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Timestamp:
2008-01-03T12:10:08+01:00 (13 years ago)
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smasson
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doc update, see ticket:1

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  • trunk/DOC/BETA/Chapters/Chap_LBC.tex

    r707 r781  
    128128\label{LBC_north_fold} 
    129129 
    130 The north fold boundary condition have been introduced in order to handle the north boundary of an three-polar ORCA grid. Such a grid has two poles in the northern hemisphere. \colorbox{yellow}{to be completed...} 
     130The 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...} 
    131131 
    132132%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
     
    137137%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    138138 
    139 % ================================================================ 
    140 % Exchanged with neighbouring processors  
    141 % ================================================================ 
    142 \section{Exchanged with neighbouring processors (\mdl{lbclnk}, \mdl{lib\_mpp})} 
     139% ==================================================================== 
     140% Sub-Domain:Exchange with neighbouring processors  
     141% ==================================================================== 
     142\section{Exchange with neighbouring processors (\mdl{lbclnk}, \mdl{lib\_mpp})} 
    143143\label{LBC_mpp} 
    144144 
     
    184184 
    185185 
    186    The OPA model computes equation terms with the help of mask arrays ( 0 onto land points and 1 onto sea points). It is easily readable and very efficient in the context of the vectorial architecture. But in the case of scalar processor, computations over the land regions becomes more expensive in term of CPU time. It is all the more 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 allows to search in the mpp domain decomposition strategy if a splitting can be found with a maximum number of only land points processors which could be eliminated (mppini2 program). This optimisation is made with the knowledge of the specific bathymetry in a first time and after, the OPA model, in its initialization part, take account only processors with a sea region. For that, one must indicate in the parameter file the initial cutting along i- and j-axes with jpni and jpnjand the ocean processor number jpnij < jpni x jpnj. Each processor name and neighbour parameters (nbound, nono, noea,...) are modified by an algorithm in the inimpp2.F subroutine. 
    187186 
    188187   The OPA model computes equation terms with the help of mask arrays (0 onto land points and 1 onto sea points). It is easily readable and very efficient in the context of the vectorial architecture. But in the case of scalar processor, computations over the land regions becomes more expensive in term of CPU time. It is all the more so 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 allows to search in the mpp domain decomposition strategy if a splitting can be found with a maximum number of only land points processors which could be eliminated: the mpp\_optimiz tools (available from the DRAKKAR web site). This optimisation is made with the knowledge of the specific bathymetry. The user 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 set each processor name and neighbour parameters (nbound, nono, noea,...) so that the land processors are not taken into account.  
     
    205204\section{Open Boundary Conditions (\key{obc})} 
    206205\label{LBC_obc} 
    207  
    208 %gm%  to be update from documentation already written by J.M. Molines + Mercator input 
    209  
    210 % ================================================================ 
     206%-----------------------------------------nam_obc  ------------------------------------------- 
     207%-    nobc_dta    =    0     !  = 0 the obc data are equal to the initial state 
     208%-                           !  = 1 the obc data are read in 'obc   .dta' files 
     209%-    rdpein      =    1.    !  ??? 
     210%-    rdpwin      =    1.    !  ??? 
     211%-    rdpnin      =   30.    !  ??? 
     212%-    rdpsin      =    1.    !  ??? 
     213%-    rdpeob      = 1500.    !  time relaxation (days) for the east  open boundary 
     214%-    rdpwob      =   15.    !    "        "           for the west  open boundary 
     215%-    rdpnob      =  150.    !    "        "           for the north open boundary 
     216%-    rdpsob      =   15.    !    "        "           for the south open boundary 
     217%-    zbsic1      =  140.e+6 !  barotropic stream function on first  isolated coastline 
     218%-    zbsic2      =    1.e+6 !    "                   "    on second isolated coastline 
     219%-    zbsic3      =    0.    !    "                   "    on thrid  isolated coastline 
     220%-    ln_obc_clim = .true.   !  climatological obc data files (default T) 
     221%-    ln_vol_cst  = .true.   !  total volume conserved 
     222\namdisplay{namobc}  
     223 
     224It 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 oceans enter the model and should support inflow and outflow conditions.  
     225 
     226The open boundary package OBC is the first open boundary option developed in NEMO (originally in OPA8.2). It allows the user to  
     227\begin{itemize} 
     228\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, and other necessary modifications; 
     229\item impose values of tracers and velocities at that boundary (values which may be taken from a climatology): this is the``fixed OBC'' option.  
