release & Author(s) & Modifications \\
    \hline
    {\em   4.0} & {\em ...} & {\em ...} \\
    {\em   3.6} & {\em ...} & {\em ...} \\
    {\em   3.4} & {\em ...} & {\em ...} \\
    {\em <=3.4} & {\em ...} & {\em ...}
  \end{tabularx}
}

\clearpage

Using the representation described in \autoref{chap:DOM},
several semi-discrete space forms of the dynamical equations are available depending on
the vertical coordinate used and on the conservation properties of the vorticity term.
In all the equations presented here, the masking has been omitted for simplicity.
One must be aware that all the quantities are masked fields and
that each time an average or difference operator is used, the resulting field is multiplied by a mask.

The prognostic ocean dynamics equation can be summarized as follows:
\[
  \text{NXT} = \dbinom	{\text{VOR} + \text{KEG} + \text {ZAD} }
  {\text{COR} + \text{ADV}                       }
  + \text{HPG} + \text{SPG} + \text{LDF} + \text{ZDF}
\]
NXT stands for next, referring to the time-stepping.
The first group of terms on the rhs of this equation corresponds to the Coriolis and advection terms that
are decomposed into either a vorticity part (VOR), a kinetic energy part (KEG) and
a vertical advection part (ZAD) in the vector invariant formulation,
or a Coriolis and advection part (COR+ADV) in the flux formulation.
The terms following these are the pressure gradient contributions
(HPG, Hydrostatic Pressure Gradient, and SPG, Surface Pressure Gradient);
and contributions from lateral diffusion (LDF) and vertical diffusion (ZDF),
which are added to the rhs in the \mdl{dynldf} and \mdl{dynzdf} modules.
The vertical diffusion term includes the surface and bottom stresses.
The external forcings and parameterisations require complex inputs
(surface wind stress calculation using bulk formulae, estimation of mixing coefficients)
that are carried out in modules SBC, LDF and ZDF and are described in
\autoref{chap:SBC}, \autoref{chap:LDF} and \autoref{chap:ZDF}, respectively.

In the present chapter we also describe the diagnostic equations used to compute the horizontal divergence,
curl of the velocities (\emph{divcur} module) and the vertical velocity (\emph{wzvmod} module).

The different options available to the user are managed by namelist variables.
For term \textit{ttt} in the momentum equations, the logical namelist variables are \textit{ln\_dynttt\_xxx},
where \textit{xxx} is a 3 or 4 letter acronym corresponding to each optional scheme.
%If a CPP key is used for this term its name is \key{ttt}.
The corresponding code can be found in the \textit{dynttt\_xxx} module in the DYN directory,
and it is usually computed in the \textit{dyn\_ttt\_xxx} subroutine.

The user has the option of extracting and outputting each tendency term from the 3D momentum equations
(\texttt{trddyn?} defined), as described in \autoref{chap:MISC}.
Furthermore, the tendency terms associated with the 2D barotropic vorticity balance (when \texttt{trdvor?} is defined)
can be derived from the 3D terms.
\cmtgm{STEVEN: not quite sure I've got the sense of the last sentence.
  Does MISC correspond to "extracting tendency terms" or "vorticity balance"?}

%% =================================================================================================
\section{Sea surface height and diagnostic variables ($\eta$, $\zeta$, $\chi$, $w$)}
\label{sec:DYN_divcur_wzv}

%% =================================================================================================
\subsection[Horizontal divergence and relative vorticity (\textit{divhor.F90})]{Horizontal divergence and relative vorticity (\protect\mdl{divcur})}
\label{subsec:DYN_divcur}

The vorticity is defined at an $f$-point (\ie\ corner point) as follows (used in ):
\begin{equation}
  \label{eq:DYN_divcur_cur}
  \zeta =\frac{1}{e_{1f}\,e_{2f} }\left( {\;\delta_{i+1/2} \left[ {e_{2v}\;v} \right]
      -\delta_{j+1/2} \left[ {e_{1u}\;u} \right]\;} \right)
\end{equation}

The horizontal divergence is defined at a $T$-point.
It is given by:
\[
  % \label{eq:DYN_divcur_div}
  \chi =\frac{1}{e_{1t}\,e_{2t}\,e_{3t} }
  \left( {\delta_i \left[ {e_{2u}\,e_{3u}\,u} \right]
      +\delta_j \left[ {e_{1v}\,e_{3v}\,v} \right]} \right)
\]

Note that although the vorticity has the same discrete expression in $z$- and $s$-coordinates,
its physical meaning is not identical.
$\zeta$ is a pseudo vorticity along $s$-surfaces
(only pseudo because $(u,v)$ are still defined along geopotential surfaces,
but are not necessarily defined at the same depth).

The vorticity and divergence at the \textit{before} step are used in the computation of
the horizontal diffusion of momentum.
Note that because they have been calculated prior to the Asselin filtering of the \textit{before} velocities,
the \textit{before} vorticity and divergence arrays must be included in the restart file to
ensure perfect restartability.
The vorticity and divergence at the \textit{now} time step are used for the computation of
the nonlinear advection and of the vertical velocity respectively.

%% =================================================================================================
\subsection[Horizontal divergence and relative vorticity (\textit{sshwzv.F90})]{Horizontal divergence and relative vorticity (\protect\mdl{sshwzv})}
\label{subsec:DYN_sshwzv}

The sea surface height is given by:
\begin{equation}
  \label{eq:DYN_spg_ssh}
  \begin{aligned}
    \frac{\partial \eta }{\partial t}
    &\equiv    \frac{1}{e_{1t} e_{2t} }\sum\limits_k { \left\{  \delta_i \left[ {e_{2u}\,e_{3u}\;u} \right]
        +\delta_j \left[ {e_{1v}\,e_{3v}\;v} \right]  \right\} }
    -    \frac{\textit{emp}}{\rho_w }   \\
    &\equiv    \sum\limits_k {\chi \ e_{3t}}  -  \frac{\textit{emp}}{\rho_w }
  \end{aligned}
\end{equation}
where \textit{emp} is the surface freshwater budget (evaporation minus precipitation),
expressed in Kg/m$^2$/s (which is equal to mm/s),
and $\rho_w$=1,035~Kg/m$^3$ is the reference density of sea water (Boussinesq approximation).
If river runoff is expressed as a surface freshwater flux (see \autoref{chap:SBC}) then
\textit{emp} can be written as the evaporation minus precipitation, minus the river runoff.
In the presence of ice-shelf (\np[=.true.]{ln_isf}{ln\_isf}) 
\textit{emp} can be written as the evaporation minus precipitation, minus the ice-shelf water flux.
In the leapfrog case, the sea-surface height is evaluated using exactly the same time stepping scheme as
the tracer equation \autoref{eq:TRA_nxt}:
a leapfrog scheme in combination with an Asselin time filter,
\ie\ the velocity appearing in \autoref{eq:DYN_spg_ssh} is centred in time (\textit{now} velocity).
This is of paramount importance.
Replacing $T$ by the number $1$ in the tracer equation and summing over the water column must lead to
the sea surface height equation otherwise tracer content will not be conserved
\citep{griffies.pacanowski.ea_MWR01, leclair.madec_OM09}.

