source: branches/DEV_r1826_DOC/DOC/TexFiles/Chapters/Chap_TRA.tex @ 2195

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

% missing/update 
% traqsr: need to coordinate with SBC module

%STEVEN :  is the use of the word "positive" to describe a scheme enough, or should it be "positive definite"? I added a comment to this effect on some instances of this below

%\newpage
\vspace{2.cm}
%$\ $\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 
equations 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: penetrative Solar Radiation, and BBC: 
Bottom Boundary Condition), the contribution from the bottom boundary Layer 
(BBL) parametrisation, and an internal damping (DMP) term. The terms QSR, 
BBC, BBL and DMP are optional. The external forcings and parameterisations 
require complex inputs and complex calculations (e.g. bulk formulae, estimation 
of mixing coefficients) that are carried out in the SBC, LDF and ZDF modules 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).
%%%
\gmcomment{change the position of eosbn2 in the reference code}
%%%

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 \textit{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 for output (\key{trdtra} is defined), as described in Chap.~\ref{MISC}.

$\ $\newline    % force a new ligne
% ================================================================
% Tracer Advection
% ================================================================
\section  [Tracer Advection (\textit{traadv})]
      {Tracer Advection (\mdl{traadv})}
\label{TRA_adv}
%------------------------------------------namtra_adv-----------------------------------------------------
\namdisplay{namtra_adv}
%-------------------------------------------------------------------------------------------------------------

The advection tendency of a tracer 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}{b_t} \left( 
\;\delta _i \left[ e_{2u}\,e_{3u} \;  u\; \tau _u  \right]
+\delta _j \left[ e_{1v}\,e_{3v}  \;  v\; \tau _v  \right] \; \right)
-\frac{1}{e_{3t}} \;\delta _k \left[ w\; \tau _w \right]
\end{equation}
where $\tau$ is either T or S, and $b_t= e_{1t}\,e_{2t}\,e_{3t}$ is the volume of $T$-cells. 
In pure $z$-coordinate (\key{zco} is defined), it reduces to :
\begin{equation} \label{Eq_tra_adv_zco}
ADV_\tau = - \frac{1}{e_{1t}\,e_{2t}} \left( \; \delta_i \left[ e_{2u} \;u \;\tau_u \right] 
                                                                 + \delta_j \left[ e_{1v} \;v \;\tau_v  \right] \; \right)
                   -  \frac{1}{e_{3t}}                      \delta_k \left[             w \;\tau_w \right]
\end{equation}
since the vertical scale factors are functions of $k$ only, and thus 
$e_{3u} =e_{3v} =e_{3t} $. The flux form in \eqref{Eq_tra_adv} 
requires implicitly the use of the continuity equation. Indeed, it is obtained
by using the following equality : $\nabla \cdot \left( \vect{U}\,T \right)=\vect{U} \cdot \nabla T$ 
which results from the use of the continuity equation, $\nabla \cdot \vect{U}=0$ or 
$\partial _t e_3 + e_3\;\nabla \cdot \vect{U}=0$ in constant (default option) 
or variable (\key{vvl} defined) volume case, respectively. 
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 the conservation properties of the continuous equations. In other words, 
by substituting $\tau$ by 1 in (\ref{Eq_tra_adv}) we recover the 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]{./TexFiles/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 the black and dark grey areas are exchanged. Monotonic upstream 
scheme for conservative laws (muscl):  a parabolic interpolation is used and black, 
dark grey and grey areas are exchanged. Second order scheme (cen2): the mean 
value is used and black, dark grey, grey and 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 key difference between the advection schemes used in \NEMO is the choice 
made in space and time interpolation to define the value of the tracer at the 
velocity points (Fig.~\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: 
\begin{description}
\item [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 
$\left. {\tau _w } \right|_{k=1/2} =T_{k=1} $, $i.e.$ 
the product of surface velocity (at $z=0$) by the first level tracer value.
\item [non-linear free surface:] (\key{vvl} is defined) 
convergence/divergence in the first ocean level moves the free surface 
up/down. There is no tracer advection through it so that the advective 
fluxes through the surface are also zero 
\end{description}
In all cases, this boundary condition retains local conservation of tracer. 
Global conservation is obtained in both rigid-lid and non-linear free surface 
cases, but not in the linear free surface case. Nevertheless, in the latter
case, it 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{Roullet_Madec_JGR00,Griffies_al_MWR01,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 Chap.~\ref{DYN}). 
When eddy induced velocity (\textit{eiv}) parameterisation is used it is the \textit{now} 
\textit{effective} velocity ($i.e.$ the sum of the eulerian and eiv velocities) which is used.

The choice of an advection scheme is made in the \textit{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 the \textit{traadv\_xxx.F90} module, where 
\textit{xxx} is a 3 or 4 letter acronym corresponding to 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
\footnote{negative values can appear in an initially strictly positive tracer field 
which is advected}
, implying that false extrema are permitted. Their use is not recommended on passive tracers ; 
(3) It is highly recommended that the same advection-diffusion scheme is 
used on both active and passive tracers. Indeed, if a source or sink of a 
passive tracer depends on an 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})]
         {$2^{nd}$ order centred scheme (cen2) (\np{ln\_traadv\_cen2}=true)}
\label{TRA_adv_cen2}

In the centred second order formulation, the tracer at velocity points is 
evaluated as the mean of the two neighbouring $T$-point values. 
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. The centered second 
order advection is computed in the \mdl{traadv\_cen2} module. In this module,
it is also proposed to combine the \textit{cen2} scheme with an upstream scheme
in specific areas which requires a strong diffusion in order to avoid the generation 
of false extrema. These areas are the vicinity of large river mouths, some straits 
with coarse resolution, and the vicinity of ice cover area ($i.e.$ when the ocean 
temperature is close to the freezing point).

Note that using the 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. Note also that 

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

In the $4^{th}$ order formulation (to be implemented), tracer values are 
evaluated at velocity points as a $4^{th}$ order interpolation, and thus uses 
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 \eqref{Eq_tra_adv} is kept at $2^{nd}$ order. The phrase ``$4^{th}$ 
order scheme'' 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 terms.  

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 using 
\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 a $T$-grid cell adjacent 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 a reduction to a second order scheme for these near boundary 
grid points.

