source: NEMO/branches/NERC/dev_release-3.4_NEMOTAM_consolidated/DOC/TexFiles/Chapters/Chap_SBC.tex @ 10771

% ================================================================
% 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 $\left( {\textit{emp},\;\textit{emp}_S } \right)$
\end{itemize}
plus an optional field:
\begin{itemize}
   \item the atmospheric pressure at the ocean surface $\left( p_a \right)$
\end{itemize}

Five different ways to provide the first 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), CLIO (\np{ln\_clio}~=~true) or MFS
\footnote { Note that MFS bulk formulae compute fluxes only for the ocean component}
(\np{ln\_mfs}~=~true) bulk formulae) and a coupled 
formulation (exchanges with a atmospheric model via the OASIS coupler) 
(\np{ln\_cpl}~=~true). When used, the atmospheric pressure forces both 
ocean and ice dynamics (\np{ln\_apr\_dyn}~=~true).
The frequency at which the six or seven fields have to be updated is the \np{nn\_fsbc} 
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", see \S\ref{SBC_iof}).
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 or lateral inflow (\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); the 
transformation of the solar radiation (if provided as daily mean) into a diurnal 
cycle (\np{ln\_dm2dc}~=~true); and a neutral drag coefficient can be read from an external wave 
model (\np{ln\_cdgw}~=~true). The latter option is possible only in case core or mfs bulk formulas are selected.

In this chapter, we first discuss where the surface boundary condition appears in the
model equations. Then we present the five ways of providing the surface boundary condition, 
followed by the description of the atmospheric pressure and the river runoff. 
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: \textit{emp} and $\textit{emp}_S$ 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, \textit{emp} (\textit{dynspg\_xxx} modules), and in the salinity time evolution equations 
as a concentration/dilution effect, 
$\textit{emp}_{S}$ (\mdl{trasbc} module). 
\begin{equation} \label{Eq_trasbc_emp}
\begin{aligned}
&\frac{\partial \eta }{\partial t}\equiv \cdots \;+\;\textit{emp}\quad  \\ 
\\
 &\frac{\partial S}{\partial t}\equiv \cdots \;+\left. {\frac{\textit{emp}_S \;S}{e_{3t} }} \right|_{k=1} \\ 
 \end{aligned}
\end{equation} 

In the real ocean, $\textit{emp}=\textit{emp}_S$ and the ocean salt content is conserved, 
but it exist several numerical reasons why this equality should be broken. 
For example, when the 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, 
$\textit{emp}_{S}$ take into account both concentration/dilution effect associated with 
freezing/melting and the salt flux between ice and ocean, while \textit{emp} is 
only the volume flux. In addition, in the current version of \NEMO, the sea-ice is 
assumed to be above the ocean (the so-called levitating sea-ice). Freezing/melting does 
not change the ocean volume (no impact on \textit{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{Roullet_Madec_JGR00}).

%\colorbox{yellow}{Miss: }
%
%A extensive description of all namsbc namelist (parameter that have to be 
%created!)
%
%Especially the \np{nn\_fsbc}, 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]   \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{  \label{Tab_ssm}   
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 nn{\_}fsbc ET nn{\_}fsbc*rdt de sorte de reinitialiser la moyenne si on change la frequence ou le pdt


% ================================================================
%       Input Data 
% ================================================================
\section{Input Data generic interface}
\label{SBC_input}

A generic interface has been introduced to manage the way input data (2D or 3D fields, 
like surface forcing or ocean T and S) are specify in \NEMO. This task is archieved by fldread.F90. 
The module was design with four main objectives in mind: 
\begin{enumerate}  
\item optionally provide a time interpolation of the input data at model time-step, 
whatever their input frequency is, and according to the different calendars available in the model.
\item optionally provide an on-the-fly space interpolation from the native input data grid to the model grid.
\item make the run duration independent from the period cover by the input files.
\item provide a simple user interface and a rather simple developer interface by limiting the 
 number of prerequisite information. 
\end{enumerate}  

As a results the user have only to fill in for each variable a structure in the namelist file 
to defined the input data file and variable names, the frequency of the data (in hours or months), 
whether its is climatological data or not, the period covered by the input file (one year, month, week or day), 
and two additional parameters for on-the-fly interpolation. When adding a new input variable, 
the developer has to add the associated structure in the namelist, read this information 
by mirroring the namelist read in \rou{sbc\_blk\_init} for example, and simply call \rou{fld\_read} 
to obtain the desired input field at the model time-step and grid points.

The only constraints are that the input file is a NetCDF file, the file name follows a nomenclature 
(see \S\ref{SBC_fldread}), the period it cover is one year, month, week or day, and, if on-the-fly 
interpolation is used, a file of weights must be supplied (see \S\ref{SBC_iof}).

