source: trunk/NEMO/DOC/BETA/Chapters/Chap_TRA.tex @ 705

% ================================================================
% Chapter 1 Ñ Ocean Tracers (TRA)
% ================================================================
\chapter{Ocean Tracers (TRA)}
\label{TRA}
\minitoc

% missing/update 
% traqsr: need to coordinate with SBC module
% trabbl : advective case to be discussed
%        diffusive case : add : only the bottom ocean cell is concerned
%        ==> addfigure on bbl


$\ $\newline    % force a new ligne

Using the representation described in Chap.~\ref{DOM}, several semi-discrete space 
forms of the tracer equations are available depending on the vertical 
coordinate used and on the physics used. 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 mean or difference 
operator is used, the resulting field is multiplied by a mask.

The two active tracers are potential temperature and salinity. Their prognostic equation can be summarized as follows:
\begin{equation*}
\text{NXT} = \text{ADV}+\text{LDF}+\text{ZDF}+\text{SBC}
                   \ (+\text{QSR})\ (+\text{BBC})\ (+\text{BBL})\ (+\text{DMP})
\end{equation*}

NXT stands for next, referring to the time-stepping. From left to right, the terms on the rhs of the tracer equations are the advection (ADV), the lateral diffusion (LDF), the vertical diffusion (ZDF), the contributions from the external forcings (SBC: Surface Boundary Condition, QSR: Solar Radiation penetration, and BBC: Bottom Boundary Condition), the contribution from the bottom boundary Layer (BBL) parametrisation, and an internal damping (DMP) term. The last four have been put inside brackets as they are optional. The external forcings and parameterizations require complex inputs and calculations (bulk formulae, estimation of mixing coefficients) that are carried out in modules of the SBC, LDF and ZDF categories and described in chapters \S\ref{SBC}, \S\ref{LDF} and  \S\ref{ZDF}, respectively. Note that \mdl{tranpc}, the non-penetrative convection module,  although (temporarily) located in the NEMO/OPA//TRA directory, is described with the model vertical physics (ZDF).

In the present chapter we also describe the diagnostic equations used to compute the sea-water properties (density, Brunt-Vais\"{a}l\"{a} frequency, specific heat and freezing point) although the associated modules ($i.e.$ \mdl{eosbn2}, \mdl{ocfzpt} and \mdl{phycst}) are (temporarily) located in the NEMO/OPA directory.

The different options available to the user are managed by namelist logical or CPP keys. For each equation term ttt, the namelist logicals are \textit{ln\_trattt\_xxx}, where \textit{xxx} is a 3 or 4 letter acronym accounting for each optional scheme. The CPP key (when it exists) is \textbf{key\_trattt}. The corresponding code can be found in the \textit{trattt} or \textit{trattt\_xxx} module, in the NEMO/OPA/TRA directory.

The user has the option of extracting each tendency term on the rhs of the tracer equation (\key {trdtra} defined), as described in Chap.~\ref{MISC}.

% ================================================================
% Tracer Advection
% ================================================================
\section{Tracer Advection (\mdl{traadv})}
\label{TRA_adv}
%------------------------------------------nam_traadv-----------------------------------------------------
\namdisplay{nam_traadv}
%-------------------------------------------------------------------------------------------------------------

The advection tendency in flux form is the divergence of the advective 
fluxes. Its discrete expression is given by :
\begin{equation} \label{Eq_tra_adv}
ADV_\tau =-\frac{1}{e_{1T} {\kern 1pt}e_{2T} {\kern 1pt}e_{3T} }\left( 
{\;\delta _i \left[ {e_{2u} {\kern 1pt}e_{3u} {\kern 1pt}\;u\;\tau _u } 
\right]+\delta _j \left[ {e_{1v} {\kern 1pt}e_{3v} {\kern 1pt}v\;\tau _v } 
\right]\;} \right)-\frac{1}{\mathop e\nolimits_{3T} }\delta _k \left[ 
{w\;\tau _w } \right]
\end{equation}
which, in pure z-coordinate (\key{zco} defined), reduces to :
\begin{equation} \label{Eq_tra_adv_zco}
ADV_\tau =-\frac{1}{e_{1T} {\kern 1pt}e_{2T} {\kern 1pt}}\left( {\;\delta _i 
\left[ {e_{2u} {\kern 1pt}{\kern 1pt}\;u\;\tau _u } \right]+\delta _j \left[ 
{e_{1v} {\kern 1pt}v\;\tau _v } \right]\;} \right)-\frac{1}{\mathop 
e\nolimits_{3T} }\delta _k \left[ {w\;\tau _w } \right]
\end{equation}
as the vertical scale factors are function of $k$ only, and thus $e_{3u} 
=e_{3v} =e_{3T} $.

The flux form requires implicitly the use of the continuity equation: 
$\nabla \cdot \left( \vect{U}\,T \right)=\vect{U} \cdot \nabla T$ 
using $\nabla \cdot \vect{U}=0)$ or $\partial _t e_3 +\nabla \cdot \vect{U}=0$ in variable volume case ($i.e.$ \key{vvl} defined). Therefore it is of 
paramount importance to design the discrete analogue of the advection 
tendency so that it is consistent with the continuity equation in order to 
enforce conservation properties of the continuous equations. In other words, 
by substituting $\tau$ by 1 in (\ref{Eq_tra_adv}) we recover discrete form of the 
continuity equation which is used to calculate the vertical velocity.
%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
\begin{figure}[!t] \label{Fig_adv_scheme}  \begin{center}
\includegraphics[width=0.9\textwidth]{./Figures/Fig_adv_scheme.pdf}
\caption{Schematic representation of some ways used to evaluate the tracer value at $u$-point and the amount of tracer exchanged between two neighbouring grid points. Upsteam biased scheme (ups): the upstream value is used and the black area is exchanged. Piecewise parabolic method (ppm): a parabolic interpolation is used and black + dark grey areas is exchanged. Monotonic upstream scheme for conservative laws (muscl):  a parabolic interpolation is used and black + dark grey + grey areas are exchanged. Second order scheme (cen2): the mean value is used and black + dark grey + grey + light grey areas are exchanged. Note that this illustration does not include the flux limiter used in ppm and muscl schemes.}
\end{center}   \end{figure}
%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
The advection schemes used in OPA differ by the choice made in space and 
time interpolation to define the value of the tracer at the velocity points (\ref{Fig_adv_scheme}). 
Along solid lateral and bottom boundaries a zero tracer flux is naturally 
specified, since the normal velocity is zero there. At the sea surface the 
boundary condition depends on the type of sea surface chosen: (1) in 
rigid-lid formulation, $w=0$ at the surface, so the advective fluxes through the 
surface is zero ; (2) in non-linear free surface (variable volume case, 
\key{vvl} defined), convergence/divergence in the first ocean level moves 
up/down the free surface: there is no tracer advection through it so that 
the advective fluxes through the surface is also zero ; (3) in the linear 
free surface, the first level thickness is constant in time. The vertical 
boundary condition is applied at the fixed surface $z=0$ rather than on the 
moving surface $z=\eta$. There is a non-zero advective flux which is set for all 
advection schemes as the product of surface velocity (at $z=0$) by the first level 
tracer value: $\left. {\tau _w } \right|_{k=1 \mathord{\left/ {\vphantom {1 
2}} \right. \kern-\nulldelimiterspace} 2} =T_{k=1} $. This boundary 
condition retains local conservation of tracer. Strict global conservation 
is not possible in linear free surface but is achieved to a good 
approximation since the non-conservative term is the product of the time 
derivative of the tracer and the free surface height, two quantities that are not correlated (see \S\ref{PE_free_surface}, and also \citet{Roullet2000,Griffies2001,Campin2004}).

The velocity field that appears in (\ref{Eq_tra_adv}) and (\ref{Eq_tra_adv_zco}) is the centred (\textit{now}) \textit{eulerian} ocean velocity (see \S\ref{DYN}). Nevertheless, when advective bottom boundary layer (\textit{bbl}) and/or eddy induced velocity (\textit{eiv}) parameterisations are used it is the \textit{now} \textit{effective} velocity (i.e. the sum of the eulerian, the bbl and/or the eiv velocities) which is used.

