% ================================================================
% Chapter Ñ Lateral Boundary Condition (LBC) 
% ================================================================
\chapter{Lateral Boundary Condition (LBC) }
\label{LBC}
\minitoc

\newpage
$\ $\newline    % force a new ligne


%gm% add here introduction to this chapter

% ================================================================
% Boundary Condition at the Coast
% ================================================================
\section{Boundary Condition at the Coast (\np{shlat})}
\label{LBC_coast}
%--------------------------------------------nam_lbc-------------------------------------------------------
\namdisplay{nam_lbc} 
%--------------------------------------------------------------------------------------------------------------

%The lateral ocean boundary conditions contiguous to coastlines are Neumann conditions for heat and salt (no flux across boundaries) and Dirichlet conditions for momentum (ranging from free-slip to "strong" no-slip). They are handled automatically by the mask system (see \S\ref{DOM_msk}). 

%OPA allows land and topography grid points in the computational domain due to the presence of continents or islands, and includes the use of a full or partial step representation of bottom topography. The computation is performed over the whole domain, i.e. we do not try to restrict the computation to ocean-only points. This choice has two motivations. Firstly, working on ocean only grid points overloads the code and harms the code readability. Secondly, and more importantly, it drastically reduces the vector portion of the computation, leading to a dramatic increase of CPU time requirement on vector computers.  The current section describes how the masking affects the computation of the various terms of the equations with respect to the boundary condition at solid walls. The process of defining which areas are to be masked is described in \S\ref{DOM_msk}.

The discrete representation of a domain with complex boundaries (coastlines and 
bottom topography) leads to arrays that include large portions where a computation 
is not required as the model variables remain at zero. Nevertheless, vectorial 
supercomputers are far more efficient when computing over a whole array, and the 
readability of a code is greatly improved when boundary conditions are applied in 
an automatic way rather than by a specific computation before or after each 
computational loop. An efficient way to work over the whole domain while specifying 
the boundary conditions, is to use multiplication by mask arrays in the computation. 
A mask array is a matrix whose elements are $1$ in the ocean domain and $0$ 
elsewhere. A simple multiplication of a variable by its own mask ensures that it will 
remain zero over land areas. Since most of the boundary conditions consist of a 
zero flux across the solid boundaries, they can be simply applied by multiplying 
variables by the correct mask arrays, $i.e.$ the mask array of the grid point where 
the flux is evaluated. For example, the heat flux in the \textbf{i}-direction is evaluated 
at $u$-points. Evaluating this quantity as,

\begin{equation} \label{Eq_lbc_aaaa}
\frac{A^{lT} }{e_1 }\frac{\partial T}{\partial i}\equiv \frac{A_u^{lT} 
}{e_{1u} } \; \delta _{i+1 / 2} \left[ T \right]\;\;mask_u 
\end{equation}
(where mask$_{u}$ is the mask array at a $u$-point) ensures that the heat flux is 
zero inside land and at the boundaries, since mask$_{u}$ is zero at solid boundaries 
which in this case are defined at $u$-points (normal velocity $u$ remains zero at 
the coast) (Fig.~\ref{Fig_LBC_uv}). 

%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
\begin{figure}[!t] \label{Fig_LBC_uv}  \begin{center}
\includegraphics[width=0.90\textwidth]{./TexFiles/Figures/Fig_LBC_uv.pdf}
\caption {Lateral boundary (thick line) at T-level. The velocity normal to the 
		 boundary is set to zero.}
\end{center}   \end{figure}
%>>>>>>>>>>>>>>>>>>>>>>>>>>>>

For momentum the situation is a bit more complex as two boundary conditions 
must be provided along the coast (one each for the normal and tangential velocities). 
The boundary of the ocean in the C-grid is defined by the velocity-faces. 
For example, at a given $T$-level, the lateral boundary (a coastline or an intersection 
with the bottom topography) is made of segments joining $f$-points, and normal 
velocity points are located between two $f-$points (Fig.~\ref{Fig_LBC_uv}). 
The boundary condition on the normal velocity (no flux through solid boundaries) 
can thus be easily implemented using the mask system. The boundary condition 
on the tangential velocity requires a more specific treatment. This boundary 
condition influences the relative vorticity and momentum diffusive trends, and is 
required in order to compute the vorticity at the coast. Four different types of 
lateral boundary condition are available, controlled by the value of the \np{shlat} 
namelist parameter. (The value of the mask$_{f}$ array along the coastline is set 
equal to this parameter.) These are:

%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
\begin{figure}[!p] \label{Fig_LBC_shlat}  \begin{center}
\includegraphics[width=0.90\textwidth]{./TexFiles/Figures/Fig_LBC_shlat.pdf}
\caption {lateral boundary condition (a) free-slip ($shlat=0$) ; (b) no-slip ($shlat=2$) 
; (c) "partial" free-slip ($0<shlat<2$) and (d) "strong" no-slip ($2<shlat$). 
Implied "ghost" velocity inside land area is display in grey. }
\end{center}   \end{figure}
%>>>>>>>>>>>>>>>>>>>>>>>>>>>>

\begin{description}

\item[free-slip boundary condition (\np{shlat}=0): ]  the tangential velocity at the 
coastline is equal to the offshore velocity, $i.e.$ the normal derivative of the 
tangential velocity is zero at the coast, so the vorticity: mask$_{f}$ array is set 
to zero inside the land and just at the coast (Fig.~\ref{Fig_LBC_shlat}-a).