Note that when an input data is archived on a disc which is accessible directly 
from the workspace where the code is executed, then the use can set the \np{cn\_dir} 
to the pathway leading to the data. By default, the data are assumed to have been 
copied so that cn\_dir='./'.

% -------------------------------------------------------------------------------------------------------------
% Input Data specification (\mdl{fldread})
% -------------------------------------------------------------------------------------------------------------
\subsection{Input Data specification (\mdl{fldread})}
\label{SBC_fldread}

The structure associated with an input variable contains the following information:
\begin{alltt}  {{\tiny    
\begin{verbatim}
!  file name  ! frequency (hours) ! variable  ! time interp. !  clim  ! 'yearly'/ ! weights  ! rotation ! 
!             !  (if <0  months)  !   name    !   (logical)  !  (T/F) ! 'monthly' ! filename ! pairing  !
\end{verbatim}
}}\end{alltt} 
where 
\begin{description}  
\item[File name]: the stem name of the NetCDF file to be open. 
This stem will be completed automatically by the model, with the addition of a '.nc' at its end 
and by date information and possibly a prefix (when using AGRIF). 
Tab.\ref{Tab_fldread} provides the resulting file name in all possible cases according to whether 
it is a climatological file or not, and to the open/close frequency (see below for definition). 

%--------------------------------------------------TABLE--------------------------------------------------
\begin{table}[htbp] 
\begin{center}
\begin{tabular}{|l|c|c|c|}
\hline
                         & daily or weekLLL          & monthly                   &   yearly          \\   \hline
clim = false   & fn\_yYYYYmMMdDD  &   fn\_yYYYYmMM   &   fn\_yYYYY  \\   \hline
clim = true       & not possible                  &  fn\_m??.nc             &   fn                \\   \hline
\end{tabular}
\end{center}
\caption{ \label{Tab_fldread}   naming nomenclature for climatological or interannual input file, 
as a function of the Open/close frequency. The stem name is assumed to be 'fn'. 
For weekly files, the 'LLL' corresponds to the first three letters of the first day of the week ($i.e.$ 'sun','sat','fri','thu','wed','tue','mon'). The 'YYYY', 'MM' and 'DD' should be replaced by the 
actual year/month/day, always coded with 4 or 2 digits. Note that (1) in mpp, if the file is split 
over each subdomain, the suffix '.nc' is replaced by '\_PPPP.nc', where 'PPPP' is the 
process number coded with 4 digits; (2) when using AGRIF, the prefix
'\_N' is added to files, 
where 'N'  is the child grid number.}
\end{table}
%--------------------------------------------------------------------------------------------------------------
  

\item[Record frequency]: the frequency of the records contained in the input file. 
Its unit is in hours if it is positive (for example 24 for daily forcing) or in months if negative 
(for example -1 for monthly forcing or -12 for annual forcing). 
Note that this frequency must really be an integer and not a real. 
On some computers, seting it to '24.' can be interpreted as 240!

\item[Variable name]: the name of the variable to be read in the input NetCDF file.

\item[Time interpolation]: a logical to activate, or not, the time interpolation. If set to 'false', 
the forcing will have a steplike shape remaining constant during each forcing period. 
For example, when using a daily forcing without time interpolation, the forcing remaining 
constant from 00h00'00'' to 23h59'59". If set to 'true', the forcing will have a broken line shape. 
Records are assumed to be dated the middle of the forcing period. 
For example, when using a daily forcing with time interpolation, linear interpolation will 
be performed between mid-day of two consecutive days. 

\item[Climatological forcing]: a logical to specify if a input file contains climatological forcing 
which can be cycle in time, or an interannual forcing which will requires additional files 
if the period covered by the simulation exceed the one of the file. See the above the file 
naming strategy which impacts the expected name of the file to be opened. 

\item[Open/close frequency]: the frequency at which forcing files must be opened/closed. 
Four cases are coded: 'daily', 'weekLLL' (with 'LLL' the first 3 letters of the first day of the week), 
'monthly' and 'yearly' which means the forcing files will contain data for one day, one week, 
one month or one year. Files are assumed to contain data from the beginning of the open/close period. 
For example, the first record of a yearly file containing daily data is Jan 1st even if the experiment 
is not starting at the beginning of the year. 

\item[Others]: 'weights filename' and 'pairing rotation' are associted with on-the-fly interpolation 
which is described in \S\ref{SBC_iof}.

