source: trunk/DOC/BETA/Chapters/Chap_DYN.tex @ 817

% ================================================================
% Chapter Ñ Ocean Dynamics (DYN)
% ================================================================
\chapter{Ocean Dynamics (DYN)}
\label{DYN}
\minitoc

% add a figure for  dynvor ens, ene latices


$\ $\newline      %force an empty line

Using the representation described in Chap.\ref{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 a 
average or difference operator is used, the resulting field is multiplied by a mask.

The prognostic ocean dynamics equation can be summarized as follows:
\begin{equation*}
\text{NXT} = \dbinom {\text{VOR} + \text{KEG} + \text {ZAD} }
                  {\text{COR} + \text{ADV}                       }
         + \text{HPG} + \text{SPG} + \text{LDF} + \text{ZDF}
\end{equation*}

NXT stands for next, referring to the time-stepping. The first group of terms on 
the rhs of the momentum equations corresponds to the Coriolis and advection 
terms that are decomposed into 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 Chapters \ref{SBC}, \ref{LDF} and \ref{ZDF}, respectively. 

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

The different options available to the user are managed by namelist variables. 
For equation term \textit{ttt}, 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 \textbf{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 each tendency term of both the rhs of the 
3D momentum equation (\key{trddyn} defined) for output, as described in 
Chap.\ref{MISC}.  Furthermore, the tendency terms associated to the 2D 
barotropic vorticity balance (\key{trdvor} defined) can be derived on-line from the 
3D terms.
%%%
\gmcomment{STEVEN: not quite sure I've got the sense of the last sentence. does MISC correspond to "extracting tendency terms" or "vorticity balance"?}

% ================================================================
% Coriolis and Advection terms: vector invariant form
% ================================================================
\section{Coriolis and Advection: vector invariant form}
\label{DYN_adv_cor_vect}
%-----------------------------------------nam_dynadv----------------------------------------------------
\namdisplay{nam_dynadv} 
%-------------------------------------------------------------------------------------------------------------

The vector invariant form of the momentum equations is the one most 
often used in applications of \NEMO ocean model. The flux form option 
(see next section) has been recently introduced in version $2$. 
Coriolis and momentum 
advection terms are evaluated using a leapfrog scheme, $i.e.$ the velocity 
appearing in these expressions is centred in time (\textit{now} velocity). 
At the lateral boundaries either free slip, no slip or partial slip boundary 
conditions are applied following Chap.\ref{LBC}.

% -------------------------------------------------------------------------------------------------------------
%        Vorticity term 
% -------------------------------------------------------------------------------------------------------------
\subsection   [Vorticity term (\textit{dynvor}) ]
         {Vorticity term (\mdl{dynvor})}
\label{DYN_vor}
%------------------------------------------nam_dynvor----------------------------------------------------
\namdisplay{nam_dynvor} 
%-------------------------------------------------------------------------------------------------------------

Different discretisations of the vorticity term (\textit{ln\_dynvor\_xxx}=.true.) are 
available: conserving potential enstrophy of horizontally non-divergent flow; 
conserving horizontal kinetic energy; or conserving potential enstrophy for the 
relative vorticity term and horizontal kinetic energy for the planetary vorticity term 
(see  Appendix~\ref{Apdx_C}). The vorticity terms are given below for the general 
case, but note that in the full step $z$-coordinate (\key{zco} is defined), 
$e_{3u} =e_{3v} =e_{3f}$ so that the vertical scale factors disappear.

%-------------------------------------------------------------
%                 enstrophy conserving scheme
%-------------------------------------------------------------
\subsubsection{Enstrophy conserving scheme (\np{ln\_dynvor\_ens}=.true.)}
\label{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 ($i.e.$ $\chi=0$), but does not conserve the total kinetic energy. It is given by:
\begin{equation} \label{Eq_dynvor_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 (\np{ln\_dynvor\_ene}=.true.)}
\label{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_dynvor_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 (\np{ln\_dynvor\_mix}=.true.) }
\label{DYN_vor_mix}

The mixed energy/enstrophy conserving scheme (MIX scheme), a mixture of the 
two previous schemes is used. It consists of the ENS scheme (\ref{Eq_dynvor_ens}) 
to the relative vorticity term, and of the ENE scheme (\ref{Eq_dynvor_ene}) applied 
to the planetary vorticity term.
\begin{equation} \label{Eq_dynvor_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.
\end{equation} 

%-------------------------------------------------------------
%                 energy and enstrophy conserving scheme
%-------------------------------------------------------------
\subsubsection{Energy and enstrophy conserving scheme (\np{ln\_dynvor\_een}=.true.) }
\label{DYN_vor_een}

In the energy and enstrophy conserving scheme (EEN scheme), the vorticity term 
is  evaluated using the vorticity advection scheme of \citet{Arakawa1990}. 
This scheme conserves both total energy and potential enstrophy in the limit of 
horizontally nondivergent flow ($i.e. \ \chi=0$). While EEN is more complicated 
than ENS or ENE and does not conserve potential enstrophy and total energy in 
general flow, it tolerates arbitrarily thin layers. This feature is essential for 
$z$-coordinate with partial step. 
%%%
\gmcomment{gm :   it actually conserve kinetic energy  !   show that in appendix C }
%%%

