source: trunk/DOC/TexFiles/Chapters/Chap_SBC.tex @ 1320

% ================================================================
% Chapter Ñ Surface Boundary Condition (SBC) 
% ================================================================
\chapter{Surface Boundary Condition (SBC) }
\label{SBC}
\minitoc

\newpage
$\ $\newline    % force a new ligne
%---------------------------------------namsbc--------------------------------------------------
\namdisplay{namsbc}
%--------------------------------------------------------------------------------------------------------------
$\ $\newline    % force a new ligne

The ocean needs six fields as surface boundary condition:
\begin{itemize}
\item the two components of the surface ocean stress $\left( {\tau _u \;,\;\tau _v} \right)$
\item the incoming solar and non solar heat fluxes $\left( {Q_{ns} \;,\;Q_{sr} } \right)$
\item the surface freshwater budget $\left( {\text{EMP},\;\text{EMP}_S } \right)$
\end{itemize}

Four different ways to provide those six fields to the ocean are available which 
are controlled by namelist variables: an analytical formulation (\np{ln\_ana}=true), 
a flux formulation (\np{ln\_flx}=true), a bulk formulae formulation (CORE 
(\np{ln\_core}=true) or CLIO (\np{ln\_clio}=true) bulk formulae) and a coupled 
formulation (exchanges with a atmospheric model via the OASIS coupler) 
(\np{ln\_cpl}=true). The frequency at which the six fields have to be updated is
the  \np{nf\_sbc} namelist parameter. 
When the fields are supplied from data files (flux and bulk formulations), the input fields 
need not be supplied on the model grid.  Instead a file of coordinates and weights can be supplied which
maps the data from the supplied grid to the model points (so called "Interpolation on the Fly").
In addition, the resulting fields can be further modified using 
several namelist options. These options control  the rotation of vector components
supplied relative to an east-north coordinate system onto the local grid directions in the model;
the addition of a surface restoring 
term to observed SST and/or SSS (\np{ln\_ssr}=true); the modification of fluxes 
below ice-covered areas (using observed ice-cover or a sea-ice model) 
(\np{nn\_ice}=0,1, 2 or 3); the addition of river runoffs as surface freshwater 
fluxes (\np{ln\_rnf}=true); the addition of a freshwater flux adjustment in 
order to avoid a mean sea-level drift (\np{nn\_fwb}= 0, 1 or 2); and the 
transformation of the solar radiation (if provided as daily mean) into a diurnal 
cycle (\np{ln\_dm2dc}=true).

In this chapter, we first discuss where the surface boundary condition 
appears in the model equations. Then we present the four ways of providing 
the surface boundary condition. Next the scheme for interpolation on the fly is described.
Finally, the different options that further modify
the fluxes applied to the ocean are discussed.


% ================================================================
% Surface boundary condition for the ocean
% ================================================================
\section{Surface boundary condition for the ocean}
\label{SBC_general}


The surface ocean stress is the stress exerted by the wind and the sea-ice 
on the ocean. The two components of stress are assumed to be interpolated 
onto the ocean mesh, $i.e.$ resolved onto the model (\textbf{i},\textbf{j}) direction 
at $u$- and $v$-points They are applied as a surface boundary condition of the 
computation of the momentum vertical mixing trend (\mdl{dynzdf} module) :
\begin{equation} \label{Eq_sbc_dynzdf}
\left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{z=1}
    = \frac{1}{\rho _o} \binom{\tau _u}{\tau _v }
\end{equation}
where $(\tau _u ,\;\tau _v )=(utau,vtau)$ are the two components of the wind 
stress vector in the $(\textbf{i},\textbf{j})$ coordinate system.

