/[MITgcm]/manual/s_phys_pkgs/text/obcs.tex

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-revision 1.5 by jmc,
Mon Aug 30 23:09:21 2010 UTC
+revision 1.17 by mlosch,
Wed Apr 27 09:45:04 2016 UTC
 Line 11 
 Alistair Adcroft, Patrick Heimbach, Sama
  \subsubsection{Introduction
  \label{sec:pkg:obcs:intro}}
+ The OBCS-package is fundamental to regional ocean modelling with the
+ MITgcm, but there are so many details to be considered in
+ regional ocean modelling that this package cannot accomodate all
+ imaginable and possible options. Therefore, for a regional simulation
+ with very particular details, it is recommended to familiarize oneself
+ not only with the compile- and runtime-options of this package, but
+ also with the code itself. In many cases it will be necessary to adapt
+ the obcs-code (in particular \code{S/R OBCS\_CALC}) to the application
+ in question; in these cases the obcs-package (together with the
+ rbcs-package, section \ref{sec:pkg:rbcs}) is a very
+ useful infrastructure for implementing special regional models.
  %----------------------------------------------------------------------
-Line 24 
 at compile time
+Line 34 
 at compile time
  \begin{itemize}
  %
  \item
- using the \texttt{packages.conf} file by adding \texttt{obcs} to it,
+ using the \code{packages.conf} file by adding \code{obcs} to it,
  %
  \item
- or using \texttt{genmake2} adding
+ or using \code{genmake2} adding
- \texttt{-enable=obcs} or \texttt{-disable=obcs} switches
+ \code{-enable=obcs} or \code{-disable=obcs} switches
  %
  \item
  \textit{Required packages and CPP options:} \\
-Line 37 
 To alternatives are available for prescr
+Line 47 
 To alternatives are available for prescr
  which differ in the way how OB's are treated in time:
  A simple time-management (e.g. constant in time, or cyclic with
  fixed fequency) is provided through
- S/R \texttt{obcs\_external\_fields\_load}.
+ S/R \code{obcs\_external\_fields\_load}.
  More sophisticated ``real-time'' (i.e. calendar time) management is
- available through \texttt{obcs\_prescribe\_read}.
+ available through \code{obcs\_prescribe\_read}.
  The latter case requires
- packages \texttt{cal} and \texttt{exf} to be enabled.
+ packages \code{cal} and \code{exf} to be enabled.
  %
  \end{itemize}
  (see also Section \ref{sec:buildingCode}).
  Parts of the OBCS code can be enabled or disabled at compile time
  via CPP preprocessor flags. These options are set in
- \texttt{OBCS\_OPTIONS.h}. Table \ref{tab:pkg:obcs:cpp} summarizes them.
+ \code{OBCS\_OPTIONS.h}. Table \ref{tab:pkg:obcs:cpp} summarizes them.
  \begin{table}[!ht]
  \centering
-Line 58 
 via CPP preprocessor flags. These option
+Line 68 
 via CPP preprocessor flags. These option
        \hline
        \textbf{CPP option}  &  \textbf{Description}  \\
        \hline \hline
-         \texttt{ALLOW\_OBCS\_NORTH} &
+         \code{ALLOW\_OBCS\_NORTH} &
            enable Northern OB \\
-         \texttt{ALLOW\_OBCS\_SOUTH} &
+         \code{ALLOW\_OBCS\_SOUTH} &
            enable Southern OB \\
-         \texttt{ALLOW\_OBCS\_EAST} &
+         \code{ALLOW\_OBCS\_EAST} &
            enable Eastern OB \\
-         \texttt{ALLOW\_OBCS\_WEST} &
+         \code{ALLOW\_OBCS\_WEST} &
            enable Western OB \\
        \hline
-         \texttt{ALLOW\_OBCS\_PRESCRIBE} &
+         \code{ALLOW\_OBCS\_PRESCRIBE} &
            enable code for prescribing OB's \\
-         \texttt{ALLOW\_OBCS\_SPONGE} &
+         \code{ALLOW\_OBCS\_SPONGE} &
            enable sponge layer code \\
-         \texttt{ALLOW\_OBCS\_BALANCE} &
+         \code{ALLOW\_OBCS\_BALANCE} &
            enable code for balancing transports through OB's \\
-         \texttt{ALLOW\_ORLANSKI} &
+         \code{ALLOW\_ORLANSKI} &
            enable Orlanski radiation conditions at OB's \\
+         \code{ALLOW\_OBCS\_STEVENS} &
+           enable Stevens (1990) boundary conditions at OB's \\
+         & (currently only implemented for eastern and western \\
+         &  boundaries and NOT for ptracers) \\
        \hline
      \end{tabular}
    }
-Line 88 
 via CPP preprocessor flags. These option
+Line 102 
 via CPP preprocessor flags. These option
  \label{sec:pkg:obcs:runtime}}
  Run-time parameters are set in files
- \texttt{data.pkg}, \texttt{data.obcs}, and \texttt{data.exf}
+ \code{data.pkg}, \code{data.obcs}, and \code{data.exf}
  if ``real-time'' prescription is requested
- (i.e. package \texttt{exf} enabled).