     230\item calculate boundary values by a sophisticated algorithm combining radiation and relaxation (``radiative OBC'' option) 
     231\end{itemize} 
     232 
     233The 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. 
     234 
     235%---------------------------------------------- 
     236\subsection{Boundary geometry} 
     237\label{OBC_geom} 
     238% 
     239First 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; those names refer to the grid geometry (not to the direction of the geographical ``west'', ``east'', etc). 
     240 
     241The 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}). 
     242 
     243 
     244%--------------------------------------------------TABLE-------------------------------------------------- 
     245 
     246\begin{table}[htbp]  \label{Tab_obc_param} 
     247\begin{center} 
     248\begin{tabular}{|l|c|c|c|} 
     249\hline 
     250Boundary and  & Constant index  & Starting index (d\'{e}but) & Ending index (fin) \\ 
     251Logical flag  &                 &                            &                     \\ 
     252\hline 
     253West          & \jp{jpiwob} $>= 2$         &  \jp{jpjwd}$>= 2$          &  \jp{jpjwf}<= \jp{jpjglo}-1 \\ 
     254lp\_obc\_west & $i$-index of a $u$ point   & $j$ of a $T$ point   &$j$ of a $T$ point \\ 
     255\hline 
     256East            & \jp{jpieob}$<=$\jp{jpiglo}-2&\jp{jpjed} $>= 2$         & \jp{jpjef}$<=$ \jp{jpjglo}-1 \\ 
     257 lp\_obc\_east  & $i$-index of a $u$ point    & $j$ of a $T$ point & $j$ of a $T$ point \\ 
     258\hline 
     259South           & \jp{jpjsob} $>= 2$         & \jp{jpisd} $>= 2$          & \jp{jpisf}$<=$\jp{jpiglo}-1 \\ 
     260lp\_obc\_south  & $j$-index of a $v$ point   & $i$ of a $T$ point   & $i$ of a $T$ point \\ 
     261\hline 
     262North           & \jp{jpjnob} $<=$ \jp{jpjglo}-2& \jp{jpind} $>= 2$        & \jp{jpinf}$<=$\jp{jpiglo}-1 \\ 
     263lp\_obc\_north  & $j$-index of a $v$ point      & $i$  of a $T$ point & $i$ of a $T$ point \\ 
     264\hline 
     265\end{tabular} 
     266\end{center} 
     267\caption{Names of different indices concerning the open boundaries. In the case of a completely open ocean domain with four ocean boundaries, the parameters take exactly the values indicated.} 
     268\end{table} 
     269 
     270The 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 or \jp{jpiglo}, \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, 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.  
     271 
     272 
     273The starting and ending indices are to be thought 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 the example for 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. 
     274 
     275%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
     276\begin{figure}[!t] \label{Fig_obc_north}  \begin{center} 
     277\includegraphics[width=0.70\textwidth]{./Figures/Fig_obc_north.pdf} 
     278\caption {Localization of the North open boundary points.} 
     279\end{center}   
     280\end{figure} 
     281%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
     282 
     283Although not absolutely 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 mdoel variables next to the boundary, is calculated in a consistent way. On Fig.\ref{Fig_obc_north} we indicate by a $=$ symbol, the points which should have the same depth. It means that in the 4 points near the boundary, the bathymetry is 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). 
     284 
     285%---------------------------------------------- 
     286\subsection{Boundary data} 
     287\label{OBC_data} 
     288 
     289It is necessary to provide information at the boundaries. The simple case happens 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 of 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.  
     290Open 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 on the horizontal and \jp{jpk} on the vertical.  
     291 
     292When 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 are available, it is possible to provide information about all the variables (including the tangential velocity) so that the specification of four variables at each boundaries will soon become standard. Regarding the sea surface height, one must distinguish the filtered free surface case from the time-splitting or explicit treatment of the free surface. 
     293 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}.  
     294No information other than the total velocity needs to be provided at the open boundaries in that case. In the other two cases (time stplitting or explicit free surface), the user must provides 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 used but remains in NEMO V2.3. 
     295 
     296One 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 command on the global files: ncks -F $-d\;x,ib,ie$ $-d\;y,jb,je$.  
     297The open boundary files can be constructed using ncks commands, following table~\ref{Tab_obc_ind}.  
