--- manual/s_phys_pkgs/text/obcs.tex	2011/03/16 10:39:25	1.11
+++ manual/s_phys_pkgs/text/obcs.tex	2016/04/27 08:54:55	1.16
@@ -233,9 +233,9 @@
 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.
@@ -356,35 +356,60 @@
 in \code{data}, see \code{verification/exp4} for an example.
 
 \paragraph{OBCS\_CALC\_STEVENS:} ~ \\
-(THE IMPLEMENTATION OF THESE BOUNDARY CONDITIONS IS NOT COMPLETE. SO
-FAR ONLY EASTERN AND WESTERN BOUNDARIES ARE SUPPORTED.) \\
+(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. 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):
+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 $u^{n+1}(y,z) = \bar{u}_{ob}(y) + u'(y,z)$, where $u_{n}'$ is the
-  deviation from the vertically averaged velocity one grid point
-  inward from the boundary.
+\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}.
-\item If $u^{n+1}$ is directed out of the model domain, the tracer is
-  advected out of the domain with $u^{n+1}+c$, where $c$ is a phase
-  velocity estimated as
-  $\frac{1}{2}\frac{\partial\chi}{\partial{t}}/\frac{\partial\chi}{\partial{x}}$.
+  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
+  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.
+  \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:} ~ \\
 %