The choice of an advection scheme is made in the \np{nam\_traadv} namelist, by 
setting to \textit{true} one and only one of the logicals \textit{ln\_traadv\_xxx}. The 
corresponding code can be found in \textit{traadv\_xxx.F90} module, where 
\textit{xxx} is a 3 or 4 letter acronym accounting for each scheme. Details 
of the advection schemes are given below. The choice of an advection scheme 
is a complex matter which depends on the model physics, model resolution, 
type of tracer, as well as the issue of numerical cost. 

Note that (1) cen2, cen4 and TVD schemes require an explicit diffusion 
operator while the other schemes are diffusive enough so that they do not 
require additional diffusion ; (2) cen2, cen4, MUSCL2, and UBS are not 
\textit{positive} schemes, meaning false extrema are permitted. It is not recommended to use 
them on passive tracers ; (3) It is highly recommended to use the same 
advection-diffusion scheme on both active and passive tracers. In 
particular, if a source or sink of a passive tracer depends on a active one, 
the difference of treatment of active and passive tracers can create very 
nice-looking frontal structures that are pure numerical artefacts. 


% -------------------------------------------------------------------------------------------------------------
%        2nd order centred scheme  
% -------------------------------------------------------------------------------------------------------------
\subsection{$2^{nd}$ order centred scheme (cen2) (\np{ln\_traadv\_cen2}=T)}
\label{TRA_adv_cen2}

In the centred second order formulation, the tracer at velocity points is 
evaluated as the mean of the two neighbouring $T$-points. For example, in the 
$i$-direction :
\begin{equation} \label{Eq_tra_adv_cen2}
\tau _u^{cen2} =\overline T ^{i+1/2}
\end{equation}

The scheme is non diffusive ($i.e.$ it conserves the tracer variance, $\tau^2)$ 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 $T$ in (\ref{Eq_tra_adv_cen2}) is the \textit{now} tracer value.

Note that using cen2 scheme, the overall tracer advection is of second order accuracy since both (\ref{Eq_tra_adv}) and (\ref{Eq_tra_adv_cen2}) have this order of accuracy.

% -------------------------------------------------------------------------------------------------------------
%        4nd order centred scheme  
% -------------------------------------------------------------------------------------------------------------
\subsection{$4^{nd}$ order centred scheme (cen4) (\np{ln\_traadv\_cen4}=T)}
\label{TRA_adv_cen4}

In the $4^{th}$ order formulation (to be implemented), tracer is evaluated 
at velocity points as the $4^{th}$ order interpolation of $T$, and thus use the 
four neighbouring $T$-points. For example, in the $i$-direction:
\begin{equation} \label{Eq_tra_adv_cen4}
\tau _u^{cen4} 
=\overline{   T - \frac{1}{6}\,\delta _i \left[ \delta_{i+1/2}[T] \,\right]   }^{\,i+1/2}
\end{equation}

Strictly speaking, the cen4 scheme is not a $4^{th}$ order advection scheme 
but a $4^{th}$ order evaluation of advective fluxes since the divergence of 
advective fluxes, (\ref{Eq_tra_adv}), is kept at $2^{nd}$ order. The ``$4^{th}$ order 
scheme'' denomination used in oceanographic literature is usually associated 
with the scheme presented here. Introducing a \textit{true} $4^{th}$ order advection 
scheme is feasible but, for consistency reasons, it requires changes in the 
discretisation of the tracer advection together with changes in both the 
continuity equation and the momentum advection. 

A direct consequence of the pseudo-fourth order nature of the scheme is that 
it is not non-diffusive, i.e. the global variance of a tracer is not 
preserved through \textit{cen4}. Furthermore, it must be used in conjunction with an 
explicit diffusion operator to produce a sensible solution. The 
time-stepping is also performed using a leapfrog scheme in conjunction with 
an Asselin time-filter, so $T$ in (\ref{Eq_tra_adv_cen4}) is the \textit{now} tracer.

At $T$-grid cell abutted to a boundary (coastline, bottom and surface), an 
additional hypothesis must be made to evaluate $\tau _u^{cen4}$. This 
hypothesis usually reduces the order of the scheme. Here we choose to set 
the gradient of $T$ across the boundary to zero. Alternative conditions can be 
specified such as the reduction to a second order scheme for near boundary 
grid point.

% -------------------------------------------------------------------------------------------------------------
%        TVD scheme  
% -------------------------------------------------------------------------------------------------------------
\subsection{Total Variance Dissipation scheme (TVD) (\np{ln\_traadv\_tvd}=T)}
\label{TRA_adv_tvd}

In the Total Variance Dissipation (TVD) formulation, the tracer at velocity 
points is evaluated as a combination of upstream and centred scheme. For 
example, in the $i$-direction :
\begin{equation} \label{Eq_tra_adv_tvd}
\begin{split}
\tau _u^{ups}&= \begin{cases}
               T_{i+1}  & \text{if $\ u_{i+1/2} <     0$} \hfill \\
               T_i         & \text{if $\ u_{i+1/2} \geq 0$} \hfill \\
              \end{cases}     \\
\\
\tau _u^{tvd}&=\tau _u^{ups} +c_u \;\left( {\tau _u^{cen2} -\tau _u^{ups} } \right)
\end{split}
\end{equation}
where $c_u$ is a flux limiter function taking values between 0 and 1. There 
exists many ways to define $c_u$., each correcponding to a different total 
variance decreasing scheme. The one chosen in OPA is described in \citet{Zalesak1979}. $c_u$ only departs from $1$ when the advective term produces a local 
extremum in the tracer field. The resulting scheme is quite expensive but 
\emph{positive}. It can be used on both active and passive tracers. This scheme is tested 
and compared with MUSCL and the MPDATA scheme in \citet{Levy2001}; note that 
in this paper it is referred to as "FCT" (Flux corrected transport) 
rather than TVD.

For stability reasons in (\ref{Eq_tra_adv_tvd}) $\tau _u^{cen2}$ is evaluated using the 
\textit{now} velocity (leap-frog environment: centred in time) while $\tau _u^{ups}$ is 
evaluated using the \textit{before} velocity (diffusive part: forward in time). 

% -------------------------------------------------------------------------------------------------------------
%        MUSCL scheme  
% -------------------------------------------------------------------------------------------------------------
\subsection[MUSCL scheme  (\np{ln\_traadv\_muscl}=T)]
   {Monotone Upstream Scheme for Conservative Laws (MUSCL) (\np{ln\_traadv\_muscl}=T)}
\label{TRA_adv_muscl}

The Monotone Upstream Scheme for Conservative Laws (MUSCL) has been 
implemented by \citet{Levy2001}. In its formulation, the tracer at 
velocity points is evaluated assuming a linear tracer variation between two 
$T$-points (Fig.\ref{Fig_adv_scheme}). For example, in the $i$-direction :
\begin{equation} \label{Eq_tra_adv_muscl}
   \tau _u^{mus} = \left\{      \begin{aligned}
         &\tau _i  &+ \frac{1}{2} \;\left( 1-\frac{u_{i+1/2} \;\Delta t}{e_{1u}} \right)
         &\ \widetilde{\partial _i \tau}  & \quad \text{if }\;u_{i+1/2} \geqslant 0      \\
         &\tau _{i+1/2} &+\frac{1}{2}\;\left( 1+\frac{u_{i+1/2} \;\Delta t}{e_{1u} } \right)
         &\ \widetilde{\partial_{i+1/2} \tau } & \text{if }\;u_{i+1/2} <0
   \end{aligned}    \right.
\end{equation}
where $\widetilde{\partial _i \tau}$ is the slope of the tracer 
on which a limitation is imposed to ensure the \textit{positive} character of the scheme.

The time stepping is performed using a forward scheme, that is the \textit{before} tracer 
field is used to evaluate $\tau _u^{mus}$.

For an ocean grid point abutted to land and where the ocean velocity is 
toward land, two choices are available: use of an upstream flux 
(\np{ln\_traadv\_muscl}=T) or use of second order flux 
(\np{ln\_traadv\_muscl2}=T). Note that the latter choice does not insure the 
\textit{positive} character of the scheme. Only the former can be used on both active and 
passive tracers.