     298 
     299%--------------------------------------------------TABLE-------------------------------------------------- 
     300 
     301\begin{table}[htbp]  \label{Tab_obc_ind} 
     302\begin{center} 
     303\begin{tabular}{|l|c|c|c|c|c|} 
     304\hline 
     305OBC  & Variable   & file name      & Index  & Start  & end  \\ 
     306West &  T,S       &   obcwest\_TS.nc &  $ib$+1     &   $jb$+1 &  $je-1$  \\ 
     307     &    U       &   obcwest\_U.nc  &  $ib$+1     &   $jb$+1 &  $je-1$  \\  
     308     &    V       &   obcwest\_V.nc  &  $ib$+1     &   $jb$+1 &  $je-1$  \\        
     309\hline 
     310East &  T,S       &   obceast\_TS.nc &  $ie$-1     &   $jb$+1 &  $je-1$  \\ 
     311     &    U       &   obceast\_U.nc  &  $ie$-2     &   $jb$+1 &  $je-1$  \\  
     312     &    V       &   obceast\_V.nc  &  $ie$-1     &   $jb$+1 &  $je-1$  \\        
     313\hline          
     314South &  T,S      &   obcsouth\_TS.nc &  $jb$+1     &  $ib$+1 &  $ie-1$  \\ 
     315      &    U      &   obcsouth\_U.nc  &  $jb$+1     &  $ib$+1 &  $ie-1$  \\  
     316      &    V      &   obcsouth\_V.nc  &  $jb$+1     &  $ib$+1 &  $ie-1$  \\     
     317\hline 
     318North &  T,S      &   obcnorth\_TS.nc &  $je$-1     &  $ib$+1 &  $ie-1$  \\ 
     319      &    U      &   obcnorth\_U.nc  &  $je$-1     &  $ib$+1 &  $ie-1$  \\  
     320      &    V      &   obcnorth\_V.nc  &  $je$-2     &  $ib$+1 &  $ie-1$  \\   
     321\hline 
     322\end{tabular} 
     323\end{center} 
     324\caption{Indications 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 $-F$ $-d\;y,je-2$  $-d\;x,ib+1,ie-1$ }  
     325\end{table} 
     326 
     327It 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} = .T.). The case of an arbitrary number of time frames is not yet implemented correctly; the user is supposed to write his own code in the module \mdl{obc\_dta} to deal with this situation.  
     328 
     329\subsection{Radiation algorithm} 
     330\label{OBC_rad} 
     331 
     332The art of open boundary management consists in applying a constraint strong enough so that the inner domain "feels" the rest of the ocean, but not too strong  
     333in order to allow perturbations to leave the domain with minimum false reflexions of energy. The cosntraint to the specified boundary data is set by time scales  
     334specified separately for each boundary and for ``inflow'' and `outflow'' conditions as defined below. These 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, for example  
     335\jp{lp\_obc\_west}=.T., \np{rdpwob}=0, \np{rdpwin}=0, this means that the boundary in question (western boundary here) is a ``fixed '' boundary where the solution is set exactly by the boundary data. This is not recommanded, excepted in compination with increased viscosity in a ``sponge'' layer next to the boundary in order to avois spurious reflexions.   
     336 
     337 
     338The radiation\/relaxation algorithm is applied when either relxation time (for ``inflow'' or `outflow'') is nonzero. It has been developed and tested in the SPEM model and its successor ROMS \citep{Barnier1996, Marchesiello2001}, a $s$-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 skepticism \citep{Durran2001, Blayo2005}. 
     339 
     340The first part of the algorithm consists in calculating a phase 
     341velocity to determine whether perturbations tend to propagate toward, 
     342or away from, the boundary.  
     343Let us consider a model variable $\phi$.  
     344The phase velocity ($C_{\phi x}$,$C_{\phi y}$) for the variable  
     345$\phi$, in the directions normal and tangential to 
     346the boundary is 
     347\begin{equation} \label{Eq_obc_cphi} 
     348C_{\phi x} = \frac{ -\phi_{t} }{ ( \phi_{x}^{2} + \phi_{y}^{2}) } \phi_{x}  
     349\;\;\;\;\; \;\;\;  
     350C_{\phi y} = \frac{ -\phi_{t} }{ ( \phi_{x}^{2} + \phi_{y}^{2}) } \phi_{y}.  
     351\end{equation} 
     352Following \citet{Treguier2001} and \citet{Marchesiello2001} retain only the normal 
     353projection of the total velocity, $C_{\phi x}$, setting 
     354$C_{\phi y} =0$ (but unlike the original Orlanski radiation algorithm we retain $\phi_{y}$ in the expression 
     355for $C_{\phi x}$).   
     356 
     357The discrete form of (\ref{Eq_obc_cphi}), described by \citet{Barnier1998}, 
     358takes into account the two rows of grid points situated inside the domain  
     359next to the boundary, and the three previous time steps ($n$, $n-1$, 
     360and $n-2$).  