The surface heat flux is decomposed into two parts, a non solar and a solar heat 
flux, $Q_{ns}$ and $Q_{sr}$, respectively. The former is the non penetrative part 
of the heat flux ($i.e.$ the sum of sensible, latent and long wave heat fluxes). 
It is applied as a surface boundary condition trend of the first level temperature 
time evolution equation (\mdl{trasbc} module). 
\begin{equation} \label{Eq_sbc_trasbc_q}
\frac{\partial T}{\partial t}\equiv \cdots \;+\;\left. {\frac{Q_{ns} }{\rho 
_o \;C_p \;e_{3T} }} \right|_{k=1} \quad
\end{equation}
$Q_{sr}$ is the penetrative part of the heat flux. It is applied as a 3D 
trends of the temperature equation (\mdl{traqsr} module) when \np{ln\_traqsr}=True.

\begin{equation} \label{Eq_sbc_traqsr}
\frac{\partial T}{\partial t}\equiv \cdots \;+\frac{Q_{sr} }{\rho _o C_p 
\,e_{3T} }\delta _k \left[ {I_w } \right]
\end{equation}
where $I_w$ is a non-dimensional function that describes the way the light 
penetrates inside the water column. It is generally a sum of decreasing 
exponentials (see \S\ref{TRA_qsr}).

The surface freshwater budget is provided by fields: EMP and EMP$_S$ which 
may or may not be identical. Indeed, a surface freshwater flux has two effects: 
it changes the volume of the ocean and it changes the surface concentration of 
salt (and other tracers). Therefore it appears in the sea surface height as a volume 
flux, EMP (\textit{dynspg\_xxx} modules), and in the salinity time evolution equations 
as a concentration/dilution effect, 
EMP$_{S}$ (\mdl{trasbc} module). 
\begin{equation} \label{Eq_trasbc_emp}
\begin{aligned}
&\frac{\partial \eta }{\partial t}\equiv \cdots \;+\;\text{EMP}\quad  \\ 
\\
 &\frac{\partial S}{\partial t}\equiv \cdots \;+\left. {\frac{\text{EMP}_S \;S}{e_{3T} }} \right|_{k=1} \\ 
 \end{aligned}
\end{equation} 

In the real ocean, EMP$=$EMP$_S$ and the ocean salt content is conserved, 
but it exist several numerical reasons why this equality should be broken. 
For example:

When the rigid-lid assumption is made, the ocean volume becomes constant and 
thus, EMP$=$0, not EMP$_{S }$.

When the ocean is coupled to a sea-ice model, the water exchanged between ice and 
ocean is slightly salty (mean sea-ice salinity is $\sim $\textit{4 psu}). In this case, 
EMP$_{S}$ take into account both concentration/dilution effect associated with 
freezing/melting and the salt flux between ice and ocean, while EMP is 
only the volume flux. In addition, in the current version of \NEMO, the 
sea-ice is assumed to be above the ocean. Freezing/melting does not change 
the ocean volume (no impact on EMP) but it modifies the SSS.
%gm  \colorbox{yellow}{(see {\S} on LIM sea-ice model)}.

Note that SST can also be modified by a freshwater flux. Precipitation (in 
particular solid precipitation) may have a temperature significantly different from 
the SST. Due to the lack of information about the temperature of 
precipitation, we assume it is equal to the SST. Therefore, no 
concentration/dilution term appears in the temperature equation. It has to 
be emphasised that this absence does not mean that there is no heat flux 
associated with precipitation! Precipitation can change the ocean volume and thus the
ocean heat content. It is therefore associated with a heat flux (not yet  
diagnosed in the model) \citep{Roullet2000}).

%\colorbox{yellow}{Miss: }
%
%A extensive description of all namsbc namelist (parameter that have to be 
%created!)
%
%Especially the \np{nf\_sbc}, the \mdl{sbc\_oce} module (fluxes + mean sst sss ssu 
%ssv) i.e. information required by flux computation or sea-ice
%
%\mdl{sbc\_oce} containt the definition in memory of the 7 fields (6+runoff), add 
%a word on runoff: included in surface bc or add as lateral obc{\ldots}.
%
%Sbcmod manage the ``providing'' (fourniture) to the ocean the 7 fields
%
%Fluxes update only each nf{\_}sbc time step (namsbc) explain relation 
%between nf{\_}sbc and nf{\_}ice, do we define nf{\_}blk??? ? only one 
%nf{\_}sbc
%
%Explain here all the namlist namsbc variable{\ldots}.
%
%\colorbox{yellow}{End Miss }

The ocean model provides the surface currents, temperature and salinity 
averaged over \np{nf\_sbc} time-step (\ref{Tab_ssm}).The computation of the 
mean is done in \mdl{sbcmod} module.