The vertical velocity is computed by an upward integration of the horizontal divergence starting at the bottom,
taking into account the change of the thickness of the levels:
\begin{equation}
  \label{eq:DYN_wzv}
  \left\{
    \begin{aligned}
      &\left. w \right|_{k_b-1/2} \quad= 0    \qquad \text{where } k_b \text{ is the level just above the sea floor }  	\\
      &\left. w \right|_{k+1/2}     = \left. w \right|_{k-1/2}  +  \left. e_{3t} \right|_{k}\;  \left. \chi \right|_k
      - \frac{1} {2 \rdt} \left(  \left. e_{3t}^{t+1}\right|_{k} - \left. e_{3t}^{t-1}\right|_{k}\right)
    \end{aligned}
  \right.
\end{equation}

In the case of a non-linear free surface (\key{qco}), the top vertical velocity is $-\textit{emp}/\rho_w$,
as changes in the divergence of the barotropic transport are absorbed into the change of the level thicknesses,
re-orientated downward.
\cmtgm{not sure of this...  to be modified with the change in emp setting}
In the case of a linear free surface(\key{linssh}), the time derivative in \autoref{eq:DYN_wzv} disappears.
The upper boundary condition applies at a fixed level $z=0$.
The top vertical velocity is thus equal to the divergence of the barotropic transport
(\ie\ the first term in the right-hand-side of \autoref{eq:DYN_spg_ssh}).

Note also that whereas the vertical velocity has the same discrete expression in $z$- and $s$-coordinates,
its physical meaning is not the same:
in the second case, $w$ is the velocity normal to the $s$-surfaces.
Note also that the $k$-axis is re-orientated downwards in the \fortran\ code compared to
the indexing used in the semi-discrete equations such as \autoref{eq:DYN_wzv}
(see \autoref{subsec:DOM_Num_Index_vertical}).

When \np[=.true.]{ln_zad_Aimp}{ln\_zad\_Aimp},
a proportion of the vertical advection can be treated implicitly (see \autoref{sec:DYN_zdf}) depending on the Courant number. 
This option can be useful when the value of the timestep is limited by vertical advection \citep{lemarie.debreu.ea_OM15}.

%% =================================================================================================
\section{Coriolis and advection: vector invariant form}
\label{sec:DYN_adv_cor_vect}

\begin{listing}
  \nlst{namdyn_adv}
  \caption{\forcode{&namdyn_adv}}
  \label{lst:namdyn_adv}
\end{listing}

The vector invariant form of the momentum equations is the one most often used in
applications of the \NEMO\ ocean model.
The flux form option (see next section) has been present since version $2$.
Options are defined through the \nam{dyn_adv}{dyn\_adv} namelist variables Coriolis and
momentum advection terms are evaluated either using a leapfrog scheme or a RK3 scheme.
In the leapfrog case the velocity appearing in these expressions is centred in time (\textit{now} velocity).
In the RK3 case the velocity appearing in these expressions is forward in time (\textit{before} velocity) at stage 1, 
it is is centred in time (\textit{now} velocity) at stage 2 and 3. 
At the lateral boundaries either free slip, no slip or partial slip boundary conditions are applied following
\autoref{chap:LBC}.

%% =================================================================================================
\subsection[Vorticity term (\textit{dynvor.F90})]{Vorticity term (\protect\mdl{dynvor})}
\label{subsec:DYN_vor}

\begin{listing}
  \nlst{namdyn_vor}
  \caption{\forcode{&namdyn_vor}}
  \label{lst:namdyn_vor}
\end{listing}

Options are defined through the \nam{dyn_vor}{dyn\_vor} namelist variables.
Four discretisations of the vorticity term (\texttt{ln\_dynvor\_xxx}\forcode{=.true.}) are available:
conserving potential enstrophy of horizontally non-divergent flow (ENS scheme);
conserving horizontal kinetic energy (ENE scheme);
conserving potential enstrophy for the relative vorticity term and
horizontal kinetic energy for the planetary vorticity term (MIX scheme);
or conserving both the potential enstrophy of horizontally non-divergent flow and horizontal kinetic energy
(EEN scheme) (see \autoref{subsec:INVARIANTS_vorEEN}).
In the case of ENS, ENE or MIX schemes the land sea mask may be slightly modified to ensure the consistency of
vorticity term with analytical equations (\np[=.true.]{ln_dynvor_con}{ln\_dynvor\_con}).
The vorticity terms are all computed in dedicated routines that can be found in the \mdl{dynvor} module.

%                 enstrophy conserving scheme
%% =================================================================================================
\subsubsection[Enstrophy conserving scheme (\forcode{ln_dynvor_ens})]{Enstrophy conserving scheme (\protect\np{ln_dynvor_ens}{ln\_dynvor\_ens})}
\label{subsec:DYN_vor_ens}

In the enstrophy conserving case (ENS scheme),
the discrete formulation of the vorticity term provides a global conservation of the enstrophy
($ [ (\zeta +f ) / e_{3f} ]^2 $ in $s$-coordinates) for a horizontally non-divergent flow (\ie\ $\chi$=$0$),
but does not conserve the total kinetic energy.
It is given by:
\begin{equation}
  \label{eq:DYN_vor_ens}
  \left\{
    \begin{aligned}
      {+\frac{1}{e_{1u} } } & {\overline {\left( { \frac{\zeta +f}{e_{3f} }} \right)} }^{\,i}
      & {\overline{\overline {\left( {e_{1v}\,e_{3v}\;v} \right)}} }^{\,i, j+1/2}    \\
      {- \frac{1}{e_{2v} } } & {\overline {\left( {\frac{\zeta +f}{e_{3f} }} \right)} }^{\,j}
      & {\overline{\overline {\left( {e_{2u}\,e_{3u}\;u} \right)}} }^{\,i+1/2, j}
    \end{aligned}
  \right.
\end{equation}

%                 energy conserving scheme
%% =================================================================================================
\subsubsection[Energy conserving scheme (\forcode{ln_dynvor_ene})]{Energy conserving scheme (\protect\np{ln_dynvor_ene}{ln\_dynvor\_ene})}
\label{subsec:DYN_vor_ene}

The kinetic energy conserving scheme (ENE scheme) conserves the global kinetic energy but not the global enstrophy.
It is given by:
\begin{equation}
  \label{eq:DYN_vor_ene}
  \left\{
    \begin{aligned}
      {+\frac{1}{e_{1u}}\; {\overline {\left( {\frac{\zeta +f}{e_{3f} }} \right)
            \;  \overline {\left( {e_{1v}\,e_{3v}\;v} \right)} ^{\,i+1/2}} }^{\,j} }    \\
      {- \frac{1}{e_{2v}}\; {\overline {\left( {\frac{\zeta +f}{e_{3f} }} \right)
            \;  \overline {\left( {e_{2u}\,e_{3u}\;u} \right)} ^{\,j+1/2}} }^{\,i} }
    \end{aligned}
  \right.
\end{equation}

%                 mix energy/enstrophy conserving scheme
%% =================================================================================================
\subsubsection[Mixed energy/enstrophy conserving scheme (\forcode{ln_dynvor_mix})]{Mixed energy/enstrophy conserving scheme (\protect\np{ln_dynvor_mix}{ln\_dynvor\_mix})}
\label{subsec:DYN_vor_mix}

For the mixed energy/enstrophy conserving scheme (MIX scheme), a mixture of the two previous schemes is used.
It consists of the ENS scheme (\autoref{eq:DYN_vor_ens}) for the relative vorticity term,
and of the ENE scheme (\autoref{eq:DYN_vor_ene}) applied to the planetary vorticity term.
\[
  % \label{eq:DYN_vor_mix}
  \left\{ {
      \begin{aligned}
        {+\frac{1}{e_{1u} }\; {\overline {\left( {\frac{\zeta }{e_{3f} }} \right)} }^{\,i}
          \; {\overline{\overline {\left( {e_{1v}\,e_{3v}\;v} \right)}} }^{\,i,j+1/2} -\frac{1}{e_{1u} }
          \; {\overline {\left( {\frac{f}{e_{3f} }} \right)
              \;\overline {\left( {e_{1v}\,e_{3v}\;v} \right)} ^{\,i+1/2}} }^{\,j} } \\
        {-\frac{1}{e_{2v} }\; {\overline {\left( {\frac{\zeta }{e_{3f} }} \right)} }^j
          \; {\overline{\overline {\left( {e_{2u}\,e_{3u}\;u} \right)}} }^{\,i+1/2,j} +\frac{1}{e_{2v} }
          \; {\overline {\left( {\frac{f}{e_{3f} }} \right)
              \;\overline {\left( {e_{2u}\,e_{3u}\;u} \right)} ^{\,j+1/2}} }^{\,i} } \hfill
      \end{aligned}
    } \right.
\]