     361The same equation can then be discretized at the boundary at 
     362time steps $n$ and $n+1$ in order to extrapolate the new boundary 
     363value $\phi^{n+1}$.  
     364 
     365In the present open boundary algorithm, the new boundary values are  
     366updated differently according to the sign of $C_{\phi x}$. 
     367Let us take an East boundary as an example. The solution for variable $\phi$ at the boundary is given from a generalized wave equation  
     368with phase velocity $C_{\phi}$, with the addition of  a relaxation term: 
     369\begin{eqnarray} 
     370\phi_{t} &  =  & -C_{\phi x} \phi_{x} + \frac{1}{\tau_{o}} (\phi_{c}-\phi)  
     371                        \;\;\; \;\;\; \;\;\; (C_{\phi x} > 0), \label{Eq_obc_rado} \\ 
     372\phi_{t} &  =  & \frac{1}{\tau_{i}} (\phi_{c}-\phi)  
     373\;\;\; \;\;\; \;\;\;\;\;\; (C_{\phi x} < 0), \label{Eq_obc_radi} 
     374\end{eqnarray} 
     375where $\phi_{c}$ is the estimate of $\phi$ at the boundary, provided as boundary data.  
     376Note that in (\ref{Eq_obc_rado}), $C_{\phi x}$ 
     377is bounded by the ratio $\delta x/\delta t$ for stability reasons.  
     378When $C_{\phi x}$ is eastward (outward propagation),  
     379the radiation condition (\ref{Eq_obc_rado}) is used.  
     380When  $C_{\phi x}$ is westward (inward propagation), (\ref{Eq_obc_radi}) is used with 
     381a strong relaxation to climatology (usually $\tau_{i}=\np{rdpein}=$1~day). 
     382The time derivative in (\ref{Eq_obc_radi}) is calculated with a Euler time-stepping 
     383scheme. It results from this choice that setting 
     384$\tau_{i}$ smaller than, or equal to the time step is in fact equivalent to a fixed boundary condition; 
     385a time scale of one day is usually a good compromise which guarantees that the inflow conditions remain close to  
     386climatology while ensuring numerical stability.  
     387 
     388In  the case of a west boundary located in the Eastern Atlantic, \citet{Penduff2000} have been able to implement the radiation algorithm  
     389without any boundary data, using persistence from the previous time step instead. This solution did not work in other cases \citep{Treguier2001} so that the use of boundary data is recommended.   
     390Even in the outflow condition (\ref{Eq_obc_rado}), we have found desireable to maintain a weak relaxation to climatology.  
     391The time step is usually chosen so as to be larger than typical turbulent scales (of order 1000~days). 
     392 
     393The radiation condition is applied to the different model variables: temperature, salinity, tangential and normal velocities.  
     394For normal and tangential velocities $u$ and $v$ radiation is applied with phase velocities calculated from $u$ and $v$ respectively.  
     395For the radiation of tracers, we use the phase velocity calculated from the tangential velocity, to avoid calculating too many independent  
     396radiation velocities and because tangential velocities and tracers have the same position along the boundary on a C grid.   
     397 
     398\subsection{Domain decomposition (\key{mpp\_mpi})} 
     399\label{OBC_mpp} 
     400When \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 a 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 a mpp subdomain boundary cuts the open boundary perpendicularly. Those geometry constraints must be checked by the user (there is no safeguard in the code).   
     401The general principle for the open boundary mpp code is that loops over the open boundaries are performed on local indices (nie0, nie1, nje0, nje1 for the 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. 
     402  
     403Arrays of climatological data that are read in 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 to spare memory in the CLIPPER model, where the eastern boundary crossed 8 processors so that \jp{jpj} was must smaller than (\jp{jpjef}-\jp{jpjed}+1).  
     404 
     405\subsection{Volume conservation} 
     406\label{OBC_vol} 
     407 
     408It is necessary to control the volume inside a domain with 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}. 
     409 
     410 
     411%---------------------------------------- EXTRAS 
     412\colorbox{yellow}{Explain obc\_vol{\ldots}} 
     413 
     414\colorbox{yellow}{OBC algorithm for update, OBC restart, list of routines where obc key appears{\ldots}} 
     415 
     416\colorbox{yellow}{OBC rigid lid? {\ldots}} 
     417 
     418 
     419 
     420 
     421% ==================================================================== 
    211422% Flow Relaxation Scheme  
    212 % ================================================================ 
     423% ==================================================================== 
    213424\section{Flow Relaxation Scheme (???)} 
    214425\label{LBC_bdy} 
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