\end{description}

Additional remarks:\\
(1) The time interpolation is a simple linear interpolation between two consecutive records of 
the input data. The only tricky point is therefore to specify the date at which we need to do 
the interpolation and the date of the records read in the input files. 
Following \citet{Leclair_Madec_OM09}, the date of a time step is set at the middle of the 
time step. For example, for an experiment starting at 0h00'00" with a one hour time-step, 
a time interpolation will be performed at the following time: 0h30'00", 1h30'00", 2h30'00", etc.
However, for forcing data related to the surface module, values are not needed at every 
time-step but at every \np{nn\_fsbc} time-step. For example with \np{nn\_fsbc}~=~3, 
the surface module will be called at time-steps 1, 4, 7, etc. The date used for the time interpolation 
is thus redefined to be at the middle of \np{nn\_fsbc} time-step period. In the previous example, 
this leads to: 1h30'00", 4h30'00", 7h30'00", etc. \\ 
(2) For code readablility and maintenance issues, we don't take into account the NetCDF input file 
calendar. The calendar associated with the forcing field is build according to the information 
provided by user in the record frequency, the open/close frequency and the type of temporal interpolation. 
For example, the first record of a yearly file containing daily data that will be interpolated in time 
is assumed to be start Jan 1st at 12h00'00" and end Dec 31st at 12h00'00". \\
(3) If a time interpolation is requested, the code will pick up the needed data in the previous (next) file 
when interpolating data with the first (last) record of the open/close period. 
For example, if the input file specifications are ''yearly, containing daily data to be interpolated in time'', 
the values given by the code between 00h00'00" and 11h59'59" on Jan 1st will be interpolated values 
between Dec 31st 12h00'00" and Jan 1st 12h00'00". If the forcing is climatological, Dec and Jan will 
be keep-up from the same year. However, if the forcing is not climatological, at the end of the 
open/close period the code will automatically close the current file and open the next one. 
Note that, if the experiment is starting (ending) at the beginning (end) of an open/close period 
we do accept that the previous (next) file is not existing. In this case, the time interpolation 
will be performed between two identical values. For example, when starting an experiment on 
Jan 1st of year Y with yearly files and daily data to be interpolated, we do accept that the file 
related to year Y-1 is not existing. The value of Jan 1st will be used as the missing one for 
Dec 31st of year Y-1. If the file of year Y-1 exists, the code will read its last record. 
Therefore, this file can contain only one record corresponding to Dec 31st, a useful feature for 
user considering that it is too heavy to manipulate the complete file for year Y-1.


% -------------------------------------------------------------------------------------------------------------
% Interpolation on the Fly
% -------------------------------------------------------------------------------------------------------------
\subsection [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 
\href{http://climate.lanl.gov/Software/SCRIP}{here} under a copyright agreement).
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.

\subsubsection{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.

\subsubsection{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{equation*} \begin{split}
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{split}
\end{equation*}
The gradients here are taken with respect to the horizontal indices and not distances since the spatial dependency has been absorbed into the weights.

\subsubsection{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 determines whether the input data grid should be treated 
as cyclical or not by inspecting a global attribute stored in the weights input file.
This attribute must be called "ew\_wrap" and be of integer type.
If it is negative, the input non-model grid is assumed not to be cyclic.
If zero or greater, then the value represents the number of columns that overlap.
$E.g.$ if the input grid has columns at longitudes 0, 1, 2, .... , 359, then ew\_wrap should be set to 0;
if longitudes are 0.5, 2.5, .... , 358.5, 360.5, 362.5, ew\_wrap should be 2.
If the model does not find attribute ew\_wrap, then a value of -999 is assumed.
In this case the \rou{fld\_read} routine defaults ew\_wrap to value 0 and therefore the grid 
is assumed to be cyclic with no overlapping columns.
(In fact this only matters when bicubic interpolation is required.)
Note that no testing is done to check the validity in the model, since there is no way 
of knowing the name used for the longitude variable,
so it is up to the user to make sure his or her data is correctly represented.

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 few columns on the opposite side of the grid (cyclical case).

\subsubsection{Limitations}
\label{SBC_iof_lim}

\begin{enumerate}  
\item  The case where input data grids are not logically rectangular has not been tested.
\item  This code is not guaranteed to produce positive definite answers from positive definite inputs
          when a bicubic interpolation method is used.
\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{enumerate}

\subsubsection{Utilities}
\label{SBC_iof_util}

% to be completed
A set of utilities to create a weights file for a rectilinear input grid is available 
(see the directory NEMOGCM/TOOLS/WEIGHTS).


% ================================================================
% 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} ($\textit{emp}=\textit{emp}_S$). 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. See \S\ref{SBC_fldread} for a more detailed description of the parameters.

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}, \textit{sbcblk\_clio} or \textit{sbcblk\_mfs}) ]
      {Bulk formulation \small{(\mdl{sbcblk\_core} \mdl{sbcblk\_clio} \mdl{sbcblk\_mfs} modules)} }
\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. Three bulk formulations 
are available : the CORE, the CLIO and the MFS bulk formulea. The choice is made by setting to true
one of the following namelist variable : \np{ln\_core} ; \np{ln\_clio} or  \np{ln\_mfs}.