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

In the Total Variance Dissipation (TVD) formulation, the tracer at velocity 
points is evaluated using a combination of an upstream and a 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 exist many ways to define $c_u$, each correcponding to a different 
total variance decreasing scheme. The one chosen in \NEMO is described in 
\citet{Zalesak_JCP79}. $c_u$ 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{Levy_al_GRL01}; note that in this paper it is referred to as "FCT" (Flux corrected 
transport) rather than TVD. The TVD scheme is computed in the \mdl{traadv\_tvd} module.

For stability reasons (see \S\ref{DOM_nxt}), in (\ref{Eq_tra_adv_tvd}) 
$\tau _u^{cen2}$ is evaluated using the \textit{now} tracer while $\tau _u^{ups}$ 
is evaluated using the \textit{before} tracer. In other words, the advective part of 
the scheme is time stepped with a leap-frog scheme while a forward scheme is 
used for the diffusive part. 

% -------------------------------------------------------------------------------------------------------------
%        MUSCL scheme  
% -------------------------------------------------------------------------------------------------------------
\subsection[MUSCL scheme  (\np{ln\_traadv\_muscl})]
   {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{Levy_al_GRL01}. 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} \;\rdt}{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} \;\rdt}{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 adjacent to land and where the ocean velocity is 
directed toward land, two choices are available: an upstream flux 
(\np{ln\_traadv\_muscl}=true) or a second order flux 
(\np{ln\_traadv\_muscl2}=true). Note that the latter choice does not ensure 
the \textit{positive} character of the scheme. Only the former can be used 
on both active and passive tracers. The two MUSCL schemes are computed 
in the \mdl{traadv\_tvd} and \mdl{traadv\_tvd2} modules.

% -------------------------------------------------------------------------------------------------------------
%        UBS scheme  
% -------------------------------------------------------------------------------------------------------------
\subsection   [Upstream-Biased Scheme (UBS) (\np{ln\_traadv\_ubs})]
         {Upstream-Biased Scheme (UBS) (\np{ln\_traadv\_ubs}=true)}
\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 the 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{Shchepetkin_McWilliams_OM05}. 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 that false extrema are permitted, 
but the amplitude of such are significantly reduced over the centred second 
order method. Nevertheless it is not recommended that it should be applied 
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. 
Therefore the vertical flux is evaluated using the TVD scheme when 
\np{ln\_traadv\_ubs}=true.

For stability reasons  (see \S\ref{DOM_nxt}), in \eqref{Eq_tra_adv_ubs}, 
the first term (which corresponds to a second order centred scheme) 
is evaluated using the \textit{now} tracer (centred in time) while the 
second term (which is the diffusive part of the scheme), is 
evaluated using the \textit{before} tracer (forward in time). 
This choice is discussed by \citet{Webb_al_JAOT98} in the context of the 
QUICK advection scheme. UBS and QUICK schemes only differ 
by one coefficient. Replacing 1/6 with 1/8 in \eqref{Eq_tra_adv_ubs} 
leads to the QUICK advection scheme \citep{Webb_al_JAOT98}. 
This option is not available through a namelist parameter, since the 
1/6 coefficient is hard coded. Nevertheless it is quite easy to make the 
substitution in the \mdl{traadv\_ubs} module and obtain a QUICK scheme.

Note that :

(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. Such a technique has been 
implemented and validated in ORCA05 with 301 levels. It is not available 
in the current reference version. 

(2) In a forthcoming release four options will be available 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 the \citet{Shchepetkin_McWilliams_OM05} 
implementation of UBS in ROMS, or  \textit{(d)} a UBS. The $3^{rd}$ case 
has dispersion properties similar to an eighth-order accurate conventional scheme.

(3) It is straightforward to rewrite \eqref{Eq_tra_adv_ubs} as follows:
\begin{equation} \label{Eq_traadv_ubs2}
\tau _u^{ubs} = \tau _u^{cen4} + \frac{1}{12} \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}
or equivalently 
\begin{equation} \label{Eq_traadv_ubs2b}
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_traadv_ubs2} has several advantages. Firstly, it clearly reveals 
that the UBS scheme is based on the fourth order scheme to which an 
upstream-biased diffusion term is added. Secondly, this emphasises that the 
$4^{th}$ order part (as well as the $2^{nd}$ order part as stated above) has 
to be evaluated at the \emph{now} time step using \eqref{Eq_tra_adv_ubs}. 
Thirdly, the diffusion term is in fact a biharmonic operator with an eddy 
coefficient which is simply proportional to the velocity:
 $A_u^{lm}= - \frac{1}{12}\,{e_{1u}}^3\,|u|$. Note that NEMO v2.3 still uses 
 \eqref{Eq_tra_adv_ubs}, not \eqref{Eq_traadv_ubs2}. This should be 
 changed in forthcoming release.
 %%%
 \gmcomment{the change in UBS scheme has to be done}
 %%%

% -------------------------------------------------------------------------------------------------------------
%        QCK scheme  
% -------------------------------------------------------------------------------------------------------------
\subsection   [QUICKEST scheme (QCK) (\np{ln\_traadv\_qck})]
         {QUICKEST scheme (QCK) (\np{ln\_traadv\_qck}=true)}
\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 the ULTIMATE QUICKEST 
limiter \citep{Leonard1991}. It has been implemented in NEMO by G. Reffray 
(MERCATOR-ocean) and can be found in the \mdl{traadv\_qck} module.
The resulting scheme is quite expensive but \emph{positive}. 
It can be used on both active and passive tracers. 
Nevertheless, the intrinsic diffusion of QCK makes its use risky in the vertical 
direction where the control of artificial diapycnal fluxes is of paramount importance. 
Therefore the vertical flux is evaluated using the CEN2 scheme. 
This no more ensure the positivity of the scheme. The use of TVD in the vertical 
direction as for the UBS case should be implemented to maintain the property.


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

The Piecewise Parabolic Method (PPM) proposed by Colella and Woodward (1984) 
is based on a quadradic piecewise rebuilding. Like the QCK scheme, it is associated 
with the ULTIMATE QUICKEST limiter \citep{Leonard1991}. It has been implemented 
in \NEMO by G. Reffray (MERCATOR-ocean) but is not yet offered in the reference 
version 3.0.