% -------------------------------------------------------------------------------------------------------------
%        UBS scheme  
% -------------------------------------------------------------------------------------------------------------
\subsection{Upstream Biased Scheme (UBS) (\np{ln\_traadv\_ubs}=T)}
\label{TRA_adv_ubs}

The UBS advection scheme is an upstream biased third order scheme based on 
an upstream-biased parabolic interpolation. It is also known as Cell 
Averaged QUICK scheme (Quadratic Upstream Interpolation for Convective 
Kinematics). For example, in the $i$-direction :
\begin{equation} \label{Eq_tra_adv_ubs}
   \tau _u^{ubs} =\overline T ^{i+1/2}-\;\frac{1}{6} \left\{      
   \begin{aligned}
         &\tau"_i          & \quad \text{if }\ u_{i+1/2} \geqslant 0      \\
         &\tau"_{i+1}   & \quad \text{if }\ u_{i+1/2}       <       0
   \end{aligned}    \right.
\end{equation}
where $\tau "_i =\delta _i \left[ {\delta _{i+1/2} \left[ \tau \right]} \right]$.

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 \cite{Farrow1995}. 
It is a relatively good compromise between accuracy and smoothness. It is 
not a \emph{positive} scheme meaning false extrema are permitted but the 
amplitude of such are significantly reduced over the centred second order 
method. Nevertheless it is not recommended to apply it to a passive tracer 
that requires positivity. 

The intrinsic diffusion of UBS makes its use risky in the vertical direction 
where the control of artificial diapycnal fluxes is of paramount importance. 
It has therefore been preferred to evaluate the vertical flux using the TVD 
scheme when \np{ln\_traadv\_ubs}=T.

For stability reasons, in \eqref{Eq_tra_adv_ubs}, the first term 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. UBS and QUICK 
schemes only differ by one coefficient. Substituting 1/6 with 1/8 in 
(\ref{Eq_tra_adv_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{traadv\_ubs} module and obtain a QUICK scheme

NB 1: When a high vertical resolution $O(1m)$ is used, the model stability can 
be controlled by vertical advection (not vertical diffusion which is usually 
solved using an implicit scheme). Computer time can be saved by using a 
time-splitting technique on vertical advection. This possibility have been 
implemented and validated in ORCA05-L301. It is not currently offered in the 
current reference version. 

NB 2 : In a forthcoming release four options will be proposed for the 
vertical component used in the UBS scheme. $\tau _w^{ubs}$ will be 
evaluated using either \textit{(a)} a centred $2^{nd}$ order scheme , or  \textit{(b)} a TVD 
scheme, or  \textit{(c)} an interpolation based on conservative parabolic splines 
following \citet{Sacha2005} implementation of UBS in ROMS, 
or  \textit{(d)} an UBS. The $3^{rd}$ case has dispersion properties similar to an 
eight-order accurate conventional scheme.

NB 3 : It is straight forward to rewrite \eqref{Eq_tra_adv_ubs} as follows:
\begin{equation} \label{Eq_tra_adv_ubs2}
\tau _u^{ubs} = \left\{  \begin{aligned}
   & \tau _u^{cen4} + \frac{1}{12} \tau"_i      & \quad \text{if }\ u_{i+1/2} \geqslant 0 \\
   & \tau _u^{cen4} - \frac{1}{12} \tau"_{i+1}  & \quad \text{if }\ u_{i+1/2}       <       0
                   \end{aligned}    \right.
\end{equation}
or equivalently 
\begin{equation} \label{Eq_tra_adv_ubs2}
u_{i+1/2} \ \tau _u^{ubs} 
=u_{i+1/2} \ \overline{ T - \frac{1}{6}\,\delta _i\left[ \delta_{i+1/2}[T] \,\right] }^{\,i+1/2}
- \frac{1}{2} |u|_{i+1/2} \;\frac{1}{6} \;\delta_{i+1/2}[\tau"_i]
\end{equation}
\eqref{Eq_tra_adv_ubs2} has several advantages. First it clearly evidence that the UBS scheme is based on the fourth order scheme to which is added an upstream biased diffusive term. Second, this emphasises that the $4^{th}$ order part have to be evaluated at \emph{now} time step, not only the $2^{th}$ order part as stated above using \eqref{Eq_tra_adv_ubs} and also as it is coded in NEMO v2.3. Third, the diffusive term is in fact a biharmonic operator with a eddy coefficient with is simply proportional to the velocity: $A_u^{lm}= - \frac{1}{12}\,{e_{1u}}^3\,|u|$. Note that the current version of NEMO uses \eqref{Eq_tra_adv_ubs}, not \eqref{Eq_tra_adv_ubs2}. 


% -------------------------------------------------------------------------------------------------------------
%        QCK scheme  
% -------------------------------------------------------------------------------------------------------------
\subsection{QUICKEST scheme (QCK) (\np{ln\_traadv\_qck}=T)}
\label{TRA_adv_qck}

The Quadratic Upstream Interpolation for Convective Kinematics with 
Estimated Streaming Terms (QUICKEST) scheme proposed by \citet{Leonard1979} is 
the third order Godunov scheme. It is associated with ULTIMATE QUICKEST 
limiter \citep{Leonard1991}. It has been implemented in NEMO by G. Reffray 
(MERCATOR-ocean).

The resulting scheme is quite expensive but \emph{positive}. It can be used on both active 
and passive tracers.

% -------------------------------------------------------------------------------------------------------------
%        PPM scheme  
% -------------------------------------------------------------------------------------------------------------
\subsection{Piecewise Parabolic Method (PPM) (\np{ln\_traadv\_ppm}=T)}
\label{TRA_adv_ppm}

The Piecewise Parabolic Method (PPM) proposed by Colella and Woodward (1984) 
is based on a quadradic piecewise rebuilding. As QCK scheme, it is associated with 
ULTIMATE QUICKEST limiter \citep{Leonard1991}. It has been implemented in \NEMO by G. Reffray (MERCATOR-ocean) but is not yet offered in the current reference 
version.

% ================================================================
% Tracer Lateral Diffusion
% ================================================================
\section{Tracer Lateral Diffusion (\mdl{traldf})}
\label{TRA_ldf}
%-----------------------------------------nam_traldf------------------------------------------------------
\namdisplay{nam_traldf}
%-------------------------------------------------------------------------------------------------------------
 
The options available for lateral diffusion are laplacian (rotated or not) 
or biharmonic operators, that latter being more scale-selective (more 
diffusive at small scales). The specification of eddy diffusivity 
coefficients (either constant, variable in space and time) as well as the 
computation of the slope along which the operators act are performed in 
\mdl{ldftra} and \mdl{ldfslp} modules, respectively. This is described in Chap.~\ref{LDF}. The lateral diffusion of tracers is evaluated using a forward scheme, 
i.e. the tracers appearing in its expression are the \textit{before} tracers 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}) 

% -------------------------------------------------------------------------------------------------------------
%        Iso-level laplacian operator
% -------------------------------------------------------------------------------------------------------------
\subsection{Iso-level laplacian operator (\mdl{traldf\_lap}, \np{ln\_traldf\_lap}) }
\label{TRA_ldf_lap}

A laplacian diffusive operator (i.e. a harmonic operator) acting along the model surfaces is given by: 
\begin{equation} \label{Eq_tra_ldf_lap}
\begin{split}
D_T^{lT} =\frac{1}{e_{1T} \; e_{2T}\;  e_{3T} } &\left[ {\quad \delta _i 
\left[ {A_u^{lT} \left( {\frac{e_{2u} e_{3u} }{e_{1u} }\;\delta _{i+1/2} 
\left[ T \right]} \right)} \right]} \right.
\\
&\ \left. {+\; \delta _j \left[ 
{A_v^{lT} \left( {\frac{e_{1v} e_{3v} }{e_{2v} }\;\delta _{j+1/2} \left[ T 
\right]} \right)} \right]\quad } \right]
\end{split}
\end{equation}

This lateral operator is a \emph{horizontal} one (i.e. acting along geopotential surfaces) in 
$z$-coordinate with or without partial step, but it is simply an iso-level operator in $s$-coordinate. 
It is thus used when, in addition to \np{ln\_traldf\_lap}=T, we have \np{ln\_traldf\_level}=T, or both \np{ln\_traldf\_hor}=T and \np{ln\_zco}=F. In both cases, it significantly contributes to diapycnal mixing. It is therefore not recommended to use it.