+ (i.e. package \code{exf} enabled).
- These parameter files are read in S/R
+ vThese parameter files are read in S/R
- \texttt{packages\_readparms.F}, \texttt{obcs\_readparms.F}, and
+ \code{packages\_readparms.F}, \code{obcs\_readparms.F}, and
- \texttt{exf\_readparms.F}, respectively.
+ \code{exf\_readparms.F}, respectively.
  Run-time parameters may be broken into 3 categories:
  (i) switching on/off the package at runtime,
  (ii) OBCS package flags and parameters,
- (iii) additional timing flags in \texttt{data.exf}, if selected.
+ (iii) additional timing flags in \code{data.exf}, if selected.
  \paragraph{Enabling the package}
  ~ \\
  %
  The OBCS package is switched on at runtime by setting
- \texttt{useOBCS = .TRUE.} in \texttt{data.pkg}.
+ \code{useOBCS = .TRUE.} in \code{data.pkg}.
  \paragraph{Package flags and parameters}
  ~ \\
  %
  Table \ref{tab:pkg:obcs:runtime_flags} summarizes the
- runtime flags that are set in \texttt{data.obcs}, and
+ runtime flags that are set in \code{data.obcs}, and
  their default values.
  \begin{table}[!ht]
-Line 119 
 their default values.
+Line 133 
 their default values.
        \hline
        \textbf{Flag/parameter} & \textbf{default} &  \textbf{Description}  \\
        \hline \hline
-          \multicolumn{3}{|c|}{\textit{basic flags \& parameters} } \\
+          \multicolumn{3}{|c|}{\textit{basic flags \& parameters} (OBCS\_PARM01) } \\
           \hline
          OB\_Jnorth & 0 &
             Nx-vector of J-indices (w.r.t. Ny) of Northern OB
-Line 133 
 their default values.
+Line 147 
 their default values.
          OB\_Iwest & 0 &
             Ny-vector of I-indices (w.r.t. Nx) of Western OB
             at each J-position (w.r.t. Ny) \\
-         useOBCSprescribe & \texttt{.FALSE.} &
+         useOBCSprescribe & \code{.FALSE.} &
             ~ \\
-         useOBCSsponge & \texttt{.FALSE.} &
+         useOBCSsponge & \code{.FALSE.} &
             ~ \\
-         useOBCSbalance & \texttt{.FALSE.} &
+         useOBCSbalance & \code{.FALSE.} &
             ~ \\
+            OBCS\_balanceFacN/S/E/W & 1 & factor(s) determining the details
+            of the balaning code \\
+         useOrlanskiNorth/South/EastWest & \code{.FALSE.} &
+            turn on Orlanski boundary conditions for individual boundary\\
+         useStevensNorth/South/EastWest & \code{.FALSE.} &
+            turn on Stevens boundary conditions for individual boundary\\
          OB\textbf{X}\textbf{y}File & ~ &
             file name of OB field \\
          ~ & ~ &
-Line 146 
 their default values.
+Line 166 
 their default values.