%-------------------------------------------------TABLE---------------------------------------------------
\begin{table}[tb]  \label{Tab_ssm}
\begin{center}
\begin{tabular}{|l|l|l|l|}
\hline
Variable description             & Model variable  & Units  & point \\  \hline
i-component of the surface current  & ssu\_m & $m.s^{-1}$   & U \\   \hline
j-component of the surface current  & ssv\_m & $m.s^{-1}$   & V \\   \hline
Sea surface temperature          & sst\_m & \r{}$K$      & T \\   \hline
Sea surface salinty              & sss\_m & $psu$        & T \\   \hline
\end{tabular}
\caption{Ocean variables provided by the ocean to the surface module (SBC). 
The variable are averaged over nf{\_}sbc time step, $i.e.$ the frequency of 
computation of surface fluxes.}
\end{center}
\end{table}
%--------------------------------------------------------------------------------------------------------------



%\colorbox{yellow}{Penser a} mettre dans le restant l'info nf{\_}sbc ET nf{\_}sbc*rdt de sorte de reinitialiser la moyenne si on change la frequence ou le pdt


% ================================================================
% Analytical formulation (sbcana module) 
% ================================================================
\section  [Analytical formulation (\textit{sbcana}) ]
      {Analytical formulation (\mdl{sbcana} module) }
\label{SBC_ana}

%---------------------------------------namsbc_ana--------------------------------------------------
\namdisplay{namsbc_ana}
%--------------------------------------------------------------------------------------------------------------


The analytical formulation of the surface boundary condition is the default scheme.
In this case, all the six fluxes needed by the ocean are assumed to 
be uniform in space. They take constant values given in the namelist 
namsbc{\_}ana by the variables \np{rn\_utau0}, \np{rn\_vtau0}, \np{rn\_qns0}, 
\np{rn\_qsr0}, and \np{rn\_emp0} (EMP$=$EMP$_S$). The runoff is set to zero. 
In addition, the wind is allowed to reach its nominal value within a given number 
of time steps (\np{nn\_tau000}).

If a user wants to apply a different analytical forcing, the \mdl{sbcana} 
module can be modified to use another scheme. As an example, 
the \mdl{sbc\_ana\_gyre} routine provides the analytical forcing for the 
GYRE configuration (see GYRE configuration manual, in preparation).


% ================================================================
% Flux formulation 
% ================================================================
\section  [Flux formulation (\textit{sbcflx}) ]
      {Flux formulation (\mdl{sbcflx} module) }
\label{SBC_flx}
%------------------------------------------namsbc_flx----------------------------------------------------
\namdisplay{namsbc_flx} 
%-------------------------------------------------------------------------------------------------------------

In the flux formulation (\np{ln\_flx}=true), the surface boundary 
condition fields are directly read from input files. The user has to define 
in the namelist namsbc{\_}flx the name of the file, the name of the variable 
read in the file, the time frequency at which it is given (in hours), and a logical 
setting whether a time interpolation to the model time step is required 
for this field). (fld\_i namelist structure).

\textbf{Caution}: when the frequency is set to --12, the data are monthly 
values. These are assumed to be climatological values, so time interpolation 
between December the 15$^{th}$ and January the 15$^{th}$ is done using 
records 12 and 1

When higher frequency is set and time interpolation is demanded, the model 
will try to read the last (first) record of previous (next) year in a file 
having the same name but a suffix {\_}prev{\_}year ({\_}next{\_}year) being 
added (e.g. "{\_}1989"). These files must only contain a single record. If they don't exist, 
the model assumes that the last record of the previous year is equal to the first 
record of the current year, and similarly, that the first record of the 
next year is equal to the last record of the current year. This will cause 
the forcing to remain constant over the first and last half fld\_frequ hours.

Note that in general, a flux formulation is used in associated with a 
restoring term to observed SST and/or SSS. See \S\ref{SBC_ssr} for its 
specification.


% ================================================================
% Bulk formulation
% ================================================================
\section  [Bulk formulation (\textit{sbcblk\_core} or \textit{sbcblk\_clio}) ]
      {Bulk formulation \small{(\mdl{sbcblk\_core} or \mdl{sbcblk\_clio} module)} }
\label{SBC_blk}

In the bulk formulation, the surface boundary condition fields are computed 
using bulk formulae and atmospheric fields and ocean (and ice) variables. 