\item[no-slip boundary condition (\np{shlat}=2): ] the tangential velocity vanishes 
at the coastline. Assuming that the tangential velocity decreases linearly from 
the closest ocean velocity grid point to the coastline, the normal derivative is 
evaluated as if the velocities at the closest land velocity gridpoint and the closest 
ocean velocity gridpoint were of the same magnitude but in the opposite direction 
(Fig.~\ref{Fig_LBC_shlat}-b). Therefore, the vorticity along the coastlines is given by: 

\begin{equation*}
\zeta \equiv 2 \left(\delta_{i+1/2} \left[e_{2v} v \right] - \delta_{j+1/2} \left[e_{1u} u \right] \right) / \left(e_{1f} e_{2f} \right) \ ,
\end{equation*}
where $u$ and $v$ are masked fields. Setting the mask$_{f}$ array to $2$ along 
the coastline provides a vorticity field computed with the no-slip boundary condition, 
simply by multiplying it by the mask$_{f}$ :
\begin{equation} \label{Eq_lbc_bbbb}
\zeta \equiv \frac{1}{e_{1f} {\kern 1pt}e_{2f} }\left( {\delta _{i+1/2} 
\left[ {e_{2v} \,v} \right]-\delta _{j+1/2} \left[ {e_{1u} \,u} \right]} 
\right)\;\mbox{mask}_f 
\end{equation}

\item["partial" free-slip boundary condition (0$<$\np{shlat}$<$2): ] the tangential 
velocity at the coastline is smaller than the offshore velocity, $i.e.$ there is a lateral 
friction but not strong enough to make the tangential velocity at the coast vanish 
(Fig.~\ref{Fig_LBC_shlat}-c). This can be selected by providing a value of mask$_{f}$ 
strictly inbetween $0$ and $2$.

\item["strong" no-slip boundary condition (2$<$\np{shlat}): ] the viscous boundary 
layer is assumed to be smaller than half the grid size (Fig.~\ref{Fig_LBC_shlat}-d). 
The friction is thus larger than in the no-slip case.

\end{description}

Note that when the bottom topography is entirely represented by the $s$-coor-dinates 
(pure $s$-coordinate), the lateral boundary condition on tangential velocity is of much 
less importance as it is only applied next to the coast where the minimum water depth 
can be quite shallow.

The alternative numerical implementation of the no-slip boundary conditions for an 
arbitrary coast line of \citet{Shchepetkin1996} is also available through the 
\key{noslip\_accurate} CPP key. It is based on a fourth order evaluation of the shear at the 
coast which, in turn, allows a true second order scheme in the interior of the domain 
($i.e.$ the numerical boundary scheme simulates the truncation error of the numerical 
scheme used in the interior of the domain). \citet{Shchepetkin1996} found that such a 
technique considerably improves the quality of the numerical solution. In \NEMO, such 
spectacular improvements have not been found in the half-degree global ocean 
(ORCA05), but significant reductions of numerically induced coastal upwellings were 
found in an eddy resolving simulation of the Alboran Sea \citep{OlivierPh2001}. 
Nevertheless, since a no-slip boundary condition is not recommended in an eddy 
permitting or resolving simulation \citep{Penduff2007}, the use of this option is also 
not recommended.

In practice, the no-slip accurate option changes the way the curl is evaluated at the 
coast (see \mdl{divcur} module), and requires the nature of each coastline grid point 
(convex or concave corners, straight north-south or east-west coast) to be specified.  
This is performed in routine \rou{dom\_msk\_nsa} in the \mdl{domask} module.

% ================================================================
% Boundary Condition around the Model Domain
% ================================================================
\section{Model Domain Boundary Condition (\jp{jperio})}
\label{LBC_jperio}

At the model domain boundaries several choices are offered: closed, cyclic east-west, 
south symmetric across the equator, a north-fold, and combination closed-north fold 
or cyclic-north-fold. The north-fold boundary condition is associated with the 3-pole ORCA mesh. 

% -------------------------------------------------------------------------------------------------------------
%        Closed, cyclic, south symmetric (\jp{jperio} = 0, 1 or 2) 
% -------------------------------------------------------------------------------------------------------------
\subsection{Closed, cyclic, south symmetric (\jp{jperio} = 0, 1 or 2)}
\label{LBC_jperio012}

The choice of closed, cyclic or symmetric model domain boundary condition is made 
by setting \jp{jperio} to 0, 1 or 2 in file \mdl{par\_oce}. Each time such a boundary 
condition is needed, it is set by a call to routine \mdl{lbclnk}. The computation of 
momentum and tracer trends proceeds from $i=2$ to $i=jpi-1$ and from $j=2$ to 
$j=jpj-1$, $i.e.$ in the model interior. To choose a lateral model boundary condition 
is to specify the first and last rows and columns of the model variables. 

\begin{description}

\item[For closed boundary (\textit{jperio=0})], solid walls are imposed at all model 
boundaries: first and last rows and columns are set to zero.

\item[For cyclic east-west boundary (\textit{jperio=1})], first and last rows are set 
to zero (closed) whilst the first column is set to the value of the last-but-one column 
and the last column to the value of the second one (Fig.~\ref{Fig_LBC_jperio}-a). 
Whatever flows out of the eastern (western) end of the basin enters the western 
(eastern) end. Note that there is no option for north-south cyclic or for doubly 
cyclic cases.