Note : in forced mode, when a sea-ice model is used, a bulk formulation (CLIO or CORE) have to be used. 
Therefore the two bulk (CLIO and CORE) formulea 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{Large_Yeager_Rep04}. 
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.
This \citet{Large_Yeager_Rep04} dataset is available through the 
\href{http://nomads.gfdl.noaa.gov/nomads/forms/mom4/CORE.html}{GFDL web site}. 

Note that substituting ERA40 to NCEP reanalysis fields 
does not require changes in the bulk formulea themself. 
This is the so-called DRAKKAR Forcing Set (DFS) \citep{Brodeau_al_OM09}. 

The required 8 input fields are:

%--------------------------------------------------TABLE--------------------------------------------------
\begin{table}[htbp]   \label{Tab_CORE}
\begin{center}
\begin{tabular}{|l|c|c|c|}
\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 (see \S\ref{SBC_fldread}). 

% -------------------------------------------------------------------------------------------------------------
%        MFS Bulk formulae
% -------------------------------------------------------------------------------------------------------------
\subsection    [MFS Bulk formulea (\np{ln\_mfs}=true)]
            {MFS Bulk formulea (\np{ln\_mfs}=true, \mdl{sbcblk\_mfs})}
\label{SBC_blk_mfs}
%------------------------------------------namsbc_mfs----------------------------------------------------
\namdisplay{namsbc_mfs} 
%----------------------------------------------------------------------------------------------------------

The MFS (Mediterranean Forecasting System) bulk formulae have been developed by
 \citet{Castellari_al_JMS1998}. 
They have been designed to handle the ECMWF operational data and are currently 
in use in the MFS operational system \citep{Tonani_al_OS08}, \citep{Oddo_al_OS09}.
The wind stress computation uses a drag coefficient computed according to \citet{Hellerman_Rosenstein_JPO83}.
The surface boundary condition for temperature involves the balance between surface solar radiation,
net long-wave radiation, the latent and sensible heat fluxes.
Solar radiation is dependent on cloud cover and is computed by means of
an astronomical formula \citep{Reed_JPO77}. Albedo monthly values are from \citet{Payne_JAS72} 
as means of the values at $40^{o}N$ and $30^{o}N$ for the Atlantic Ocean (hence the same latitudinal
band of the Mediterranean Sea). The net long-wave radiation flux
\citep{Bignami_al_JGR95} is a function of
air temperature, sea-surface temperature, cloud cover and relative humidity.
Sensible heat and latent heat fluxes are computed by classical
bulk formulae parameterized according to \citet{Kondo1975}.
Details on the bulk formulae used can be found in \citet{Maggiore_al_PCE98} and \citet{Castellari_al_JMS1998}.

The required 7 input fields must be provided on the model Grid-T and  are:
\begin{itemize}
\item          Zonal Component of the 10m wind ($ms^{-1}$)  (\np{sn\_windi})
\item          Meridional Component of the 10m wind ($ms^{-1}$)  (\np{sn\_windj})
\item          Total Claud Cover (\%)  (\np{sn\_clc})
\item          2m Air Temperature ($K$) (\np{sn\_tair})
\item          2m Dew Point Temperature ($K$)  (\np{sn\_rhm})
\item          Total Precipitation ${Kg} m^{-2} s^{-1}$ (\np{sn\_prec})
\item          Mean Sea Level Pressure (${Pa}$) (\np{sn\_msl})
\end{itemize}
% -------------------------------------------------------------------------------------------------------------
% ================================================================
% 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 a frequency which is defined in the OASIS coupler, 
while sea and ice surface temperature, ocean and ice albedo, and ocean currents 
are sent to the atmospheric component.

A generalised coupled interface has been developed. It is currently interfaced with OASIS 3
(\key{oasis3}) and does not support OASIS 4
\footnote{The \key{oasis4} exist. It activates portion of the code that are still under development.}. 
It has been successfully used to interface \NEMO to most of the European atmospheric 
GCM (ARPEGE, ECHAM, ECMWF, HadAM, HadGAM, LMDz), 
as well as to \href{http://wrf-model.org/}{WRF} (Weather Research and Forecasting Model).

Note that in addition to the setting of \np{ln\_cpl} to true, the \key{coupled} have to be defined. 
The CPP key is mainly used in sea-ice to ensure that the atmospheric fluxes are 
actually recieved by the ice-ocean system (no calculation of ice sublimation in coupled mode).
When PISCES biogeochemical model (\key{top} and \key{pisces}) is also used in the coupled system, 
the whole carbon cycle is computed by defining \key{cpl\_carbon\_cycle}. In this case, 
CO$_2$ fluxes will be exchanged between the atmosphere and the ice-ocean system (and need to be activated
in namsbc{\_}cpl).

