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}
%% =================================================================================================
\section[Horizontal divergence (\textit{sshwzv.F90}, $\eta$, $w$ )]{Horizontal divergence (\protect\mdl{sshwzv})}
\label{sec: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 $\rho_w$=1,026~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[Horizontal divergence and relative vorticity (\textit{divhor.F90}, $\zeta$, $\chi$ )]{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.

%% =================================================================================================
\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[Lateral diffusion term and operators (\textit{dynldf.F90})]{Lateral diffusion term and operators (\protect\mdl{dynldf})}
\label{sec:DYN_ldf}

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

Options are defined through the \nam{dyn_ldf}{dyn\_ldf} namelist variables.
The options available for lateral diffusion are to use either laplacian (rotated or not) or biharmonic operators.
The coefficients may be constant or spatially variable;
the description of the coefficients is found in the chapter on lateral physics (\autoref{chap:LDF}).
The lateral diffusion of momentum is evaluated using a forward scheme,
\ie\ the velocity appearing in its expression is the \textit{before} velocity in time,
except for the pure vertical component that appears when a tensor of rotation is used.
This latter term is solved implicitly together with the vertical diffusion term (see \autoref{chap:TD}).

At the lateral boundaries either free slip,
no slip or partial slip boundary conditions are applied according to the user's choice (see \autoref{chap:LBC}).

\cmtgm{
  Hyperviscous operators are frequently used in the simulation of turbulent flows to
  control the dissipation of unresolved small scale features.
  Their primary role is to provide strong dissipation at the smallest scale supported by
  the grid while minimizing the impact on the larger scale features.
  Hyperviscous operators are thus designed to be more scale selective than the traditional,
  physically motivated Laplace operator.
  In finite difference methods,
  the biharmonic operator is frequently the method of choice to achieve this scale selective dissipation since
  its damping time (\ie\ its spin down time) scale like $\lambda^{-4}$ for disturbances of wavelength $\lambda$
  (so that short waves damped more rapidelly than long ones),
  whereas the Laplace operator damping time scales only like $\lambda^{-2}$.
}

%% =================================================================================================
\subsection[Iso-level laplacian (\forcode{ln_dynldf_lap})]{Iso-level laplacian operator (\protect\np{ln_dynldf_lap}{ln\_dynldf\_lap})}
\label{subsec:DYN_ldf_lap}

For lateral iso-level diffusion, the discrete operator is:
\begin{equation}
  \label{eq:DYN_ldf_lap}
  \left\{
    \begin{aligned}
      D_u^{l{\mathrm {\mathbf U}}} =\frac{1}{e_{1u} }\delta_{i+1/2} \left[ {A_T^{lm}
          \;\chi } \right]-\frac{1}{e_{2u} {\kern 1pt}e_{3u} }\delta_j \left[
        {A_f^{lm} \;e_{3f} \zeta } \right] \\ \\
      D_v^{l{\mathrm {\mathbf U}}} =\frac{1}{e_{2v} }\delta_{j+1/2} \left[ {A_T^{lm}
          \;\chi } \right]+\frac{1}{e_{1v} {\kern 1pt}e_{3v} }\delta_i \left[
        {A_f^{lm} \;e_{3f} \zeta } \right]
    \end{aligned}
  \right.
\end{equation}

As explained in \autoref{sec:DIFFOPERS_3}, this formulation (as the gradient of a divergence and curl of the vorticity) preserves 
symmetry and ensures a complete separation between the vorticity and divergence parts of the momentum diffusion.