                         \textbf{E}(ast), \textbf{W}(est) \\
          ~ & ~ &
             \textbf{y}: \textbf{t}(emperature), \textbf{s}(salinity),
-            \textbf{u}(-velocity), \textbf{v}(-velocity) \\
+            \textbf{u}(-velocity), \textbf{v}(-velocity), \\
+         ~ & ~ &
+            \textbf{w}(-velocity), \textbf{eta}(sea surface height)\\
+         ~ & ~ &
+            \textbf{a}(sea ice area), \textbf{h}(sea ice thickness),
+            \textbf{sn}(snow thickness), \textbf{sl}(sea ice salinity)\\
        \hline
-       \multicolumn{3}{|c|}{\textit{Orlanski parameters} } \\
+       \multicolumn{3}{|c|}{\textit{Orlanski parameters} (OBCS\_PARM02) } \\
        \hline
          cvelTimeScale & 2000 sec &
             averaging period for phase speed \\
-Line 156 
 their default values.
+Line 181 
 their default values.
             maximum allowable phase speed-CFL for AB-II \\
          CFIX & 0.8 m/s &
             fixed boundary phase speed \\
-         useFixedCEast & .FALSE. &
+         useFixedCEast & \code{.FALSE.} &
             ~ \\
-         useFixedCWest & .FALSE. &
+         useFixedCWest & \code{.FALSE.} &
             ~ \\
        \hline
-       \multicolumn{3}{|c|}{\textit{Sponge-layer parameters} } \\
+       \multicolumn{3}{|c|}{\textit{Sponge-layer parameters} (OBCS\_PARM03)} \\
        \hline
          spongeThickness & 0 &
             sponge layer thickness (in \# grid points) \\
-Line 177 
 their default values.
+Line 202 
 their default values.
          Vrelaxobcsbound & 0 sec &
             relaxation time scale at the
             outermost sponge layer point of a zonal OB \\
+       \hline
+       \multicolumn{3}{|c|}{\textit{Stevens parameters} (OBCS\_PARM04) } \\
+       \hline
+         T/SrelaxStevens & 0~sec & relaxation time scale for
+            temperature/salinity \\
+         useStevensPhaseVel & \code{.TRUE.} & \\
+         useStevensAdvection & \code{.TRUE.} & \\
           \hline
        \hline
      \end{tabular}
-Line 196 
 There are four open boundaries (OBs), a
+Line 228 
 There are four open boundaries (OBs), a
  Northern, Southern, Eastern, and Western.
  All OB locations are specified by their absolute
  meridional (Northern/Southern) or zonal (Eastern/Western) indices.
- Thus, for each zonal position $i=1,\ldots,Nx$ a meridional index
+ Thus, for each zonal position $i=1,\ldots,N_x$ a meridional index
  $j$ specifies the Northern/Southern OB position,
- and for each meridional position $j=1,\ldots,Ny$, a zonal index
+ and for each meridional position $j=1,\ldots,N_y$, a zonal index
  $i$ specifies the Eastern/Western OB position.
- For Northern/Southern OB this defines an $Nx$-dimensional
+ For Northern/Southern OB this defines an $N_x$-dimensional
- ``row'' array $\tt OB\_Jnorth(Ny)$ / $\tt OB\_Jsouth(Ny)$,
+ ``row'' array $\tt OB\_Jnorth(Nx)$ / $\tt OB\_Jsouth(Nx)$,
- and an $Ny$-dimenisonal
+ and an $N_y$-dimenisonal
- ``column'' array $\tt OB\_Ieast(Nx)$ / $\tt OB\_Iwest(Nx)$
+ ``column'' array $\tt OB\_Ieast(Ny)$ / $\tt OB\_Iwest(Ny)$.
  Positions determined in this way allows Northern/Southern
  OBs to be at variable $j$ (or $y$) positions, and Eastern/Western
  OBs at variable $i$ (or $x$) positions.
-Line 212 
 A zero (0) element in $\tt OB\_I\ldots$,
+Line 244 
 A zero (0) element in $\tt OB\_I\ldots$,
  means there is no corresponding OB in that column/row.
  For a Northern/Southern OB, the OB V point is to the South/North.
  For an Eastern/Western OB, the OB U point is to the West/East.