The new namelist above allows control of various aspects of the coupling fields (particularly for
vectors) and now allows for any coupling fields to have multiple sea ice categories (as required by LIM3
and CICE).  When indicating a multi-category coupling field in namsbc{\_}cpl the number of categories will be
determined by the number used in the sea ice model.  In some limited cases it may be possible to specify 
single category coupling fields even when the sea ice model is running with multiple categories - in this
case the user should examine the code to be sure the assumptions made are satisfactory.  In cases where
this is definitely not possible the model should abort with an error message.  The new code has been tested using
ECHAM with LIM2, and HadGAM3 with CICE but although it will compile with \key{lim3} additional minor code changes
may be required to run using LIM3.


% ================================================================
%        Atmospheric pressure
% ================================================================
\section   [Atmospheric pressure (\textit{sbcapr})]
         {Atmospheric pressure (\mdl{sbcapr})}
\label{SBC_apr}
%------------------------------------------namsbc_apr----------------------------------------------------
\namdisplay{namsbc_apr} 
%-------------------------------------------------------------------------------------------------------------

The optional atmospheric pressure can be used to force ocean and ice dynamics 
(\np{ln\_apr\_dyn}~=~true, \textit{namsbc} namelist ).
The input atmospheric forcing defined via \np{sn\_apr} structure (\textit{namsbc\_apr} namelist) 
can be interpolated in time to the model time step, and even in space when the 
interpolation on-the-fly is used. When used to force the dynamics, the atmospheric 
pressure is further transformed into an equivalent inverse barometer sea surface height, 
$\eta_{ib}$, using:
\begin{equation} \label{SBC_ssh_ib}
   \eta_{ib} = -  \frac{1}{g\,\rho_o}  \left( P_{atm} - P_o \right) 
\end{equation}
where $P_{atm}$ is the atmospheric pressure and $P_o$ a reference atmospheric pressure.
A value of $101,000~N/m^2$ is used unless \np{ln\_ref\_apr} is set to true. In this case $P_o$ 
is set to the value of $P_{atm}$ averaged over the ocean domain, $i.e.$ the mean value of 
$\eta_{ib}$ is kept to zero at all time step.

The gradient of $\eta_{ib}$ is added to the RHS of the ocean momentum equation 
(see \mdl{dynspg} for the ocean). For sea-ice, the sea surface height, $\eta_m$, 
which is provided to the sea ice model is set to $\eta - \eta_{ib}$ (see \mdl{sbcssr} module).
$\eta_{ib}$ can be set in the output. This can simplify altimetry data and model comparison 
as inverse barometer sea surface height is usually removed from these date prior to their distribution.

When using time-splitting and BDY package for open boundaries conditions, the equivalent
inverse barometer sea surface height $\eta_{ib}$ can be added to BDY ssh data:
\np{ln\_apr\_obc}  might be set to true.

% ================================================================
%        Tidal Potential
% ================================================================
\section   [Tidal Potential (\textit{sbctide})]
                        {Tidal Potential (\mdl{sbctide})}
\label{SBC_tide}

A module is available to use the tidal potential forcing and is activated with with \key{tide}.


%------------------------------------------nam_tide----------------------------------------------------
\namdisplay{nam_tide}
%-------------------------------------------------------------------------------------------------------------

Concerning the tidal potential, some parameters are available in namelist:

- \texttt{ln\_tide\_pot} activate the tidal potential forcing

- \texttt{nb\_harmo} is the number of constituent used

- \texttt{clname} is the name of constituent


The tide is generated by the forces of gravity ot the Earth-Moon and Earth-Sun sytem;
they are expressed as the gradient of the astronomical potential ($\vec{\nabla}\Pi_{a}$). \\

The potential astronomical expressed, for the three types of tidal frequencies
following, by : \\
Tide long period :
\begin{equation}
\Pi_{a}=gA_{k}(\frac{1}{2}-\frac{3}{2}sin^{2}\phi)cos(\omega_{k}t+V_{0k})
\end{equation}
diurnal Tide :
\begin{equation}
\Pi_{a}=gA_{k}(sin 2\phi)cos(\omega_{k}t+\lambda+V_{0k})
\end{equation}
Semi-diurnal tide:
\begin{equation}
\Pi_{a}=gA_{k}(cos^{2}\phi)cos(\omega_{k}t+2\lambda+V_{0k})
\end{equation}


$A_{k}$ is the amplitude of the wave k, $\omega_{k}$ the pulsation of the wave k, $V_{0k}$ the astronomical phase of the wave
$k$ to Greenwich.

We make corrections to the astronomical potential.
We obtain : 
\begin{equation}
\Pi-g\delta = (1+k-h) \Pi_{A}(\lambda,\phi)
\end{equation}
with $k$ a number of Love estimated to 0.6 which parametrized the astronomical tidal land,
and $h$ a number of Love to 0.3 which parametrized the parametrization due to the astronomical tidal land.