%                 energy and enstrophy conserving scheme
%% =================================================================================================
\subsubsection[Energy and enstrophy conserving scheme (\forcode{ln_dynvor_een})]{Energy and enstrophy conserving scheme (\protect\np{ln_dynvor_een}{ln\_dynvor\_een})}
\label{subsec:DYN_vor_een}

In both the ENS and ENE schemes,
it is apparent that the combination of $i$ and $j$ averages of the velocity allows for
the presence of grid point oscillation structures that will be invisible to the operator.
These structures are \textit{computational modes} that will be at least partly damped by
the momentum diffusion operator (\ie\ the subgrid-scale advection), but not by the resolved advection term.
The ENS and ENE schemes therefore do not contribute to dump any grid point noise in the horizontal velocity field.
Such noise would result in more noise in the vertical velocity field, an undesirable feature.
This is a well-known characteristic of $C$-grid discretization where
$u$ and $v$ are located at different grid points,
a price worth paying to avoid a double averaging in the pressure gradient term as in the $B$-grid.
\cmtgm{ To circumvent this, Adcroft (ADD REF HERE)
Nevertheless, this technique strongly distort the phase and group velocity of Rossby waves....}

A very nice solution to the problem of double averaging was proposed by \citet{arakawa.hsu_MWR90}.
The idea is to get rid of the double averaging by considering triad combinations of vorticity.
It is noteworthy that this solution is conceptually quite similar to the one proposed by
\citep{griffies.gnanadesikan.ea_JPO98} for the discretization of the iso-neutral diffusion operator (see \autoref{apdx:INVARIANTS}).

The \citet{arakawa.hsu_MWR90} vorticity advection scheme for a single layer is modified
for spherical coordinates as described by \citet{arakawa.lamb_MWR81} to obtain the EEN scheme.
First consider the discrete expression of the potential vorticity, $q$, defined at an $f$-point:
\[
  % \label{eq:DYN_pot_vor}
  q  = \frac{\zeta +f} {e_{3f} }
\]
where the relative vorticity is defined by (\autoref{eq:DYN_divcur_cur}),
the Coriolis parameter is given by $f=2 \,\Omega \;\sin \varphi _f $ and the layer thickness at $f$-points is:
\begin{equation}
  \label{eq:DYN_een_e3f}
  e_{3f} = \overline{\overline {e_{3t} }} ^{\,i+1/2,j+1/2}
\end{equation}

\begin{figure}[!ht]
  \centering
  \includegraphics[width=0.66\textwidth]{DYN_een_triad}
  \caption[Triads used in the energy and enstrophy conserving scheme (EEN)]{
    Triads used in the energy and enstrophy conserving scheme (EEN) for
    $u$-component (upper panel) and $v$-component (lower panel).}
  \label{fig:DYN_een_triad}
\end{figure}

A key point in \autoref{eq:DYN_een_e3f} is how the averaging in the \textbf{i}- and \textbf{j}- directions is made.
It uses the sum of masked t-point vertical scale factor divided either by the sum of the four t-point masks
(\np[=1]{nn_een_e3f}{nn\_een\_e3f}), or just by $4$ (\np[=.true.]{nn_een_e3f}{nn\_een\_e3f}).
The latter case preserves the continuity of $e_{3f}$ when one or more of the neighbouring $e_{3t}$ tends to zero and
extends by continuity the value of $e_{3f}$ into the land areas.
This case introduces a sub-grid-scale topography at f-points
(with a systematic reduction of $e_{3f}$ when a model level intercept the bathymetry)
that tends to reinforce the topostrophy of the flow
(\ie\ the tendency of the flow to follow the isobaths) \citep{penduff.le-sommer.ea_OS07}.

Next, the vorticity triads, $ {^i_j}\mathbb{Q}^{i_p}_{j_p}$ can be defined at a $T$-point as
the following triad combinations of the neighbouring potential vorticities defined at f-points
(\autoref{fig:DYN_een_triad}):
\begin{equation}
  \label{eq:DYN_Q_triads}
  _i^j \mathbb{Q}^{i_p}_{j_p}
  = \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)
\end{equation}
where the indices $i_p$ and $k_p$ take the values: $i_p = -1/2$ or $1/2$ and $j_p = -1/2$ or $1/2$.

Finally, the vorticity terms are represented as:
\begin{equation}
  \label{eq:DYN_vor_een}
  \left\{ {
      \begin{aligned}
        +q\,e_3 \, v 	&\equiv +\frac{1}{e_{1u} }   \sum_{\substack{i_p,\,k_p}}
        {^{i+1/2-i_p}_j}  \mathbb{Q}^{i_p}_{j_p}  \left( e_{1v}\,e_{3v} \;v  \right)^{i+1/2-i_p}_{j+j_p}   \\
        - q\,e_3 \, u     &\equiv -\frac{1}{e_{2v} }    \sum_{\substack{i_p,\,k_p}}
        {^i_{j+1/2-j_p}}  \mathbb{Q}^{i_p}_{j_p}  \left( e_{2u}\,e_{3u} \;u  \right)^{i+i_p}_{j+1/2-j_p}   \\
      \end{aligned}
    } \right.
\end{equation}

This EEN scheme in fact combines the conservation properties of the ENS and ENE schemes.
It conserves both total energy and potential enstrophy in the limit of horizontally nondivergent flow
(\ie\ $\chi$=$0$) (see \autoref{subsec:INVARIANTS_vorEEN}).
Applied to a realistic ocean configuration, it has been shown that it leads to a significant reduction of
the noise in the vertical velocity field \citep{le-sommer.penduff.ea_OM09}.
Furthermore, used in combination with a partial steps representation of bottom topography,
it improves the interaction between current and topography,
leading to a larger topostrophy of the flow \citep{barnier.madec.ea_OD06, penduff.le-sommer.ea_OS07}.

%% =================================================================================================
\subsection[Kinetic energy gradient term (\textit{dynkeg.F90})]{Kinetic energy gradient term (\protect\mdl{dynkeg})}
\label{subsec:DYN_keg}

As demonstrated in \autoref{apdx:INVARIANTS},
there is a single discrete formulation of the kinetic energy gradient term that,
together with the formulation chosen for the vertical advection (see below),
conserves the total kinetic energy:
\[
  % \label{eq:DYN_keg}
  \left\{
    \begin{aligned}
      -\frac{1}{2 \; e_{1u} }  & \ \delta_{i+1/2} \left[ {\overline {u^2}^{\,i} + \overline{v^2}^{\,j}} \right]   \\
      -\frac{1}{2 \; e_{2v} }  & \ \delta_{j+1/2} \left[ {\overline {u^2}^{\,i} + \overline{v^2}^{\,j}} \right]
    \end{aligned}
  \right.
\]

%% =================================================================================================
\subsection[Vertical advection term (\textit{dynzad.F90})]{Vertical advection term (\protect\mdl{dynzad})}
\label{subsec:DYN_zad}