The \citet{Arakawa1990} vorticity advection scheme for a single layer is modified 
for spherical coordinates as described by \citet{Arakawa1981} to obtain the EEN 
scheme. The potential vorticity, defined at an $f$-point, is: 
\begin{equation} \label{Eq_pot_vor}
q_f  = \frac{\zeta +f} {e_{3f} }
\end{equation}
where the relative vorticity is defined by (\ref{Eq_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_een_e3f}
e_{3f} = \overline{\overline {e_{3t} }} ^{\,i+1/2,j+1/2}
\end{equation}

%>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
\begin{figure}[!ht] \label{Fig_DYN_een_triad}
\begin{center}
\includegraphics[width=0.70\textwidth]{./Figures/Fig_DYN_een_triad.pdf}
\caption{Triads used in the energy and enstrophy conserving scheme (een) for 
$u$-component (upper panel) and $v$-component (lower panel).}
\end{center}
\end{figure}
%>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>

Note that a key point in \eqref{Eq_een_e3f} is that the averaging in \textbf{i}- and 
\textbf{j}- directions uses the masked vertical scale factor but is always divided by 
$4$, not by the sum of the mask at $T$-point. This 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 $e_{3f}$ in the land areas. 
%%%
\gmcomment{this has to be further investigate in case of several step topography}
%%%

The vorticity terms are represented as: 
\begin{equation} \label{Eq_dynvor_een}
\left\{ {
\begin{aligned}
 +q\,e_3 \, v  &\equiv +\frac{1}{e_{1u} }  \left[ 
{{\begin{array}{*{20}c}
      {\,\ \ a_{j+1/2}^{i   } \left( {e_{1v} e_{3v} \ v} \right)_{j+1}^{i+1/2} } 
   {\,+\,b_{j+1/2}^{i   } \left( {e_{1v} e_{3v} \ v} \right)_{j+1}^{i-1/2}  } \\
 \\
     {  +\,c_{j-1/2}^{i   }  \left( {e_{1v} e_{3v} \ v} \right)_{j    }^{i+1/2}         } 
   {\,+\,d_{j+1/2}^{i   } \left( {e_{1v} e_{3v} \ v} \right)_{j+1}^{i+1/2} } \\
\end{array} }} \right] \\ 
\\
-q\,e_3 \,u       &\equiv -\frac{1}{e_{2v} }  \left[ 
{{\begin{array}{*{20}c}
   {\,\ \ a_{j-1/2}^{i   }  \left( {e_{2u} e_{3v} \ u} \right)_{j+1}^{i+1/2} } 
   {\,+\,b_{j-1/2}^{i+1}  \left( {e_{2u} e_{3v} \ u} \right)_{j+1/2}^{i+1} } \\
 \\
      {  +\,c_{j+1/2}^{i+1} \left( {e_{2u} e_{3v} \ u} \right)_{j+1/2}^{i+1} } 
   {\,+\,d_{j+1/2}^{i   }  \left( {e_{2u} e_{3v} \ u} \right)_{j+1/2}^{i   } } \\
\end{array} }} \right]
\end{aligned} 
} \right.
\end{equation} 
where $a$, $b$, $c$ and $d$ are triad combinations of the neighbouring 
potential vorticities (Fig. \ref{Fig_DYN_een_triad}): 
\begin{equation} \label{Eq_een_triads}
\left\{ 
\begin{aligned}
 a_{\,j+1/2}^i & = \frac{1}  {12} \left( q_{j+1/2}^{i+1} + q_{j+1 /2}^i + q_{j-1/2}^i  \right)    \\ 
 \\
 b_{\,j+1/2}^i & = \frac{1}  {12} \left( q_{j+1/2}^{i-1} +q_{j+1/2}^i +q_{j-1/2}^i   \right)     \\ 
\\
 c_{\,j+1/2}^i & = \frac{1}  {12} \left( q_{j-1/2}^{i-1} +q_{j+1/2}^i +q_{j-1/2}^i   \right)     \\ 
\\
 d_{\,j+1/2}^i & = \frac{1}  {12} \left( q_{j-1/2}^{i+1} +q_{j+1/2}^i +q_{j-1/2}^i  \right)     \\ 
\end{aligned} 
\right.
\end{equation}

%--------------------------------------------------------------------------------------------------------------
%           Kinetic Energy Gradient term
%--------------------------------------------------------------------------------------------------------------
\subsection   [Kinetic Energy Gradient term (\textit{dynkeg})]
         {Kinetic Energy Gradient term (\mdl{dynkeg})}
\label{DYN_keg}

As demonstarted in Appendix~\ref{Apdx_C}, 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:
\begin{equation} \label{Eq_dynkeg}
\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.
\end{equation} 

%--------------------------------------------------------------------------------------------------------------
%           Vertical advection term
%--------------------------------------------------------------------------------------------------------------
\subsection   [Vertical advection term (\textit{dynzad}) ]
         {Vertical advection term (\mdl{dynzad}) }
\label{DYN_zad}

The discrete formulation of the vertical advection, together 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 Appendix~\ref{Apdx_C}).
\begin{equation} \label{Eq_dynzad}
\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.
\end{equation} 

% ================================================================
% Coriolis and Advection : flux form
% ================================================================
\section{Coriolis and Advection: flux form}
\label{DYN_adv_cor_flux}
%------------------------------------------nam_dynadv----------------------------------------------------
\namdisplay{nam_dynadv} 
%-------------------------------------------------------------------------------------------------------------

In the flux form (as in the vector invariant form), the Coriolis and momentum 
advection terms are evaluated using a leapfrog scheme, $i.e.$ the velocity 
appearing in their expressions is centred in time (\textit{now} velocity). At the 
lateral boundaries either free slip, no slip or partial slip boundary conditions 
are applied following Chap.\ref{LBC}.


