--- 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:} ~ \\ %