The atmospheric fields used depend on the bulk formulae used. Two bulk formulations 
are available : the CORE and CLIO bulk formulea. The choice is made by setting to true
one of the following namelist variable : \np{ln\_core} and \np{ln\_clio}.

Note : in forced mode, when a sea-ice model is used, a bulk formulation have to be used. 
Therefore the two bulk formulea provided include the computation of the fluxes over both 
an ocean and an ice surface. 

% -------------------------------------------------------------------------------------------------------------
%        CORE Bulk formulea
% -------------------------------------------------------------------------------------------------------------
\subsection    [CORE Bulk formulea (\np{ln\_core}=true)]
            {CORE Bulk formulea (\np{ln\_core}=true, \mdl{sbcblk\_core})}
\label{SBC_blk_core}
%------------------------------------------namsbc_core----------------------------------------------------
\namdisplay{namsbc_core} 
%-------------------------------------------------------------------------------------------------------------

The CORE bulk formulae have been developed by \citet{LargeYeager2004}. 
They have been designed to handle the CORE forcing, a mixture of NCEP 
reanalysis and satellite data. They use an inertial dissipative method to compute 
the turbulent transfer coefficients (momentum, sensible heat and evaporation) 
from the 10 metre wind speed, air temperature and specific humidity.

Note that substituting ERA40 to NCEP reanalysis fields 
does not require changes in the bulk formulea themself. 

The required 8 input fields are:

%--------------------------------------------------TABLE--------------------------------------------------
\begin{table}[htbp]   \label{Tab_CORE}
\begin{center}
\begin{tabular}{|l|l|l|l|}
\hline
Variable desciption              & Model variable  & Units   & point \\    \hline
i-component of the 10m air velocity & utau      & $m.s^{-1}$         & T  \\  \hline
j-component of the 10m air velocity & vtau      & $m.s^{-1}$         & T  \\  \hline
10m air temperature              & tair      & \r{}$K$            & T   \\ \hline
Specific humidity             & humi      & \%              & T \\      \hline
Incoming long wave radiation     & qlw    & $W.m^{-2}$         & T \\      \hline
Incoming short wave radiation    & qsr    & $W.m^{-2}$         & T \\      \hline
Total precipitation (liquid + solid)   & precip & $Kg.m^{-2}.s^{-1}$ & T \\   \hline
Solid precipitation              & snow      & $Kg.m^{-2}.s^{-1}$ & T \\   \hline
\end{tabular}
\end{center}
\end{table}
%--------------------------------------------------------------------------------------------------------------

Note that the air velocity is provided at a tracer ocean point, not at a velocity ocean point ($u$- and $v$-points). It is simpler and faster (less fields to be read), but it is not the recommended method when the ocean grid
size is the same or larger than the one of the input atmospheric fields.

% -------------------------------------------------------------------------------------------------------------
%        CLIO Bulk formulea
% -------------------------------------------------------------------------------------------------------------
\subsection    [CLIO Bulk formulea (\np{ln\_clio}=true)]
            {CLIO Bulk formulea (\np{ln\_clio}=true, \mdl{sbcblk\_clio})}
\label{SBC_blk_clio}
%------------------------------------------namsbc_clio----------------------------------------------------
\namdisplay{namsbc_clio} 
%-------------------------------------------------------------------------------------------------------------

The CLIO bulk formulae were developed several years ago for the 
Louvain-la-neuve coupled ice-ocean model (CLIO, \cite{Goosse_al_JGR99}). 
They are simpler bulk formulae. They assume the stress to be known and 
compute the radiative fluxes from a climatological cloud cover. 