%--------------------------------------------------------------------------------------------------------------
%           Coriolis plus curvature metric terms
%--------------------------------------------------------------------------------------------------------------
\subsection   [Coriolis plus curvature metric terms (\textit{dynvor}) ]
         {Coriolis plus curvature metric terms (\mdl{dynvor}) }
\label{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_dyncor_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} 

Any of the (\ref{Eq_dynvor_ens}), (\ref{Eq_dynvor_ene}) and (\ref{Eq_dynvor_een}) 
schemes can be used to compute the product of the Coriolis parameter and the 
vorticity. However, the energy-conserving scheme  (\ref{Eq_dynvor_een}) has 
exclusively been used to date. This term is evaluated using a leapfrog scheme, 
$i.e.$ the velocity is centred in time (\textit{now} velocity).

%--------------------------------------------------------------------------------------------------------------
%           Flux form Advection term
%--------------------------------------------------------------------------------------------------------------
\subsection   [Flux form Advection term (\textit{dynadv}) ]
         {Flux form Advection term (\mdl{dynadv}) }
\label{DYN_adv_flux}

The discrete expression of the advection term is given by :
\begin{equation} \label{Eq_dynadv}
\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.
\end{equation}

Two advection schemes are available: a $2^{nd}$ order centered finite 
difference scheme, CEN2, or a $3^{rd}$ order upstream biased scheme, UBS. 
The latter is described in \citet{Sacha2005}. The schemes are selected using 
the namelist logicals \np{ln\_dynadv\_cen2} and \np{ln\_dynadv\_ubs}. 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, 
$i.e.$ at the $T$-, $f$-, and $uw$-points for $u$ and at the $f$-, $T$- and 
$vw$-points for $v$. 

%-------------------------------------------------------------
%                 2nd order centred scheme
%-------------------------------------------------------------
\subsubsection{$2^{nd}$ order centred scheme (cen2) (\np{ln\_dynadv\_cen2}=.true.)}
\label{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_dynadv_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 (i.e. conserves the kinetic energy) but dispersive 
($i.e.$ 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.

%-------------------------------------------------------------
%                 UBS scheme
%-------------------------------------------------------------
\subsubsection{Upstream Biased Scheme (UBS) (\np{ln\_dynadv\_ubs}=.true.)}
\label{DYN_adv_ubs}

The UBS advection scheme is an upstream biased third order scheme based on 
an upstream-biased parabolic interpolation. For example, the evaluation of 
$u_T^{ubs} $ is done as follows:
\begin{equation} \label{Eq_dynadv_ubs}
u_T^{ubs} =\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 ($i.e.$ hyper-diffusive) truncation error \citep{Sacha2005}. 
The overall performance of the advection scheme is similar to that reported in 
\citet{Farrow1995}. 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. 

The UBS scheme is not used in all directions. In the vertical, the centred $2^{nd}$ 
order evaluation of the advection is preferred, $i.e.$ $u_{uw}^{ubs}$ and 
$u_{vw}^{ubs}$ in \eqref{Eq_dynadv_cen2} are used. UBS is diffusive and is 
associated with vertical mixing of momentum. \gmcomment{ gm  pursue the 
sentence:Since vertical mixing of momentum is a source term of the TKE equation...  }

For stability reasons,  the first term in (\ref{Eq_dynadv_ubs}), 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 diffusive part of the scheme, 
is evaluated using the \textit{before} velocity (forward in time). This is discussed 
by \citet{Webb1998} in the context of the Quick advection scheme.

Note that the UBS and Quadratic Upstream Interpolation for Convective Kinematics 
(QUICK) schemes only differ by one coefficient. Substituting $1/6$ with $1/8$ in 
(\ref{Eq_dynadv_ubs}) leads to the QUICK advection scheme \citep{Webb1998}. 
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 
\mdl{dynadv\_ubs} module and obtain a QUICK scheme.

Note also that in the current version of \mdl{dynadv\_ubs}, there is also the 
possibility of using a $4^{th}$ order evaluation of the advective velocity as in 
ROMS. This is an error and should be suppressed soon.
%%%
\gmcomment{action :  this have to be done}
%%%

% ================================================================
%           Hydrostatic pressure gradient term
% ================================================================
\section  [Hydrostatic pressure gradient (\textit{dynhpg})]
      {Hydrostatic pressure gradient (\mdl{dynhpg})}
\label{DYN_hpg}
%------------------------------------------nam_dynhpg---------------------------------------------------
\namdisplay{nam_dynhpg} 
\namdisplay{namflg} 
%-------------------------------------------------------------------------------------------------------------
%%%
\gmcomment{Suppress the namflg namelist and incorporate it in the namhpg namelist}
%%%

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, $i.e.$ 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.