% ================================================================
%        River runoffs
% ================================================================
\section   [River runoffs (\textit{sbcrnf})]
         {River runoffs (\mdl{sbcrnf})}
\label{SBC_rnf}
%------------------------------------------namsbc_rnf----------------------------------------------------
\namdisplay{namsbc_rnf} 
%-------------------------------------------------------------------------------------------------------------

%River runoff generally enters the ocean at a nonzero depth rather than through the surface. 
%Many models, however, have traditionally inserted river runoff to the top model cell.
%This was the case in \NEMO prior to the version 3.3. The switch toward a input of runoff 
%throughout a nonzero depth has been motivated by the numerical and physical problems 
%that arise when the top grid cells are of the order of one meter. This situation is common in 
%coastal modelling and becomes more and more often open ocean and climate modelling 
%\footnote{At least a top cells thickness of 1~meter and a 3 hours forcing frequency are
%required to properly represent the diurnal cycle \citep{Bernie_al_JC05}. see also \S\ref{SBC_dcy}.}.


%To do this we need to treat evaporation/precipitation fluxes and river runoff differently in the 
%\mdl{tra\_sbc} module.  We decided to separate them throughout the code, so that the variable 
%\textit{emp} represented solely evaporation minus precipitation fluxes, and a new 2d variable 
%rnf was added which represents the volume flux of river runoff (in kg/m2s to remain consistent with 
%emp).  This meant many uses of emp and emps needed to be changed, a list of all modules which use 
%emp or emps and the changes made are below:


%Rachel:
River runoff generally enters the ocean at a nonzero depth rather than through the surface.
Many models, however, have traditionally inserted river runoff to the top model cell.
This was the case in \NEMO prior to the version 3.3, and was combined with an option 
to increase vertical mixing near the river mouth.

However, with this method numerical and physical problems arise when the top grid cells are 
of the order of one meter. This situation is common in coastal modelling and is becoming 
more common in open ocean and climate modelling 
\footnote{At least a top cells thickness of 1~meter and a 3 hours forcing frequency are
required to properly represent the diurnal cycle \citep{Bernie_al_JC05}. see also \S\ref{SBC_dcy}.}.

As such from V~3.3 onwards it is possible to add river runoff through a non-zero depth, and for the 
temperature and salinity of the river to effect the surrounding ocean.
The user is able to specify, in a NetCDF input file, the temperature and salinity of the river, along with the   
depth (in metres) which the river should be added to.

Namelist options, \np{ln\_rnf\_depth}, \np{ln\_rnf\_sal} and \np{ln\_rnf\_temp} control whether 
the river attributes (depth, salinity and temperature) are read in and used.  If these are set 
as false the river is added to the surface box only, assumed to be fresh (0~psu), and/or 
taken as surface temperature respectively.

The runoff value and attributes are read in in sbcrnf.  
For temperature -999 is taken as missing data and the river temperature is taken to be the 
surface temperatue at the river point.
For the depth parameter a value of -1 means the river is added to the surface box only, 
and a value of -999 means the river is added through the entire water column. 
After being read in the temperature and salinity variables are multiplied by the amount of runoff (converted into m/s) 
to give the heat and salt content of the river runoff.
After the user specified depth is read ini, the number of grid boxes this corresponds to is 
calculated and stored in the variable \np{nz\_rnf}.
The variable \textit{h\_dep} is then calculated to be the depth (in metres) of the bottom of the 
lowest box the river water is being added to (i.e. the total depth that river water is being added to in the model).

The mass/volume addition due to the river runoff is, at each relevant depth level, added to the horizontal divergence 
(\textit{hdivn}) in the subroutine \rou{sbc\_rnf\_div} (called from \mdl{divcur}).
This increases the diffusion term in the vicinity of the river, thereby simulating a momentum flux.
The sea surface height is calculated using the sum of the horizontal divergence terms, and so the 
river runoff indirectly forces an increase in sea surface height. 

The \textit{hdivn} terms are used in the tracer advection modules to force vertical velocities.
This causes a mass of water, equal to the amount of runoff, to be moved into the box above. 
The heat and salt content of the river runoff is not included in this step, and so the tracer 
concentrations are diluted as water of ocean temperature and salinity is moved upward out of the box 
and replaced by the same volume of river water with no corresponding heat and salt addition.

For the linear free surface case, at the surface box the tracer advection causes a flux of water 
(of equal volume to the runoff) through the sea surface out of the domain, which causes a salt and heat flux out of the model.
As such the volume of water does not change, but the water is diluted.

For the non-linear free surface case (\key{vvl}), no flux is allowed through the surface.
Instead in the surface box (as well as water moving up from the boxes below) a volume of runoff water 
is added with no corresponding heat and salt addition and so as happens in the lower boxes there is a dilution effect.
(The runoff addition to the top box along with the water being moved up through boxes below means the surface box has a large 
increase in volume, whilst all other boxes remain the same size)

In trasbc the addition of heat and salt due to the river runoff is added.
This is done in the same way for both vvl and non-vvl.
The temperature and salinity are increased through the specified depth according to the heat and salt content of the river. 