% ================================================================
% Tracer Lateral Diffusion
% ================================================================
\section  [Tracer Lateral Diffusion (\textit{traldf})]
      {Tracer Lateral Diffusion (\mdl{traldf})}
\label{TRA_ldf}
%-----------------------------------------nam_traldf------------------------------------------------------
\namdisplay{namtra_ldf}
%-------------------------------------------------------------------------------------------------------------
 
The options available for lateral diffusion are a laplacian (rotated or not) 
or a biharmonic operator, the latter being more scale-selective (more 
diffusive at small scales). The specification of eddy diffusivity 
coefficients (either constant or variable in space and time) as well as the 
computation of the slope along which the operators act, are performed in the 
\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 rotation tensor 
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 (lap) (\np{ln\_traldf\_lap})]
         {Iso-level laplacian operator (lap) (\np{ln\_traldf\_lap}=true) }
\label{TRA_ldf_lap}

A laplacian diffusion operator ($i.e.$ a harmonic operator) acting along the model 
surfaces is given by: 
\begin{equation} \label{Eq_tra_ldf_lap}
D_T^{lT} =\frac{1}{b_tT} \left( \;
   \delta _{i}\left[ A_u^{lT} \; \frac{e_{2u}\,e_{3u}}{e_{1u}} \;\delta _{i+1/2} [T] \right] 
+ \delta _{j}\left[ A_v^{lT} \;  \frac{e_{1v}\,e_{3v}}{e_{2v}} \;\delta _{j+1/2} [T] \right]  \;\right)
\end{equation}
where  $b_t$=$e_{1t}\,e_{2t}\,e_{3t}$  is the volume of $T$-cells. 
It can be found in the \mdl{traadv\_lap} module.

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

Note that 
(a) In pure $z$-coordinate (\key{zco} is defined), $e_{3u}$=$e_{3v}$=$e_{3t}$, 
so that the vertical scale factors disappear from (\ref{Eq_tra_ldf_lap}) ; 
(b) In partial step $z$-coordinate (\np{ln\_zps}=true), tracers in horizontally 
adjacent cells are located at different depths in the 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 the \mdl{zpshde} module, 
described in \S\ref{TRA_zpshde}.

% -------------------------------------------------------------------------------------------------------------
%        Rotated laplacian operator
% -------------------------------------------------------------------------------------------------------------
\subsection   [Rotated laplacian operator (iso) (\np{ln\_traldf\_lap})]
         {Rotated laplacian operator (iso) (\np{ln\_traldf\_lap}=true)}
\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}{b_t}   & \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} [T] 
        - e_{1v}\,r_{2v} \,\overline{\overline{ \delta_{k+1/2} [T] }}^{\,j+1/2,k} 
                                                    \right)   \right]                 \\ 
& +\delta_k \left[ A_w^{lT} \left( 
       -\;e_{2w}\,r_{1w} \,\overline{\overline{ \delta_{i+1/2} [T] }}^{\,i,k+1/2}
                                                    \right.   \right.                 \\ 
& \qquad \qquad \quad 
        - e_{1w}\,r_{2w} \,\overline{\overline{ \delta_{j+1/2} [T] }}^{\,j,k+1/2}     \\
& \left. {\left. {   \qquad \qquad \ \ \ \left. {
        +\;\frac{e_{1w}\,e_{2w}}{e_{3w}} \,\left( r_{1w}^2 + r_{2w}^2 \right)
           \,\delta_{k+1/2} [T] } \right) } \right] \quad } \right\} 
 \end{split}
 \end{equation}
where $b_t$=$e_{1t}\,e_{2t}\,e_{3t}$  is the volume of $T$-cells, 
$r_1$ and $r_2$ are the slopes between the surface of computation 
($z$- or $s$-surfaces) and the surface along which the diffusion operator 
acts ($i.e.$ horizontal or iso-neutral surfaces).  It is thus used when, 
in addition to \np{ln\_traldf\_lap}= true, we have \np{ln\_traldf\_iso}=true, 
or both \np{ln\_traldf\_hor}=true and \np{ln\_zco}=true. 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 that used in the vertical 
physics (see \S\ref{TRA_zdf}). For computer efficiency reasons, this term 
is not computed in the \mdl{traldf\_iso} module, but in the \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 the model to run safely without any additional 
background horizontal diffusion \citep{Guilyardi_al_CD01}. An alternative scheme 
developed by \cite{Griffies_al_JPO98} which preserves both tracer and its variance 
is currently been tested in \NEMO. It should be available in a forthcoming
release.

Note that in the partial step $z$-coordinate (\np{ln\_zps}=true), the horizontal 
derivatives at the bottom level in \eqref{Eq_tra_ldf_iso} 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 (bilap) (\np{ln\_traldf\_bilap})]
         {Iso-level bilaplacian operator (bilap) (\np{ln\_traldf\_bilap}=true)}
\label{TRA_ldf_bilap}

The lateral fourth order bilaplacian operator on tracers is obtained by 
applying (\ref{Eq_tra_ldf_lap}) twice. The operator requires an additional assumption 
on boundary conditions: both first and third derivative terms normal to the 
coast are set to zero. It is used when, in addition to \np{ln\_traldf\_bilap}=true, 
we have \np{ln\_traldf\_level}=true, or both \np{ln\_traldf\_hor}=true and 
\np{ln\_zco}=false. In both cases, it can contribute diapycnal mixing, 
although less than in the laplacian case. It is therefore not recommended.

Note that in the code, the bilaplacian routine does not call the laplacian 
routine twice but is rather a separate routine that can be found in the
\mdl{traldf\_bilap} module. This is due to the fact that we introduce the 
eddy diffusivity coefficient, A, in the operator as: 
$\nabla \cdot \nabla \left( {A\nabla \cdot \nabla T} \right)$, 
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 (bilapg) (\np{ln\_traldf\_bilap})]
         {Rotated bilaplacian operator (bilapg) (\np{ln\_traldf\_bilap}=true)}
\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 can be found in the
\mdl{traldf\_bilapg}.

It is used when, in addition to \np{ln\_traldf\_bilap}=true, we have 
\np{ln\_traldf\_iso}= .true, or both \np{ln\_traldf\_hor}=true and \np{ln\_zco}=true. 
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 (\textit{trazdf})]
      {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, and is based on a laplacian operator. 
The vertical diffusion 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, 
$A_w^{vT}=A_w^{vS}$ except when double diffusive mixing is 
parameterised ($i.e.$ \key{zdfddm} is defined). The way these coefficients 
are evaluated is given in \S\ref{ZDF} (ZDF). Furthermore, when 
iso-neutral mixing is used, both mixing coefficients 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 
heat and salt must be specified. At the surface they are prescribed 
from the surface forcing and added in a dedicated routine (see \S\ref{TRA_sbc}), 
whilst at the bottom they are set to zero for heat and salt unless 
a geothermal flux forcing is prescribed as a bottom boundary 
condition (see \S\ref{TRA_bbc}). 