\textit{Notes} : In pure z-coordinate (\key{zco} defined), $e_{3u}=e_{3v}=e_{3T}$, so that the vertical scale factors disappear from (\ref{Eq_tra_ldf_lap}). 

\textit{Notes} : In partial step $z$-coordinate (\np{ln\_zps}=T), tracers in horizontally adjacent cells are located at different depths in vicinity of the bottom. In this case, horizontal derivatives in (\ref{Eq_tra_ldf_lap}) at the bottom level require a specific treatment. They are calculated in module zpshde, described in \S\ref{TRA_zpshde}.

% -------------------------------------------------------------------------------------------------------------
%        Rotated laplacian operator
% -------------------------------------------------------------------------------------------------------------
\subsection{Rotated laplacian operator (\mdl{traldf\_iso}, \np{ln\_traldf\_lap})}
\label{TRA_ldf_iso}

The general form of the second order lateral tracer subgrid scale physics (\ref{Eq_PE_zdf}) takes the following semi-discrete space form in $z$- and $s$-coordinates:

\begin{equation} \label{Eq_tra_ldf_iso}
\begin{split}
 D_T^{lT} =& \frac{1}{e_{1T}\,e_{2T}\,e_{3T} }
 \\
& \left\{ {\delta _i \left[ {A_u^{lT}  \left( 
    {\frac{e_{2u} \; e_{3u} }{e_{1u} } \,\delta _{i+1/2}[T]
   -e_{2u} \; r_{1u} \,\overline{\overline {\delta _{k+1/2}[T]}}^{\,i+1/2,k}}
 \right)} \right]} \right. 
\\ 
& +\delta 
_j \left[ {A_v^{lT} \left( {\frac{e_{1v}\,e_{3v} }{e_{2v} 
}\,\delta _{j+1/2} \left[ T \right]-e_{1v}\,r_{2v} 
\,\overline{\overline {\delta _{k+1/2} \left[ T \right]}} ^{\,j+1/2,k}} 
\right)} \right] 
\\ 
& +\delta 
_k \left[ {A_w^{lT} \left( 
-e_{2w}\,r_{1w} \,\overline{\overline {\delta _{i+1/2} \left[ T \right]}} ^{\,i,k+1/2}
\right.} \right. 
\\ 
& \qquad \qquad \quad 
-e_{1w}\,r_{2w} \,\overline{\overline {\delta _{j+1/2} \left[ T \right]}} ^{\,j,k+1/2}
\\
& \left. {\left. { 
 \quad \quad \quad \left. {{\kern 
1pt}+\frac{e_{1w}\,e_{2w} }{e_{3w} }\,\left( {r_{1w} ^2+r_{2w} ^2} 
\right)\,\delta _{k+1/2} \left[ T \right]} \right)} \right]\;\;\;} \right\} 
 \end{split}
 \end{equation}
where $r_1$ and $r_2$ are the slopes between the surface of computation ($z$- or $s$-surfaces) and the surface along which the diffusive operator acts ($i.e.$ horizontal or iso-neutral surfaces). 
 It is thus used when, in addition to \np{ln\_traldf\_lap}=T, we have \np{ln\_traldf\_iso}=T, or both \np{ln\_traldf\_hor}=T and \np{ln\_zco}=T. The way these slopes are evaluated is given in \S\ref{LDF_slp}. At the surface, bottom and lateral boundaries, the turbulent fluxes of heat and salt are set to zero using the mask technique (see \S\ref{LBC_coast}). 

The operator in \eqref{Eq_tra_ldf_iso} involves both lateral and vertical derivatives. For numerical stability, the vertical second derivative must be solved using the same implicit time scheme as those used in the vertical physics (see \S\ref{TRA_zdf}). For computer efficiency reasons, this term is not computed in \mdl{traldf} module, but in \mdl{trazdf} module where, if iso-neutral mixing is used, the vertical mixing coefficient is simply increased by $\frac{e_{1w}\,e_{2w} }{e_{3w} }\ \left( {r_{1w} ^2+r_{2w} ^2} \right)$. 

This formulation conserves the tracer but does not ensure the decrease of the tracer variance. Nevertheless the treatment performed on the slopes (see \S\ref{LDF}) allows to run safely without any additional background horizontal diffusion \citep{Guily2001}. An alternate scheme \citep{Griffies1998} which preserves both tracer and its variance is currently been tested in \NEMO.

Note that in partial step $z$-coordinate (\np{ln\_zps}=T), the horizontal derivatives in \eqref{Eq_tra_ldf_iso} at the bottom level require a specific treatment. They are calculated in module zpshde, described in \S\ref{TRA_zpshde}.

% -------------------------------------------------------------------------------------------------------------
%        Iso-level bilaplacian operator
% -------------------------------------------------------------------------------------------------------------
\subsection{Iso-level bilaplacian operator (\mdl{traldf\_bilap}, \np{ln\_traldf\_bilap})}
\label{TRA_ldf_bilap}

The lateral fourth order operator formulation on tracers is obtained by applying (\ref{Eq_tra_ldf_lap}) twice. It requires an additional assumption on boundary conditions: first and third derivative terms normal to the coast are set to zero.

It is used when, in addition to \np{ln\_traldf\_bilap}=T, we have \np{ln\_traldf\_level}=T, or both \np{ln\_traldf\_hor}=T and \np{ln\_zco}=F. In both cases, it can contributes to diapycnal mixing even if it should be less than in the laplacian case. It is therefore not recommended to use it.

\textit{Notes:} In the code, the bilaplacian routine does not call twice the laplacian 
routine but is rather a specific routine. This is due to the fact that we 
introduce the eddy diffusivity coefficient, A, in the operator as: $\nabla 
\cdot \nabla \left( {A\nabla \cdot \nabla T} \right)$ and instead of 
$-\nabla \cdot a\nabla \left( {\nabla \cdot a\nabla T} \right)$ where 
$a=\sqrt{|A|}$ and $A<0$. This was a mistake: both formulations 
ensure the total variance decrease, but the former requires a larger number 
of code-lines. It will be corrected in a forthcoming release.

% -------------------------------------------------------------------------------------------------------------
%        Rotated bilaplacian operator
% -------------------------------------------------------------------------------------------------------------
\subsection{Rotated bilaplacian operator (\mdl{traldf\_bilapg}, \np{ln\_traldf\_bilap})}
\label{TRA_ldf_bilapg}

The lateral fourth order operator formulation on tracers is obtained by applying (\ref{Eq_tra_ldf_iso}) twice. It requires an additional assumption on boundary conditions: first and third derivative terms normal to the coast, the bottom and the surface are set to zero.

 It is used when, in addition to \np{ln\_traldf\_bilap}=T, we have \np{ln\_traldf\_iso}=T, or both \np{ln\_traldf\_hor}=T and \np{ln\_zco}=T. Nevertheless, this rotated bilaplacian operator has never been seriously tested. No warranties that it is neither free of bugs or correctly formulated. 
Moreover, the stability range of such an operator will be probably quite narrow, requiring a significantly smaller time-step than the one used on unrotated operator.

% ================================================================
% Tracer Vertical Diffusion
% ================================================================
\section{Tracer Vertical Diffusion (\mdl{trazdf})}
\label{TRA_zdf}
%--------------------------------------------namzdf---------------------------------------------------------
\namdisplay{namzdf}
%--------------------------------------------------------------------------------------------------------------

The formulation of the vertical subgrid scale tracer physics is the same for all the vertical coordinates, based on a laplacian operator. The vertical diffusive operator given by (\ref{Eq_PE_zdf}) takes the following semi-discrete space form:
\begin{equation} \label{Eq_tra_zdf}
\begin{split}
D^{vT}_T &= \frac{1}{e_{3T}} \; \delta_k \left[ 
\frac{A^{vT}_w}{e_{3w}}  \delta_{k+1/2}[T]   \right] 
\\
D^{vS}_T &= \frac{1}{e_{3T}} \; \delta_k \left[ 
\frac{A^{vS}_w}{e_{3w}}  \delta_{k+1/2}[S]   \right] 
\end{split}
\end{equation}

where $A_w^{vT}$ and $A_w^{vS}$ are the vertical eddy diffusivity 
coefficients on Temperature and Salinity, respectively. Generally, $A_w^{vT}=A_w^{vS}$ ecept when double diffusion mixing is parameterised (\key{zdfddm} defined). The way these coefficients can be evaluated is given in \S\ref{ZDF} (ZDF). Furthermore, when iso-neutral mixing is used, the both mixing coefficient are increased by $\frac{e_{1w}\,e_{2w} }{e_{3w} }\ \left( {r_{1w} ^2+r_{2w} ^2} \right)$ to account for the vertical second derivative of \eqref{Eq_tra_ldf_iso}. 