In the non-linear free surface case (vvl), near the end of the time step the change in sea surface height is redistrubuted 
through the grid boxes, so that the original ratios of grid box heights are restored.
In doing this water is moved into boxes below, throughout the water column, so the large volume addition to the surface box is spread between all the grid boxes.

It is also possible for runnoff to be specified as a negative value for modelling flow through straits, i.e. modelling the Baltic flow in and out of the North Sea.
When the flow is out of the domain there is no change in temperature and salinity, regardless of the namelist options used, as the ocean water leaving the domain removes heat and salt (at the same concentration) with it. 


%\colorbox{yellow}{Nevertheless, Pb of vertical resolution and 3D input : increase vertical mixing near river mouths to mimic a 3D river 

%All river runoff and emp fluxes are assumed to be fresh water (zero salinity) and at the same temperature as the sea surface.}

%\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

%\gmcomment{  word doc of runoffs:
%
%In the current \NEMO setup river runoff is added to emp fluxes, these are then applied at just the sea surface as a volume change (in the variable volume case this is a literal volume change, and in the linear free surface case the free surface is moved) and a salt flux due to the concentration/dilution effect.  There is also an option to increase vertical mixing near river mouths; this gives the effect of having a 3d river.  All river runoff and emp fluxes are assumed to be fresh water (zero salinity) and at the same temperature as the sea surface.
%Our aim was to code the option to specify the temperature and salinity of river runoff, (as well as the amount), along with the depth that the river water will affect.  This would make it possible to model low salinity outflow, such as the Baltic, and would allow the ocean temperature to be affected by river runoff.  

%The depth option makes it possible to have the river water affecting just the surface layer, throughout depth, or some specified point in between.

%To do this we need to treat evaporation/precipitation fluxes and river runoff differently in the tra_sbc module.  We decided to separate them throughout the code, so that the variable emp represented solely evaporation minus precipitation fluxes, and a new 2d variable rnf was added which represents the volume flux of river runoff (in kg/m2s to remain consistent with emp).  This meant many uses of emp and emps needed to be changed, a list of all modules which use emp or emps and the changes made are below:

%}

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

% -------------------------------------------------------------------------------------------------------------
%        Diurnal cycle
% -------------------------------------------------------------------------------------------------------------
\subsection   [Diurnal  cycle (\textit{sbcdcy})]
         {Diurnal cycle (\mdl{sbcdcy})}
\label{SBC_dcy}
%------------------------------------------namsbc_rnf----------------------------------------------------
%\namdisplay{namsbc} 
%-------------------------------------------------------------------------------------------------------------

%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
\begin{figure}[!t]    \begin{center}
\includegraphics[width=0.8\textwidth]{./TexFiles/Figures/Fig_SBC_diurnal.pdf}
\caption{ \label{Fig_SBC_diurnal}    
Example of recontruction of the diurnal cycle variation of short wave flux  
from daily mean values. The reconstructed diurnal cycle (black line) is chosen 
as the mean value of the analytical cycle (blue line) over a time step, not 
as the mid time step value of the analytically cycle (red square). From \citet{Bernie_al_CD07}.}
\end{center}   \end{figure}
%>>>>>>>>>>>>>>>>>>>>>>>>>>>>

\cite{Bernie_al_JC05} have shown that to capture 90$\%$ of the diurnal variability of 
SST requires a vertical resolution in upper ocean of 1~m or better and a temporal resolution 
of the surface fluxes of 3~h or less. Unfortunately high frequency forcing fields are rare, 
not to say inexistent. Nevertheless, it is possible to obtain a reasonable diurnal cycle 
of the SST knowning only short wave flux (SWF) at high frequency \citep{Bernie_al_CD07}.
Furthermore, only the knowledge of daily mean value of SWF is needed, 
as higher frequency variations can be reconstructed from them, assuming that 
the diurnal cycle of SWF is a scaling of the top of the atmosphere diurnal cycle 
of incident SWF. The \cite{Bernie_al_CD07} reconstruction algorithm is available
in \NEMO by setting \np{ln\_dm2dc}~=~true (a \textit{namsbc} namelist parameter) when using 
CORE bulk formulea (\np{ln\_blk\_core}~=~true) or the flux formulation (\np{ln\_flx}~=~true). 
The reconstruction is performed in the \mdl{sbcdcy} module. The detail of the algoritm used 
can be found in the appendix~A of \cite{Bernie_al_CD07}. The algorithm preserve the daily 
mean incomming SWF as the reconstructed SWF at a given time step is the mean value 
of the analytical cycle over this time step (Fig.\ref{Fig_SBC_diurnal}). 
The use of diurnal cycle reconstruction requires the input SWF to be daily 
($i.e.$ a frequency of 24 and a time interpolation set to true in \np{sn\_qsr} namelist parameter).
Furthermore, it is recommended to have a least 8 surface module time step per day,
that is  $\rdt \ \np{nn\_fsbc} < 10,800~s = 3~h$. An example of recontructed SWF 
is given in Fig.\ref{Fig_SBC_dcy} for a 12 reconstructed diurnal cycle, one every 2~hours 
(from 1am to 11pm).