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

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

% -------------------------------------------------------------------------------------------------------------
%        surface boundary condition
% -------------------------------------------------------------------------------------------------------------
\subsection   [Surface boundary condition (\textit{trasbc})]
         {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. 

Due to interactions and mass exchange with other media ($i.e.$ atmosphere, sea-ice, land),
the change in the heat and salt content of the surface layer of the ocean is due both 
to the heat and salt fluxes crossing the sea surface and not linked with $F_{mass}$, the water 
exchange with the other media, and to the heat and salt content of this water exchange.
In a forcoming release, these two parts, computed in the surface module (SBC), will included directly
in $Q_{ns}$, the surface heat flux and $F_{salt}$, the surface salt flux.
\gmcomment{  The specification of those fluxes is further detail in the SBC chapter (see \S\ref{SBC}).  }
This change will provide a same forcing formulation for any tracers (including temperature and salinity).
 
In the current version, the situation is a little bit more complicated. 
The surface module (\mdl{sbcmod}, see \S\ref{SBC}) provides the following 
forcing fields (used on tracers):

$\bullet$ $Q_{ns}$,the non solar part of the net surface heat flux that cross the sea surface 
(difference between the total surface heat flux and the fraction of the short wave flux that 
penetrates into the water column, see \S\ref{TRA_qsr})

$\bullet$ \textit{emp}, the mass flux exchanged with the atmosphere (evaporation minus precipitation)

$\bullet$ $\textit{emp}_S$, an equivalent mass flux taking into account the effect of ice-ocean mass exchanged

$\bullet$ \textit{rnf}, the mass flux associated with runoff (see \S\ref{SBC_rnf} for further detail of how it acts on temperature and salinity tendencies)

The $\textit{emp}_S$ field is not simply the budget evaporation-precipitation+freezing-melting because 
the sea-ice is not currently embedded in the ocean but levitates above it. There is not mass
exchanged between the sea-ice and the ocean. Instead we only take into account the salt
flux link to the fact that sea-ice has a non-sero salinity, and the concentration/dilution effect
due to the freezing/melting (F/M) process. These two parts of the forcing are then converted into 
a equivalent mass flux given by $\textit{emp}_S - \textit{emp}$. As a result of this mess, 
the surface boundary condition on temperature and salinity is applied as follows:

In the nonlinear free surface case (\key{vvl} is defined, \jp{lk\_vvl}=true):
\begin{equation} \label{Eq_tra_sbc}
\begin{aligned}
 &F^T = \frac{ 1 }{\rho _o \;C_p \,\left. e_{3t} \right|_{k=1} }   
           &\overline{ \left( Q_{ns} - \textit{emp}\;C_p\,\left. T \right|_{k=1} \right) }^t  & \\ 
%
& F^S =\frac{ 1 }{\rho _o \,\left. e_{3t} \right|_{k=1} } 
           &\overline{ \left( (\textit{emp}_S - \textit{emp})\;\left. S \right|_{k=1}  \right) }^t   & \\   
 \end{aligned}
\end{equation} 

In the linear free surface case (\key{vvl} not defined, , \jp{lk\_vvl}=false):
\begin{equation} \label{Eq_tra_sbc_lin}
\begin{aligned}
 &F^T = \frac{ 1 }{\rho _o \;C_p \,\left. e_{3t} \right|_{k=1} }  &\overline{ Q_{ns} }^t  & \\ 
%
& F^S =\frac{ 1 }{\rho _o \,\left. e_{3t} \right|_{k=1} } 
           &\overline{ \left( \textit{emp}_S\;\left. S \right|_{k=1}  \right) }^t   & \\   
 \end{aligned}
\end{equation} 
where $\overline{x }^t$ means that $x$ is averaged over two consecutive time step 
($t-\rdt/2$ and $t+\rdt/2$). Such a time averaged prevents the excitation of the 
divergence of odd and even time step (see \S\ref{STP}).

The two set of equations, \eqref{Eq_tra_sbc} and \eqref{Eq_tra_sbc_lin}, are obtained 
by assuming that the temperature of precipitation and evaporation are equal to
the ocean surface temperature while their salinity is zero. Therefore, the heat content
of \textit{emp} budget must be added to the temperature equation in variable volume case, 
while it does not appear in constant volume. Similarly, the \textit{emp} budget affects 
the ocean surface salinity in constant volume case (through the concentration dilution effect)
while it does not appears explicitly in variable volume as salinity change will be
induced by volume change. In both constant and variable volume, surface salinity 
will change with ice-ocean salt flux and F/M flux without mass exchanges 
($\textit{emp}_S - \textit{emp}$).

Note that concentration/dilution effect due to F/M is computed using
a constant ice salinity as well as a constant ocean salinity. 
This approximation suppresses the correlation between \textit{SSS} 
and F/M flux, allowing the ice-ocean salt exchanges to be conservative.
Indeed, if this approximation is not made, even if the F/M budget is zero 
on average over the whole ocean domain and over the seasonal cycle, 
the associated salt flux is not, since sea-surface salinity and F/M flux are 
intrinsically correlated (high \textit{SSS} are found where freezing is 
strong whilst low \textit{SSS} is usually associated with high melting areas.

Even using this approximation, an exact conservation of heat and salt content 
is only achieved in the variable volume case. In the constant volume case, 
there is a small unbalance associated with the product $(\partial_t\eta - \textit{emp}) * \textit{SSS}$.
Nevertheless, the salt content variation is quite small and will not induce
a long term drift as there is no physical reason that $(\partial_t\eta - \textit{emp})$ 
and \textit{SSS} are correlated \citep{Roullet_Madec_JGR00}. 
Note that, while quite small, the unbalance in constant volume case is larger 
than the unbalance associated with the Asselin time filter \citep{Leclair_Madec_OM09}. 
This is the reason why the modified filter is not applied in constant volume case.