At the surface and bottom boundaries, the turbulent fluxes of momentum, heat and salt must be specified. At the surface they are prescribed from the surface forcing (see \S\ref{TRA_sbc}), while at the bottom they are set to zero for heat and salt, unless a geothermal flux forcing is prescribed as a bottom boundary condition (\S\ref{TRA_bbc}). 

The large eddy coefficient found in the mixed layer together with high 
vertical resolution implies a too restrictive constraint on the time step in 
explicit time stepping case (\np{ln\_zdfexp}=True). Therefore, the default implicit time stepping is generally preferred for the vertical diffusion as it overcomes the stability 
constraint. A forward time differencing scheme (\np{ln\_zdfexp}=T) using a time splitting technique (\np{n\_zdfexp} $> 1$) is provided as an alternative. Namelist variables 
\np{ln\_zdfexp} and \np{n\_zdfexp} apply to both tracers and dynamics. 

% ================================================================
% External Forcing
% ================================================================
\section{External Forcing}
\label{TRA_sbc_qsr_bbc}

% -------------------------------------------------------------------------------------------------------------
%        surface boundary condition
% -------------------------------------------------------------------------------------------------------------
\subsection{surface boundary condition (\mdl{trasbc})}
\label{TRA_sbc}

The surface boundary condition for tracers is implemented in a separate 
module (\mdl{trasbc}) instead of entering as a boundary condition on the vertical 
diffusion operator (as in the case of momentum). This has been found to 
enhance readability of the code. The two formulations are completely 
equivalent; the forcing terms in trasbc are the surface fluxes divided by 
the thickness of the top model layer. Following \citet{Roullet2000} the 
forcing on an ocean tracer, $c$, can be split into two parts: $F_{ext}^C $, the 
flux of tracer crossing the sea surface and not linked with the water 
exchange d at the surface with the atmosphere, and $F_{wf}^C$ the forcing 
on the concentration associated with the water flux. The latter forcing has 
also two components: a direct effect of change in concentration associated 
with the tracer carried by the water flux, and an indirect concentration/dilution effect :
\begin{equation*}
\begin{split}
 F^C &= F_{ext} + F_{wf}^d                                          +F_{wf}^i    \\
\\ 
        &= F_{ext} - \left( c_E \, E - c_p \,P - c_R \,R \right) +c\left( E-P-R \right)
\end{split}
\end{equation*} 

Two cases must be distinguished, the nonlinear free surface case (\key{vvl} defined) and the linear free surface case. The first case is simpler, because the indirect concentration/dilution effect is naturally taken into 
account by letting the vertical scale factors vary in time. The salinity of 
water exchanged at the surface is assumed to be zero, so there is no salt 
flux at the free surface, excepted in the presence of sea ice. The heat flux 
at the free surface is the sum of $F_{ext}$, the direct heating/cooling 
(by the total non-penetrative heat flux) and $F_{wf}^e$ the heat carried by 
the water exchanged through the surface (evaporation, precipitation, 
runoff). The temperature of precipitations is not well known. In the model 
we assume that this water has the same temperature as the sea surface 
temperature, The resulting forcing terms for temperature T and salinity S 
are: 
\begin{equation} \label{Eq_tra_forcing}
\begin{aligned}
 &F^T =\frac{Q_{ns} }{\rho _o \;C_p \,e_{3T} }-\frac{\text{EMP}\;\left. T 
\right|_{k=1} }{e_{3T} }  & \\ 
\\
& F^S =\frac{\text{EMP}_S\;\left. S \right|_{k=1} }{e_{3T} }   &
 \end{aligned}
\end{equation} 

where EMP is the freshwater budget (evaporation minus precipitation minus river runoff) which forces the ocean volume, $Q_{ns}$ is the non-penetrative part of the net surface heat flux (difference between the total surface heat flux and the fraction of the short wave flux that penetrates in the water column), the product EMP$_S\;.\left. S \right|_{k=1}$ is  the ice-ocean salt flux, and $\left. S\right|_{k=1}$ is the sea surface salinity (\textit{SSS}). The total salt content is conserved in this formulation (excepted for the effect of the Asselin filter).
%AMT note: the ice-ocean flux had been forgotten in the first release of the key_vvl option, 
%AMT  has this been corrected in the code? 

In the second case (linear free surface), the vertical scale factors are fixed in time so that the concentration/dilution effect must be added in trasbc. Because of the hypothesis made for the temperature of precipitation and runoffs, for temperature $F_{wf}^e +F_{wf}^i =0$. The resulting forcing term for temperature is: 

\begin{equation} \label{Eq_tra_forcing_q}
F^T=\frac{Q_{ns} }{\rho _o \;C_p \,e_{3T} }
\end{equation} 

The salinity forcing is still given by \eqref{Eq_tra_forcing} but the definition of EMP$_S$ is 
different: it is the total surface freshwater budget (evaporation minus 
precipitation minus river runoff plus the rate of change of the sea ice 
thickness). The total salt content is not exactly conserved (\citet{Roullet2000}, see also 
\S\ref{PE_free_surface}).

In the case of the rigid lid approximation, the surface salinity forcing $F^s$ 
is also expressed by \eqref{Eq_tra_forcing} but now the global integral of the product 
EMP*S is not compensated by the advection of fluid through the top level: in 
the rigid lid case (contrary to the linear free surface), because \textit{w(k=1) = 0}. As a 
result, even if the budget of \textit{EMP} is zero in average over the whole ocean 
domain, the associated salt flux is not, as sea-surface salinity and \textit{EMP} are 
intrinsically correlated (high \textit{SSS} are found where evaporation is strong while 
low \textit{SSS} is usually associated with high precipitation or river runoff input).

The $Q_{ns} $ and \textit{EMP} fields are defined and updated in \mdl{sbcmod} module (see 
\S\ref{SBC}).

% -------------------------------------------------------------------------------------------------------------
%        Solar Radiation Penetration 
% -------------------------------------------------------------------------------------------------------------
\subsection{Solar Radiation Penetration (\mdl{traqsr})}
\label{TRA_qsr}
%--------------------------------------------namqsr---------------------------------------------------------
\namdisplay{namqsr}
%--------------------------------------------------------------------------------------------------------------

When the penetrative solar radiation option is used (\np{ln\_flxqsr}=T, the solar radiation penetrates the top few meters of the ocean, otherwise all the heat flux is absorbed in the first ocean level (\np{ln\_flxqsr}=F). A term is thus added to the time evolution equation of temperature \eqref{Eq_PE_tra_T} while the surface boundary condition is modified to take into account only the non-penetrative part of the surface heat flux:
\begin{equation} \label{Eq_PE_qsr}
\begin{split}
\frac{\partial T}{\partial t} &= {\ldots} + \frac{1}{\rho_o\, C_p \,e_3} \; \frac{\partial I}{\partial k}   \\
Q_{ns} &= Q_\text{Total} - Q_{sr}
\end{split}
\end{equation}

where $I$ is the downward irradiance. The additional term in \eqref{Eq_PE_qsr} is discretized as follows:
\begin{equation} \label{Eq_tra_qsr}
\frac{1}{\rho_o\, C_p \,e_3} \; \frac{\partial I}{\partial k} \equiv \frac{1}{\rho_o\, C_p\, e_{3T}} \delta_k \left[ I_w \right]
\end{equation}