The hydrostatic pressure gradient term is evaluated either using a leapfrog scheme, 
$i.e.$ the density appearing in its expression is centred in time (\emph{now} rho), or 
a semi-implcit scheme. At the lateral boundaries either free slip, no slip or partial slip 
boundary conditions are applied.

%--------------------------------------------------------------------------------------------------------------
%           z-coordinate with full step
%--------------------------------------------------------------------------------------------------------------
\subsection   [$z$-coordinate with full step (\np{ln\_dynhpg\_zco}) ]
         {$z$-coordinate with full step (\np{ln\_dynhpg\_zco}=.true.)}
\label{DYN_hpg_zco}

The hydrostatic pressure can be obtained by integrating the hydrostatic equation 
vertically from the surface. However, the pressure is large at great depth 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_dynhpg_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_dynhpg_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} 
+    \frac{1}{2}\;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} 
+    \frac{1}{2}\;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 (\ref{Eq_dynhpg_zco_surf}) is adequate because of 
the definition of $e_{3w}$ as the vertical derivative of the scale factor at the surface 
level ($z=0)$. 

%--------------------------------------------------------------------------------------------------------------
%           z-coordinate with partial step
%--------------------------------------------------------------------------------------------------------------
\subsection   [$z$-coordinate with partial step (\np{ln\_dynhpg\_zps})]
         {$z$-coordinate with partial step (\np{ln\_dynhpg\_zps}=.true.)}
\label{DYN_hpg_zps}

With partial bottom cells, tracers in horizontally adjacent cells generally live at 
different depths. Before taking horizontal gradients between these tracer points, 
a linear interpolation is used to approximate the deeper tracer as if it actually lived 
at the depth of the shallower tracer point. 

Apart from this modification, the horizontal hydrostatic pressure gradient evaluated 
in the $z$-coordinate with partial step is exactly as in the pure $z$-coordinate case. 
As explained in detail in section \S\ref{TRA_zpshde}, the nonlinearity of pressure 
effects in the equation of state is such that it is better to interpolate temperature and 
salinity vertically before computing the density. Horizontal gradients of temperature 
and salinity are needed for the TRA modules, which is the reason why the horizontal 
gradients of density at the deepest model level are computed in module \mdl{zpsdhe} 
located in the TRA directory and described in \S\ref{TRA_zpshde}.

%--------------------------------------------------------------------------------------------------------------
%           s- and s-z-coordinates
%--------------------------------------------------------------------------------------------------------------
\subsection{$s$- and $z$-$s$-coordinates}
\label{DYN_hpg_sco}

Pressure gradient formulations in $s$-coordinate have been the subject of a vast 
literature ($e.g.$, \citet{Song1998, Sacha2003}). A number of different pressure 
gradient options are coded, but they are not yet fully documented or tested. 

$\bullet$ Traditional coding (see for example \citet{Madec1996}: (\np{ln\_dynhpg\_sco}=.true., 
\np{ln\_dynhpg\_hel}=.true.)
\begin{equation} \label{Eq_dynhpg_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 
\eqref{Eq_dynhpg_zco_surf} - \eqref{Eq_dynhpg_zco}, and $z_T$ is the depth of 
the $T$-point evaluated from the sum of the vertical scale factors at the $w$-point 
($e_{3w}$). The version \np{ln\_dynhpg\_hel}=.true. has been added by Aike 
Beckmann and involves a redefinition of the relative position of $T$-points relative 
to $w$-points. 

$\bullet$ Weighted density Jacobian (WDJ) \citep{Song1998} (\np{ln\_dynhpg\_wdj}=.true.)

$\bullet$ Density Jacobian with cubic polynomial scheme (DJC) \citep{Sacha2003} 
(\np{ln\_dynhpg\_djc}=.true.)

$\bullet$ Rotated axes scheme (rot) \citep{Thiem2006} (\np{ln\_dynhpg\_rot}=.true.)

Note that expression \eqref{Eq_dynhpg_sco} is used when the variable volume 
formulation is activated (\key{vvl}) because in that case, even with a flat bottom, 
the coordinate surfaces are not horizontal but follow the free surface 
\citep{Levier2007}. The other pressure gradient options are not yet available.

%--------------------------------------------------------------------------------------------------------------
%           Time-scheme
%--------------------------------------------------------------------------------------------------------------
\subsection   [Time-scheme (\np{ln\_dynhpg\_imp}) ]
         {Time-scheme (\np{ln\_dynhpg\_imp}=.true./.false.)}
\label{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 phenomenum 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{Brown1978, 
Maltrud1998}. It involves the evaluation of the hydrostatic pressure gradient as an 
average over the three time levels $t-\Delta t$, $t$, and $t+\Delta t$ ($i.e.$  
\textit{before},  \textit{now} and  \textit{after} time-steps), rather than at central 
time level $t$ only, as in the standard leapfrog scheme. 