\item[For symmetric boundary condition across the equator (\textit{jperio=2})], 
last rows, and first and last columns are set to zero (closed). The row of symmetry 
is chosen to be the $u$- and $T-$points equator line ($j=2$, i.e. at the southern 
end of the domain). For arrays defined at $u-$ or $T-$points, the first row is set 
to the value of the third row while for most of $v$- and $f$-point arrays ($v$, $\zeta$, 
$j\psi$, but \gmcomment{not sure why this is "but"} scalar arrays such as eddy coefficients) 
the first row is set to minus the value of the second row (Fig.~\ref{Fig_LBC_jperio}-b). 
Note that this boundary condition is not yet available for the case of a massively 
parallel computer (\textbf{key{\_}mpp} defined).

\end{description}

%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
\begin{figure}[!t] \label{Fig_LBC_jperio}  \begin{center}
\includegraphics[width=1.0\textwidth]{./TexFiles/Figures/Fig_LBC_jperio.pdf}
\caption {setting of (a) east-west cyclic  (b) symmetric across the equator boundary conditions.}
\end{center}   \end{figure}
%>>>>>>>>>>>>>>>>>>>>>>>>>>>>

% -------------------------------------------------------------------------------------------------------------
%        North fold (\textit{jperio = 3 }to $6)$ 
% -------------------------------------------------------------------------------------------------------------
\subsection{North-fold (\textit{jperio = 3 }to $6)$ }
\label{LBC_north_fold}

The north fold boundary condition has been introduced in order to handle the north 
boundary of a three-polar ORCA grid. Such a grid has two poles in the northern hemisphere. 
\colorbox{yellow}{to be completed...}

%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
\begin{figure}[!t] \label{Fig_North_Fold_T}  \begin{center}
\includegraphics[width=0.90\textwidth]{./TexFiles/Figures/Fig_North_Fold_T.pdf}
\caption {North fold boundary with a $T$-point pivot and cyclic east-west boundary condition 
($jperio=4$), as used in ORCA 2, 1/4, and 1/12. Pink shaded area corresponds to the inner 
domain mask (see text). }
\end{center}   \end{figure}
%>>>>>>>>>>>>>>>>>>>>>>>>>>>>

% ====================================================================
% Exchange with neighbouring processors 
% ====================================================================
\section  [Exchange with neighbouring processors (\textit{lbclnk}, \textit{lib\_mpp})]
		{Exchange with neighbouring processors (\mdl{lbclnk}, \mdl{lib\_mpp})}
\label{LBC_mpp}

For massively parallel processing (mpp), a domain decomposition method is used. 
The basic idea of the method is to split the large computation domain of a numerical 
experiment into several smaller domains and solve the set of equations by addressing 
independent local problems. Each processor has its own local memory and computes 
the model equation over a subdomain of the whole model domain. The subdomain 
boundary conditions are specified through communications between processors 
which are organized by explicit statements (message passing method). 

A big advantage is that the method does not need many modifications of the initial 
FORTRAN code. From the modeller's point of view, each sub domain running on 
a processor is identical to the "mono-domain" code. In addition, the programmer 
manages the communications between subdomains, and the code is faster when 
the number of processors is increased. The porting of OPA code on an iPSC860 
was achieved during Guyon's PhD [Guyon et al. 1994, 1995] in collaboration with 
CETIIS and ONERA. The implementation in the operational context and the studies 
of performance on a T3D and T3E Cray computers have been made in collaboration 
with IDRIS and CNRS. The present implementation is largely inspired by Guyon's 
work  [Guyon 1995].

The parallelization strategy is defined by the physical characteristics of the 
ocean model. Second order finite difference schemes lead to local discrete 
operators that depend at the very most on one neighbouring point. The only 
non-local computations concern the vertical physics (implicit diffusion, 1.5 
turbulent closure scheme, ...) (delocalization over the whole water column), 
and the solving of the elliptic equation associated with the surface pressure 
gradient computation (delocalization over the whole horizontal domain). 
Therefore, a pencil strategy is used for the data sub-structuration 
\gmcomment{no idea what this means!}
: the 3D initial domain is laid out on local processor 
memories following a 2D horizontal topological splitting. Each sub-domain 
computes its own surface and bottom boundary conditions and has a side 
wall overlapping interface which defines the lateral boundary conditions for 
computations in the inner sub-domain. The overlapping area consists of the 
two rows at each edge of the sub-domain. After a computation, a communication 
phase starts: each processor sends to its neighbouring processors the update 
values of the points corresponding to the interior overlapping area to its 
neighbouring sub-domain (i.e. the innermost of the two overlapping rows). 
The communication is done through message passing. Usually the parallel virtual 
language, PVM, is used as it is a standard language available on  nearly  all 
MPP computers. More specific languages (i.e. computer dependant languages) 
can be easily used to speed up the communication, such as SHEM on a T3E 
computer. The data exchanges between processors are required at the very 
place where lateral domain boundary conditions are set in the mono-domain 
computation (\S III.10-c): the lbc\_lnk routine which manages such conditions 
is substituted by mpplnk.F or mpplnk2.F routine when running on an MPP 
computer (\key{mpp\_mpi} defined). It has to be pointed out that when using 
the MPP version of the model, the east-west cyclic boundary condition is done 
implicitly, whilst the south-symmetric boundary condition option is not available.