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

When the penetrative solar radiation option is used (\np{ln\_flxqsr}=true), 
the solar radiation penetrates the top few 10 meters of the ocean, otherwise 
all the heat flux is absorbed in the first ocean level (\np{ln\_flxqsr}=false). 
Thus, in the former case a term is added to the time evolution equation of 
temperature \eqref{Eq_PE_tra_T} whilst 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 $Q_{sr}$ is the penetrative part of the surface heat flux ($i.e.$ the shortwave radiation) 
and $I$ is the downward irradiance ($\left. I \right|_{z=\eta}=Q_{sr}$). 
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}

The shortwave radiation,  $Q_{sr}$, consists of energy distributed across a wide spectral range. 
The ocean is strongly absorbing for wavelengths longer than 700~nm and these 
wavelengths contribute to heating the upper few 10 centimetres. The fraction of $Q_{sr}$ 
that resides in these almost non-penetrative wavebands, $R$, is $\sim 58\%$ (specified 
through namelist parameter \np{rn\_abs}).  It is assumed to penetrate the ocean 
following a decreasing exponential profile, with an e-folding depth scale, $\xi_0$, 
of a few 10 centimetres (typically $\xi_0=0.35~m$ set as \np{rn\_si0} in the namtra\_qsr namlist).
For shorter wavelengths (400-700~nm), the ocean is more transparent, and solar energy 
propagates to depths where it contributes to a penetrating flux of solar energy and thus 
to local heating below the surface. 
The way this second part of the solar energy penetrates in the ocean depends on 
which formulation is chosen. In the simple 2-wavebands light penetration  (\np{ln\_qsr\_2bd}=true) 
a chlorophyll-independent monochromatic formulation is also chosen for the shorter wavelengths, 
leading to the following expression  \citep{Paulson1977}:
\begin{equation} \label{Eq_traqsr_iradiance}
I(z) = Q_{sr} \left[Re^{-z / \xi_0} + \left( 1-R\right) e^{-z / \xi_1} \right]
\end{equation}
where $\xi_1$ is the second extinction length scales associated with the shorter wavebands.  
It is usually chosen to be 23~m through \np{rn\_si0} namelist parameter. 
The set of default values ($\xi_0$, $\xi_1$, $R$) corresponds to a Type I water in 
Jerlov's (1968) classification (oligotrophic waters).

Such assumptions have been shown to provide a very crude and simplistic 
representation of observed light penetration profiles (\cite{Morel_JGR88}, see also 
Fig.\ref{Fig_traqsr_irradiance}). Light absorption in the ocean depends on the 
particules concentration and it is spectrally selective. \cite{Morel_JGR88} has shown 
that an accurate representation of light penetration can be provided by a 61 waveband 
formulation. Unfortunately, such a model is very computationally expensive. 
Thus, \cite{Lengaigne_al_CD07} have constructed a simplified version of this 
formulation in which visible light is splitted into three wavebands: blue (400-500 nm), 
green (500-600 nm) and red (600-700nm). For each wave-band, the chlorophyll-dependant 
attenuation coefficient is fitted to the coefficients computed from the full spectral model 
of \cite{Morel_JGR88} (as modified by \cite{Morel_Maritorena_JGR01}) assuming 
the same power-law expression. As shown on Fig.\ref{Fig_traqsr_irradiance}, 
this formulation, called RGB (Reed-Green-Blue), reproduces quite closely 
the light penetration profiles predicted by the full spectal model with much faster 
computing efficiently, in contrast with the 2-bands formulation. 

The RGB formulation is used when \np{ln\_qsr\_rgb}=true. The RGB attenuation coefficients
($i.e.$ the inverse of the extinction length scales) are tabulated over 61 nonuniform 
chlorophyll classes ranging from 0.01 to 10 g.Chl/L (see the routine \rou{trc\_oce\_rgb} 
in \mdl{trc\_oce} module). Three type of chlorophyll can be used in the RGB formulation:
(1) a constant 0.05 g.Chl/L value everywhere (\np{nn\_chdta}=0) ; (2) observed 
time varying chlorophyll (\np{nn\_chdta}=0) ; (3) simulated time varying chlorophyll
by TOP biogeochemical model (\np{ln\_qsr\_bio}=true). In the later case, the RGB 
formulation is used to calculated both the phytoplankton light limitation in PISCES 
or LOBSTER and the oceanic heating rate. 

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 is modified in routine \mdl{traqsr}. 

When $z$-coordinate is preferred to $s$-coordinate, the depth of $w-$levels does 
not significantly vary with location. The level at which the light has been totally 
absorbed ($i.e.$ it is 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. 
Finally, note that when the ocean is shallow ($<$ 200~m), part of the 
solar radiation can reach the ocean floor. In this case, we have 
chosen that all remaining radiation is absorbed in the last ocean 
level ($i.e.$ $I$ is masked). 

%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
\begin{figure}[!t] \label{Fig_traqsr_irradiance}  \begin{center}
\includegraphics[width=1.0\textwidth]{./TexFiles/Figures/Fig_TRA_Irradiance.pdf}
\caption{Penetration profile of the Downward solar irradiance 
calculated by four models. Two wavebands chlorophyll-independant formulation (blue), 
a chlorophyll-dependant monochromatic formulation (green), 4 waveband RGB formulation (red), 
61 waveband Morel (1988) formulation (black) for a chlorophyll concentration of 
(a) Chl=0.05 mg/m$^3$ and (b) Chl=0.5 mg/m$^3$. From \citet{Lengaigne_al_CD07}.}
\end{center}   \end{figure}
%>>>>>>>>>>>>>>>>>>>>>>>>>>>>

% -------------------------------------------------------------------------------------------------------------
%        Bottom Boundary Condition
% -------------------------------------------------------------------------------------------------------------
\subsection   [Bottom Boundary Condition (\textit{trabbc} - \key{bbc})]
         {Bottom Boundary Condition (\mdl{trabbc} - \key{bbc})}
\label{TRA_bbc}
%--------------------------------------------nambbc--------------------------------------------------------
\namdisplay{namtra_bbc}
%--------------------------------------------------------------------------------------------------------------
%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
\begin{figure}[!t] \label{Fig_geothermal}  \begin{center}
\includegraphics[width=1.0\textwidth]{./TexFiles/Figures/Fig_TRA_geoth.pdf}
\caption{Geothermal Heat flux (in $mW.m^{-2}$) used by \cite{Emile-Geay_Madec_OS09}.
It is inferred from the age of the sea floor and the formulae of \citet{Stein_Stein_Nat92}.}
\end{center}   \end{figure}
%>>>>>>>>>>>>>>>>>>>>>>>>>>>>

Usually it is assumed that there is no exchange of heat or 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. However, there is a 
non-zero heat flux across the seafloor that is associated with solid 
earth cooling. This flux is weak compared to surface fluxes (a mean 
global value of $\sim0.1\;W/m^2$ \citep{Stein_Stein_Nat92}), but it is 
systematically positive and acts on the densest water masses. 
Taking this flux into account in a global ocean model increases
the deepest overturning cell ($i.e.$ the one associated with the Antarctic 
Bottom Water) by a few Sverdrups  \citep{Emile-Geay_Madec_OS09}. 