$\bullet$ leapfrog scheme (\np{ln\_dynhpg\_imp}=.true.):

\begin{equation} \label{Eq_dynhpg_lf}
\frac{u^{t+\Delta t}-u^{t-\Delta t}}{2\Delta t}
=\;\cdots \;-\frac{1}{\rho _o \,e_{1u} }\delta _{i+1/2} \left[ {p_h^t } \right]
\end{equation}

$\bullet$ semi-implicit scheme (\np{ln\_dynhpg\_imp}=.true.):
\begin{equation} \label{Eq_dynhpg_imp}
\frac{u^{t+\Delta t}-u^{t-\Delta t}}{2\Delta t}
=\;\cdots \;-\frac{1}{\rho _o \,e_{1u} } \delta _{i+1/2} \left[ \frac{ p_h^{t+\Delta t} +2p_h^t 
+p_h^{t-\Delta t} } { 4 }  \right]
\end{equation}

The semi-implicit time scheme \eqref{Eq_dynhpg_imp} is made possible without 
significant additional computation since the density can be updated to time level 
$t+\Delta t$ before computing the horizontal hydrostatic pressure gradient. It can 
be easily shown that the stability limit associated with the hydrostatic pressure 
gradient doubles using \eqref{Eq_dynhpg_imp} compared to that using the 
standard leapfrog scheme \eqref{Eq_dynhpg_lf}. Note that \eqref{Eq_dynhpg_imp} 
is equivalent to applying a time filter to the pressure gradient to eliminate high 
frequency IGWs. Obviously, when using \eqref{Eq_dynhpg_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{ln\_dynhpg\_imp}=.true.. 
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:
\begin{equation} \label{Eq_rho_flt}
   \rho^t = \rho( \widetilde{T},\widetilde {S},z_T)
 \quad     \text{with}  \quad 
   \widetilde{\,\cdot\,} = \frac{  \,\cdot\,^{t+\Delta t} +2 \,\,\cdot\,^t + \,\cdot\,^{t-\Delta t}  } {4}
\end{equation}
\gmcomment{STEVEN: bullets look odd in this, could use X}

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 the namelist parameter 
\np{nn\_dynhpg\_rst}=.true..

% ================================================================
% Surface Pressure Gradient
% ================================================================
\section  [Surface pressure gradient (\textit{dynspg}) ]
      {Surface pressure gradient (\mdl{dynspg})}
\label{DYN_hpg_spg}
%-----------------------------------------nam_dynspg----------------------------------------------------
\namdisplay{nam_dynspg} 
%------------------------------------------------------------------------------------------------------------

The form of the surface pressure gradient term is dependent on the representation 
of the free surface (\S\ref{PE_hor_pg}). The main distinction is between the fixed 
volume case (linear free surface or rigid lid) and the variable volume case 
(nonlinear free surface, \key{vvl} is defined). In the linear free surface case 
(\S\ref{PE_free_surface}) and the rigid lid case (\S\ref{PE_rigid_lid}), the vertical 
scale factors $e_{3}$ are fixed in time, whilst in the nonlinear case 
(\S\ref{PE_free_surface}) they are time-dependent. With both linear and nonlinear 
free surface, external gravity waves are allowed in the equations, which imposes 
a very small time step when an explicit time stepping is used. Two methods are 
proposed to allow a longer time step for the three-dimensional equations: the 
filtered free surface method, which involves a modification of the continuous 
equations (see \eqref{Eq_PE_flt}), and the split-explicit free surface method 
described below. The extra term introduced in the filtered method is calculated 
implicitly, so that the update of the $next$ velocities is done in module 
\mdl{dynspg\_flt} and not in \mdl{dynnxt}.

%--------------------------------------------------------------------------------------------------------------
% Linear free surface formulation
%--------------------------------------------------------------------------------------------------------------
\subsection{Linear free surface formulation (\key{exp} or \textbf{\_ts} or \textbf{\_flt})}
\label{DYN_spg_linear}

In the linear free surface formulation, the sea surface height is assumed to be 
small compared to the thickness of the ocean levels, so that $(a)$ the time 
evolution of the sea surface height becomes a linear equation, and $(b)$ the 
thickness of the ocean levels is assumed to be constant with time. 
As mentioned in (\S\ref{PE_free_surface}) the linearization affects the 
conservation of tracers. 

%-------------------------------------------------------------
% Explicit
%-------------------------------------------------------------
\subsubsection{Explicit (\key{dynspg\_exp})}
\label{DYN_spg_exp}

In the explicit free surface formulation, the model time step is chosen to be 
small enough to describe the external gravity waves (typically a few tens of 
seconds). The sea surface height is given by :
\begin{equation} \label{Eq_dynspg_ssh}
\frac{\partial \eta }{\partial t}\equiv \frac{\text{EMP}}{\rho _w }+\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)} 
\end{equation}

where EMP is the surface freshwater budget, expressed in $Kg.m^{-2}.s^{-1}$, 
and $\rho _w =1,000\,Kg.m^{-3}$ is the volumic mass of pure water. If river 
runoff is expressed as a surface freshwater flux, see \S\ref{SBC}, then EMP 
can be written as the evaporation minus precipitation, minus the river runoff. 
The sea-surface height is evaluated using a leapfrog scheme in combination 
with an Asselin time filter, $i.e.$ the velocity appearing in 
\eqref{Eq_dynspg_ssh} is centred in time (\textit{now} velocity). 