A formulation including extinction coefficients is assumed for the downward irradiance $I$ 
\citep{Paulson1977}:
\begin{equation} \label{Eq_traqsr_iradiance}
I(z) = Q_{sr} \left[Re^{-z / \xi_1} + \left( 1-R\right) e^{-z / \xi_2} \right]
\end{equation}
where $Q_{sr}$ is the penetrative part of the surface heat flux, 
$\xi_1$ and $\xi_2$ are two extinction length scales and $R$ determines the relative 
contribution of the two terms. The default values used correspond to a Type 
I water in Jerlov's [1968] classification: $\xi_1 = 0.35m$, $\xi_2 = 0.23m$ and 
$R = 0.58$ ((corresponding to \np{xsi1}, \np{xsi2} and \np{rabs} namelist parameters, respectively). $I$ is masked (no flux through the ocean bottom), so all the solar radiation that reaches the last ocean level is absorbed in that level. The trend in \eqref{Eq_tra_qsr} associated with the penetration of the solar radiation is added to the temperature trend and the surface heat flux modified in routine \mdl{traqsr}. Note that in $z$-coordinates, the depth of $T-$levels depends on the single variable $k$. A one dimensional array of the coefficients $gdsr(k) = Re^{-z_w (k)/\xi_1} + (1-R)e^{-z_w (k)/\xi_2}$ can then be computed once and saved in central memory. Moreover \textit{nksr}, the level at which $gdrs$ becomes negligible (less than the computer precision) is computed once and the trend associated with the penetration of the solar radiation is only added until that level. At last, note that when the ocean is shallow (< 200~m), the part of the solar radiation can reach the ocean floor. In this case, we have chosen that all the radiation is absorbed at the last ocean level ($i.e.$ $I_w$ is masked). 

When coupling with a biology model (PISCES or LOBSTER), it is possible to calculate the light attenuation using information from the biology model. At the time of this writing, reading the light attenuation from a file is not implemented yet in the reference version.

\colorbox{yellow}{case 4 bands and bio-coupling to add !!!}

% -------------------------------------------------------------------------------------------------------------
%        Bottom Boundary Condition
% -------------------------------------------------------------------------------------------------------------
\subsection{Bottom Boundary Condition (\mdl{trabbc} + \key{bbc})}
\label{TRA_bbc}
%--------------------------------------------nambbc--------------------------------------------------------
\namdisplay{nambbc}
%--------------------------------------------------------------------------------------------------------------
%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
\begin{figure}[!t] \label{Fig_geothermal}  \begin{center}
\includegraphics[width=1.0\textwidth]{./Figures/Fig_TRA_geoth.pdf}
\caption{Geothermal Heat flux (in $mW.m^{-2}$) as inferred from the age of the sea floor and the formulae of \citet{Stein1992}.}
\end{center}   \end{figure}
%>>>>>>>>>>>>>>>>>>>>>>>>>>>>

Usually it is considered that there is no exchange of heat nor salt through 
the ocean bottom, i.e. a no flux boundary condition is applied on active 
tracers at the bottom. This is the default option in NEMO, and it is 
implemented using the masking technique. Nevertheless, there exists a 
non-zero heat flux across the seafloor that is associated with the solid 
earth cooling. This flux is weak compared with surface fluxes --- a mean 
global value of $\sim0.1\;W/m^2$ \citep{Stein1992} --- but it is 
systematically positive and it acts only on the densest water masses. Taking 
this flux into account in a global ocean model increases by a few Sverdrups 
the deepest overturning cell (i.e. the one associated with the Antarctic 
Bottom Water). 

The presence or not of a geothermal heating is controlled by the namelist 
parameter  \np{ngeo\_flux}. Set to 1, a constant geothermal heatingis 
introducted which value is given by the \np{ngeo\_flux\_const}, also a 
namelist parameter. Set to 2, a spatially varying geothermal heat flux is 
introducted which is provided in the geothermal\_heating.nc NetCDF 
file (Fig.\ref{Fig_geothermal}).

% ================================================================
% Bottom Boundary Layer
% ================================================================
\section{Bottom Boundary Layer (\mdl{trabbl} + \key{bbl\_diff} or \key{bbl\_adv})}
\label{TRA_bbl}
%--------------------------------------------nambbl---------------------------------------------------------
\namdisplay{nambbl}
%--------------------------------------------------------------------------------------------------------------

In z-coordinate configuration, the bottom topography is represented as a 
series of discrete steps. This is not adequate to represent gravity driven 
downslope flows. Such flows arise downstream of sills such as the Strait of 
Gibraltar, Bab El Mandeb, or Denmark Strait, where dense water formed in 
marginal seas flows into a basin filled with less dense water. The amount of 
entrainment that occurs in those gravity plumes is critical to determine the 
density and volume flux of the densest waters of the ocean, such as the 
Antarctic Bottom water, or the North Atlantic Deep Water. $z$-coordinate 
models tend to overestimate the entrainment because the gravity flow is 
mixed down vertically by convection as it goes ``downstairs'' following the 
step topography, sometimes over a thickness much larger than the thickness 
of the observed gravity plume. A similar problem occurs in $s$-coordinate when 
the thickness of the bottom level varies in large proportions downstream of 
a sill \citep{Willebrand2001}, and the thickness of the plume is not 
resolved. 

The idea of the bottom boundary layer parameterization first introduced by 
\citet{BeckDos1998} is to allow a direct communication between 
two adjacent bottom cells at varying level, whenever the densest water is 
located above the less dense water. The communication can be by diffusive 
fluxes (diffusive BBL), advective fluxes (advective BBL) or both. Only 
tracers are modified, not the velocities. Implementing a BBL 
parameterization for momentum is a more complex problem because of the 
pressure gradient errors. 

% -------------------------------------------------------------------------------------------------------------
%        Diffusive BBL
% -------------------------------------------------------------------------------------------------------------
\subsection{Diffusive Bottom Boundary layer (\mdl{trabbl})}
\label{TRA_bbl_diff}

The lateral diffusivity $A_l^\sigma $ in the BBL can be prescribed with a 
spatial dependence, e.g., in the conditional form
\begin{equation} \label{Eq_tra_bbl}
A_l^\sigma (i,j,t)=\left\{ {\begin{array}{l}
 \mbox{large}\quad if\;\nabla \rho \cdot \nabla H<0 \\ 
 \\
 0\quad \quad \;\,\mbox{otherwise} \\ 
 \end{array}} \right.
\end{equation} 

The large value of the coefficient when the diffusive BBL is active is given 
by the namelist parameter \np{atrbbl.}

% -------------------------------------------------------------------------------------------------------------
%        Advective BBL
% -------------------------------------------------------------------------------------------------------------
\subsection{Advective Bottom Boundary Layer (\mdl{trabb\_adv})}
\label{TRA_bbl_adv}

Implemented in NEMO v2. 

\colorbox{yellow} {Documentation to be added here }

% ================================================================
% Tracer damping
% ================================================================
\section{Tracer damping (\mdl{tradmp})}
\label{TRA_dmp}
%--------------------------------------------namdmp-----------------------------------------------------
\namdisplay{namdmp}
%--------------------------------------------------------------------------------------------------------------

In some applications it can be useful to add a Newtonian damping term in the 
temperature and salinity equations:
\begin{equation} \label{Eq_tra_dmp}
\begin{split}
 \frac{\partial T}{\partial t}=\;\cdots \;-\gamma \,\left( {T-T_o } \right)  \\
\\ 
 \frac{\partial S}{\partial t}=\;\cdots \;-\gamma \,\left( {S-S_o } \right) 
 \end{split}
 \end{equation} 
where $\gamma$ is the inverse of a time scale, and $T_o$ and $S_o$ are given temperature and salinity fields (usually a climatology). The restoring term is added when \key{tradmp} is defined. It also requires that both \key{temdta} and \key{saldta} are defined ($i.e.$ that $T_o$ and $S_o$ are read). The restoring coefficient $S_o$ is a three-dimensional array initialized by the user in 
\rou{dtacof} routine also located in  \mdl{tradmp}. 