The presence or not of geothermal heating is controlled by the namelist 
parameter  \np{nn\_geoflx}. When this parameter is set to 1, a constant 
geothermal heating is introduced whose value is given by the 
\np{nn\_geoflx\_cst}, which is also a namelist parameter. When it is set to 2, 
a spatially varying geothermal heat flux is introduced which is provided 
in the \ifile{geothermal\_heating} NetCDF file (Fig.\ref{Fig_geothermal}).

% ================================================================
% Bottom Boundary Layer
% ================================================================
\section  [Bottom Boundary Layer (\mdl{trabbl} - \key{trabbl})]
      {Bottom Boundary Layer (\mdl{trabbl} - \key{trabbl})}
\label{TRA_bbl}
%--------------------------------------------nambbl---------------------------------------------------------
\namdisplay{nambbl}
%--------------------------------------------------------------------------------------------------------------

In a $z$-coordinate configuration, the bottom topography is represented by a 
series of discrete steps. This is not adequate to represent gravity driven 
downslope flows. Such flows arise either downstream of sills such as the Strait of 
Gibraltar or Denmark Strait, where dense water formed in marginal seas flows 
into a basin filled with less dense water, or along the continental slope when dense 
water masses are formed on a continental shelf. The amount of entrainment 
that occurs in these gravity plumes is critical in determining the density 
and volume flux of the densest waters of the ocean, such as Antarctic Bottom Water, 
or 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 the $s$-coordinate when the thickness of the bottom level varies in large 
proportions downstream of a sill \citep{Willebrand_al_PO01}, and the thickness 
of the plume is not resolved. 

The idea of the bottom boundary layer (BBL) parameterisation, first introduced by 
\citet{Beckmann_Doscher1997}, is to allow a direct communication between 
two adjacent bottom cells at different levels, whenever the densest water is 
located above the less dense water. The communication can be by a diffusive 
(diffusive BBL), advective fluxes (advective BBL), or both. In the current 
implementation of the BBL, only the tracers are modified, not the velocities. 
Furthermore, it only connects ocean bottom cells, and therefore does not include 
one of the improvment introduced by \citet{Campin_Goosse_Tel99}.

% -------------------------------------------------------------------------------------------------------------
%        Diffusive BBL
% -------------------------------------------------------------------------------------------------------------
\subsection{Diffusive Bottom Boundary layer (\np{nn\_bbl\_ldf}=1)}
\label{TRA_bbl_diff}

When applying sigma-diffusion (\key{trabbl} defined and \np{nn\_bbl\_ldf} set to 1), 
the diffusive flux between two adjacent cells living at the ocean bottom is given by 
\begin{equation} \label{Eq_tra_bbl_diff}
{\rm {\bf F}}_\sigma=A_l^\sigma \; \nabla_\sigma T
\end{equation} 
with $\nabla_\sigma$ the lateral gradient operator taken between bottom cells, 
and  $A_l^\sigma$ the lateral diffusivity in the BBL. Following \citet{Beckmann_Doscher1997}, 
the latter is prescribed with a spatial dependence, $e.g.$ in the conditional form
\begin{equation} \label{Eq_tra_bbl_coef}
A_l^\sigma (i,j,t)=\left\{ {\begin{array}{l}
 A_{bbl}  \quad \quad   \mbox{if}  \quad   \nabla_\sigma \rho  \cdot  \nabla H<0 \\ 
 \\
 0\quad \quad \;\,\mbox{otherwise} \\ 
 \end{array}} \right.
\end{equation} 
where $A_{bbl}$ is the BBL diffusivity coefficient, given by the namelist 
parameter \np{rn\_ahtbbl} and usually set to a value much larger 
than the one used on lateral mixing in open ocean. The constraint in \eqref{Eq_tra_bbl_coef} 
implies that sigma-like diffusion only occurs when density above the sea floor, at the top of 
the slope, is larger than in the deeper ocean (see green arrow in Fig.\ref{Fig_bbl}). 
In practice, this constraint is applied separately in the two horizontal directions, 
and the density gradient in \eqref{Eq_tra_bbl_coef} is evaluated with the log gradient formulation: 
\begin{equation} \label{Eq_tra_bbl_Drho}
   \nabla_\sigma \rho / \rho = \alpha \,\nabla_\sigma T + \beta   \,\nabla_\sigma S
\end{equation} 
where $\rho$, $\alpha$ and $\beta$ are function of $\overline{T}^\sigma$, 
$\overline{S}^\sigma$, $\overline{H}^\sigma$, the along bottom mean temperature, 
salinity and depth, respectively.

% -------------------------------------------------------------------------------------------------------------
%        Advective BBL
% -------------------------------------------------------------------------------------------------------------
\subsection   {Advective Bottom Boundary Layer  (\np{nn\_bbl\_adv}= 1 or 2)}
\label{TRA_bbl_adv}


%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
\begin{figure}[!t] \label{Fig_bbl}  \begin{center}
\includegraphics[width=0.7\textwidth]{./TexFiles/Figures/Fig_BBL_adv.pdf}
\caption{Advective/diffusive Bottom Boundary Layer. The BBl parameterisation is 
activated when $\rho^i_{kup}$ is larger than $\rho^{i+1}_{kdnw}$. 
Red arrows indicate the additional overturning circulation due to the advective BBL. 
The transport of the downsloping flow is defined either as the transport of the bottom 
ocean cell (black arrow), or as a function of the along slope density gradient. 
The green arrow indicates the diffusive BBL flux connecting directly $kup$ and $kdwn$
ocean bottom cells.
connection}
\end{center}   \end{figure}
%>>>>>>>>>>>>>>>>>>>>>>>>>>>>


%!!      nn_bbl_adv = 1   use of the ocean velocity as bbl velocity
%!!      nn_bbl_adv = 2   follow Campin and Goosse (1999) implentation
%!!        i.e. transport proportional to the along-slope density gradient

%%%gmcomment   :  this section has to be really written

When applying an advective BBL (\np{nn\_bbl\_adv} = 1 or 2), an overturning 
circulation is added which connects two adjacent bottom grid-points only if dense 
water overlies less dense water on the slope. The density difference causes dense 
water to move down the slope. 