The discrete formulation of the vertical advection, t
ogether with the formulation chosen for the gradient of kinetic energy (KE) term,
conserves the total kinetic energy.
Indeed, the change of KE due to the vertical advection is exactly balanced by
the change of KE due to the gradient of KE (see \autoref{apdx:INVARIANTS}).
\[
  % \label{eq:DYN_zad}
  \left\{
    \begin{aligned}
      -\frac{1} {e_{1u}\,e_{2u}\,e_{3u}} &\ \overline{\ \overline{ e_{1t}\,e_{2t}\;w } ^{\,i+1/2}  \;\delta_{k+1/2} \left[ u \right]\  }^{\,k}  \\
      -\frac{1} {e_{1v}\,e_{2v}\,e_{3v}}  &\ \overline{\ \overline{ e_{1t}\,e_{2t}\;w } ^{\,j+1/2}  \;\delta_{k+1/2} \left[ u \right]\  }^{\,k}
    \end{aligned}
  \right.
\]

%% =================================================================================================
\section{Coriolis and advection: flux form}
\label{sec:DYN_adv_cor_flux}

Options are defined through the \nam{dyn_adv}{dyn\_adv} namelist variables.
In the flux form (as in the vector invariant form),
the Coriolis and momentum advection terms are evaluated using either a leapfrog scheme or a RK3 scheme.
In the leapfrog case the velocity appearing in these expressions is centred in time (\textit{now} velocity).
In the RK3 case the velocity appearing in these expressions is forward in time (\textit{before} velocity) at stage 1, 
it is is centred in time (\textit{now} velocity) at stage 2 and 3.
At the lateral boundaries either free slip,
no slip or partial slip boundary conditions are applied following \autoref{chap:LBC}.

%% =================================================================================================
\subsection[Coriolis plus curvature metric terms (\textit{dynvor.F90})]{Coriolis plus curvature metric terms (\protect\mdl{dynvor})}
\label{subsec:DYN_cor_flux}

In flux form, the vorticity term reduces to a Coriolis term in which the Coriolis parameter has been modified to account for the "metric" term.
This altered Coriolis parameter is thus discretised at $f$-points.
It is given by:
\begin{multline*}
  % \label{eq:DYN_cor_metric}
  f+\frac{1}{e_1 e_2 }\left( {v\frac{\partial e_2 }{\partial i}  -  u\frac{\partial e_1 }{\partial j}} \right)  \\
  \equiv   f + \frac{1}{e_{1f} e_{2f} } \left( { \ \overline v ^{i+1/2}\delta_{i+1/2} \left[ {e_{2u} } \right]
      -  \overline u ^{j+1/2}\delta_{j+1/2} \left[ {e_{1u} } \right]  }  \ \right)
\end{multline*}

%                 energy conserving scheme at T-point
%% =================================================================================================
\subsubsection[Energy conserving scheme  (\forcode{ln_dynvor_enT})]{Energy conserving scheme (\protect\np{ln_dynvor_enT}{ln\_dynvor\_enT})}
\label{subsec:DYN_vor_enT}

The kinetic energy conserving scheme at T-point (ENT scheme) conserves the global kinetic energy but not the global enstrophy.
It is given by:
\begin{equation}
  \label{eq:DYN_vor_enT}
  \left\{
    \begin{aligned}
      &+\frac{1}{e_{1u}\,e_{2u}\,e_{3u}} \, \overline{ \left( f^T + \overline{\overline{ \zeta }}^{\,i,j} \right)
            e_{1t}\,e_{2t}\,e_{3t} \,  \overline{v}^{\,j}}^{\,i+1/2} \\
      &-\frac{1}{e_{1v}\,e_{2v}\,e_{3v}} \, \overline{ \left( f^T + \overline{\overline{ \zeta }}^{\,i,j} \right)
            e_{1t}\,e_{2t}\,e_{3t} \,  \overline{u}^{\,i}}^{\,j+1/2}
    \end{aligned}
  \right.
\end{equation}


Any of the (\autoref{eq:DYN_vor_ens}), (\autoref{eq:DYN_vor_ene}), (\autoref{eq:DYN_vor_enT}) and (\autoref{eq:DYN_vor_een}) 
schemes can be used to
compute the product of the Coriolis parameter and the vorticity.
However, the energy-conserving schemes (\autoref{eq:DYN_vor_een} and \autoref{eq:DYN_vor_enT}) 
have exclusively been used to date.
This term is evaluated using either a leapfrog scheme or a RK3 scheme.
In the leapfrog case it is centred in time (\textit{now} velocity).
In the RK3 case it is forward in time (\textit{before} velocity) at stage 1, 
it is is centred in time (\textit{now} velocity) at stage 2 and 3. 

%% =================================================================================================
\subsection[Flux form advection term (\textit{dynadv.F90})]{Flux form advection term (\protect\mdl{dynadv})}
\label{subsec:DYN_adv_flux}

The discrete expression of the advection term is given by:
\[
  % \label{eq:DYN_adv}
  \left\{
    \begin{aligned}
      \frac{1}{e_{1u}\,e_{2u}\,e_{3u}}
      \left(      \delta_{i+1/2} \left[ \overline{e_{2u}\,e_{3u}\;u }^{i       }  \ u_t      \right]
        + \delta_{j       } \left[ \overline{e_{1u}\,e_{3u}\;v }^{i+1/2}  \ u_f      \right] \right.  \ \;   \\
      \left.   + \delta_{k      } \left[ \overline{e_{1w}\,e_{2w}\;w}^{i+1/2}  \ u_{uw} \right] \right)   \\
      \\
      \frac{1}{e_{1v}\,e_{2v}\,e_{3v}}
      \left(     \delta_{i       } \left[ \overline{e_{2u}\,e_{3u }\;u }^{j+1/2} \ v_f       \right]
        + \delta_{j+1/2} \left[ \overline{e_{1u}\,e_{3u }\;v }^{i       } \ v_t       \right] \right.  \ \, \, \\
      \left.  + \delta_{k      } \left[ \overline{e_{1w}\,e_{2w}\;w}^{j+1/2} \ v_{vw}  \right] \right) \\
    \end{aligned}
  \right.
\]

Two advection schemes are available:
a $2^{nd}$ order centered finite difference scheme, CEN2,
or a $3^{rd}$ order upstream biased scheme, UP3.
The latter is described in \citet{shchepetkin.mcwilliams_OM05}.
The schemes are selected using the namelist logicals \np{ln_dynadv_cen2}{ln\_dynadv\_cen2} and \np{ln_dynadv_up3}{ln\_dynadv\_up3}.
In flux form, the schemes differ by the choice of a space and time interpolation to define the value of
$u$ and $v$ at the centre of each face of $u$- and $v$-cells, \ie\ at the $T$-, $f$-,
and $uw$-points for $u$ and at the $f$-, $T$- and $vw$-points for $v$.

%                 2nd order centred scheme
%% =================================================================================================
\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})}
\label{subsec:DYN_adv_cen2}

In the centered $2^{nd}$ order formulation, the velocity is evaluated as the mean of the two neighbouring points:
\begin{equation}
  \label{eq:DYN_adv_cen2}
  \left\{
    \begin{aligned}
      u_T^{cen2} &=\overline u^{i }       \quad &  u_F^{cen2} &=\overline u^{j+1/2}  \quad &  u_{uw}^{cen2} &=\overline u^{k+1/2}   \\
      v_F^{cen2} &=\overline v ^{i+1/2} \quad & v_F^{cen2} &=\overline v^j		\quad &  v_{vw}^{cen2} &=\overline v ^{k+1/2}  \\
    \end{aligned}
  \right.
\end{equation}

The scheme is non diffusive (\ie\ conserves the kinetic energy) but dispersive (\ie\ it may create false extrema).
It is therefore notoriously noisy and must be used in conjunction with an explicit diffusion operator to
produce a sensible solution.
The associated time-stepping is performed using a leapfrog scheme in conjunction with an Asselin time-filter,
so $u$ and $v$ are the \emph{now} velocities.