The required 7 input fields are:

%--------------------------------------------------TABLE--------------------------------------------------
\begin{table}[htbp]   \label{Tab_CLIO}
\begin{center}
\begin{tabular}{|l|l|l|l|}
\hline
Variable desciption           & Model variable  & Units           & point \\  \hline
i-component of the ocean stress     & utau         & $N.m^{-2}$         & U \\   \hline
j-component of the ocean stress     & vtau         & $N.m^{-2}$         & V \\   \hline
Wind speed module             & vatm         & $m.s^{-1}$         & T \\   \hline
10m air temperature              & tair         & \r{}$K$            & T \\   \hline
Specific humidity                & humi         & \%              & T \\   \hline
Cloud cover                   &           & \%              & T \\   \hline
Total precipitation (liquid + solid)   & precip    & $Kg.m^{-2}.s^{-1}$ & T \\   \hline
Solid precipitation              & snow         & $Kg.m^{-2}.s^{-1}$ & T \\   \hline
\end{tabular}
\end{center}
\end{table}
%--------------------------------------------------------------------------------------------------------------

As for the flux formulation, information about the input data required by the 
model is provided in the namsbc\_blk\_core or namsbc\_blk\_clio 
namelist (via the structure fld\_i). The first and last record assumption is also made 
(see \S\ref{SBC_flx})

% ================================================================
% Coupled formulation
% ================================================================
\section  [Coupled formulation (\textit{sbccpl}) ]
      {Coupled formulation (\mdl{sbccpl} module)}
\label{SBC_cpl}
%------------------------------------------namsbc_cpl----------------------------------------------------
\namdisplay{namsbc_cpl} 
%-------------------------------------------------------------------------------------------------------------

In the coupled formulation of the surface boundary condition, the fluxes are 
provided by the OASIS coupler at each \np{nf\_cpl} time-step, while sea and ice 
surface temperature, ocean and ice albedo, and ocean currents are sent to 
the atmospheric component.

The generalised coupled interface is under development. It should be available
in summer 2008. It will include the ocean interface for most of the European 
atmospheric GCM (ARPEGE, ECHAM, ECMWF, HadAM, LMDz).

% ================================================================
% Interpolation on the Fly
% ================================================================

\section [Interpolation on the Fly] {Interpolation on the Fly}
\label{SBC_iof}

Interpolation on the Fly allows the user to supply input files required
for the surface forcing on grids other than the model grid.
To do this he or she must supply, in addition to the source data file,
a file of weights to be used to interpolate from the data grid to the model
grid.
The original development of this code used the SCRIP package (freely available 
under a copyright agreement from http://climate.lanl.gov/Software/SCRIP).
In principle, any package can be used to generate the weights, but the
variables in the input weights file must have the same names and meanings as
assumed by the model.
Two methods are currently available: bilinear and bicubic interpolation.

\subsection{Bilinear Interpolation}
\label{SBC_iof_bilinear}

The input weights file in this case has two sets of variables: src01, src02,
src03, src04 and wgt01, wgt02, wgt03, wgt04.
The "src" variables correspond to the point in the input grid to which the weight
"wgt" is to be applied. Each src value is an integer corresponding to the index of a 
point in the input grid when written as a one dimensional array.  For example, for an input grid
of size 5x10, point (3,2) is referenced as point 8, since (2-1)*5+3=8.
There are four of each variable because bilinear interpolation uses the four points defining
the grid box containing the point to be interpolated.
All of these arrays are on the model grid, so that values src01(i,j) and
wgt01(i,j) are used to generate a value for point (i,j) in the model.

Symbolically, the algorithm used is:

\begin{equation}
f_{m}(i,j) = f_{m}(i,j) + \sum_{k=1}^{4} {wgt(k)f(idx(src(k)))}
\end{equation}
where function idx() transforms a one dimensional index src(k) into a two dimensional index,
and wgt(1) corresponds to variable "wgt01" for example.

\subsection{Bicubic Interpolation}
\label{SBC_iof_bicubic}

Again there are two sets of variables: "src" and "wgt".
But in this case there are 16 of each.
The symbolic algorithm used to calculate values on the model grid is now:

\begin{multline*}
f_{m}(i,j) = f_{m}(i,j) + \sum_{k=1}^{4} {wgt(k)f(idx(src(k)))}
\\
                        + \sum_{k=5}^{8} {wgt(k)\left.\frac{\partial f}{\partial i}\right| _{idx(src(k))} }
\\
                        + \sum_{k=9}^{12} {wgt(k)\left.\frac{\partial f}{\partial j}\right| _{idx(src(k))} }
\\
                        + \sum_{k=13}^{16} {wgt(k)\left.\frac{\partial ^2 f}{\partial i \partial j}\right| _{idx(src(k))} }
\end{multline*}
The gradients here are taken with respect to the horizontal indices and not distances since the spatial dependency has been absorbed into the weights.

