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

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-revision 1.9 by mlosch,
Thu Mar  3 07:46:09 2011 UTC
+revision 1.18 by mlosch,
Wed Jun 15 13:40:53 2016 UTC
 Line 153 
 their default values.
             ~ \\
          useOBCSbalance & \code{.FALSE.} &
             ~ \\
-            OBCS\_balanceFacN/S/E/W & 0 & factor(s) determining the details
+            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\\
 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 268 
 points relative to the Northern/Eastern
  eg. $\tt OB\_Jnorth(3)=-1$  means that the point $\tt (3,Ny)$
  is a northern OB.
+ \noindent\textbf{Simple examples:} For a model grid with $ N_{x}\times
+ N_{y} = 120\times144$ horizontal grid points with four open boundaries
+ along the four egdes of the domain, the simplest way of specifying the
+ boundary points in \code{data.obcs} is:
+ \begin{verbatim}
+   OB_Ieast = 144*-1,
+ # or OB_Ieast = 144*120,
+   OB_Iwest = 144*1,
+   OB_Jnorth = 120*-1,
+ # or OB_Jnorth = 120*144,
+   OB_Jsouth = 120*1,
+ \end{verbatim}
+ If only the first $50$ grid points of the southern boundary are
+ boundary points:
+ \begin{verbatim}
+   OB_Jsouth(1:50) = 50*1,
+ \end{verbatim}
  \noindent
  \textsf{Add special comments for case \#define NONLIN\_FRSURF,
  see obcs\_ini\_fixed.F}
-Line 279 
 see obcs\_ini\_fixed.F}
+Line 297 
 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}
+Line 374 
 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:} ~ \\
+ \paragraph{OBCS\_BALANCE\_FLOW:} ~ \\
  %
- This is not (yet) a separate routine in the code, but it may become
+ When turned on (\code{ALLOW\_OBCS\_BALANCE}
- one to make this code more transparent. The code is part of
- \code{S/R~OBCS\_CALC}. When turned on (\code{ALLOW\_OBCS\_BALANCE}
  defined in \code{OBCS\_OPTIONS.h} and \code{useOBCSbalance=.true.} in
- \code{data.obcs/OBCS\_PARM01}), the normal velocities across each of
+ \code{data.obcs/OBCS\_PARM01}), this routine balances the net flow
- the four boundaries are modified separately, so that the net volume
+ across the open boundaries. By default the net flow across the
- transport across \emph{each} boundary is zero. For example, for the
+ boundaries is computed and all normal velocities on boundaries are
- western boundary at $i=i_{b}$, the modified velocity is:
+ 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}
+ 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 to
+ This also ensures a net total inflow of zero through all boundaries,
- make it a useful flag to prevent infinite sea-level change within the
+ but this combination of flags is \emph{not} useful if you want to
- domain, but the flag is \emph{not} useful if you want to simulate,
+ simulate, say, a sector of the Southern Ocean with a strong ACC
- say, a sector of the Southern Ocean with a strong ACC entering through
+ entering through the western and leaving through the eastern boundary,
- the western and leaving through the eastern boundary, because this
+ because the value of ``\code{-1}'' for these flags will make sure that
- flag will make sure that the strong inflow is removed. It is
+ the strong inflow is removed. Clearly, gobal balancing with
- recommended that this part of the code is adapted to the particular
+ \code{OBCS\_balanceFacE/W/N/S} $\ge0$ is the preferred method.
- needs of the simulation in question.
  \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} ~ \\
  %

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