%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
\begin{figure}[!t] \label{Fig_mpp}  \begin{center}
\includegraphics[width=0.90\textwidth]{./TexFiles/Figures/Fig_mpp.pdf}
\caption {Positioning of a sub-domain when massively parallel processing is used. }
\end{center}   \end{figure}
%>>>>>>>>>>>>>>>>>>>>>>>>>>>>

In the standard version of the OPA model, the splitting is regular and arithmetic.
 the i-axis is divided by \jp{jpni} and the j-axis by \jp{jpnj} for a number of processors 
 \jp{jpnij} most often equal to $jpni \times jpnj$ (model parameters set in 
 \mdl{par\_oce}). Each processor is independent and without message passing 
 or synchronous process 
 \gmcomment{how does a synchronous process relate to this?}, 
 programs run alone and access just its own local memory. For this reason, the 
 main model dimensions are now the local dimensions of the subdomain (pencil) 
 that are named \jp{jpi}, \jp{jpj}, \jp{jpk}. These dimensions include the internal 
 domain and the overlapping rows. The number of rows to exchange (known as 
 the halo) is usually set to one (\jp{jpreci}=1, in \mdl{par\_oce}). The whole domain 
 dimensions are named \jp{jpiglo}, \jp{jpjglo} and \jp{jpk}. The relationship between 
 the whole domain and a sub-domain is:
\begin{eqnarray} 
      jpi & = & ( jpiglo-2*jpreci + (jpni-1) ) / jpni + 2*jpreci  \nonumber \\
      jpj & = & ( jpjglo-2*jprecj + (jpnj-1) ) / jpnj + 2*jprecj  \label{Eq_lbc_jpi}
\end{eqnarray}
where \jp{jpni}, \jp{jpnj} are the number of processors following the i- and j-axis.

\colorbox{yellow}{Figure IV.3: example of a domain splitting with 9 processors and 
no east-west cyclic boundary conditions.}

One also defines variables nldi and nlei which correspond to the internal 
domain bounds, and the variables nimpp and njmpp which are the position 
of the (1,1) grid-point in the global domain. An element of $T_{l}$, a local array 
(subdomain) corresponds to an element of $T_{g}$, a global array 
(whole domain) by the relationship: 
\begin{equation} \label{Eq_lbc_nimpp}
T_{g} (i+nimpp-1,j+njmpp-1,k) = T_{l} (i,j,k),
\end{equation}
with  $1 \leq i \leq jpi$, $1  \leq j \leq jpj $ , and  $1  \leq k \leq jpk$.

Processors are numbered from 0 to $jpnij-1$, the number is saved in the variable 
nproc. In the standard version, a processor has no more than four neighbouring 
processors named nono (for north), noea (east), noso (south) and nowe (west) 
and two variables, nbondi and nbondj, indicate the relative position of the processor 
\colorbox{yellow}{(see Fig.IV.3)}:
\begin{itemize}
\item 		nbondi = -1 	an east neighbour, no west processor,
\item 		nbondi =  0	an east neighbour, a west neighbour,
\item 		nbondi =  1 	no east processor, a west neighbour,
\item 		nbondi =  2 	no splitting following the i-axis.
\end{itemize}
During the simulation, processors exchange data with their neighbours. 
If there is effectively a neighbour, the processor receives variables from this 
processor on its overlapping row, and sends the data issued from internal 
domain corresponding to the overlapping row of the other processor.
		 
\colorbox{yellow}{Figure IV.4: pencil splitting with the additional outer halos }


The \NEMO model computes equation terms with the help of mask arrays (0 on land 
points and 1 on sea points). It is easily readable and very efficient in the context of 
a computer with vectorial architecture. However, in the case of a scalar processor, 
computations over the land regions become more expensive in terms of CPU time. 
It is worse when we use a complex configuration with a realistic bathymetry like the 
global ocean where more than 50 \% of points are land points. For this reason, a 
pre-processing tool can be used to choose the mpp domain decomposition with a 
maximum number of only land points processors, which can then be eliminated. 
(For example, the mpp\_optimiz tools, available from the DRAKKAR web site.) 
This optimisation is dependent on the specific bathymetry employed. The user 
then chooses optimal parameters \jp{jpni}, \jp{jpnj} and \jp{jpnij} with 
$jpnij < jpni \times jpnj$, leading to the elimination of $jpni \times jpnj - jpnij$ 
land processors. When those parameters are specified in module \mdl{par\_oce}, 
the algorithm in the \rou{inimpp2} routine sets each processor's parameters (nbound, 
nono, noea,...) so that the land-only processors are not taken into account. 

\colorbox{yellow}{Note that the inimpp2 routine is general so that the original inimpp 
routine should be suppressed from the code.}

When land processors are eliminated, the value corresponding to these locations in 
the model output files is zero. Note that this is a problem for a mesh output file written 
by such a model configuration, because model users often divide by the scale factors 
($e1t$, $e2t$, etc) and do not expect the grid size to be zero, even on land. It may be 
best not to eliminate land processors when running the model especially to write the 
mesh files as outputs (when \np{nmsh} namelist parameter differs from 0).
\gmcomment{Steven : dont understand this, no land processor means no output file 
covering this part of globe; its only when files are stitched together into one that you 
can leave a hole}

%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
\begin{figure}[!ht] \label{Fig_mppini2}  \begin{center}
\includegraphics[width=0.90\textwidth]{./TexFiles/Figures/Fig_mppini2.pdf}
\caption {Example of Atlantic domain defined for the CLIPPER projet. Initial grid is 
composed of 773 x 1236 horizontal points. (a) the domain is split onto 9 \time 20 
subdomains (jpni=9, jpnj=20). 52 subdomains are land areas. (b) 52 subdomains 
are eliminated (white rectangles) and the resulting number of processors really 
used during the computation is jpnij=128.}
\end{center}   \end{figure}
%>>>>>>>>>>>>>>>>>>>>>>>>>>>>