The two main cases in which \eqref{Eq_tra_dmp} is used are \textit{(a)} the specification of 
the boundary conditions along artificial walls of a limited domain basin and 
\textit{(b)} the computation of the velocity field associated with a given $T$-$S$ field 
(for example to build the initial state of a prognostic simulation, or to 
use the resulting velocity field for a passive tracer study). The first case 
applies to regional models that have artificial walls instead of open 
boundaries. In the vicinity of these walls, $S_o$ takes large values 
(equivalent to a few day time scale) whereas it is zero in the interior of 
the model domain. The second case corresponds to the use of the robust 
diagnostic method \citep{Sarmiento1982}. It allows to find the velocity 
field consistent with the model dynamics while having a $T$-$S$ field close to a 
given climatology field ($T_o -S_o$). The time scale associated with 
$S_o$ is generally not a constant but spatially varying in order to respect 
some considerations. For example, it is usually set to zero in the mixed 
layer (defined either on a density or $S_o$ criterion) \citep{Madec1996} and in the equatorial region \citep{Reverdin1991, Fujio1991, MartiTh1992} as those two regions have a small time scale of adjustment, 
while smaller $S_o$ are used in the deep ocean where the typical time scale 
is long \citep{Sarmiento1982}. In addition it is reduced (and even 
zero) along the western boundary to allow the model to reconstruct its own 
western boundary structure in equilibrium with its physics. The choice of a 
Newtonian damping acting in the mixed layer or not is controlled by \np{nmldmp} 
(\textbf{namelist} \np{nmldmp}parameter).

The robust diagnostic method is very efficient to prevent the temperature drift in intermediate waters but it produces artificial sources of heat and salt within the ocean. It has also undesirable effects on the ocean convection. It tends to prevent deep convection and subsequent deep-water formation by stabilising too much the water columns.

An example of computation of $S_o$ for robust diagnostic experiments with the ORCA2 model is provided in the \mdl{tradmp} module (subroutines \rou{dtacof} and \rou{cofdis} which compute coefficient and the distance to the bathymetry, respectively). Those routines are provided as examples and can be customised by the user. 

% ================================================================
% Tracer time evolution
% ================================================================
\section{Tracer time evolution (\mdl{tranxt})}
\label{TRA_nxt}
%--------------------------------------------namdom-----------------------------------------------------
\namdisplay{namdom}
%--------------------------------------------------------------------------------------------------------------

The general framework of dynamics time stepping is a leap-frog scheme, $i.e.$ a three level centred time scheme associated with a Asselin time filter (cf. \S\ref{DOM_nxt}):
\begin{equation} \label{Eq_tra_nxt}
\begin{split}
T^{t+\Delta t} &= T^{t-\Delta t} + 2 \, \Delta t  \ \text{RHS}_T^t   \\
\\
T_f^t  \;\ \quad &= T^t \;\quad +\gamma \,\left[ {T_f^{t-\Delta t} -2T^t+T^{t+\Delta t}} \right]
\end{split}
\end{equation} 

where $\text{RHS}_T$ is the right hand side of the temperature equation, the subscript $f$ denotes 
filtered values and $\gamma$ is the Asselin coefficient. $\gamma$ is initialized as \np{atfp} (\textbf{namelist} parameter). Its default value is \np{atfp=0.1}. 

When the vertical mixing is solved implicitly, the update of the next tracer 
fields is done in module \mdl{trazdf}. In that case only the swap of arrays 
and the Asselin filtering is done in \mdl{tranxt} module.

In order to prepare the computation of the next time step, a swap of tracer arrays is performed: $T^{t-\Delta t} = T^t$ and $T^t = T_f$. 

% ================================================================
% Equation of State (eosbn2) 
% ================================================================
\section{Equation of State (\mdl{eosbn2}) }
\label{TRA_eosbn2}
%--------------------------------------------nameos-----------------------------------------------------
\namdisplay{nameos}
%--------------------------------------------------------------------------------------------------------------

% -------------------------------------------------------------------------------------------------------------
%        Equation of State
% -------------------------------------------------------------------------------------------------------------
\subsection{Equation of State (\np{neos} = 0, 1 or 2)}
\label{TRA_eos}

It is necessary to know the equation of state for the ocean very accurately to determine stability properties (especially the Brunt-Vais\"{a}l\"{a} frequency), particularly in the deep ocean. The ocean density is a non linear empirical function of \textit{in situ }temperature, salinity and pressure. The reference is the equation of state defined by the Joint Panel on Oceanographic Tables 
and Standards \citep{UNESCO1983}. It was the standard equation of state used in early releases of OPA. Even though this computation is fully vectorised, it is quite time consuming ($15$ to $20${\%} of the total CPU time) as it requires the prior computation of the \textit{in situ} temperature from the model \textit{potential} temperature using the \citep{Bryden1973} polynomial for adiabatic lapse rate and a $4^th$ order Runge-Kutta integration scheme. Since OPA6, we have chosen the \citet{JackMcD1995} equation of state for seawater. It allows the computation of the \textit{in situ} ocean density directly as a function of \textit{potential} temperature relative to the sea surface (an OPA variable), the practical salinity (another OPA variable) and the pressure (assuming no pressure variation along geopotential surfaces, i.e. the pressure in decibars is 
approximated by the depth in meters). Both the \citet{UNESCO1983} and \citet{JackMcD1995} equations of state have the same expression except that the values of the various coefficients have been adjusted by \citet{JackMcD1995} in order to use directly the \textit{potential} temperature instead of the \textit{in situ} one. This reduces the CPU time of the in situ density computation to about $3${\%} of the total CPU time, while maintaining a quite accurate equation of state.

In the computer code, a \textit{true} density, $d$, is computed, i.e. the ratio of seawater volumic mass over $\rho_o$, a reference volumic mass (\textit{rau0} defined in \mdl{phycst}, usually $rau0= 1,020~Kg/m^3$). The default option (\np{neos}=0) is the \citet{JackMcD1995} equation of state. It is highly recommended to use it. Nevertheless, for process studies, it is often convenient to use a linear approximation of the density$^{\ast}$\footnote{$^{\ast }$ With the linear equation of state there is no longer 
a distinction between \textit{in situ} and \textit{potential} density. Cabling and thermobaric effects are also removed.}. Two linear formulations are available: a function of $T$ only (\np{neos}=1) and a function of both $T$ and $S$ (\np{neos}=2):
\begin{equation} \label{Eq_tra_eos_linear}
\begin{aligned}
 d(T)    &= {\rho (T)} / {\rho _0 } &&= 1.028 - \alpha \;T     \\ 
 d(T,S) &= {\rho (T,S)}                &&= \ \ \ \beta \;S - \alpha \;T 
\end{aligned}
\end{equation} 
where $\alpha $ and $\beta $ are the thermal and haline expansion coefficients, and $\rho_o$, the reference volumic mass, $rau0$. $\alpha $ and $\beta$ can be modified through \np{ralpha} and \np{rbeta} namelist parameters). Note that when $d$ is a function of $T$ only (\np{neos}=1), the salinity is a passive tracer and can be used as such.


% -------------------------------------------------------------------------------------------------------------
%        Brunt-Vais\"{a}l\"{a} Frequency
% -------------------------------------------------------------------------------------------------------------
\subsection{Brunt-Vais\"{a}l\"{a} Frequency (\np{neos} = 0, 1 or 2)}
\label{TRA_bn2}

An accurate computation of the ocean stability (i.e. of $N$, the brunt-Vais\"{a}l\"{a} frequency) is of paramount importance as it is used in several ocean parameterisations (namely TKE, KPP, Richardson number dependent vertical diffusion, enhanced vertical diffusion, non-penetrative convection, iso-neutral diffusion). In particular, one must be aware that $N^2$ has to be computed with an \textit{in situ} reference. The expression of $N^2$ depends on the type of equation of state used (\np{neos} namelist parameter).

For \np{neos}=0 (\citet{JackMcD1995} equation of state), the \citet{McDougall1987} 
polynomial expression is used with the pressure in decibar approximated by 
the depth in meters: 
\begin{multline} \label{Eq_tra_bn2}
N^2 = \frac{g}{e_{3w}} \; \beta ( \overline{T}^{\,k+1/2},\widetilde{S},z_w )   \\
      \left\{  \alpha / \beta ( \overline{T}^{\,k+1/2},\widetilde{S},z_w )
         \ \delta_{k+1/2}[T]     - \delta_{k+1/2}[S]
       \right\}
\end{multline} 
where $T$ is the \textit{potential} temperature, $\widetilde{S}=\overline{S}^{\,k+1/2} - 35.$ a salinity anomaly, and $\alpha$ ($\beta\,$) the thermal (haline) expansion coefficient. Both $\alpha$ and $\beta$ depend on \textit{potential} temperature, salinity which are averaged at $w$-points prior to the computation.