+ For example,
- \begin{verbatim}
+ \begin{tabbing}
-  For example
+   \code{OB\_Jnorth(3)=34} \=  means that:  \= \\
-      OB_Jnorth(3)=34  means that:
+   \> \code{T(3,34)} \> is a an OB point  \\
-           T( 3 ,34) is a an OB point
+   \> \code{U(3,34)} \> is a an OB point \\
-           U(3:4,34) is a an OB point
+   \> \code{V(3,34)} \> is a an OB point \\
-           V( 4 ,34) is a an OB point
+   \code{OB\_Jsouth(3)=1} \> means that: \\
-  while
+   \> \code{T(3,1)} \> is a an OB point \\
-      OB_Jsouth(3)=1  means that:
+   \> \code{U(3,1)} \> is a an OB point \\
-           T( 3 ,1) is a an OB point
+   \> \code{V(3,2)} \> is a an OB point \\
-           U(3:4,1) is a an OB point
+   \code{OB\_Ieast(10)=69} \>  means that:  \>  \\
-           V( 4 ,2) is a an OB point
+   \> \code{T(69,10)} \> is a an OB point \\
- \end{verbatim}
+   \> \code{U(69,10)} \> is a an OB point \\
+   \> \code{V(69,10)} \> is a an OB point \\
- For convenience, negative values for Jnorth/Ieast refer to
+   \code{OB\_Iwest(10)=1} \>  means that:  \>  \\
+   \> \code{T(1,10)} \> is a an OB point \\
+   \> \code{U(2,10)} \> is a an OB point \\
+   \> \code{V(1,10)} \> is a an OB point
+ \end{tabbing}
+ For convenience, negative values for \code{Jnorth}/\code{Ieast} refer to
  points relative to the Northern/Eastern edges of the model
  eg. $\tt OB\_Jnorth(3)=-1$  means that the point $\tt (3,Ny)$
  is a northern OB.
-Line 242 
 see obcs\_ini\_fixed.F}
+Line 279 
 see obcs\_ini\_fixed.F}
  \paragraph{OBCS\_READPARMS:} ~ \\
  Set OB positions through arrays
- {\tt OB\_Jnorth(Ny), OB\_Jsouth(Ny), OB\_Ieast(Nx), OB\_Iwest(Nx)},
+ {\tt OB\_Jnorth(Nx), OB\_Jsouth(Nx), OB\_Ieast(Ny), OB\_Iwest(Ny)},
  and runtime flags (see Table \ref{tab:pkg:obcs:runtime_flags}).
  \paragraph{OBCS\_CALC:} ~ \\
-Line 267 
 $T,S,U,V$ values determined via Orlanski
+Line 304 
 $T,S,U,V$ values determined via Orlanski
  \item
  prescribed time-constant or time-varying fields (see below).
  %
+ \item
+ use prescribed boundary fields to compute Stevens boundary conditions.
  \end{itemize}
+ \paragraph{ORLANSKI:} ~ \\
- \paragraph{ORLANSKI} ~ \\
- %
- Orlanski radiation conditions \citep{orl:76}
- \paragraph{OBCS\_PRESCRIBE\_READ} Setting OB fields and updates \\
  %
- ~
+ Orlanski radiation conditions \citep{orl:76}, examples can be found in
+ \code{verification/dome} and
+ \code{verification/tutorial\_plume\_on\_slope}
+ (\ref{sec:eg-gravityplume}).
+ \paragraph{OBCS\_PRESCRIBE\_READ:} ~ \\
+ %
+ When \code{useOBCSprescribe = .TRUE.} the model tries to read
+ temperature, salinity, u- and v-velocities from files specified in the
+ runtime parameters \code{OB[N/S/E/W][t/s/u/v]File}. These files are
+ the usual IEEE, big-endian files with dimensions of a section along an
+ open boundary:
+ \begin{itemize}
+ \item For North/South boundary files the dimensions are
+   $(N_x\times N_r\times\mbox{time levels})$, for East/West boundary
+   files the dimensions are $(N_y\times N_r\times\mbox{time levels})$.
+ \item If a non-linear free surface is used
+   (\ref{sec:nonlinear-freesurface}), additional files
+   \code{OB[N/S/E/W]etaFile} for the sea surface height $\eta$ with
+   dimension $(N_{x/y}\times\mbox{time levels})$ may be specified.