%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
\begin{figure}[!t]  \begin{center}
\includegraphics[width=0.7\textwidth]{./TexFiles/Figures/Fig_SBC_dcy.pdf}
\caption{ \label{Fig_SBC_dcy}   
Example of recontruction of the diurnal cycle variation of short wave flux  
from daily mean values on an ORCA2 grid with a time sampling of 2~hours (from 1am to 11pm). 
The display is on (i,j) plane. }
\end{center}   \end{figure}
%>>>>>>>>>>>>>>>>>>>>>>>>>>>>

Note also that the setting a diurnal cycle in SWF is highly recommended  when 
the top layer thickness approach 1~m or less, otherwise large error in SST can 
appear due to an inconsistency between the scale of the vertical resolution 
and the forcing acting on that scale.

% -------------------------------------------------------------------------------------------------------------
%        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 (\np{ln\_flx}~=~true), 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}
\textit{emp} = \textit{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 $\textit{emp}_{o }$ 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}, \key{lim3} or \key{cice}). 
The activation automatically overwrites the read value of nn{\_}ice to its appropriate 
value ($i.e.$ $2$ for LIM-2, $3$ for LIM-3 or $4$ for CICE).
\end{description}

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

\subsection   [Interface to CICE (\textit{sbcice\_cice})]
         {Interface to CICE (\mdl{sbcice\_cice})}
\label{SBC_cice}

It is now possible to couple a global NEMO configuration (without AGRIF) to the CICE sea-ice
model by using \key{cice}.  The CICE code can be obtained from 
\href{http://oceans11.lanl.gov/trac/CICE/}{LANL} and the additional 'hadgem3' drivers will be required, 
even with the latest code release.  Input grid files consistent with those used in NEMO will also be needed, 
and CICE CPP keys \textbf{ORCA\_GRID}, \textbf{CICE\_IN\_NEMO} and \textbf{coupled} should be used (seek advice from UKMO 
if necessary).  Currently the code is only designed to work when using the CORE forcing option for NEMO (with
\textit{calc\_strair~=~true} and \textit{calc\_Tsfc~=~true} in the CICE name-list), or alternatively when NEMO 
is coupled to the HadGAM3 atmosphere model (with \textit{calc\_strair~=~false} and \textit{calc\_Tsfc~=~false}).
The code is intended to be used with \np{nn\_fsbc} set to 1 (although coupling ocean and ice less frequently 
should work, it is possible the calculation of some of the ocean-ice fluxes needs to be modified slightly - the
user should check that results are not significantly different to the standard case).

There are two options for the technical coupling between NEMO and CICE.  The standard version allows
complete flexibility for the domain decompositions in the individual models, but this is at the expense of global
gather and scatter operations in the coupling which become very expensive on larger numbers of processors. The
alternative option (using \key{nemocice\_decomp} for both NEMO and CICE) ensures that the domain decomposition is
identical in both models (provided domain parameters are set appropriately, and 
\textit{processor\_shape~=~square-ice} and \textit{distribution\_wght~=~block} in the CICE name-list) and allows
much more efficient direct coupling on individual processors.  This solution scales much better although it is at 
the expense of having more idle CICE processors in areas where there is no sea ice.


% -------------------------------------------------------------------------------------------------------------
%        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 \textit{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}

% -------------------------------------------------------------------------------------------------------------
%        Neutral Drag Coefficient from external wave model
% -------------------------------------------------------------------------------------------------------------
\subsection   [Neutral drag coefficient from external wave model (\textit{sbcwave})]
                        {Neutral drag coefficient from external wave model (\mdl{sbcwave})}
\label{SBC_wave}
%------------------------------------------namwave----------------------------------------------------
\namdisplay{namsbc_wave}
%-------------------------------------------------------------------------------------------------------------
\begin{description}

\item [??] In order to read a neutral drag coeff, from an external data source (i.e. a wave model), the 
logical variable \np{ln\_cdgw}
 in $namsbc$ namelist must be defined ${.true.}$. 
The \mdl{sbcwave} module containing the routine \np{sbc\_wave} reads the
namelist ${namsbc\_wave}$ (for external data names, locations, frequency, interpolation and all 
the miscellanous options allowed by Input Data generic Interface see \S\ref{SBC_input}) 
and a 2D field of neutral drag coefficient. Then using the routine 
TURB\_CORE\_1Z or TURB\_CORE\_2Z, and starting from the neutral drag coefficent provided, the drag coefficient is computed according 
to stable/unstable conditions of the air-sea interface following \citet{Large_Yeager_Rep04}.

\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.