\subsection{Implementation}
\label{SBC_iof_imp}

To activate this option, a non-empty string should be supplied in the weights filename column of the relevant namelist;
if this is left as an empty string no action is taken.
In the model, weights files are read in and stored in a structured type (WGT) in the fldread module, as and when they are first required.
This initialisation procedure tries to determine whether the input data grid should be treated as cyclical or not.
(In fact this only matters when bicubic interpolation is required.)
To do this the model looks in the input data file (i.e. the data to which the weights are to be applied) for a variable with name "nav\_lon" or "lon".
If found, it checks the difference between the first and last values of longitude along a single row.
If the absolute value of this difference is close to 360 degrees or less than twice the maximum spacing from 360 degrees, the grid is assumed to be cyclical, and the difference determines whether the first column is a repeat of the last one or not.
If neither "nav\_lon" or "lon" can be found, the model resorts to looking at the first and last columns of data.
If the sum of the absolute values of the differences between the columns is very small, then the grid is assumed to be cyclical with coincident first and last columns.
If both of these tests fail, the grid is assumed not to be cyclical.

Next the routine reads in the weights.
Bicubic interpolation is assumed if it finds a variable with name "src05", otherwise bilinear interpolation is used.
The WGT structure includes dynamic arrays both for the storage of the weights (on the model grid), and when required, for reading in the variable to be interpolated (on the input data grid).
The size of the input data array is determined by examining the values in the "src" arrays to find the minimum and maximum i and j values required.
Since bicubic interpolation requires the calculation of gradients at each point on the grid, the corresponding arrays are dimensioned with a halo of width one grid point all the way around.
When the array of points from the data file is adjacent to an edge of the data grid, the halo is either a copy of the row/column next to it (non-cyclical case), or is a copy of one from the first two rows/columns on the opposite side of the grid (cyclical case with coincident end rows/columns, or cyclical case with non-coincident end rows/columns).

\subsection{Limitations}
\label{SBC_iof_lim}

\begin{description}
\item
Input data grids must be logically rectangular.
\item
This code is not guaranteed to produce positive definite answers from positive definite inputs.
\item
The cyclic condition is only applied on left and right columns, and not to top and bottom rows.
\item
The gradients across the ends of a cyclical grid assume that the grid spacing between the two columns involved are consistent with the weights used.
\item
Neither interpolation scheme is conservative.
(There is a conservative scheme available in SCRIP, but this has not been implemented.)
\end{description}

\subsection{Utilities}
\label{SBC_iof_util}

% to be completed
A set of utilities to create a weights file for a rectilinear input grid is available.

% ================================================================
% Miscellanea options
% ================================================================
\section{Miscellaneous options}
\label{SBC_misc}

% -------------------------------------------------------------------------------------------------------------
%        Rotation of vector pairs onto the model grid directions
% -------------------------------------------------------------------------------------------------------------
\subsection{Rotation of vector pairs onto the model grid directions}
\label{SBC_rotation}

When using a flux (\np{ln\_flx}=true) or bulk (\np{ln\_clio}=true or \np{ln\_core}=true) formulation, 
pairs of vector components can be rotated from east-north directions onto the local grid directions.  
This is particularly useful when interpolation on the fly is used since here any vectors are likely to be defined 
relative to a rectilinear grid.
To activate this option a non-empty string is supplied in the rotation pair column of the relevant namelist.
The eastward component must start with "U" and the northward component with "V".  
The remaining characters in the strings are used to identify which pair of components go together.
So for example, strings "U1" and "V1" next to "utau" and "vtau" would pair the wind stress components together
and rotate them on to the model grid directions; "U2" and "V2" could be used against a second pair of components, 
and so on.
The extra characters used in the strings are arbitrary.
The rot\_rep routine from the \mdl{geo2ocean} module is used to perform the rotation.