The surface pressure gradient, also evaluated using a leap-frog scheme, is 
then simply given by :
\begin{equation} \label{Eq_dynspg_exp}
\left\{ \begin{aligned}
 - \frac{1}                      {e_{1u}} \; \delta _{i+1/2} \left[  \,\eta\,  \right]    \\
 \\
 - \frac{1}                      {e_{2v}} \; \delta _{j+1/2} \left[  \,\eta\,  \right]  
\end{aligned} \right.
\end{equation} 

Consistent with the linearization, a factor of $\left. \rho \right|_{k=1} / \rho _o$ 
is omitted in \eqref{Eq_dynspg_exp}. 

%-------------------------------------------------------------
% Split-explicit time-stepping
%-------------------------------------------------------------
\subsubsection{Split-explicit time-stepping (\key{dynspg\_ts})}
\label{DYN_spg_ts}
%--------------------------------------------namdom----------------------------------------------------
\namdisplay{namdom} 
%--------------------------------------------------------------------------------------------------------------

The split-explicit free surface formulation used in \NEMO follows the one 
proposed by \citet{Griffies2004}. The general idea is to solve the free surface 
equation with a small time step \np{rdtbt}, while the three dimensional 
prognostic variables are solved with a longer time step that is a multiple of \np{rdtbt} 
(Fig.\ref {Fig_DYN_dynspg_ts}). 

%>   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >
\begin{figure}[!t] \label{Fig_DYN_dynspg_ts}
\begin{center}
\includegraphics[width=0.90\textwidth]{./Figures/Fig_DYN_dynspg_ts.pdf}
\caption{Schematic of the split-explicit time stepping scheme for the barotropic 
and baroclinic modes, after \citet{Griffies2004}. Time increases to the right. 
Baroclinic time steps are denoted by $t-\Delta t$, $t, t+\Delta t$, and $t+2\Delta t$. 
The curved line represents a leap-frog time step, and the smaller barotropic time 
steps $N \Delta t=2\Delta t$ are denoted by the zig-zag line. The vertically 
integrated forcing \textbf{M}(t) computed at the baroclinic time step $t$ 
represents the interaction between the barotropic and baroclinic motions. 
While keeping the total depth, tracer, and freshwater forcing fields fixed, a 
leap-frog integration carries the surface height and vertically integrated velocity 
from $t$ to $t+2 \Delta t$ using N barotropic time steps of length $\Delta t$. 
Time averaging the barotropic fields over the N+1 time steps (endpoints 
included) centers the vertically integrated velocity at the baroclinic timestep 
$t+\Delta t$. A baroclinic leap-frog time step carries the surface height to 
$t+\Delta t$ using the convergence of the time averaged vertically integrated 
velocity taken from baroclinic time step t. }
%%%
\gmcomment{STEVEN: what does convergence mean in this context?}
%%%
\end{center}
\end{figure}
%>   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >

The split-explicit formulation has a damping effect on external gravity waves, 
which is weaker damping than for the filtered free surface but still significant as 
shown by \citet{Levier2007} in the case of an analytical barotropic Kelvin wave. 

%-------------------------------------------------------------
% Filtered formulation 
%-------------------------------------------------------------
\subsubsection{Filtered formulation (\key{dynspg\_flt})}
\label{DYN_spg_flt}

The filtered formulation follows the \citet{Roullet2000} implementation. The extra 
term introduced in the equations (see {\S}I.2.2) is solved implicitly. The elliptic 
solvers available in the code are documented in \S\ref{MISC}. The amplitude of 
the extra term is given by the namelist variable \np{rnu}. The default value is 1, 
as recommended by \citet{Roullet2000}

\gmcomment{\np{rnu}=1 to be suppressed from namelist !}

%-------------------------------------------------------------
% Non-linear free surface formulation 
%-------------------------------------------------------------
\subsection{Non-linear free surface formulation (\key{vvl})}
\label{DYN_spg_vvl}

In the non-linear free surface formulation, the variations of volume are fully 
taken into account. This option is presented in a report \citep{Levier2007} 
available on the \NEMO web site. The three time-stepping methods (explicit, 
split-explicit and filtered) are the same as in \S\ref{DYN_spg_linear} except 
that the ocean depth is now time-dependent. In particular, this means that 
in the filtered case, the matrix to be inverted has to be recomputed at each 
time-step.

%--------------------------------------------------------------------------------------------------------------
%           Rigid-lid formulation
%--------------------------------------------------------------------------------------------------------------
\subsection{Rigid-lid formulation (\key{dynspg\_rl})}
\label{DYN_spg_rl}

With the rigid lid formulation, an elliptic equation has to be solved for 
the barotropic streamfunction. For consistency this equation is obtained by 
taking the discrete curl of the discrete vertical sum of the discrete 
momentum equation: 
\begin{equation}\label{Eq_dynspg_rl}
\frac{1}{\rho _o }\nabla _h p_s \equiv \left( {{\begin{array}{*{20}c}
 {\overline M_u +\frac{1}{H\;e_2 } \delta_ j \left[ \partial_t \psi \right]}      \\
 \\
 {\overline M_v -\frac{1}{H\;e_1 }  \delta_i  \left[ \partial_t \psi \right]}        \\
\end{array} }} \right)
\end{equation}