%                 UP3 scheme
%% =================================================================================================
\subsubsection[UP3: Upstream Biased Scheme (\forcode{ln_dynadv_up3})]{UP3: Upstream Biased Scheme (\protect\np{ln_dynadv_up3}{ln\_dynadv\_up3})}
\label{subsec:DYN_adv_up3}

The UP3 advection scheme is an upstream biased third order scheme based on
an upstream-biased parabolic interpolation.
For example, the evaluation of $u_T^{up3} $ is done as follows:
\begin{equation}
  \label{eq:DYN_adv_up3}
  u_T^{up3} =\overline u ^i-\;\frac{1}{6}
  \begin{cases}
    u"_{i-1/2}& 	\text{if $\ \overline{e_{2u}\,e_{3u} \ u}^i  \geqslant 0$ } 	\\
    u"_{i+1/2}& 	\text{if $\ \overline{e_{2u}\,e_{3u} \ u}^i  < 0$ }
  \end{cases}
\end{equation}
where $u"_{i+1/2} =\delta_{i+1/2} \left[ {\delta_i \left[ u \right]} \right]$.
This results in a dissipatively dominant (\ie\ hyper-diffusive) truncation error
\citep{shchepetkin.mcwilliams_OM05}.
The overall performance of the advection scheme is similar to that reported in \citet{farrow.stevens_JPO95}.
It is a relatively good compromise between accuracy and smoothness.
It is not a \emph{positive} scheme, meaning that false extrema are permitted.
But the amplitudes of the false extrema are significantly reduced over those in the centred second order method.
As the scheme already includes a diffusion component, it can be used without explicit lateral diffusion on momentum
(\ie\ \np[=]{ln_dynldf_lap}{ln\_dynldf\_lap}\np[=.false.]{ln_dynldf_bilap}{ln\_dynldf\_bilap}),
and it is recommended to do so.

The UP3 scheme is used in all directions.
UP3 is diffusive and is associated with vertical mixing of momentum. \cmtgm{ gm  pursue the
sentence:Since vertical mixing of momentum is a source term of the TKE equation...  }

For stability reasons, the first term in (\autoref{eq:DYN_adv_up3}),
which corresponds to a second order centred scheme, is evaluated using the \textit{now} velocity (centred in time),
while the second term, which is the diffusion part of the scheme,
is evaluated using the \textit{before} velocity (forward in time).
This is discussed by \citet{webb.de-cuevas.ea_JAOT98} in the context of the Quick advection scheme.

Note that the UP3 and QUICK (Quadratic Upstream Interpolation for Convective Kinematics) schemes only differ by
one coefficient.
Replacing $1/6$ by $1/8$ in (\autoref{eq:DYN_adv_up3}) leads to the QUICK advection scheme \citep{webb.de-cuevas.ea_JAOT98}.
This option is not available through a namelist parameter, since the $1/6$ coefficient is hard coded.
Nevertheless it is quite easy to make the substitution in the \mdl{dynadv\_up3} module and obtain a QUICK scheme.

%% =================================================================================================
\section[Hydrostatic pressure gradient (\textit{dynhpg.F90})]{Hydrostatic pressure gradient (\protect\mdl{dynhpg})}
\label{sec:DYN_hpg}

\begin{listing}
  \nlst{namdyn_hpg}
  \caption{\forcode{&namdyn_hpg}}
  \label{lst:namdyn_hpg}
\end{listing}

%Options are defined through the \nam{dyn_hpg}{dyn\_hpg} namelist variables.
%The key distinction between the different algorithms used for
%the hydrostatic pressure gradient is the vertical coordinate used,
%since HPG is a \emph{horizontal} pressure gradient, \ie\ computed along geopotential surfaces.
%As a result, any tilt of the surface of the computational levels will require a specific treatment to
%compute the hydrostatic pressure gradient.

%In v5.0 partial cells are considered as penalized cells. A penalized cell has a solid fraction and 
%a liquid fraction, both are distributed homogeneouly across the cell. Therefore adjacent cells are not
%expected to live at distinct depth because of their content differences. 

NEMO offers a selection of different algorithms to compute the hydrostatic pressure gradient (HPG) term in the momentum equation. 
Options are defined through the \nam{dyn_hpg}{dyn\_hpg} namelist variables. 

Since HPGs are computed along geopotential surfaces, a key distinction between the various algorithms is the type of vertical 
coordinate they target. In particular, NEMO offers a number of options to compute HPGs with generalised $s$-coordinates that 
may be not aligned with geopotentials.


The hydrostatic pressure gradient term is evaluated either using a leapfrog scheme,
\ie\ the density appearing in its expression is centred in time (\emph{now} $\rho$),
or a RK3 scheme \ie\ the density appearing in its expression is forward in time (\emph{before} $\rho$),
it is centred in time (\emph{now} $\rho$) at stage 2 and 3.
At the lateral boundaries either free slip, no slip or partial slip boundary conditions are applied.

%% =================================================================================================
\subsection[Full step $Z$-coordinate (\forcode{ln_dynhpg_zco})]{Full step $Z$-coordinate (\protect\np{ln_dynhpg_zco}{ln\_dynhpg\_zco})}
\label{subsec:DYN_hpg_zco}

When using standard geopotential coordinates (\np[=.true.]{ln_zco}{ln\_zco}), the hydrostatic pressure can be directly obtained 
by vertically integrating the hydrostatic equation from the surface to the bottom. However, pressure is large at great depths while 
its horizontal gradient is several orders of magnitude smaller. This may lead to large truncation errors in the pressure gradient terms.
Thus, the two horizontal components of the hydrostatic pressure gradient are computed directly as follows:

for $k=km$ (surface layer, $jk=1$ in the code)
\begin{equation}
  \label{eq:DYN_hpg_zco_surf}
  \left\{
    \begin{aligned}
      \left. \delta_{i+1/2} \left[  p^h 			 \right] \right|_{k=km}
      &= \frac{1}{2} g \ 	\left. \delta_{i+1/2} \left[  e_{3w} \ \rho \right] \right|_{k=km}   \\
      \left. \delta_{j+1/2} \left[  p^h  			 \right] \right|_{k=km}
      &= \frac{1}{2} g \ 	\left. \delta_{j+1/2} \left[  e_{3w} \ \rho \right] \right|_{k=km}   \\
    \end{aligned}
  \right.
\end{equation}

for $1<k<km$ (interior layer)
\begin{equation}
  \label{eq:DYN_hpg_zco}
  \left\{
    \begin{aligned}
      \left. \delta_{i+1/2} \left[  p^h 			 \right] \right|_{k}
      &=					\left. \delta_{i+1/2} \left[  p^h 			 \right] \right|_{k-1}
      +    g\;	\left. \delta_{i+1/2} \left[  e_{3w} \ \overline {\rho}^{k+1/2} \right] \right|_{k}   \\
      \left. \delta_{j+1/2} \left[  p^h  			 \right] \right|_{k}
      &=     				\left. \delta_{j+1/2} \left[  p^h  			 \right] \right|_{k-1}
      +    g\;	\left. \delta_{j+1/2} \left[  e_{3w} \ \overline {\rho}^{k+1/2} \right] \right|_{k}   \\
    \end{aligned}
  \right.
\end{equation}

Note that the $1/2$ factor in (\autoref{eq:DYN_hpg_zco_surf}) is adequate because of the definition of $e_{3w}$ as
the vertical derivative of the scale factor at the surface level ($z=0$).