+ \item If non-hydrostatic dynamics are used
+   (\ref{sec:non-hydrostatic}), additional files
+   \code{OB[N/S/E/W]wFile} for the vertical velocity $w$ with
+   dimensions $(N_{x/y}\times N_r\times\mbox{time levels})$ can be
+   specified.
+ \item If \code{useSEAICE=.TRUE.} then additional files
+   \code{OB[N/S/E/W][a,h,sl,sn,uice,vice]} for sea ice area, thickness
+   (\code{HEFF}), seaice salinity, snow and ice velocities
+   $(N_{x/y}\times\mbox{time levels})$ can be specified.
+ \end{itemize}
+ As in \code{S/R external\_fields\_load} or the \code{exf}-package, the
+ code reads two time levels for each variable, e.g.\ \code{OBNu0} and
+ \code{OBNu1}, and interpolates linearly between these time levels to
+ obtain the value \code{OBNu} at the current model time (step). When the
+ \code{exf}-package is used, the time levels are controlled for each
+ boundary separately in the same way as the \code{exf}-fields in
+ \code{data.exf}, namelist \code{EXF\_NML\_OBCS}. The runtime flags
+ follow the above naming conventions, e.g. for the western boundary the
+ corresponding flags are \code{OBCWstartdate1/2} and
+ \code{OBCWperiod}. Sea-ice boundary values are controlled separately
+ with \code{siobWstartdate1/2} and \code{siobWperiod}.  When the
+ \code{exf}-package is not used, the time levels are controlled by the
+ runtime flags \code{externForcingPeriod} and \code{externForcingCycle}
+ in \code{data}, see \code{verification/exp4} for an example.
+ \paragraph{OBCS\_CALC\_STEVENS:} ~ \\
+ (THE IMPLEMENTATION OF THESE BOUNDARY CONDITIONS IS NOT
+ COMPLETE. PASSIVE TRACERS, SEA ICE AND NON-LINEAR FREE SURFACE ARE NOT
+ SUPPORTED PROPERLY.) \\
+ The boundary conditions following \citet{stevens:90} require the
+ vertically averaged normal velocity (originally specified as a stream
+ function along the open boundary) $\bar{u}_{ob}$ and the tracer fields
+ $\chi_{ob}$ (note: passive tracers are currently not implemented and
+ the code stops when package \code{ptracers} is used together with this
+ option). Currently, the code vertically averages the normal velocity
+ as specified in \code{OB[E,W]u} or \code{OB[N,S]v}. From these
+ prescribed values the code computes the boundary values for the next
+ timestep $n+1$ as follows (as an example, we use the notation for an
+ eastern or western boundary):
+ \begin{itemize}
+ \item $u^{n+1}(y,z) = \bar{u}_{ob}(y) + (u')^{n}(y,z)$, where
+   $(u')^{n}$ is the deviation from the vertically averaged velocity at
+   timestep $n$ on the boundary. $(u')^{n}$ is computed in the previous
+   time step $n$ from the intermediate velocity $u^*$ prior to the
+   correction step (see section \ref{sec:time_stepping}, e.g.,
+   eq.\,(\ref{eq:ustar-backward-free-surface})).
+   % and~(\ref{eq:vstar-backward-free-surface})).
+   (This velocity is not
+   available at the beginning of the next time step $n+1$, when
+   S/R~OBCS\_CALC/OBCS\_CALC\_STEVENS are called, therefore it needs to
+   be saved in S/R~DYNAMICS by calling S/R~OBCS\_SAVE\_UV\_N and also
+   stored in a separate restart files
+   \verb+pickup_stevens[N/S/E/W].${iteration}.data+)
+ %  Define CPP-flag OBCS\_STEVENS\_USE\_INTERIOR\_VELOCITY to use the
+ %  velocity one grid point inward from the boundary.
+ \item If $u^{n+1}$ is directed into the model domain, the boudary
+   value for tracer $\chi$ is restored to the prescribed values:
+   \[\chi^{n+1} =   \chi^{n} + \frac{\Delta{t}}{\tau_\chi} (\chi_{ob} -
+   \chi^{n}),\] where $\tau_\chi$ is the relaxation time
+   scale \texttt{T/SrelaxStevens}. The new $\chi^{n+1}$ is then subject
+   to the advection by $u^{n+1}$.