In v5.0 a symetrical lateral iso-level operator has been introduced : 
\begin{equation}
  \label{eq:DYN_ldf_lap_sym}
  \left\{
    \begin{aligned}
      D_u^{l{\mathrm {\mathbf U}}} &= \frac{1}{e_{1u}\,e_{2u}\,e_{3u} } \left( 
      \frac{1}{e_{2u}} \delta_{i+1/2} \left[ e_{2t}\,e_{2t}\,e_{3t}\, A_T^{lm} \,\epsilon_T \right] 
      - \frac{1}{e_{1u}} \delta_{j+1/2} \left[ e_{1f}\,e_{1f}\,e_{3f}\,A_F^{lm} \epsilon_F \right] \right) \\ \\
      D_v^{l{\mathrm {\mathbf U}}} &= \frac{1}{e_{1v}\,e_{2v}\,e_{3v} } \left( 
      \frac{1}{e_{2v}} \delta_{j+1/2} \left[ e_{2f}\,e_{2f}\,e_{3f}\, A_F^{lm} \,\epsilon_F \right] 
      - \frac{1}{e_{1v}} \delta_{i+1/2} \left[ e_{1t}\,e_{1t}\,e_{3t}\,A_T^{lm} \epsilon_T \right] \right)
    \end{aligned}
  \right.
\end{equation}

Where $\epsilon_F$ and $\epsilon_T$ are respectively the shearing stress component (F-point) and the 
tension stress component (T-point)  defined as : 
\begin{equation}
  \label{eq:DYN_ldf_lap_sheten}
  \left\{
    \begin{aligned}
      \epsilon_F &= \frac{e_{1f}}{e_{2f}}\; \delta_{j+1/2} \left[ \frac{u}{e_{1u}} \right] + \frac{e_{2f}}{e_{1f}}\; 
        \delta_{i+1/2} \left[ \frac{v}{e_{2v}} \right] \\ \\
      \epsilon_T &= \frac{e_{2t}}{e_{1t}}\; \delta_{i} \left[ \frac{u}{e_{2u}} \right] - \frac{e_{1t}}{e_{2t}}\; 
        \delta_j \left[ \frac{v}{e_{1v}} \right]
    \end{aligned}
  \right.
\end{equation}

%% =================================================================================================
\subsection[Rotated laplacian (\forcode{ln_dynldf_iso})]{Rotated laplacian operator (\protect\np{ln_dynldf_iso}{ln\_dynldf\_iso})}
\label{subsec:DYN_ldf_iso}

