--- manual/s_phys_pkgs/text/obcs.tex 2011/03/01 07:53:21 1.8 +++ manual/s_phys_pkgs/text/obcs.tex 2011/10/26 08:29:48 1.14 @@ -12,7 +12,7 @@ \label{sec:pkg:obcs:intro}} The OBCS-package is fundamental to regional ocean modelling with the -MITgcm, but because there are so many details to be considered in +MITgcm, but there are so many details to be considered in regional ocean modelling that this package cannot accomodate all imaginable and possible options. Therefore, for a regional simulation with very particular details, it is recommended to familiarize oneself @@ -153,7 +153,7 @@ ~ \\ 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\\ @@ -357,64 +357,137 @@ \paragraph{OBCS\_CALC\_STEVENS:} ~ \\ (THE IMPLEMENTATION OF THESE BOUNDARY CONDITIONS IS NOT COMPLETE. SO -FAR ONLY EASTERN AND WESTERN BOUNDARIES ARE SUPPORTED.) \\ +FAR ONLY EASTERN AND WESTERN BOUNDARIES ARE SUPPORTED. PASSIVE TRACERS +AND NON-LINEAR FREE SURFACE ARE NOT SUPPORTED.) \\ 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:} ~ \\ +\paragraph{OBCS\_BALANCE\_FLOW:} ~ \\ % -This is not (yet) a separate routine in the code, but it may become -one to make this code more transparent. The code is part of -\code{S/R~OBCS\_CALC}. When turned on (\code{ALLOW\_OBCS\_BALANCE} +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 -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: +\code{data.obcs/OBCS\_PARM01}), this routine balances the net flow +across the open boundaries. By default the net flow across the +boundaries is computed and all normal velocities on boundaries are +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 -make it a useful flag to prevent infinite sea-level change within the -domain, but the flag is \emph{not} useful if you want to simulate, -say, a sector of the Southern Ocean with a strong ACC entering through -the western and leaving through the eastern boundary, because this -flag will make sure that the strong inflow is removed. It is -recommended that this part of the code is adapted to the particular -needs of the simulation in question. +This also ensures a net total inflow of zero through all boundaries, +but this combination of flags is \emph{not} useful if you want to +simulate, say, a sector of the Southern Ocean with a strong ACC +entering through the western and leaving through the eastern boundary, +because the value of ``\code{-1}'' for these flags will make sure that +the strong inflow is removed. Clearly, gobal balancing with +\code{OBCS\_balanceFacE/W/N/S} $\ge0$ is the preferred method. \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} ~ \\ %