%% =================================================================================================
\subsection{Generalised $S$-coordinates}
\label{subsec:DYN_hpg_sco}

Pressure gradient formulations with a generalised $s(x,y,z,t)$ coordinate have been the subject of a vast number of papers
(\eg, \citet{song_MWR98, shchepetkin.mcwilliams_jgro03}). A number of different pressure gradient options are available in NEMO:

\begin{itemize}
\item
\textbf{Traditional coding} (\np[=.true.]{ln_hpg_sco}{ln\_hpg\_sco}, e.g. \citet{madec.delecluse.ea_JPO96}):
%\end{itemize}

\begin{equation}
  \label{eq:DYN_hpg_sco}
  \left\{
    \begin{aligned}
      - \frac{1}    					{\rho_o \, e_{1u}} \;	\delta_{i+1/2} \left[  p^h  \right]
      + \frac{g\; \overline {\rho}^{i+1/2}}	{\rho_o \, e_{1u}} \;	\delta_{i+1/2} \left[  z_t   \right],    \\
      - \frac{1}    					{\rho_o \, e_{2v}} \;	\delta_{j+1/2} \left[  p^h  \right]
      + \frac{g\; \overline {\rho}^{j+1/2}}	{\rho_o \, e_{2v}} \;	\delta_{j+1/2} \left[  z_t   \right],    \\
    \end{aligned}
  \right.
\end{equation}

where the first term is the pressure gradient along coordinates (computed as in \autoref{eq:DYN_hpg_zco_surf} - \autoref{eq:DYN_hpg_zco})
and $z_T$ is the depth of the $T$-point evaluated from the sum of the vertical scale factors at the $W$-point ($e_{3w}$). 
Note that this scheme is not recommended when using steeply inclined computational levels (e.g., terrain-following or hybrid generalised 
vertical coordinates, i.e., \np[=.true.]{ln_sco}{ln\_sco}) -  see e.g. \citet{shchepetkin.mcwilliams_jgro03}. 
However, it should be the standard choice when using $z$-coordinates (\np[=.true.]{ln_zco}{ln\_zps} or \np[=.true.]{ln_zps}{ln\_zps}) 
with the non-linear free surface (\np[=.false.]{ln_linssh}{ln\_linssh} and \key{qco}), since in this case model levels will follow the barotropic motion of the ocean  \citep{levier.treguier.ea_trpt07}.

\item
\textbf{Traditional coding with adaptation for ice shelf cavities} (\np[=.true.]{ln_hpg_isf}{ln\_hpg\_isf}):

this scheme must be used when ice shelf cavities are activated (\np[=.true.]{ln_isfcav}{ln\_isfcav} and the inclusion of \key{isf}).

\item
\textbf{Pressure Jacobian scheme} (\np[=.true.]{ln_hpg_prj}{ln\_hpg\_prj}):

this scheme uses a constrained cubic spline to reconstruct the vertical density profile within a water column. 
This method maintains the monotonicity between the density nodes. The pressure is calculated by analytical integration of the density 
profile. For the force in the $i$-direction, it calculates the difference of the pressures on the $i+\tfrac{1}{2}$ and $i-\tfrac{1}{2}$ 
faces of the cell using pressures calculated at the same height. In grid cells just above the bathymetry, this height is higher than the 
cells’ centre. This scheme works well for moderately steep computational levels but produces large velocities in the SEAMOUNT test case 
when model levels are steeply inclined. 

\item
\textbf{Density Jacobian with cubic polynomial scheme} (\np[=.true.]{ln_hpg_djc}{ln\_hpg\_djc}, \cite{shchepetkin.mcwilliams_jgro03}:

the ROMS-like, density Jacobian with cubic polynomial method has been debugged and from vn4.2 is available as an option. 
This scheme is based on section 5 of \cite{shchepetkin.mcwilliams_jgro03} For the force in the $i$-direction, it uses constrained cubic 
splines to re-construct the density along lines of constant $s$ and constant $i$ in the $(i,s)$ plane. It calculates a line integral of 
$\rho$ and then integrates vertically to obtain the horizontal pressure gradient. The constrained cubic splines require boundary conditions
 to be specified at the upper and lower boundaries and at points where model levels encrop the model bathymetry (i.e., with geopotential 
or hybrid vertical coordinates). The user can choose between von Neumann and linear extrapolation boundary conditions via the
 \texttt{ln\_hpg\_djc\_vnh} and  \texttt{ln\_hpg\_djc\_vnv} namelist switches, respectively. This scheme can be used with any type of 
generalised  $s$-coordinates - i.e., $z$ or $z^*$, terrain-following or hybrids of these two (e.g., via the vanishing quasi-sigma or 
multi-envelope  methods, see e.g. \cite{shapiro.luneva.ea_OS13,bruciaferri.shapiro.wobus_OD18,wise.harle.ea_OM21}) - but at the moment 
can not be used with ice shelf cavities.

\end{itemize}

Starting from version 4.2, the density field used by dyn\_hpg is the density anomaly field rhd rather than $1+\mathrm{rhd}$. 
The calculation of the source term for the free surface has been adjusted to take this into account. 
The true in situ density $\rho= \rho_0 (1 + r_0(z) + rhd )$ where $r_0(z)$ accounts for the variation of density
with depth for water with a potential temperature of $4^{\circ}$C and salinity of $35.16504$g/kg 
(see (13) and (14) of \citet{roquet.madec.ea_OM15}).        

%% =================================================================================================
\subsection{Ice shelf cavity}
\label{subsec:DYN_hpg_isf}

Beneath an ice shelf, the total pressure gradient is the sum of the pressure gradient due to the ice shelf load and
the pressure gradient due to the ocean load (\np[=.true.]{ln_hpg_isf}{ln\_hpg\_isf} with the inclusion of \key{isf}).\\

The main hypothesis to compute the ice shelf load is that the ice shelf is in an isostatic equilibrium.
The top pressure is computed integrating from surface to the base of the ice shelf a reference density profile
(prescribed as density of a water at 34.4 PSU and -1.9\deg{C}) and
corresponds to the water replaced by the ice shelf.
This top pressure is constant over time.
A detailed description of this method is described in \citet{losch_JGR08}.\\

The pressure gradient due to ocean load is computed using the expression \autoref{eq:DYN_hpg_sco} described in
\autoref{subsec:DYN_hpg_sco}.