+ \item If $u^{n+1}$ is directed out of the model domain, the tracer
+   $\chi^{n+1}$ on the boundary at timestep $n+1$ is estimated from
+   advection out of the domain with $u^{n+1}+c$, where $c$ is
+   a phase velocity estimated as
+   $\frac{1}{2}\frac{\partial\chi}{\partial{t}}/\frac{\partial\chi}{\partial{x}}$. The
+   numerical scheme is (as an example for an eastern boundary):
+   \[\chi_{i_{b},j,k}^{n+1} =   \chi_{i_{b},j,k}^{n} + \Delta{t}
+   (u^{n+1}+c)_{i_{b},j,k}\frac{\chi_{i_{b},j,k}^{n}
+     - \chi_{i_{b}-1,j,k}^{n}}{\Delta{x}_{i_{b},j}^{C}}\mbox{, if }u_{i_{b},j,k}^{n+1}>0,
+   \] where $i_{b}$ is the boundary index.\\
+   For test purposes, the phase velocity contribution or the entire
+   advection can be turned off by setting the corresponding parameters
+   \texttt{useStevensPhaseVel} and \texttt{useStevensAdvection} to
+   \texttt{.FALSE.}.
+ \end{itemize}
+ See \citet{stevens:90} for details. With this boundary condition
+ specifying the exact net transport across the open boundary is simple,
+ so that balancing the flow with (S/R~OBCS\_BALANCE\_FLOW, see next
+ paragraph) is usually not necessary.
+ \paragraph{OBCS\_BALANCE\_FLOW:} ~ \\
+ %
+ When turned on (\code{ALLOW\_OBCS\_BALANCE}
+ defined in \code{OBCS\_OPTIONS.h} and \code{useOBCSbalance=.true.} in
+ \code{data.obcs/OBCS\_PARM01}), this routine balances the net flow
+ across the open boundaries. By default the net flow across the
+ boundaries is computed and all normal velocities on boundaries are
+ adjusted to obtain zero net inflow.
+ This behavior can be controlled with the runtime flags
+ \code{OBCS\_balanceFacN/S/E/W}. The values of these flags determine
+ how the net inflow is redistributed as small correction velocities
+ between the individual sections. A value ``\code{-1}'' balances an
+ individual boundary, values $>0$ determine the relative size of the
+ correction. For example, the values
+ \begin{tabbing}
+  \code{OBCS\_balanceFacE}\code{ = 1.,} \\
+  \code{OBCS\_balanceFacW}\code{ = -1.,} \\
+  \code{OBCS\_balanceFacN}\code{ = 2.,} \\
+  \code{OBCS\_balanceFacS}\code{ = 0.,}
+ \end{tabbing}
+ make the model
+ \begin{itemize}
+ \item correct Western \code{OBWu} by substracting a uniform velocity to
+ ensure zero net transport through the Western open boundary;
+ \item correct Eastern and Northern normal flow, with the Northern
+   velocity correction two times larger than the Eastern correction, but
+   \emph{not} the Southern normal flow, to ensure that the total inflow through
+   East, Northern, and Southern open boundary is balanced.
+ \end{itemize}
- \paragraph{OBCS\_BALANCE} ~ \\
+ The old method of balancing the net flow for all sections individually
- %
+ can be recovered by setting all flags to -1. Then the normal
- ~
+ velocities across each of the four boundaries are modified separately,
+ so that the net volume transport across \emph{each} boundary is
+ zero. For example, for the western boundary at $i=i_{b}$, the modified
+ velocity is:
+ \[
+ u(y,z) - \int_{\mbox{western boundary}}u\,dy\,dz \approx OBNu(j,k) - \sum_{j,k}
+ OBNu(j,k) h_{w}(i_{b},j,k)\Delta{y_G(i_{b},j)}\Delta{z(k)}.