% -------------------------------------------------------------------------------------------------------------
%        Surface restoring to observed SST and/or SSS
% -------------------------------------------------------------------------------------------------------------
\subsection    [Surface restoring to observed SST and/or SSS (\textit{sbcssr})]
         {Surface restoring to observed SST and/or SSS (\mdl{sbcssr})}
\label{SBC_ssr}
%------------------------------------------namsbc_ssr----------------------------------------------------
\namdisplay{namsbc_ssr} 
%-------------------------------------------------------------------------------------------------------------

In forced mode using a flux formulation (default option or \key{flx} defined), a 
feedback term \emph{must} be added to the surface heat flux $Q_{ns}^o$:
\begin{equation} \label{Eq_sbc_dmp_q}
Q_{ns} = Q_{ns}^o + \frac{dQ}{dT} \left( \left. T \right|_{k=1} - SST_{Obs} \right)
\end{equation}
where SST is a sea surface temperature field (observed or climatological), $T$ is 
the model surface layer temperature and $\frac{dQ}{dT}$ is a negative feedback 
coefficient usually taken equal to $-40~W/m^2/K$. For a $50~m$ 
mixed-layer depth, this value corresponds to a relaxation time scale of two months. 
This term ensures that if $T$ perfectly matches the supplied SST, then $Q$ is 
equal to $Q_o$. 

In the fresh water budget, a feedback term can also be added. Converted into an 
equivalent freshwater flux, it takes the following expression :

\begin{equation} \label{Eq_sbc_dmp_emp}
EMP = EMP_o + \gamma_s^{-1} e_{3t}  \frac{  \left(\left.S\right|_{k=1}-SSS_{Obs}\right)}
                                             {\left.S\right|_{k=1}}
\end{equation}

where EMP$_{o }$ is a net surface fresh water flux (observed, climatological or an
atmospheric model product), \textit{SSS}$_{Obs}$ is a sea surface salinity (usually a time 
interpolation of the monthly mean Polar Hydrographic Climatology \citep{Steele2001}), 
$\left.S\right|_{k=1}$ is the model surface layer salinity and $\gamma_s$ is a negative 
feedback coefficient which is provided as a namelist parameter. Unlike heat flux, there is no 
physical justification for the feedback term in \ref{Eq_sbc_dmp_emp} as the atmosphere 
does not care about ocean surface salinity \citep{Madec1997}. The SSS restoring 
term should be viewed as a flux correction on freshwater fluxes to reduce the 
uncertainties we have on the observed freshwater budget.

% -------------------------------------------------------------------------------------------------------------
%        Handling of ice-covered area
% -------------------------------------------------------------------------------------------------------------
\subsection{Handling of ice-covered area  (\textit{sbcice\_...})}
\label{SBC_ice-cover}

The presence at the sea surface of an ice covered area modifies all the fluxes 
transmitted to the ocean. There are several way to handle sea-ice in the system 
depending on the value of the \np{nn{\_}ice} namelist parameter.  
\begin{description}
\item[nn{\_}ice = 0]  there will never be sea-ice in the computational domain. 
This is a typical namelist value used for tropical ocean domain. The surface fluxes 
are simply specified for an ice-free ocean. No specific things is done for sea-ice.
\item[nn{\_}ice = 1]  sea-ice can exist in the computational domain, but no sea-ice model 
is used. An observed ice covered area is read in a file. Below this area, the SST is 
restored to the freezing point and the heat fluxes are set to $-4~W/m^2$ ($-2~W/m^2$) 
in the northern (southern) hemisphere. The associated modification of the freshwater 
fluxes are done in such a way that the change in buoyancy fluxes remains zero. 
This prevents deep convection to occur when trying to reach the freezing point 
(and so ice covered area condition) while the SSS is too large. This manner of 
managing sea-ice area, just by using si IF case, is usually referred as the \textit{ice-if} 
model. It can be found in the \mdl{sbcice{\_}if} module.
\item[nn{\_}ice = 2 or more]  A full sea ice model is used. This model computes the 
ice-ocean fluxes, that are combined with the air-sea fluxes using the ice fraction of 
each model cell to provide the surface ocean fluxes. Note that the activation of a 
sea-ice model is is done by defining a CPP key (\key{lim2} or \key{lim3}). 
The activation automatically ovewrite the read value of nn{\_}ice to its appropriate 
value ($i.e.$ $2$ for LIM-2 and $3$ for LIM-3).
\end{description}

% {Description of Ice-ocean interface to be added here or in LIM 2 and 3 doc ?}

% -------------------------------------------------------------------------------------------------------------
%        Addition of river runoffs
% -------------------------------------------------------------------------------------------------------------
\subsection   [Addition of river runoffs (\textit{sbcrnf})]
         {Addition of river runoffs (\mdl{sbcrnf})}
\label{SBC_rnf}
%------------------------------------------namsbc_rnf----------------------------------------------------
\namdisplay{namsbc_rnf} 
%-------------------------------------------------------------------------------------------------------------

The river runoffs 

It is convenient to introduce the river runoff in the model as a surface 
fresh water flux. 