% ================================================================
% Open Boundary Conditions 
% ================================================================
\section{Open Boundary Conditions (\key{obc})}
\label{LBC_obc}
%-----------------------------------------nam_obc  -------------------------------------------
%-    nobc_dta    =    0     !  = 0 the obc data are equal to the initial state
%-                           !  = 1 the obc data are read in 'obc   .dta' files
%-    rdpein      =    1.    !  ???
%-    rdpwin      =    1.    !  ???
%-    rdpnin      =   30.    !  ???
%-    rdpsin      =    1.    !  ???
%-    rdpeob      = 1500.    !  time relaxation (days) for the east  open boundary
%-    rdpwob      =   15.    !    "        "           for the west  open boundary
%-    rdpnob      =  150.    !    "        "           for the north open boundary
%-    rdpsob      =   15.    !    "        "           for the south open boundary
%-    zbsic1      =  140.e+6 !  barotropic stream function on first  isolated coastline
%-    zbsic2      =    1.e+6 !    "                   "    on second isolated coastline
%-    zbsic3      =    0.    !    "                   "    on thrid  isolated coastline
%-    ln_obc_clim = .true.   !  climatological obc data files (default T)
%-    ln_vol_cst  = .true.   !  total volume conserved
\namdisplay{namobc} 

It is often necessary to implement a model configuration limited to an oceanic 
region or a basin, which communicates with the rest of the global ocean through 
''open boundaries''. As stated by \citet{Roed1986}, an open boundary is a 
computational border where the aim of the calculations is to allow the perturbations 
generated inside the computational domain to leave it without deterioration of the 
inner model solution. However, an open boundary also has to let information from 
the outer ocean enter the model and should support inflow and outflow conditions. 

The open boundary package OBC is the first open boundary option developed in 
NEMO (originally in OPA8.2). It allows the user to 
\begin{itemize}
\item tell the model that a boundary is ''open'' and not closed by a wall, for example 
by modifying the calculation of the divergence of velocity there;
\item impose values of tracers and velocities at that boundary (values which may 
be taken from a climatology): this is the``fixed OBC'' option. 
\item calculate boundary values by a sophisticated algorithm combining radiation 
and relaxation (``radiative OBC'' option)
\end{itemize}

The package resides in the OBC directory. It is described here in four parts: the 
boundary geometry (parameters to be set in \mdl{obc\_par}), the forcing data at 
the boundaries (module \mdl{obcdta}),  the radiation algorithm involving the 
namelist and module \mdl{obcrad}, and a brief presentation of boundary update 
and restart files.

%----------------------------------------------
\subsection{Boundary geometry}
\label{OBC_geom}
%
First one has to realize that open boundaries may not necessarily be located 
at the extremities of the computational domain. They may exist in the middle 
of the domain, for example at Gibraltar Straits if one wants to avoid including 
the Mediterranean in an Atlantic domain. This flexibility has been found necessary 
for the CLIPPER project \citep{Treguier2001}. Because of the complexity of the 
geometry of ocean basins, it may even be necessary to have more than one 
''west'' open boundary, more than one ''north'', etc. This is not possible with 
the OBC option: only one open boundary of each kind, west, east, south and 
north is allowed; these names refer to the grid geometry (not to the direction 
of the geographical ''west'', ''east'', etc).

The open boundary geometry is set by a series of parameters in the module 
\mdl{obc\_par}. For an eastern open boundary, parameters are \jp{lp\_obc\_east} 
(true if an east open boundary exists), \jp{jpieob} the $i$-index along which 
the eastern open boundary (eob) is located, \jp{jpjed} the $j$-index at which 
it starts, and \jp{jpjef} the $j$-index where it ends (note $d$ is for ''d\'{e}but'' 
and $f$ for ''fin'' in French). Similar parameters exist for the west, south and 
north cases (Table~\ref{Tab_obc_param}).


%--------------------------------------------------TABLE--------------------------------------------------

\begin{table}[htbp]  \label{Tab_obc_param}
\begin{center}
\begin{tabular}{|l|c|c|c|}
\hline
Boundary and  & Constant index  & Starting index (d\'{e}but) & Ending index (fin) \\
Logical flag  &                 &                            &                     \\
\hline
West          & \jp{jpiwob} $>= 2$         &  \jp{jpjwd}$>= 2$          &  \jp{jpjwf}<= \jp{jpjglo}-1 \\
lp\_obc\_west & $i$-index of a $u$ point   & $j$ of a $T$ point   &$j$ of a $T$ point \\
\hline
East            & \jp{jpieob}$<=$\jp{jpiglo}-2&\jp{jpjed} $>= 2$         & \jp{jpjef}$<=$ \jp{jpjglo}-1 \\
 lp\_obc\_east  & $i$-index of a $u$ point    & $j$ of a $T$ point & $j$ of a $T$ point \\
\hline
South           & \jp{jpjsob} $>= 2$         & \jp{jpisd} $>= 2$          & \jp{jpisf}$<=$\jp{jpiglo}-1 \\
lp\_obc\_south  & $j$-index of a $v$ point   & $i$ of a $T$ point   & $i$ of a $T$ point \\
\hline
North           & \jp{jpjnob} $<=$ \jp{jpjglo}-2& \jp{jpind} $>= 2$        & \jp{jpinf}$<=$\jp{jpiglo}-1 \\
lp\_obc\_north  & $j$-index of a $v$ point      & $i$  of a $T$ point & $i$ of a $T$ point \\
\hline
\end{tabular}
\end{center}
\caption{Names of different indices relating to the open boundaries. In the case 
of a completely open ocean domain with four ocean boundaries, the parameters 
take exactly the values indicated.}
\end{table}