%% =================================================================================================
%% \subsection[Time-scheme (\forcode{ln_dynhpg_imp})]{Time-scheme (\protect\np{ln_dynhpg_imp}{ln\_dynhpg\_imp})}
%% \label{subsec:DYN_hpg_imp}
%% 
%% The default time differencing scheme used for the horizontal pressure gradient is a leapfrog scheme and
%% therefore the density used in all discrete expressions given above is the  \textit{now} density,
%% computed from the \textit{now} temperature and salinity.
%% In some specific cases
%% (usually high resolution simulations over an ocean domain which includes weakly stratified regions)
%% the physical phenomenon that controls the time-step is internal gravity waves (IGWs).
%% A semi-implicit scheme for doubling the stability limit associated with IGWs can be used
%% \citep{brown.campana_MWR78, maltrud.smith.ea_JGR98}.
%% It involves the evaluation of the hydrostatic pressure gradient as
%% an average over the three time levels $t-\rdt$, $t$, and $t+\rdt$
%% (\ie\ \textit{before}, \textit{now} and  \textit{after} time-steps),
%% rather than at the central time level $t$ only, as in the standard leapfrog scheme.
%% 
%% $\bullet$ leapfrog scheme (\np[=.true.]{ln_dynhpg_imp}{ln\_dynhpg\_imp}):
%% 
%% \begin{equation}
%%   \label{eq:DYN_hpg_lf}
%%   \frac{u^{t+\rdt}-u^{t-\rdt}}{2\rdt} = \;\cdots \;
%%   -\frac{1}{\rho_o \,e_{1u} }\delta_{i+1/2} \left[ {p_h^t } \right]
%% \end{equation}
%% 
%% $\bullet$ semi-implicit scheme (\np[=.true.]{ln_dynhpg_imp}{ln\_dynhpg\_imp}):
%% \begin{equation}
%%   \label{eq:DYN_hpg_imp}
%%   \frac{u^{t+\rdt}-u^{t-\rdt}}{2\rdt} = \;\cdots \;
%%   -\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]
%% \end{equation}
%% 
%% The semi-implicit time scheme \autoref{eq:DYN_hpg_imp} is made possible without
%% significant additional computation since the density can be updated to time level $t+\rdt$ before
%% computing the horizontal hydrostatic pressure gradient.
%% It can be easily shown that the stability limit associated with the hydrostatic pressure gradient doubles using
%% \autoref{eq:DYN_hpg_imp} compared to that using the standard leapfrog scheme \autoref{eq:DYN_hpg_lf}.
%% Note that \autoref{eq:DYN_hpg_imp} is equivalent to applying a time filter to the pressure gradient to
%% eliminate high frequency IGWs.
%% Obviously, when using \autoref{eq:DYN_hpg_imp},
%% the doubling of the time-step is achievable only if no other factors control the time-step,
%% such as the stability limits associated with advection or diffusion.
%% 
%% In practice, the semi-implicit scheme is used when \np[=.true.]{ln_dynhpg_imp}{ln\_dynhpg\_imp}.
%% In this case, we choose to apply the time filter to temperature and salinity used in the equation of state,
%% instead of applying it to the hydrostatic pressure or to the density,
%% so that no additional storage array has to be defined.
%% The density used to compute the hydrostatic pressure gradient (whatever the formulation) is evaluated as follows:
%% \[
%%   % \label{eq:DYN_rho_flt}
%%   \rho^t = \rho( \widetilde{T},\widetilde {S},z_t)
%%   \quad	  \text{with}	\quad
%%   \widetilde{X} = 1 / 4 \left(  X^{t+\rdt} +2 \,X^t + X^{t-\rdt}  \right)
%% \]
%% 
%% Note that in the semi-implicit case, it is necessary to save the filtered density,
%% an extra three-dimensional field, in the restart file to restart the model with exact reproducibility.
%% This option is controlled by  \np{nn_dynhpg_rst}{nn\_dynhpg\_rst}, a namelist parameter.
%% 
%% =================================================================================================
\section[Surface pressure gradient (\textit{dynspg.F90})]{Surface pressure gradient (\protect\mdl{dynspg})}
\label{sec:DYN_spg}

\begin{listing}
  \nlst{namdyn_spg}
  \caption{\forcode{&namdyn_spg}}
  \label{lst:namdyn_spg}
\end{listing}

Options are defined through the \nam{dyn_spg}{dyn\_spg} namelist variables.
The surface pressure gradient term is related to the representation of the free surface (\autoref{sec:MB_hor_pg}).
The main distinction is between the fixed volume case (linear free surface, \ie\ with the inclusion of \key{linssh}) and
the variable volume case (nonlinear free surface, \key{qco}).
In the linear free surface case (\autoref{subsec:MB_free_surface})
the vertical scale factors $e_{3}$ are fixed in time,
while they are time-dependent in the nonlinear case (\autoref{subsec:MB_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 (\np[=.true.]{ln_dynspg_exp}{ln\_dynspg\_exp}).
To allow a longer time step for the three-dimensional equations, one can use a split-explicit free surface 
(\np[=.true.]{ln_dynspg_ts}{ln\_dynspg\_ts}). In that case, only a quasi-linear
 form of 2d barotropic equations is substepped with a small time increment.

%% =================================================================================================
\subsection[Explicit free surface (\forcode{ln_dynspg_exp})]{Explicit free surface (\protect\np{ln_dynspg_exp}{ln\_dynspg\_exp})}
\label{subsec:DYN_spg_exp}

In the explicit free surface formulation (\np{ln_dynspg_exp}{ln\_dynspg\_exp} set to true),
the model time step is chosen to be small enough to resolve the external gravity waves
(typically a few tens of seconds).
The surface pressure gradient, evaluated using a leap-frog scheme (\ie\ centered in time),
is thus simply given by :
\begin{equation}
  \label{eq:DYN_spg_exp}
  \left\{
    \begin{aligned}
      - \frac{1}{e_{1u}\,\rho_o} \;	\delta_{i+1/2} \left[  \,\rho \,\eta\,  \right] 	\\
      - \frac{1}{e_{2v}\,\rho_o} \;	\delta_{j+1/2} \left[  \,\rho \,\eta\,  \right]
    \end{aligned}
  \right.
\end{equation}

Note that this option is not yet compatible with RK3 time-stepping.

%% =================================================================================================
\subsection[Split-explicit free surface (\forcode{ln_dynspg_ts})]{Split-explicit free surface (\protect\np{ln_dynspg_ts}{ln\_dynspg\_ts})}
\label{subsec:DYN_spg_ts}
%\nlst{namsplit}

The split-explicit free surface formulation used in \NEMO\ (\np{ln_dynspg_ts}{ln\_dynspg\_ts} set to true),
also called the time-splitting formulation, follows the one proposed by \citet{shchepetkin.mcwilliams_OM05}.
The general idea is to solve the free surface equation and the associated barotropic velocity equations with
a smaller time step than $\rdt$, the time step used for the three dimensional prognostic variables
(\autoref{fig:DYN_spg_ts}).
The size of the small time step, $\rdt_e$ (the external mode or barotropic time step) is provided through
the \np{nn_e}{nn\_e} namelist parameter as: $\rdt_e = \rdt / nn\_e$.
This parameter can be optionally defined automatically (\np[=.true.]{ln_bt_auto}{ln\_bt\_auto}) considering that
the stability of the barotropic system is essentially controled by external waves propagation.
Maximum Courant number is in that case time independent, and easily computed online from the input bathymetry.
Therefore, $\rdt_e$ is adjusted so that the Maximum allowed Courant number is smaller than \np{rn_bt_cmax}{rn\_bt\_cmax}.

The barotropic mode solves the following equations:
% \begin{subequations}
%  \label{eq:DYN_BT}
\begin{equation}
  \label{eq:DYN_BT_dyn}
  \frac{\partial {\mathrm \overline{{\mathbf U}}_h} }{\partial t}=
  -f\;{\mathrm {\mathbf k}}\times {\mathrm \overline{{\mathbf U}}_h}
  -g\nabla _h \eta -\frac{c_b^{\textbf U}}{H+\eta} \mathrm {\overline{{\mathbf U}}_h} + \mathrm {\overline{\mathbf G}}
\end{equation}
\[
  % \label{eq:DYN_BT_ssh}
  \frac{\partial \eta }{\partial t}=-\nabla \cdot \left[ {\left( {H+\eta } \right) \; {\mathrm{\mathbf \overline{U}}}_h \,} \right]+P-E
\]
% \end{subequations}
where $\mathrm {\overline{\mathbf G}}$ is a forcing term held constant, containing coupling term between modes,
surface atmospheric forcing as well as slowly varying barotropic terms not explicitly computed to gain efficiency.
The third term on the right hand side of \autoref{eq:DYN_BT_dyn} represents the bottom stress
(see section \autoref{sec:ZDF_drg}), explicitly accounted for at each barotropic iteration.
Temporal discretization of the system above follows a three-time step Generalized Forward Backward algorithm
detailed in \citet{shchepetkin.mcwilliams_OM05}.
AB3-AM4 coefficients used in \NEMO\ follow the second-order accurate,
"multi-purpose" stability compromise as defined in \citet{shchepetkin.mcwilliams_ibk09}
(see their figure 12, lower left).