When a linear equation of state is used (\np{neos}=1 or 2, \eqref{Eq_tra_bn2} reduces to:
\begin{equation} \label{Eq_tra_bn2_linear}
N^2 = \frac{g}{e_{3w}} \left(   \beta \;\delta_{k+1/2}[S] - \alpha \;\delta_{k+1/2}[T]   \right)
\end{equation} 
where $\alpha$ and $\beta $ are the constant coefficients used to defined the linear 
equation of state \eqref{Eq_tra_eos_linear}.

% -------------------------------------------------------------------------------------------------------------
%        Specific Heat
% -------------------------------------------------------------------------------------------------------------
\subsection{Specific Heat (\textit{rcp}, \mdl{phycst})}
\label{TRA_adv_ldf}

The specific heat of sea water, $C_p$, is a function of temperature, salinity and pressure \citep{UNESCO1983}. It is only used in the model to convert surface heat fluxes into surface temperature increase, thus the pressure dependence is neglected. The dependence on $T$ and $S$ is weak. For example, with $S=35~psu$, $C_p$ increases from $3989$ to $4002$ when $T$ varies from -2~\degres C to 31~\degres C. Therefore, $C_p$ has been chosen as a constant: $C_p=4.10^3~J\,Kg^{-1}\,\degres K^{-1}$. Its computer name is \textit{rcp} and its value is set in \mdl{phycst} module.

% -------------------------------------------------------------------------------------------------------------
%        Freezing Point of Seawater
% -------------------------------------------------------------------------------------------------------------
\subsection{Freezing Point of Seawater (\mdl{ocfzpt})}
\label{TRA_fzp}

The freezing point of seawater is a function of salinity and pressure \citep{UNESCO1983}:
\begin{equation} \label{Eq_tra_eos_fzp}
   \begin{split}
T_f (S,p) &= \left( -0.0575 + 1.710523 \;10^{-3} \, \sqrt{S} 
                       -  2.154996 \;10^{-4} \,S  \right) \ S    \\
               & - 7.53\,10^{-3}\,p 
   \end{split}
\end{equation}

\eqref{Eq_tra_eos_fzp} is only used to compute the potential freezing point of sea water 
($i.e.$ referenced to the surface $p=0$), thus the pressure dependent terms in \eqref{Eq_tra_eos_fzp} (last term) has been dropped. The \textit{before} and \textit{now} surface freezing point is introduced in the code as $fzptb$ and $fzptn$ 2D arrays together with a  \textit{now} mask (\textit{freezn}) which takes 0 or 1 whether the ocean temperature is above or at the freezing point. Caution: do not confuse \textit{freezn} with the fraction of lead (\textit{frld}) defined in LIM. 

% ================================================================
% Horizontal Derivative in zps-coordinate 
% ================================================================
\section{Horizontal Derivative in \textit{zps}-coordinate (\mdl{zpshde})}
\label{TRA_zpshde}

With partial bottom cells (\np{ln\_zps}=T), tracers in horizontally adjacent cells generally live at different depths. Horizontal gradients of tracers are needed for horizontal diffusion (\mdl{traldf} module) and for the hydrostatic pressure gradient (\mdl{dynhpg} module). Before taking horizontal gradients between the tracers next to the bottom, a linear interpolation is used to approximate the deeper tracer as if it actually lived at the depth of the shallower tracer point (Fig.~\ref{Fig_Partial_step_scheme}). For example on temperature in the i-direction, the needed interpolated temperature, $\widetilde{T}$, is:
%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
\begin{figure}[!p] \label{Fig_Partial_step_scheme}  \begin{center}
\includegraphics[width=0.9\textwidth]{./Figures/Partial_step_scheme.pdf}
\caption{ Discretisation of horizontal derivative and mean of tracers in z-partial step coordinate (\np{ln\_zps}=T) in the case $( e3w_k^{i+1} - e3w_k^i  )>0$. A linear interpolation is used to estimate $\widetilde{T}_k^{i+1}$, the tracer value at the depth of the shallower tracer point of the two adjacent bottom $T$-points. The horizontal derivative is then given by: $\delta _{i+1/2} T_k=  \widetilde{T}_k^{\,i+1} -T_k^{\,i}$ and the mean by: $\overline{T}_k^{\,i+1/2}= ( \widetilde{T}_k^{\,i+1/2} - T_k^{\,i} ) / 2$.  }
\end{center}   \end{figure}
%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
\begin{equation*}
\widetilde{T}= \left\{  \begin{aligned}  
&T^{\,i+1}      -\frac{ \left( e_{3w}^{i+1} -e_{3w}^i \right)}{ e_{3w}^{i+1} }\;\delta _k T^{i+1}  
                        && \quad\text{if  $\ e_{3w}^{i+1} \geq e_{3w}^i$   }  \\
                              \\
&T^{\,i} \ \ \ \,+\frac{ \left( e_{3w}^{i+1} -e_{3w}^i \right) }{e_{3w}^i       }\;\delta _k T^{i+1}
                        && \quad\text{if  $\ e_{3w}^{i+1}    <   e_{3w}^i$   } 
            \end{aligned}   \right.
\end{equation*}
and the resulting formulation of horizontal derivative and horizontal mean value of $T$ at $U$-point are: 
\begin{equation} \label{Eq_zps_hde}
\begin{aligned}
 \delta _{i+1/2} T=  \begin{cases}
\ \ \ \widetilde {T}\quad\ -T^i     & \ \ \quad\quad\text{if  $\ e_{3w}^{i+1} \geq e_{3w}^i$ } \\
                              \\
\ \ \ T^{\,i+1}-\widetilde{T}    & \ \ \quad\quad\text{if  $\ e_{3w}^{i+1}    <   e_{3w}^i$   } 
                  \end{cases}     \\
\\
\overline {T}^{\,i+1/2} \ =   \begin{cases}
( \widetilde {T}\ \ \;\,-T^{\,i})    / 2  & \;\ \ \quad\text{if  $\ e_{3w}^{i+1} \geq e_{3w}^i$ } \\
                              \\
( T^{\,i+1}-\widetilde{T} ) / 2     & \;\ \ \quad\text{if  $\ e_{3w}^{i+1}    <   e_{3w}^i$   } 
            \end{cases}
\end{aligned}
\end{equation}

The computation of horizontal derivative of tracers as well as of density is performed once for all at each time step in \mdl{zpshde} module and stored in shared arrays to be used when needed. It has to be emphasized that the computation of the interpolated density, $\widetilde{\rho}$, is not identical to the one of $T$ and $S$. Instead of forming a linear approximation of density, we compute $\widetilde{\rho }$ from the interpolated value of $T$ and $S$, and the pressure of at $u$-point (in the equation of state pressure is approximated by depth, see \S\ref{TRA_eos} ) : 
\begin{equation} \label{Eq_zps_hde_rho}
\widetilde{\rho } = \rho ( {\widetilde{T},\widetilde {S},z_u }) 
\quad \text{where }\  z_u = \min \left( {z_T^{i+1} ,z_T^i } \right)
\end{equation} 

This is a much better approximation as the variation of $\rho$ with depth (and thus pressure) is highly non-linear with a true equation of state and thus is badly approximated with a linear interpolation. This approximation is used to compute both the horizontal pressure gradient (\S\ref{DYN_hpg}) and the slopes of neutral surfaces (\S\ref{LDF_slp})

\textit{Notes}: in almost all the advection schemes presented in this Chapter, both mean and derivative operators appear. Yet, it has been chosen not to use \eqref{Eq_zps_hde} in those schemes.: contrary to diffusion and pressure gradient computation, no correction for partial steps is applied for advection.The main motivation was to preserve the domain averaged mean variance of the field advected when using $2^{nd}$ order centred scheme. Sensitivity of the advection schemes to the way horizontal means are performed in the vicinity of partial cells should be further investigated in a near future.