+ \]
+ This also ensures a net total inflow of zero through all boundaries,
+ but this combination of flags is \emph{not} useful if you want to
+ simulate, say, a sector of the Southern Ocean with a strong ACC
+ entering through the western and leaving through the eastern boundary,
+ because the value of ``\code{-1}'' for these flags will make sure that
+ the strong inflow is removed. Clearly, gobal balancing with
+ \code{OBCS\_balanceFacE/W/N/S} $\ge0$ is the preferred method.
  \paragraph{OBCS\_APPLY\_*:} ~ \\
  ~
- \paragraph{OBCS\_SPONGE} Setting sponge layer characteristics \\
+ \paragraph{OBCS\_SPONGE:} ~ \\
  %
- ~
+ The sponge layer code (turned on with \code{ALLOW\_OBCS\_SPONGE} and
+ \code{useOBCSsponge}) adds a relaxation term to the right-hand-side of
+ the momentum and tracer equations. The variables are relaxed towards
+ the boundary values with a relaxation time scale that increases
+ linearly with distance from the boundary
+ \[
+ G_{\chi}^{\mbox{(sponge)}} =
+ - \frac{\chi - [( L - \delta{L} ) \chi_{BC} + \delta{L}\chi]/L}
+ {[(L-\delta{L})\tau_{b}+\delta{L}\tau_{i}]/L}
+ = - \frac{\chi - [( 1 - l ) \chi_{BC} + l\chi]}
+ {[(1-l)\tau_{b}+l\tau_{i}]}
+ \]
+ where $\chi$ is the model variable (U/V/T/S) in the interior,
+ $\chi_{BC}$ the boundary value, $L$ the thickness of the sponge layer
+ (runtime parameter \code{spongeThickness} in number of grid points),
+ $\delta{L}\in[0,L]$ ($\frac{\delta{L}}{L}=l\in[0,1]$) the distance from the boundary (also in grid points), and
+ $\tau_{b}$ (runtime parameters \code{Urelaxobcsbound} and
+ \code{Vrelaxobcsbound}) and $\tau_{i}$ (runtime parameters
+ \code{Urelaxobcsinner} and \code{Vrelaxobcsinner}) the relaxation time
+ scales on the boundary and at the interior termination of the sponge
+ layer. The parameters \code{Urelaxobcsbound/inner} set the relaxation
+ time scales for the Eastern and Western boundaries,
+ \code{Vrelaxobcsbound/inner} for the Northern and Southern boundaries.
  \paragraph{OB's with nonlinear free surface} ~ \\
  %
-Line 336 
 Table \ref{tab:pkg:obcs:diagnostics}.
+Line 536 
 Table \ref{tab:pkg:obcs:diagnostics}.
  %----------------------------------------------------------------------
  \subsubsection{Reference experiments}
+ In the directory \code{verifcation}, the following experiments use
+ \code{obcs}:
+ \begin{itemize}
+ \item \code{exp4}: box with 4 open boundaries, simulating flow over a
+   Gaussian bump based on \citet{adcroft:97}, also tests
+   Stevens-boundary conditions;
+ \item \code{dome}: based on the project ``Dynamics of Overflow Mixing
+   and Entrainment''
+   (\url{http://www.rsmas.miami.edu/personal/tamay/DOME/dome.html}), uses
+   Orlanski-BCs;
+ \item \code{internal\_wave}: uses a heavily modified \code{S/R~OBCS\_CALC}
+ \item \code{seaice\_obcs}: simple example who to use the sea-ice
+   related code, based on \code{lab\_sea};
+ \item \code{tutorial\_plume\_on\_slope}: uses Orlanski-BCs, see also
+   section~\ref{sec:eg-gravityplume}.
+ \end{itemize}
-Line 347 
 Table \ref{tab:pkg:obcs:diagnostics}.
+Line 563 
 Table \ref{tab:pkg:obcs:diagnostics}.
  \label{sec:pkg:obcs:experiments}
  \begin{itemize}
- \item{Ocean experiment in exp4 verification directory. }
+ \item \code{tutorial\_plume\_on\_slope} (section~\ref{sec:eg-gravityplume})
  \end{itemize}
+ %%% Local Variables:
+ %%% mode: latex
+ %%% TeX-master: "../../manual"
+ %%% End:

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