\begin{figure}[!t]
  \centering
  \includegraphics[width=0.66\textwidth]{DYN_dynspg_ts}
  \caption[Split-explicit time stepping scheme for the external and internal modes]{
    Schematic of the split-explicit time stepping scheme for the external and internal modes.
    Time increases to the right.
    In this particular exemple,
    a boxcar averaging window over \np{nn_e}{nn\_e} barotropic time steps is used
    (\np[=1]{nn_bt_flt}{nn\_bt\_flt}) and \np[=5]{nn_e}{nn\_e}.
    Internal mode time steps (which are also the model time steps) are denoted by
    $t-\rdt$, $t$ and $t+\rdt$.
    Variables with $k$ superscript refer to instantaneous barotropic variables,
    $< >$ and $<< >>$ operator refer to time filtered variables using respectively primary
    (red vertical bars) and secondary weights (blue vertical bars).
    The former are used to obtain time filtered quantities at $t+\rdt$ while
    the latter are used to obtain time averaged transports to advect tracers.
    a) Forward time integration:
    \protect\np[=1]{nn_bt_flt}{nn\_bt\_flt}.
    b) Centred time integration:
    \protect\np[=2]{nn_bt_flt}{nn\_bt\_flt}.
    c) Forward time integration with no time filtering (POM-like scheme):
    \protect\np[=3]{nn_bt_flt}{nn\_bt\_flt}.}
  \label{fig:DYN_spg_ts}
\end{figure}

In the default case (\protect\np[=1]{nn_bt_flt}{nn\_bt\_flt}),
the external mode is integrated between \textit{now} and \textit{after} baroclinic time-steps
(\autoref{fig:DYN_spg_ts}a).
To avoid aliasing of fast barotropic motions into three dimensional equations,
time filtering is eventually applied on barotropic quantities.
In that case, the integration is extended slightly beyond \textit{after} time step to
provide time filtered quantities.
These are used for the subsequent initialization of the barotropic mode in the following baroclinic step.
Since external mode equations written at baroclinic time steps finally follow a forward time stepping scheme,
asselin filtering is not applied to barotropic quantities.\\
Alternatively, one can choose to integrate barotropic equations starting from \textit{before} time step
(\protect\np[=2]{nn_bt_flt}{nn\_bt\_flt}).
Although more computationaly expensive ( \np{nn_e}{nn\_e} additional iterations are indeed necessary),
the baroclinic to barotropic forcing term given at \textit{now} time step become centred in
the middle of the integration window.
It can easily be shown that this property removes part of splitting errors between modes,
which increases the overall numerical robustness.
%references to Patrick Marsaleix' work here. Also work done by SHOM group.


As far as tracer conservation is concerned,
barotropic velocities used to advect tracers must also be updated at \textit{now} time step.
This implies to change the traditional order of computations in \NEMO:
most of momentum trends (including the barotropic mode calculation) updated first, tracers' after.
%%This \textit{de facto} makes semi-implicit hydrostatic pressure gradient
%%(see section \autoref{subsec:DYN_hpg_imp})
%%and time splitting not compatible.
Advective barotropic velocities are obtained by using a secondary set of filtering weights,
uniquely defined from the filter coefficients used for the time averaging (\citet{shchepetkin.mcwilliams_OM05}).
Consistency between the time averaged continuity equation and the time stepping of tracers is here the key to
obtain exact conservation.


One can eventually choose to feedback instantaneous values by not using any time filter
(\protect\np[=3]{nn_bt_flt}{nn\_bt\_flt}).
In that case, external mode equations are continuous in time,
\ie\ they are not re-initialized when starting a new sub-stepping sequence.
This is the method used in the POM model for example, the stability being maintained by
refreshing at (almost) each barotropic time step advection and horizontal diffusion terms.
Since the latter terms have not been added in \NEMO\ for computational efficiency,
removing time filtering would be inevitably unstable. One can however add some dissipation, but in the time domain, by slightly modifying the barotropic time stepping coefficients (\citet{demange_JCP19}). This is implemented here through an additional parameter (\np{rn_bt_alpha}{rn\_bt\_alpha}), which controls the amount of temporal diffusion.

\cmtgm{               %%% copy from griffies Book

\textbf{title: Time stepping the barotropic system }

Assume knowledge of the full velocity and tracer fields at baroclinic time $\tau$.
Hence, we can update the surface height and vertically integrated velocity with a leap-frog scheme using
the small barotropic time step $\rdt$.
We have

\[
  % \label{eq:DYN_spg_ts_eta}
  \eta^{(b)}(\tau,t_{n+1}) - \eta^{(b)}(\tau,t_{n+1}) (\tau,t_{n-1})
  = 2 \rdt \left[-\nabla \cdot \textbf{U}^{(b)}(\tau,t_n) + \text{EMP}_w(\tau) \right]
\]
\begin{multline*}
  % \label{eq:DYN_spg_ts_u}
  \textbf{U}^{(b)}(\tau,t_{n+1}) - \textbf{U}^{(b)}(\tau,t_{n-1})  \\
  = 2\rdt \left[ - f \textbf{k} \times \textbf{U}^{(b)}(\tau,t_{n})
    - H(\tau) \nabla p_s^{(b)}(\tau,t_{n}) +\textbf{M}(\tau) \right]
\end{multline*}
\

In these equations, araised (b) denotes values of surface height and vertically integrated velocity updated with
the barotropic time steps.
The $\tau$ time label on $\eta^{(b)}$ and $U^{(b)}$ 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 that sets the barotropic time steps via
\[
  % \label{eq:DYN_spg_ts_t}
  t_n=\tau+n\rdt
\]
with $n$ an integer.
The density scaled surface pressure is evaluated via
\[
  % \label{eq:DYN_spg_ts_ps}
  p_s^{(b)}(\tau,t_{n}) =
  \begin{cases}
    g \;\eta_s^{(b)}(\tau,t_{n}) \;\rho(\tau)_{k=1}) / \rho_o  &      \text{non-linear case} \\
    g \;\eta_s^{(b)}(\tau,t_{n})  &      \text{linear case}
  \end{cases}
\]
To get started, we assume the following initial conditions
\[
  % \label{eq:DYN_spg_ts_eta}
  \begin{split}
    \eta^{(b)}(\tau,t_{n=0}) &= \overline{\eta^{(b)}(\tau)}    \\
    \eta^{(b)}(\tau,t_{n=1}) &= \eta^{(b)}(\tau,t_{n=0}) + \rdt \ \text{RHS}_{n=0}
  \end{split}
\]
with
\[
  % \label{eq:DYN_spg_ts_etaF}
  \overline{\eta^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N \eta^{(b)}(\tau-\rdt,t_{n})
\]
the time averaged surface height taken from the previous barotropic cycle.
Likewise,
\[
  % \label{eq:DYN_spg_ts_u}
  \textbf{U}^{(b)}(\tau,t_{n=0}) = \overline{\textbf{U}^{(b)}(\tau)}	\\ \\
  \textbf{U}(\tau,t_{n=1}) = \textbf{U}^{(b)}(\tau,t_{n=0}) + \rdt \ \text{RHS}_{n=0}
\]
with
\[
  % \label{eq:DYN_spg_ts_u}
  \overline{\textbf{U}^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau-\rdt,t_{n})
\]
the time averaged vertically integrated transport.
Notably, there is no Robert-Asselin time filter used in the barotropic portion of the integration.