Here ${\rm {\bf M}}= \left( M_u,M_v \right)$ represents the collected 
contributions of nonlinear, viscous and hydrostatic pressure gradient terms in 
\eqref{Eq_PE_dyn} and the overbar indicates a vertical average over the 
whole water column (i.e. from $z=-H$, the ocean bottom, to $z=0$, the rigid-lid). 
The time derivative of $\psi$ is the solution of an elliptic equation:
\begin{multline} \label{Eq_bsf}
   \delta_{i+1/2} \left[ \frac{e_{2v}}{H_v\;e_{1v}} \delta_{i} \left[  \partial_t \psi \right] \right]
+ \delta_{j+1/2} \left[ \frac{e_{1u}}{H_u\;e_{2u}} \delta_{j} \left[  \partial_t \psi \right] \right]
\\ =
  \delta_{i+1/2} \left[ e_{2v} M_v  \right]
- \delta_{j+1/2} \left[ e_{1u} M_u  \right]
\end{multline}

The elliptic solvers available in the code are documented in \S\ref{MISC}). 
The boundary conditions must be given on all separate landmasses (islands), 
which is done by integrating the vorticity equation around each island. This 
requires identifying each island in the bathymetry file, a cumbersome task. 
This explains why the rigid lid option is not recommended with complex 
domains such as the global ocean. Parameters jpisl (number of islands) and 
jpnisl (maximum number of points per island) of the \hf{par\_oce} file are 
related to this option.


% ================================================================
% Lateral diffusion term
% ================================================================
\section  [Lateral diffusion term (\textit{dynldf})]
      {Lateral diffusion term (\mdl{dynldf})}
\label{DYN_ldf}
%------------------------------------------nam_dynldf----------------------------------------------------
\namdisplay{nam_dynldf} 
%-------------------------------------------------------------------------------------------------------------

The options available for lateral diffusion are for the choice of  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 lateralphysics (Chap.\ref{LDF}). The lateral diffusion of momentum is 
evaluated using a forward scheme, i.e. 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 \S\ref{DOM_nxt}) 

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

% ================================================================
\subsection   [Iso-level laplacian operator (\np{ln\_dynldf\_lap}) ]
         {Iso-level laplacian operator (\np{ln\_dynldf\_lap}=.true.)}
\label{DYN_ldf_lap}

For lateral iso-level diffusion, the discrete operator is: 
\begin{equation} \label{Eq_dynldf_lap}
\left\{ \begin{aligned}
 D_u^{l{\rm {\bf 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{\rm {\bf 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 \S\ref{PE_ldf}, 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. Note that in the full step 
$z$-coordinate (\key{zco} is defined), $e_{3u} =e_{3v} =e_{3f}$ so that they 
cancel in the rotational part of \eqref{Eq_dynldf_lap}.

%--------------------------------------------------------------------------------------------------------------
%           Rotated laplacian operator
%--------------------------------------------------------------------------------------------------------------
\subsection   [Rotated laplacian operator (\np{ln\_dynldf\_iso}) ]
         {Rotated laplacian operator (\np{ln\_dynldf\_iso}=.true.)}
\label{DYN_ldf_iso}

A rotation of the lateral momentum diffusive operator is needed in several cases: 
for iso-neutral diffusion in $z$-coordinate (\np{ln\_dynldf\_iso}=.true.) and for 
either iso-neutral (\np{ln\_dynldf\_iso}=.true.) or geopotential 
(\np{ln\_dynldf\_hor}=.true.) diffusion in $s$-coordinate. In the partial step 
case, coordinates are horizontal excepted at the deepest level and no 
rotation is performed when \np{ln\_dynldf\_hor}=.true.. The diffusive 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 
diffusive operator acts and the surface of computation ($z$- or $s$-surfaces). 
The way these slopes are evaluated is given in the lateral physics chapter 
(Chap.\ref{LDF}).

%--------------------------------------------------------------------------------------------------------------
%           Iso-level bilaplacian operator
%--------------------------------------------------------------------------------------------------------------
\subsection   [Iso-level bilaplacian operator (\np{ln\_dynldf\_bilap})]
         {Iso-level bilaplacian operator (\np{ln\_dynldf\_bilap}=.true.)}
\label{DYN_ldf_bilap}

The lateral fourth order operator formulation on momentum is obtained by 
applying \eqref{Eq_dynldf_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 Chap.\ref{LBC}).
%%%
\gmcomment{add a remark on the the change in the position of the coefficient}
%%%

% ================================================================
%           Vertical diffusion term
% ================================================================
\section  [Vertical diffusion term (\mdl{dynzdf})]
      {Vertical diffusion term (\mdl{dynzdf})}
\label{DYN_zdf}
%----------------------------------------------namzdf------------------------------------------------------
\namdisplay{namzdf} 
%-------------------------------------------------------------------------------------------------------------

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. Two time stepping schemes 
can be used for the vertical diffusion term : $(a)$ a forward time differencing 
scheme (\np{ln\_zdfexp}=.true.) using a time splitting technique 
(\np{n\_zdfexp} $>$ 1) or $(b)$ a backward (or implicit) time differencing scheme 
(\np{ln\_zdfexp}=.false.) (see \S\ref{DOM_nxt}). Note that namelist variables 
\np{ln\_zdfexp} and \np{n\_zdfexp} apply to both tracers and dynamics. 

