/[MITgcm]/manual/s_phys_pkgs/text/obcs.tex
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revision 1.13 by mlosch, Mon May 2 09:11:20 2011 UTC revision 1.14 by mlosch, Wed Oct 26 08:29:48 2011 UTC
# Line 357  in \code{data}, see \code{verification/e Line 357  in \code{data}, see \code{verification/e
357    
358  \paragraph{OBCS\_CALC\_STEVENS:} ~ \\  \paragraph{OBCS\_CALC\_STEVENS:} ~ \\
359  (THE IMPLEMENTATION OF THESE BOUNDARY CONDITIONS IS NOT COMPLETE. SO  (THE IMPLEMENTATION OF THESE BOUNDARY CONDITIONS IS NOT COMPLETE. SO
360  FAR ONLY EASTERN AND WESTERN BOUNDARIES ARE SUPPORTED.) \\  FAR ONLY EASTERN AND WESTERN BOUNDARIES ARE SUPPORTED. PASSIVE TRACERS
361    AND NON-LINEAR FREE SURFACE ARE NOT SUPPORTED.) \\
362  The boundary conditions following \citet{stevens:90} require the  The boundary conditions following \citet{stevens:90} require the
363  vertically averaged normal velocity (originally specified as a stream  vertically averaged normal velocity (originally specified as a stream
364  function along the open boundary) $\bar{u}_{ob}$ and the tracer fields  function along the open boundary) $\bar{u}_{ob}$ and the tracer fields
# Line 369  prescribed values the code computes the Line 370  prescribed values the code computes the
370  timestep $n+1$ as follows (as an example, we use the notation for an  timestep $n+1$ as follows (as an example, we use the notation for an
371  eastern or western boundary):  eastern or western boundary):
372  \begin{itemize}  \begin{itemize}
373  \item $u^{n+1}(y,z) = \bar{u}_{ob}(y) + (u')^{n}(y,z)$, where $(u')^{n}$  \item $u^{n+1}(y,z) = \bar{u}_{ob}(y) + (u')^{n}(y,z)$, where
374    is the deviation from the vertically averaged velocity at timestep    $(u')^{n}$ is the deviation from the vertically averaged velocity at
375    $n$ one grid point inward from the boundary.    timestep $n$ on the boundary. $(u')^{n}$ is computed in the previous
376      time step $n$ from the intermediate velocity $u^*$ prior to the
377      correction step (see section \ref{sec:time_stepping}, e.g.,
378      eq.\,(\ref{eq:ustar-backward-free-surface})).
379      % and~(\ref{eq:vstar-backward-free-surface})).
380      (This velocity is not
381      available at the beginning of the next time step $n+1$, when
382      S/R~OBCS\_CALC/OBCS\_CALC\_STEVENS are called, therefore it needs to
383      be saved in S/R~DYNAMICS by calling S/R~OBCS\_SAVE\_UV\_N and also
384      stored in a separate restart files
385      \verb+pickup_stevens[N/S/E/W].${iteration}.data+)
386    %  Define CPP-flag OBCS\_STEVENS\_USE\_INTERIOR\_VELOCITY to use the
387    %  velocity one grid point inward from the boundary.
388  \item If $u^{n+1}$ is directed into the model domain, the boudary  \item If $u^{n+1}$ is directed into the model domain, the boudary
389    value for tracer $\chi$ is restored to the prescribed values:    value for tracer $\chi$ is restored to the prescribed values:
390    \[\chi^{n+1} =   \chi^{n} + \frac{\Delta{t}}{\tau_\chi} (\chi_{ob} -    \[\chi^{n+1} =   \chi^{n} + \frac{\Delta{t}}{\tau_\chi} (\chi_{ob} -
# Line 380  eastern or western boundary): Line 393  eastern or western boundary):
393    to the advection by $u^{n+1}$.    to the advection by $u^{n+1}$.
394  \item If $u^{n+1}$ is directed out of the model domain, the tracer  \item If $u^{n+1}$ is directed out of the model domain, the tracer
395    $\chi^{n+1}$ on the boundary at timestep $n+1$ is estimated from    $\chi^{n+1}$ on the boundary at timestep $n+1$ is estimated from
396    advection advected out of the domain with $u^{n+1}+c$, where $c$ is    advection out of the domain with $u^{n+1}+c$, where $c$ is
397    a phase velocity estimated as    a phase velocity estimated as
398    $\frac{1}{2}\frac{\partial\chi}{\partial{t}}/\frac{\partial\chi}{\partial{x}}$. The    $\frac{1}{2}\frac{\partial\chi}{\partial{t}}/\frac{\partial\chi}{\partial{x}}$. The
399    numerical scheme is (as an example for an eastern boundary):    numerical scheme is (as an example for an eastern boundary):
400    \[\chi_{i_{b},j,k}^{n+1} =   \chi_{i_{b},j,k}^{n} + \Delta{t}    \[\chi_{i_{b},j,k}^{n+1} =   \chi_{i_{b},j,k}^{n} + \Delta{t}
401    (u^{n+1}+c)_{i_{b},j,k}\frac{\chi_{i_{b},j,k}^{n}    (u^{n+1}+c)_{i_{b},j,k}\frac{\chi_{i_{b},j,k}^{n}
402      - \chi_{i_{b}-1,j,k}^{n}}{\Delta{x}_{i_{b},j}^{C}}\mbox{, if }u_{i_{b},j,k}^{n+1}>0,      - \chi_{i_{b}-1,j,k}^{n}}{\Delta{x}_{i_{b},j}^{C}}\mbox{, if }u_{i_{b},j,k}^{n+1}>0,
403    \] where $i_{b}$ is the boundary index.    \] where $i_{b}$ is the boundary index.\\
   
404    For test purposes, the phase velocity contribution or the entire    For test purposes, the phase velocity contribution or the entire
405    advection can be turned off by setting the corresponding parameters    advection can be turned off by setting the corresponding parameters
406    \texttt{useStevensPhaseVel} and \texttt{useStevensAdvection} to    \texttt{useStevensPhaseVel} and \texttt{useStevensAdvection} to
407    \texttt{.FALSE.}.\end{itemize} See \citet{stevens:90} for details.    \texttt{.FALSE.}.
408    \end{itemize}
409    See \citet{stevens:90} for details. With this boundary condition
410    specifying the exact net transport across the open boundary is simple,
411    so that balancing the flow with (S/R~OBCS\_BALANCE\_FLOW, see next
412    paragraph) is usually not necessary.
413    
414  \paragraph{OBCS\_BALANCE\_FLOW:} ~ \\  \paragraph{OBCS\_BALANCE\_FLOW:} ~ \\
415  %  %
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