A rotation of the lateral momentum diffusion operator is needed in several cases:
for iso-neutral diffusion in the $z$-coordinate (\np[=.true.]{ln_dynldf_iso}{ln\_dynldf\_iso}) and
for either iso-neutral (\np[=.true.]{ln_dynldf_iso}{ln\_dynldf\_iso}) or
geopotential (\np[=.true.]{ln_dynldf_hor}{ln\_dynldf\_hor}) diffusion in the $s$-coordinate.
In the partial step case, coordinates are horizontal except at the deepest level and
no rotation is performed when \np[=.true.]{ln_dynldf_hor}{ln\_dynldf\_hor}.
The diffusion operator is defined simply as the divergence of down gradient momentum fluxes on
each momentum component.
It must be emphasized that this formulation ignores constraints on the stress tensor such as symmetry.
The resulting discrete representation is:
\begin{equation}
  \label{eq:DYN_ldf_iso}
  \begin{split}
    D_u^{l\textbf{U}} &= \frac{1}{e_{1u} \, e_{2u} \, e_{3u} }	\\
    &  \left\{\quad  {\delta_{i+1/2} \left[ {A_T^{lm}  \left(
              {\frac{e_{2t} \; e_{3t} }{e_{1t} } \,\delta_{i}[u]
                -e_{2t} \; r_{1t} \,\overline{\overline {\delta_{k+1/2}[u]}}^{\,i,\,k}}
            \right)} \right]} 	\right. \\
    & \qquad +\ \delta_j \left[ {A_f^{lm} \left( {\frac{e_{1f}\,e_{3f} }{e_{2f}
            }\,\delta_{j+1/2} [u] - e_{1f}\, r_{2f}
            \,\overline{\overline {\delta_{k+1/2} [u]}} ^{\,j+1/2,\,k}}
        \right)} \right] \\
    &\qquad +\ \delta_k \left[ {A_{uw}^{lm} \left( {-e_{2u} \, r_{1uw} \,\overline{\overline
              {\delta_{i+1/2} [u]}}^{\,i+1/2,\,k+1/2} }
        \right.} \right. \\
    &  \ \qquad \qquad \qquad \quad\
    - e_{1u} \, r_{2uw} \,\overline{\overline {\delta_{j+1/2} [u]}} ^{\,j,\,k+1/2} \\
    & \left. {\left. { \ \qquad \qquad \qquad \ \ \ \left. {\
                +\frac{e_{1u}\, e_{2u} }{e_{3uw} }\,\left( {r_{1uw}^2+r_{2uw}^2}
                \right)\,\delta_{k+1/2} [u]} \right)} \right]\;\;\;} \right\} \\ \\
    D_v^{l\textbf{V}} &= \frac{1}{e_{1v} \, e_{2v} \, e_{3v} } \\
    &  \left\{\quad  {\delta_{i+1/2} \left[ {A_f^{lm}  \left(
              {\frac{e_{2f} \; e_{3f} }{e_{1f} } \,\delta_{i+1/2}[v]
                -e_{2f} \; r_{1f} \,\overline{\overline {\delta_{k+1/2}[v]}}^{\,i+1/2,\,k}}
            \right)} \right]} 	\right. \\
    & \qquad +\ \delta_j \left[ {A_T^{lm} \left( {\frac{e_{1t}\,e_{3t} }{e_{2t}
            }\,\delta_{j} [v] - e_{1t}\, r_{2t}
            \,\overline{\overline {\delta_{k+1/2} [v]}} ^{\,j,\,k}}
        \right)} \right] \\
    & \qquad +\ \delta_k \left[ {A_{vw}^{lm} \left( {-e_{2v} \, r_{1vw} \,\overline{\overline
              {\delta_{i+1/2} [v]}}^{\,i+1/2,\,k+1/2} }\right.} \right. \\
    &  \ \qquad \qquad \qquad \quad\
    - e_{1v} \, r_{2vw} \,\overline{\overline {\delta_{j+1/2} [v]}} ^{\,j+1/2,\,k+1/2} \\
    & \left. {\left. { \ \qquad \qquad \qquad \ \ \ \left. {\
                +\frac{e_{1v}\, e_{2v} }{e_{3vw} }\,\left( {r_{1vw}^2+r_{2vw}^2}
                \right)\,\delta_{k+1/2} [v]} \right)} \right]\;\;\;} \right\}
  \end{split}
\end{equation}
where $r_1$ and $r_2$ are the slopes between the surface along which the diffusion operator acts and
the surface of computation ($z$- or $s$-surfaces).
The way these slopes are evaluated is given in the lateral physics chapter (\autoref{chap:LDF}).

%% =================================================================================================
\subsection[Iso-level bilaplacian (\forcode{ln_dynldf_bilap})]{Iso-level bilaplacian operator (\protect\np{ln_dynldf_bilap}{ln\_dynldf\_bilap})}
\label{subsec:DYN_ldf_bilap}

The lateral fourth order operator formulation on momentum is obtained by applying \autoref{eq:DYN_ldf_lap} twice.
It requires an additional assumption on boundary conditions:
the first derivative term normal to the coast depends on the free or no-slip lateral boundary conditions chosen,
while the third derivative terms normal to the coast are set to zero (see \autoref{chap:LBC}).
\cmtgm{add a remark on the the change in the position of the coefficient}

%% =================================================================================================
\section[Vertical diffusion term (\textit{dynzdf.F90})]{Vertical diffusion term (\protect\mdl{dynzdf})}
\label{sec:DYN_zdf}

Options are defined through the \nam{zdf}{zdf} namelist variables.
The 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.
In v5.0 only a backward (or implicit) time differencing scheme can be used for the vertical diffusion term.
(see \autoref{chap:TD}).

