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

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-revision 1.11 by mlosch,
Wed Mar 16 10:39:25 2011 UTC
+revision 1.17 by mlosch,
Wed Apr 27 09:45:04 2016 UTC
 Line 233 
 $j$ specifies the Northern/Southern OB p
  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 $N_x$-dimensional
- ``row'' array $\tt OB\_Jnorth(Ny)$ / $\tt OB\_Jsouth(Ny)$,
+ ``row'' array $\tt OB\_Jnorth(Nx)$ / $\tt OB\_Jsouth(Nx)$,
  and an $N_y$-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 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 356 
 runtime flags \code{externForcingPeriod}
  in \code{data}, see \code{verification/exp4} for an example.
  \paragraph{OBCS\_CALC\_STEVENS:} ~ \\
- (THE IMPLEMENTATION OF THESE BOUNDARY CONDITIONS IS NOT COMPLETE. SO
+ (THE IMPLEMENTATION OF THESE BOUNDARY CONDITIONS IS NOT
- FAR ONLY EASTERN AND WESTERN BOUNDARIES ARE SUPPORTED.) \\
+ 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. From these prescribed values the code computes the
+ as specified in \code{OB[E,W]u} or \code{OB[N,S]v}. From these
- boundary values for the next timestep $n+1$ as follows (as an
+ prescribed values the code computes the boundary values for the next
- example, we use the notation for an eastern or western boundary):
+ timestep $n+1$ as follows (as an example, we use the notation for an
+ eastern or western boundary):
  \begin{itemize}
- \item $u^{n+1}(y,z) = \bar{u}_{ob}(y) + u'(y,z)$, where $u_{n}'$ is the
+ \item $u^{n+1}(y,z) = \bar{u}_{ob}(y) + (u')^{n}(y,z)$, where
-   deviation from the vertically averaged velocity one grid point
+   $(u')^{n}$ is the deviation from the vertically averaged velocity at
-   inward from the boundary.
+   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}.
+   scale \texttt{T/SrelaxStevens}. The new $\chi^{n+1}$ is then subject
- \item If $u^{n+1}$ is directed out of the model domain, the tracer is
+   to the advection by $u^{n+1}$.
-   advected out of the domain with $u^{n+1}+c$, where $c$ is a phase
+ \item If $u^{n+1}$ is directed out of the model domain, the tracer
-   velocity estimated as
+   $\chi^{n+1}$ on the boundary at timestep $n+1$ is estimated from
-   $\frac{1}{2}\frac{\partial\chi}{\partial{t}}/\frac{\partial\chi}{\partial{x}}$.
+   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
+   advection can be turned off by setting the corresponding parameters
-   be turned off by setting the corresponding parameters
    \texttt{useStevensPhaseVel} and \texttt{useStevensAdvection} to
-   \texttt{.FALSE.}.\end{itemize} See \citet{stevens:90} for details.
+   \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:} ~ \\
  %

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