\np{nn\_bbl\_adv} = 1 : the downsloping velocity is chosen to be the Eulerian
ocean velocity just above the topographic step (see black arrow in Fig.\ref{Fig_bbl}) 
\citep{Beckmann_Doscher1997}. It is a \textit{conditional advection}, that is, advection
is allowed only if dense water overlies less dense water on the slope ($i.e.$ 
$\nabla_\sigma \rho  \cdot  \nabla H<0$) and if the velocity is directed towards 
greater depth ($i.e.$ $\vect{U}  \cdot  \nabla H>0$).

\np{nn\_bbl\_adv} = 2 : the downsloping velocity is chosen to be proportional to $\Delta \rho$,
the density difference between the top and down cell densities \citep{Campin_Goosse_Tel99}.
The advection is allowed only  if dense water overlies less dense water on the slope ($i.e.$ 
$\nabla_\sigma \rho  \cdot  \nabla H<0$). For example, the resulting transport of the 
downsloping flow, here in the $i$-direction (Fig.\ref{Fig_bbl}), is simply given by the 
following expression:
\begin{equation} \label{Eq_bbl_Utr}
 u^{tr}_{bbl} = \gamma \, g \frac{\Delta \rho}{\rho_o}  e_{1u} \; min \left( {e_{3u}}_{kup},{e_{3u}}_{kdwn} \right)
\end{equation}
where $\gamma$, expressed in second, is the coefficient of proportionality 
provided as \np{rn\_gambbl}, a namelist parameter, and \textit{kup} and \textit{kdwn} 
are the vertical index of the top and bottom cells, respectively.
The parameter $\gamma$ should take a different value for each bathymetric 
step. But, for simplicity, and because no direct estimation of this parameter is 
available, a uniform value has been retained. The possible values for $\gamma$ 
range between 1 and $10~s$ \citep{Campin_Goosse_Tel99}.  

The scalar properties are advected by this additional transport $( u^{tr}_{bbl}, v^{tr}_{bbl} )$ 
using the upwind scheme. Such a diffusive advective scheme has been chosen 
to mimic the entrainment between the downsloping plume and the surrounding 
water at intermediate depth. The entrainment is replaced by the vertical mixing 
included in the advection scheme. Let us consider as an example the 
case display in Fig.\ref{Fig_bbl} where the density at level $(i,kup)$ is 
larger than the one at level $(i,kdwn)$. The advective BBL scheme
modifies the tracer time tendency of the ocean cells near the 
topographic step by the downsloping flow \eqref{Eq_bbl_dw}, 
the horizontal \eqref{Eq_bbl_hor}  and the upward \eqref{Eq_bbl_up} 
return flows as follows: 
\begin{align} 
\partial_t T^{do}_{kdw} &\equiv \partial_t T^{do}_{kdw}
                                     +  \frac{u^{tr}_{bbl}}{{b_t}^{do}_{kdw}}  \left( T^{sh}_{kup} - T^{do}_{kdw} \right)  \label{Eq_bbl_dw} \\
%
\partial_t T^{sh}_{kup} &\equiv \partial_t T^{sh}_{kup} 
               + \frac{u^{tr}_{bbl}}{{b_t}^{sh}_{kup}}   \left( T^{do}_{kup} - T^{sh}_{kup} \right)   \label{Eq_bbl_hor} \\
%
\intertext{and for $k =kdw-1,\;..., \; kup$ :} 
%
\partial_t T^{do}_{k} &\equiv \partial_t S^{do}_{k}
               + \frac{u^{tr}_{bbl}}{{b_t}^{do}_{k}}   \left( T^{do}_{k+1} - T^{sh}_{k} \right)   \label{Eq_bbl_up}
\end{align}
where $b_t$ is the $T$-cell volume. 

Note that the BBL transport, $( u^{tr}_{bbl}, v^{tr}_{bbl} )$, is available in 
the model outputs. It has to be used to compute the effective velocity 
as well as the effective overturning circulation.

% ================================================================
% Tracer damping
% ================================================================
\section  [Tracer damping (\textit{tradmp})]
      {Tracer damping (\mdl{tradmp})}
\label{TRA_dmp}
%--------------------------------------------namtra_dmp-------------------------------------------------
\namdisplay{namtra_dmp}
%--------------------------------------------------------------------------------------------------------------

In some applications it can be useful to add a Newtonian damping term 
into 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 routine 
\rou{dtacof} also located in module \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 time scale of a few days) 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 us to find the velocity 
field consistent with the model dynamics whilst having a $T$-$S$ field 
close to a given climatological field ($T_o -S_o$). The time scale 
associated with $S_o$ is generally not a constant but spatially varying 
in order to respect other properties. For example, it is usually set to zero 
in the mixed layer (defined either on a density or $S_o$ criterion) 
\citep{Madec_al_JPO96} and in the equatorial region 
\citep{Reverdin1991, Fujio1991, Marti_PhD92} since these two regions 
have a short 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 the time scale is reduced (even to zero) along the western 
boundary to allow the model to reconstruct its own western boundary 
structure in equilibrium with its physics. 
The choice of the shape of the Newtonian damping is controlled by two 
namelist parameters \np{nn\_zdmp}. The former allows to specified the 
width of the equatorial band in which no damping is applied as well as a decrease 
in vicinity of the coast and a damping everywhere in the Red and Med Seas, 
whereas the latter set a damping acting in the mixed layer or not. 
The time scale associated with the damping depends on the depth as
a hyperbolic tangent, with \np{rn\_surf} as surface value, \np{rn\_bot} as 
bottom value and a transition depth of \np{rn\_dep}.  

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

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

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

The general framework for tracer time stepping is a modified leap-frog scheme 
\citep{Leclair_Madec_OM09}, $i.e.$ a three level centred time scheme associated 
with a Asselin time filter (cf. \S\ref{STP_mLF}):
\begin{equation} \label{Eq_tra_nxt}
\begin{aligned}
(e_{3t}T)^{t+\rdt} &= (e_{3t}T)_f^{t-\rdt} &+ 2 \, \rdt  \,e_{3t}^t\ \text{RHS}^t & \\
\\
(e_{3t}T)_f^t  \;\ \quad &= (e_{3t}T)^t \;\quad 
                                    &+\gamma \,\left[ {(e_{3t}T)_f^{t-\rdt} -2(e_{3t}T)^t+(e_{3t}T)^{t+\rdt}} \right] &  \\
                                 & &- \gamma\,\rdt \, \left[ Q^{t+\rdt/2} -  Q^{t-\rdt/2} \right]  &                      
\end{aligned}
\end{equation} 
where RHS is the right hand side of the temperature equation, 
the subscript $f$ denotes filtered values, $\gamma$ is the Asselin coefficient,
and $S$ is the total forcing applied on $T$ ($i.e.$ fluxes plus content in mass exchanges). 
$\gamma$ is initialized as \np{rn\_atfp} (\textbf{namelist} parameter). 
Its default value is \np{rn\_atfp}=$10^{-3}$. Note that the forcing correction term in the filter
is not applied in linear free surface (\jp{lk\_vvl}=false) (see \S\ref{TRA_sbc}.
Not also that in constant volume case, the time stepping is performed on $T$, 
not on its content, $e_{3t}T$.