The formulation of the vertical subgrid scale physics is the same whatever 
the vertical coordinate is. The vertical diffusion operators given by 
\eqref{Eq_PE_zdf} take the following semi-discrete space form:
\begin{equation} \label{Eq_dynzdf}
\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.
\end{equation} 
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 \S\ref{ZDF}).

The surface boundary condition on momentum is given by 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_dynzdf_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 Chap.\ref{SBC})

The turbulent flux of momentum at the bottom of the ocean is specified through 
a bottom friction parameterization (see \S\ref{ZDF_bfr})

% ================================================================
% External Forcing
% ================================================================
\section{External Forcings}
\label{DYN_forcing}

Besides the surface and bottom stresses (see the above section) which are 
introduced as boundary conditions on the vertical mixing, two other forcings 
enter the dynamical equations. 

One is the effect of atmospheric pressure on the ocean dynamics (to be 
introduced later). 

Another forcing term is the tidal potential, which will be introduced in the 
reference version soon. 

% ================================================================
% Time evolution term 
% ================================================================
\section  [Time evolution term (\textit{dynnxt})]
      {Time evolution term (\mdl{dynnxt})}
\label{DYN_nxt}

%----------------------------------------------namdom----------------------------------------------------
\namdisplay{namdom} 
%-------------------------------------------------------------------------------------------------------------

The general framework for dynamics time stepping is a leap-frog scheme, 
$i.e.$ a three level centred time scheme associated with an Asselin time filter 
(cf. \S\ref{DOM_nxt})
\begin{equation} \label{Eq_dynnxt}
\begin{split}
&u^{t+\Delta t} = u^{t-\Delta t} + 2 \, \Delta t  \ \text{RHS}_u^t   \\
\\
&u_f^t \;\quad = u^t+\gamma \,\left[ {u_f^{t-\Delta t} -2u^t+u^{t+\Delta t}} \right]
\end{split}
\end{equation} 
where RHS is the right hand side of the momentum equation, the subscript $f$ 
denotes filtered values and $\gamma$ is the Asselin coefficient. $\gamma$ is 
initialized as \np{atfp} (namelist parameter). Its default value is \np{atfp} = 0.1.

Note that whith the filtered free surface, the update of the \textit{next} velocities 
is done in the \mdl{dynsp\_flt} module, and only the swap of arrays 
and Asselin filtering is done in \mdl{dynnxt.}

% ================================================================
% Diagnostic variables
% ================================================================
\section{Diagnostic variables ($\zeta$, $\chi$, $w$)}
\label{DYN_divcur_wzv}

%--------------------------------------------------------------------------------------------------------------
%           Horizontal divergence and relative vorticity
%--------------------------------------------------------------------------------------------------------------
\subsection   [Horizontal divergence and relative vorticity (\textit{divcur})]
         {Horizontal divergence and relative vorticity (\mdl{divcur})}
\label{DYN_divcur}

The vorticity is defined at an $f$-point ($i.e.$ corner point) as follows:
\begin{equation} \label{Eq_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:
\begin{equation} \label{Eq_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)
\end{equation} 

Note that in the $z$-coordinate with full step (\key{zco} is defined), $e_{3u} =e_{3v} =e_{3f}$ so that they cancel in \eqref{Eq_divcur_div}.

Note also that whereas the vorticity have the same discrete expression in $z$- 
and $s$-coordinate, 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 no more necessary 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. 

%--------------------------------------------------------------------------------------------------------------
%           Vertical Velocity
%--------------------------------------------------------------------------------------------------------------
\subsection   [Vertical velocity (\textit{wzvmod})]
         {Vertical velocity (\mdl{wzvmod})}
\label{DYN_wzv}

The vertical velocity is computed by an upward integration of the horizontal 
divergence from the bottom :

\begin{equation} \label{Eq_wzv}
\left\{   \begin{aligned}
&\left. w \right|_{3/2} \quad= 0    \\
\\
&\left. w \right|_{k+1/2}     = \left. w \right|_{k+1/2}  + e_{3t}\;  \left. \chi \right|_k  
\end{aligned}   \right.
\end{equation} 

With a free surface, the top vertical velocity is non-zero, due to the 
freshwater forcing and the variations of the free surface elevation. With a 
linear free surface or with a rigid lid, the upper boundary condition 
applies at a fixed level $z=0$. Note that in the rigid-lid case (\key{dynspg\_rl} 
is defined), the surface boundary condition ($\left. w \right|_\text{surface}=0)$ is 
automatically achieved at least at computer accuracy, due to the the way the 
surface pressure gradient is expressed in discrete form (Appendix~\ref{Apdx_C}).

Note also that whereas the vertical velocity has the same discrete 
expression in $z$- and $s$-coordinate, its physical meaning is not the same: 
in the second case, $w$ is the velocity normal to the $s$-surfaces. 

With the variable volume option, the calculation of the vertical velocity is 
modified (see \citet{Levier2007}, report available on the \NEMO web site). 

% ================================================================