The open boundaries must be along coordinate lines. On the C-grid, the boundary 
itself is along a line of normal velocity points: $v$ points for a zonal open boundary 
(the south or north one), and $u$ points for a meridional open boundary (the west 
or east one). Another constraint is that there still must be a row of masked points 
all around the domain, as if the domain were a closed basin (unless periodic conditions 
are used together with open boundary conditions). Therefore, an open boundary 
cannot be located at the first/last index, namely, 1, \jp{jpiglo} or \jp{jpjglo}. Also, 
the open boundary algorithm involves calculating the normal velocity points situated 
just on the boundary, as well as the tangential velocity and temperature and salinity 
just outside the boundary. This means that for a west/south boundary, normal 
velocities and temperature are calculated at the same index \jp{jpiwob} and 
\jp{jpjsob}, respectively. For an east/north boundary, the normal velocity is 
calculated at index \jp{jpieob} and \jp{jpjnob}, but the ``outside'' temperature is 
at index \jp{jpieob}+1 and \jp{jpjnob}+1. This means that \jp{jpieob}, \jp{jpjnob} 
cannot be bigger than \jp{jpiglo}-2, \jp{jpjglo}-2. 


The starting and ending indices are to be thought of as $T$ point indices: in many 
cases they indicate the first land $T$-point, at the extremity of an open boundary 
(the coast line follows the $f$ grid points, see Fig.~\ref{Fig_obc_north} for an example 
of a northern open boundary). All indices are relative to the global domain. In the 
free surface case it is possible to have ``ocean corners'', that is, an open boundary 
starting and ending in the ocean.

%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
\begin{figure}[!t] \label{Fig_obc_north}  \begin{center}
\includegraphics[width=0.70\textwidth]{./TexFiles/Figures/Fig_obc_north.pdf}
\caption {Localization of the North open boundary points.}
\end{center}  
\end{figure}
%>>>>>>>>>>>>>>>>>>>>>>>>>>>>

Although not compulsory, it is highly recommended that the bathymetry in the 
vicinity of an open boundary follows the following rule: in the direction perpendicular 
to the open line, the water depth should be constant for 4 grid points. This is in 
order to ensure that the radiation condition, which involves model variables next 
to the boundary, is calculated in a consistent way. On Fig.\ref{Fig_obc_north} we 
indicate by an $=$ symbol, the points which should have the same depth. It means 
that at the 4 points near the boundary, the bathymetry is cylindrical \gmcomment{not sure 
why cylindrical}. The line behind the open $T$-line must be 0 in the bathymetry file 
(as shown on Fig.\ref{Fig_obc_north} for example).

%----------------------------------------------
\subsection{Boundary data}
\label{OBC_data}

It is necessary to provide information at the boundaries. The simplest case is 
when this information does not change in time and is equal to the initial conditions 
(namelist variable \np{nobc\_dta}=0). This is the case for the standard configuration 
EEL5 with open boundaries. When (\np{nobc\_dta}=1), open boundary information 
is read from netcdf files. For convenience the input files are supposed to be similar 
to the ''history'' NEMO output files, for dimension names and variable names. 
Open boundary arrays must be dimensioned according to the parameters of table~
\ref{Tab_obc_param}: for example, at the western boundary, arrays have a 
dimension of \jp{jpwf}-\jp{jpwd}+1 in the horizontal and \jp{jpk} in the vertical. 

When ocean observations are used to generate the boundary data (a hydrographic 
section for example, as in \citet{Treguier2001}) it happens often that only the velocity 
normal to the boundary is known, which is the reason why the initial OBC code 
assumes that only $T$, $S$, and the normal velocity ($u$ or $v$) needs to be 
specified. As more and more global model solutions and ocean analysis products 
become available, it will be possible to provide information about all the variables 
(including the tangential velocity) so that the specification of four variables at each 
boundaries will become standard. For the sea surface height, one must distinguish 
between the filtered free surface case and the time-splitting or explicit treatment of 
the free surface.
 In the first case, it is assumed that the user does not wish to represent high 
 frequency motions such as tides. The boundary condition is thus one of zero 
 normal gradient of sea surface height at the open boundaries, following \citet{Marchesiello2001}. 
No information other than the total velocity needs to be provided at the open 
boundaries in that case. In the other two cases (time splitting or explicit free surface), 
the user must provide barotropic information (sea surface height and barotropic 
velocities) and the use of the Flather algorithm for barotropic variables is 
recommanded. However, this algorithm has not yet been fully tested and bugs 
remain in NEMO v2.3. Users should read the code carefully before using it. Finally, 
in the case of the rigid lid approximation the barotropic streamfunction must be 
provided, as documented in \citet{Treguier2001}). This option is no longer 
recommended but remains in NEMO V2.3.

One frequently encountered case is when an open boundary domain is constructed 
from a global or larger scale NEMO configuration. Assuming the domain corresponds 
to indices $ib:ie$, $jb:je$ of the global domain, the bathymetry and forcing of the 
small domain can be created by using the following netcdf utility on the global files: 
ncks -F $-d\;x,ib,ie$ $-d\;y,jb,je$ (part of the nco series of utilities, see http://nco.sourceforge.net). 
The open boundary files can be constructed using ncks 
commands, following table~\ref{Tab_obc_ind}. 