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

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

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

% -------------------------------------------------------------------------------------------------------------
%        Equation of State
% -------------------------------------------------------------------------------------------------------------
\subsection{Equation of State (\np{nn\_eos} = 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 seawater volumic mass, $\rho$, 
abusively called density, is a non linear empirical function of \textit{in situ} 
temperature, salinity and pressure. The reference equation of state is that 
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. However, even though this computation is fully vectorised, 
it is quite time consuming ($15$ to $20${\%} of the total CPU time) since 
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 used the \citet{JackMcD1995} equation of state for 
seawater instead. It allows the computation of the \textit{in situ} ocean density 
directly as a function of \textit{potential} temperature relative to the surface 
(an \NEMO variable), the practical salinity (another \NEMO 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 exactly the same except that the values of the various coefficients have 
been adjusted by \citet{JackMcD1995} in order to directly use the \textit{potential} 
temperature instead of the \textit{in situ} one. This reduces the CPU time of the 
\textit{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 anomaly, $d_a= \rho / \rho_o - 1$, 
is computed, with $\rho_o$ a reference volumic mass. Called \textit{rau0} 
in the code, $\rho_o$ is defined in \mdl{phycst}, and a value of $1,035~Kg/m^3$. 
This is a sensible choice for the reference density used in a Boussinesq ocean 
climate model, as, with the exception of only a small percentage of the ocean, 
density in the World Ocean varies by no more than 2$\%$ from $1,035~kg/m^3$ 
\citep{Gill1982}.

The default option (namelist parameter \np{nn\_eos}=0) is the \citet{JackMcD1995} 
equation of state. Its use is highly recommended. However, for process studies, 
it is often convenient to use a linear approximation of the density.
With such an equation of state there is no longer a distinction between 
\textit{in situ} and \textit{potential} density and both cabbeling and thermobaric
effects are removed.
Two linear formulations are available: a function of $T$ only (\np{nn\_eos}=1) 
and a function of both $T$ and $S$ (\np{nn\_eos}=2):
\begin{equation} \label{Eq_tra_eos_linear}
\begin{split}
  d_a(T)       &=  \rho (T)      /  \rho_o   - 1     =  \  0.0285         -  \alpha   \;T     \\ 
  d_a(T,S)    &=  \rho (T,S)   /  \rho_o   - 1     =  \  \beta \; S       -  \alpha   \;T    
\end{split}
\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 the \np{rn\_alpha} and 
\np{rn\_beta} namelist parameters). Note that when $d_a$ is a function 
of $T$ only (\np{nn\_eos}=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{nn\_eos} = 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 for $N^2$ 
 depends on the type of equation of state used (\np{nn\_eos} namelist parameter).

For \np{nn\_eos}=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{equation} \label{Eq_tra_bn2}
N^2 = \frac{g}{e_{3w}} \; \beta   \ 
      \left(  \alpha / \beta \ \delta_{k+1/2}[T]     - \delta_{k+1/2}[S]   \right) 
\end{equation} 
where $\alpha$ and $\beta$ are the thermal and haline expansion coefficients. 
They are a function of  $\overline{T}^{\,k+1/2},\widetilde{S}=\overline{S}^{\,k+1/2} - 35.$, 
and  $z_w$, with $T$ the \textit{potential} temperature and $\widetilde{S}$ a salinity anomaly. 
Note that both $\alpha$ and $\beta$ depend on \textit{potential} 
temperature and salinity which are averaged at $w$-points prior 
to the computation instead of being computed at $T$-points and 
then averaged to $w$-points.

When a linear equation of state is used (\np{nn\_eos}=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{phycst})]
         {Specific Heat (\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 and so 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 value is set in \mdl{phycst} module. 


% -------------------------------------------------------------------------------------------------------------
%        Freezing Point of Seawater
% -------------------------------------------------------------------------------------------------------------
\subsection   [Freezing Point of Seawater]
         {Freezing Point of Seawater}
\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) have been dropped. The freezing
point is computed through \textit{tfreez}, a \textsc{Fortran} function that can be found 
in \mdl{eosbn2}.  

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

\gmcomment{STEVEN: to be consistent with earlier discussion of differencing and averaging operators, I've changed "derivative" to "difference" and "mean" to "average"}

With partial bottom cells (\np{ln\_zps}=true), in general, tracers in horizontally 
adjacent cells 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) to be active. 
\gmcomment{STEVEN from gm : question: not sure of  what -to be active- means}
Before taking horizontal gradients between the tracers next to the bottom, a linear 
interpolation in the vertical 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, for 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]{./TexFiles/Figures/Partial_step_scheme.pdf}
\caption{ Discretisation of the horizontal difference and average of tracers in the $z$-partial step coordinate (\np{ln\_zps}=true) 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 difference is then given by: $\delta _{i+1/2} T_k=  \widetilde{T}_k^{\,i+1} -T_k^{\,i}$ and the average 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 forms for the horizontal difference and the horizontal average 
value of $T$ at a $U$-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 
procedure used to compute the interpolated density, $\widetilde{\rho}$, is not 
the same as that used for $T$ and $S$. Instead of forming a linear approximation 
of density, we compute $\widetilde{\rho }$ from the interpolated values of $T$ 
and $S$, and the pressure at a $u$-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})

Note that in almost all the advection schemes presented in this Chapter, both 
averaging and differencing operators appear. Yet \eqref{Eq_zps_hde} has not 
been used in these schemes: in contrast to diffusion and pressure gradient 
computations, no correction for partial steps is applied for advection. The main 
motivation is to preserve the domain averaged mean variance of the advected 
field when using the $2^{nd}$ order centred scheme. Sensitivity of the advection 
schemes to the way horizontal averages are performed in the vicinity of partial 
cells should be further investigated in the near future.
%%%
\gmcomment{gm :   this last remark has to be done}
%%%