The formulation of the vertical subgrid scale physics is the same whatever the vertical coordinate is.
The vertical diffusion operators given by \autoref{eq:MB_zdf} take the following semi-discrete space form:
\[
  % \label{eq:DYN_zdf}
  \left\{
    \begin{aligned}
      D_u^{vm} &\equiv \frac{1}{e_{3u}} \ \delta_k \left[ \frac{A_{uw}^{vm} }{e_{3uw} }
        \ \delta_{k+1/2} [\,u\,]         \right]     \\
      \\
      D_v^{vm} &\equiv \frac{1}{e_{3v}} \ \delta_k \left[ \frac{A_{vw}^{vm} }{e_{3vw} }
        \ \delta_{k+1/2} [\,v\,]         \right]
    \end{aligned}
  \right.
\]
where $A_{uw}^{vm} $ and $A_{vw}^{vm} $ are the vertical eddy viscosity and diffusivity coefficients.
The way these coefficients are evaluated depends on the vertical physics used (see \autoref{chap:ZDF}).

The surface boundary condition on momentum is the stress exerted by the wind.
At the surface, the momentum fluxes are prescribed as the boundary condition on
the vertical turbulent momentum fluxes,
\begin{equation}
  \label{eq:DYN_zdf_sbc}
  \left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{z=1}
  = \frac{1}{\rho_o} \binom{\tau_u}{\tau_v }
\end{equation}
where $\left( \tau_u ,\tau_v \right)$ are the two components of the wind stress vector in
the (\textbf{i},\textbf{j}) coordinate system.
The high mixing coefficients in the surface mixed layer ensure that the surface wind stress is distributed in
the vertical over the mixed layer depth.
If the vertical mixing coefficient is small (when no mixed layer scheme is used)
the surface stress enters only the top model level, as a body force.
The surface wind stress is calculated in the surface module routines (SBC, see \autoref{chap:SBC}).

The turbulent flux of momentum at the bottom of the ocean is specified through a bottom friction parameterisation
(see \autoref{sec:ZDF_drg})

When activated (\np[=.true.]{ln_zad_Aimp}{ln\_zad\_Aimp}) vertical advection of momentum is 
done partly implicitly in areas where there is potential to breach the vertical CFL condition 
(see \autoref{subsec:ZDF_aimp}). The following expressions are included in the implicit solvers in 
\textit{dynzdf.F90} and \textit{trazdf.F90}:

In vector form case :
\begin{equation}
  \left\{
    \begin{aligned}
      w_i \, \delta_{k} [u] &= \frac{1}{e_{1u}\,e_{2u}} \overline{\overline{ e_{1t}\,e_{2t}\,w_i }}^{\,i,k+1/2} \, \delta_{k} \left[\frac{u}{e_{3uw}}\right]^{up1}, \\[10pt]
      w_i \, \delta_{k} [v] &= \frac{1}{e_{1v}\,e_{2v}} \overline{\overline{ e_{1t}\,e_{2t}\,w_i }}^{\,j,k+1/2} \, \delta_{k} \left[\frac{v}{e_{3vw}}\right]^{up1}.
    \end{aligned}
  \right.
\end{equation}

In flux form case :
\begin{equation}
  \left\{
    \begin{aligned}
      \delta_{k} [w_i u] &= \frac{1}{e_{1u}\,e_{2u}\,e_{3u}} \left( \left( \overline{ e_{1t}\,e_{2t}\,w_i }^{\,i} \, u \right)^{k,up1} - \left( \overline{ e_{1t}\,e_{2t}\,w_i }^{\,i} \, u \right)^{k+1,up1}\right), \\[10pt]
      \delta_{k} [w_i v] &= \frac{1}{e_{1v}\,e_{2v}\,e_{3v}} \left(\left( \overline{ e_{1t}\,e_{2t}\,w_i }^{\,j} \, v \right)^{k,up1} - \left( \overline{ e_{1t}\,e_{2t}\,w_i }^{\,j} \, v \right)^{k+1,up1}\right).
    \end{aligned}
  \right.
\end{equation}

where $w_i$ is the part of the vertical velocity to be treated implicitly