%--------------------------------------------------TABLE--------------------------------------------------

\begin{table}[htbp]  \label{Tab_obc_ind}
\begin{center}
\begin{tabular}{|l|c|c|c|c|c|}
\hline
OBC  & Variable   & file name      & Index  & Start  & end  \\
West &  T,S       &   obcwest\_TS.nc &  $ib$+1     &   $jb$+1 &  $je-1$  \\
     &    U       &   obcwest\_U.nc  &  $ib$+1     &   $jb$+1 &  $je-1$  \\ 
     &    V       &   obcwest\_V.nc  &  $ib$+1     &   $jb$+1 &  $je-1$  \\       
\hline
East &  T,S       &   obceast\_TS.nc &  $ie$-1     &   $jb$+1 &  $je-1$  \\
     &    U       &   obceast\_U.nc  &  $ie$-2     &   $jb$+1 &  $je-1$  \\ 
     &    V       &   obceast\_V.nc  &  $ie$-1     &   $jb$+1 &  $je-1$  \\       
\hline         
South &  T,S      &   obcsouth\_TS.nc &  $jb$+1     &  $ib$+1 &  $ie-1$  \\
      &    U      &   obcsouth\_U.nc  &  $jb$+1     &  $ib$+1 &  $ie-1$  \\ 
      &    V      &   obcsouth\_V.nc  &  $jb$+1     &  $ib$+1 &  $ie-1$  \\    
\hline
North &  T,S      &   obcnorth\_TS.nc &  $je$-1     &  $ib$+1 &  $ie-1$  \\
      &    U      &   obcnorth\_U.nc  &  $je$-1     &  $ib$+1 &  $ie-1$  \\ 
      &    V      &   obcnorth\_V.nc  &  $je$-2     &  $ib$+1 &  $ie-1$  \\  
\hline
\end{tabular}
\end{center}
\caption{Requirements for creating open boundary files from a global configuration, 
appropriate for the subdomain of indices $ib:ie$, $jb:je$. ``Index'' designates the 
$i$ or $j$ index along which the $u$ of $v$ boundary point is situated in the global 
configuration, starting and ending with the $j$ or $i$ indices indicated. 
For example, to generate file obcnorth\_V.nc, use the command ncks 
$-F$ $-d\;y,je-2$  $-d\;x,ib+1,ie-1$ } 
\end{table}

It is assumed that the open boundary files contain the variables for the period of 
the model integration. If the boundary files contain one time frame, the boundary 
data is held fixed in time. If the files contain 12 values, it is assumed that the input 
is a climatology for a repeated annual cycle (corresponding to the case \np{ln\_obc\_clim} 
= .True.). The case of an arbitrary number of time frames is not yet implemented 
correctly; the user is required to write his own code in the module \mdl{obc\_dta} 
to deal with this situation. 

\subsection{Radiation algorithm}
\label{OBC_rad}

The art of open boundary management consists in applying a constraint strong 
enough that the inner domain "feels" the rest of the ocean, but weak enough
that perturbations are allowed to leave the domain with minimum false reflections 
of energy. The constraints are specified separately at each boundary as time 
scales for ''inflow'' and ''outflow'' as defined below. The time scales are set (in days) 
by namelist parameters such as \np{rdpein}, \np{rdpeob} for the eastern open 
boundary for example. When both time scales are zero for a given boundary 
($e.g.$ for the western boundary, \jp{lp\_obc\_west}=.True., \np{rdpwob}=0 and 
\np{rdpwin}=0) this means that the boundary in question is a ''fixed '' boundary 
where the solution is set exactly by the boundary data. This is not recommended, 
except in combination with increased viscosity in a ''sponge'' layer next to the 
boundary in order to avoid spurious reflections.  


The radiation\/relaxation \gmcomment{the / doesnt seem to appear in the output} 
algorithm is applied when either relaxation time (for ''inflow'' or ''outflow'') is 
non-zero. It has been developed and tested in the SPEM model and its 
successor ROMS \citep{Barnier1996, Marchesiello2001}, which is an 
$s$-coordinate model on an Arakawa C-grid. Although the algorithm has 
been numerically successful in the CLIPPER Atlantic models, the physics 
do not work as expected \citep{Treguier2001}. Users are invited to consider 
open boundary conditions (OBC hereafter) with some scepticism 
\citep{Durran2001, Blayo2005}.

The first part of the algorithm calculates a phase velocity to determine 
whether perturbations tend to propagate toward, or away from, the 
boundary. Let us consider a model variable $\phi$. 
The phase velocities ($C_{\phi x}$,$C_{\phi y}$) for the variable $\phi$, 
in the directions normal and tangential to the boundary are
\begin{equation} \label{Eq_obc_cphi}
C_{\phi x} = \frac{ -\phi_{t} }{ ( \phi_{x}^{2} + \phi_{y}^{2}) } \phi_{x} 
\;\;\;\;\; \;\;\; 
C_{\phi y} = \frac{ -\phi_{t} }{ ( \phi_{x}^{2} + \phi_{y}^{2}) } \phi_{y}. 
\end{equation}
Following \citet{Treguier2001} and \citet{Marchesiello2001} we retain only 
the normal component of the velocity, $C_{\phi x}$, setting $C_{\phi y} =0$ 
(but unlike the original Orlanski radiation algorithm we retain $\phi_{y}$ in 
the expression for $C_{\phi x}$).  

The discrete form of (\ref{Eq_obc_cphi}), described by \citet{Barnier1998},
takes into account the two rows of grid points situated inside the domain 
next to the boundary, and the three previous time steps ($n$, $n-1$,
and $n-2$). The same equation can then be discretized at the boundary at
time steps $n-1$, $n$ and $n+1$ \gmcomment{since the original was three time-level} 
in order to extrapolate for the new boundary value $\phi^{n+1}$. 