%Griffies:  River runoff generally enters the ocean at a nonzero depth rather than through the surface. Many global models, however, have traditionally inserted river runoff to the top model cell. Such can become problematic numerically and physically when the top grid cells are reÞned to levels common in coastal modelling. Hence, more applications are now considering the input of runoff throughout a nonzero depth. Likewise, sea ice can melt at depth, thus necessitating a mass transport to occur within the ocean between the liquid and solid water masses.

\colorbox{yellow}{Nevertheless, Pb of vertical resolution and increase of Kz in vicinity of }

\colorbox{yellow}{river mouths{\ldots}}

%IF( ln_rnf ) THEN                                     ! increase diffusivity at rivers mouths
%        DO jk = 2, nkrnf   ;   avt(:,:,jk) = avt(:,:,jk) + rn_avt_rnf * rnfmsk(:,:)   ;   END DO
%ENDIF



% -------------------------------------------------------------------------------------------------------------
%        Freshwater budget control 
% -------------------------------------------------------------------------------------------------------------
\subsection   [Freshwater budget control (\textit{sbcfwb})]
         {Freshwater budget control (\mdl{sbcfwb})}
\label{SBC_fwb}

For global ocean simulation it can be useful to introduce a control of the mean sea 
level in order to prevent unrealistic drift of the sea surface height due to inaccuracy 
in the freshwater fluxes. In \NEMO, two way of controlling the the freshwater budget. 
\begin{description}
\item[\np{nn\_fwb}=0]  no control at all. The mean sea level is free to drift, and will 
certainly do so.
\item[\np{nn\_fwb}=1]  global mean EMP set to zero at each model time step. 
%Note that with a sea-ice model, this technique only control the mean sea level with linear free surface (\key{vvl} not defined) and no mass flux between ocean and ice (as it is implemented in the current ice-ocean coupling). 
\item[\np{nn\_fwb}=2]  freshwater budget is adjusted from the previous year annual 
mean budget which is read in the \textit{EMPave\_old.dat} file. As the model uses the 
Boussinesq approximation, the annual mean fresh water budget is simply evaluated 
from the change in the mean sea level at January the first and saved in the 
\textit{EMPav.dat} file. 
\end{description}

% Griffies doc:
% When running ocean-ice simulations, we are not explicitly representing land processes, such as rivers, catchment areas, snow accumulation, etc. However, to reduce model drift, it is important to balance the hydrological cycle in ocean-ice models. We thus need to prescribe some form of global normalization to the precipitation minus evaporation plus river runoff. The result of the normalization should be a global integrated zero net water input to the ocean-ice system over a chosen time scale. 
%How often the normalization is done is a matter of choice. In mom4p1, we choose to do so at each model time step, so that there is always a zero net input of water to the ocean-ice system. Others choose to normalize over an annual cycle, in which case the net imbalance over an annual cycle is used to alter the subsequent yearÕs water budget in an attempt to damp the annual water imbalance. Note that the annual budget approach may be inappropriate with interannually varying precipitation forcing. 
%When running ocean-ice coupled models, it is incorrect to include the water transport between the ocean and ice models when aiming to balance the hydrological cycle. The reason is that it is the sum of the water in the ocean plus ice that should be balanced when running ocean-ice models, not the water in any one sub-component. As an extreme example to illustrate the issue, consider an ocean-ice model with zero initial sea ice. As the ocean-ice model spins up, there should be a net accumulation of water in the growing sea ice, and thus a net loss of water from the ocean. The total water contained in the ocean plus ice system is constant, but there is an exchange of water between the subcomponents. This exchange should not be part of the normalization used to balance the hydrological cycle in ocean-ice models.