In the open boundary algorithm as implemented in NEMO v2.3, the new boundary 
values are updated differently depending on the sign of $C_{\phi x}$. Let us take 
an eastern boundary as an example. The solution for variable $\phi$ at the 
boundary is given by a generalized wave equation with phase velocity $C_{\phi}$, 
with the addition of a relaxation term, as:
\begin{eqnarray}
\phi_{t} &  =  & -C_{\phi x} \phi_{x} + \frac{1}{\tau_{o}} (\phi_{c}-\phi) 
                        \;\;\; \;\;\; \;\;\; (C_{\phi x} > 0), \label{Eq_obc_rado} \\
\phi_{t} &  =  & \frac{1}{\tau_{i}} (\phi_{c}-\phi) 
\;\;\; \;\;\; \;\;\;\;\;\; (C_{\phi x} < 0), \label{Eq_obc_radi}
\end{eqnarray}
where $\phi_{c}$ is the estimate of $\phi$ at the boundary, provided as boundary 
data. Note that in (\ref{Eq_obc_rado}), $C_{\phi x}$ is bounded by the ratio 
$\delta x/\delta t$ for stability reasons. When $C_{\phi x}$ is eastward (outward 
propagation), the radiation condition (\ref{Eq_obc_rado}) is used. 
When  $C_{\phi x}$ is westward (inward propagation), (\ref{Eq_obc_radi}) is 
used with a strong relaxation to climatology (usually $\tau_{i}=\np{rdpein}=$1~day).
Equation (\ref{Eq_obc_radi}) is solved with a Euler time-stepping scheme. As a 
consequence, setting $\tau_{i}$ smaller than, or equal to the time step is equivalent 
to a fixed boundary condition. A time scale of one day is usually a good compromise 
which guarantees that the inflow conditions remain close to climatology while ensuring 
numerical stability. 

In  the case of a western boundary located in the Eastern Atlantic, \citet{Penduff2000} 
have been able to implement the radiation algorithm without any boundary data, 
using persistence from the previous time step instead. This solution has not worked 
in other cases \citep{Treguier2001}, so that the use of boundary data is recommended. 
Even in the outflow condition (\ref{Eq_obc_rado}), we have found it desirable to 
maintain a weak relaxation to climatology. The time step is usually chosen so as to 
be larger than typical turbulent scales (of order 1000~days \gmcomment{or maybe seconds?}).

The radiation condition is applied to the model variables: temperature, salinity, 
tangential and normal velocities. For normal and tangential velocities, $u$ and $v$, 
radiation is applied with phase velocities calculated from $u$ and $v$ respectively.  
For the radiation of tracers, we use the phase velocity calculated from the tangential 
velocity in order to avoid calculating too many independent radiation velocities and 
because tangential velocities and tracers have the same position along the boundary 
on a C-grid.  

\subsection{Domain decomposition (\key{mpp\_mpi})}
\label{OBC_mpp}
When \key{mpp\_mpi} is active in the code, the computational domain is divided 
into rectangles that are attributed each to a different processor. The open boundary 
code is ``mpp-compatible'' up to a certain point. The radiation algorithm will not 
work if there is an mpp subdomain boundary parallel to the open boundary at the 
index of the boundary, or the grid point after (outside), or three grid points before 
(inside). On the other hand, there is no problem if an mpp subdomain boundary 
cuts the open boundary perpendicularly. These geometrical limitations must be 
checked for by the user (there is no safeguard in the code).  
The general principle for the open boundary mpp code is that loops over the open 
boundaries {not sure what this means} are performed on local indices (nie0, 
nie1, nje0, nje1 for an eastern boundary for instance) that are initialized in module 
\mdl{obc\_ini}. Those indices have relevant values on the processors that contain 
a segment of an open boundary. For processors that do not include an open 
boundary segment, the indices are such that the calculations within the loops are 
not performed.
\gmcomment{I dont understand most of the last few sentences}
 
Arrays of climatological data that are read from files are seen by all processors 
and have the same dimensions for all (for instance, for the eastern boundary, 
uedta(jpjglo,jpk,2)). On the other hand, the arrays for the calculation of radiation 
are local to each processor (uebnd(jpj,jpk,3,3) for instance).  This allowed the 
CLIPPER model for example, to save on memory where the eastern boundary 
crossed 8 processors so that \jp{jpj} was much smaller than (\jp{jpjef}-\jp{jpjed}+1). 

\subsection{Volume conservation}
\label{OBC_vol}

It is necessary to control the volume inside a domain when using open boundaries. 
With fixed boundaries, it is enough to ensure that the total inflow/outflow has 
reasonable values (either zero or a value compatible with an observed volume 
balance). When using radiative boundary conditions it is necessary to have a 
volume constraint because each open boundary works independently from the 
others. The methodology used to control this volume is identical to the one 
coded in the ROMS model \citep{Marchesiello2001}.


%---------------------------------------- EXTRAS
\colorbox{yellow}{Explain obc\_vol{\ldots}}

\colorbox{yellow}{OBC algorithm for update, OBC restart, list of routines where obc key appears{\ldots}}

\colorbox{yellow}{OBC rigid lid? {\ldots}}




% ====================================================================
% Flow Relaxation Scheme 
% ====================================================================
\section{Flow Relaxation Scheme (???)}
\label{LBC_bdy}

%gm% to be updated by Met Office
