| 233 |
and for each meridional position $j=1,\ldots,N_y$, a zonal index |
and for each meridional position $j=1,\ldots,N_y$, a zonal index |
| 234 |
$i$ specifies the Eastern/Western OB position. |
$i$ specifies the Eastern/Western OB position. |
| 235 |
For Northern/Southern OB this defines an $N_x$-dimensional |
For Northern/Southern OB this defines an $N_x$-dimensional |
| 236 |
``row'' array $\tt OB\_Jnorth(Ny)$ / $\tt OB\_Jsouth(Ny)$, |
``row'' array $\tt OB\_Jnorth(Nx)$ / $\tt OB\_Jsouth(Nx)$, |
| 237 |
and an $N_y$-dimenisonal |
and an $N_y$-dimenisonal |
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``column'' array $\tt OB\_Ieast(Nx)$ / $\tt OB\_Iwest(Nx)$. |
``column'' array $\tt OB\_Ieast(Ny)$ / $\tt OB\_Iwest(Ny)$. |
| 239 |
Positions determined in this way allows Northern/Southern |
Positions determined in this way allows Northern/Southern |
| 240 |
OBs to be at variable $j$ (or $y$) positions, and Eastern/Western |
OBs to be at variable $j$ (or $y$) positions, and Eastern/Western |
| 241 |
OBs at variable $i$ (or $x$) positions. |
OBs at variable $i$ (or $x$) positions. |
| 268 |
eg. $\tt OB\_Jnorth(3)=-1$ means that the point $\tt (3,Ny)$ |
eg. $\tt OB\_Jnorth(3)=-1$ means that the point $\tt (3,Ny)$ |
| 269 |
is a northern OB. |
is a northern OB. |
| 270 |
|
|
| 271 |
|
\noindent\textbf{Simple examples:} For a model grid with $ N_{x}\times |
| 272 |
|
N_{y} = 120\times144$ horizontal grid points with four open boundaries |
| 273 |
|
along the four egdes of the domain, the simplest way of specifying the |
| 274 |
|
boundary points in \code{data.obcs} is: |
| 275 |
|
\begin{verbatim} |
| 276 |
|
OB_Ieast = 144*-1, |
| 277 |
|
# or OB_Ieast = 144*120, |
| 278 |
|
OB_Iwest = 144*1, |
| 279 |
|
OB_Jnorth = 120*-1, |
| 280 |
|
# or OB_Jnorth = 120*144, |
| 281 |
|
OB_Jsouth = 120*1, |
| 282 |
|
\end{verbatim} |
| 283 |
|
If only the first $50$ grid points of the southern boundary are |
| 284 |
|
boundary points: |
| 285 |
|
\begin{verbatim} |
| 286 |
|
OB_Jsouth(1:50) = 50*1, |
| 287 |
|
\end{verbatim} |
| 288 |
|
|
| 289 |
\noindent |
\noindent |
| 290 |
\textsf{Add special comments for case \#define NONLIN\_FRSURF, |
\textsf{Add special comments for case \#define NONLIN\_FRSURF, |
| 291 |
see obcs\_ini\_fixed.F} |
see obcs\_ini\_fixed.F} |
| 297 |
|
|
| 298 |
\paragraph{OBCS\_READPARMS:} ~ \\ |
\paragraph{OBCS\_READPARMS:} ~ \\ |
| 299 |
Set OB positions through arrays |
Set OB positions through arrays |
| 300 |
{\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)}, |
| 301 |
and runtime flags (see Table \ref{tab:pkg:obcs:runtime_flags}). |
and runtime flags (see Table \ref{tab:pkg:obcs:runtime_flags}). |
| 302 |
|
|
| 303 |
\paragraph{OBCS\_CALC:} ~ \\ |
\paragraph{OBCS\_CALC:} ~ \\ |
| 374 |
in \code{data}, see \code{verification/exp4} for an example. |
in \code{data}, see \code{verification/exp4} for an example. |
| 375 |
|
|
| 376 |
\paragraph{OBCS\_CALC\_STEVENS:} ~ \\ |
\paragraph{OBCS\_CALC\_STEVENS:} ~ \\ |
| 377 |
(THE IMPLEMENTATION OF THESE BOUNDARY CONDITIONS IS NOT COMPLETE. SO |
(THE IMPLEMENTATION OF THESE BOUNDARY CONDITIONS IS NOT |
| 378 |
FAR ONLY EASTERN AND WESTERN BOUNDARIES ARE SUPPORTED.) \\ |
COMPLETE. PASSIVE TRACERS, SEA ICE AND NON-LINEAR FREE SURFACE ARE NOT |
| 379 |
|
SUPPORTED PROPERLY.) \\ |
| 380 |
The boundary conditions following \citet{stevens:90} require the |
The boundary conditions following \citet{stevens:90} require the |
| 381 |
vertically averaged normal velocity (originally specified as a stream |
vertically averaged normal velocity (originally specified as a stream |
| 382 |
function along the open boundary) $\bar{u}_{ob}$ and the tracer fields |
function along the open boundary) $\bar{u}_{ob}$ and the tracer fields |
| 388 |
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 |
| 389 |
eastern or western boundary): |
eastern or western boundary): |
| 390 |
\begin{itemize} |
\begin{itemize} |
| 391 |
\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 |
| 392 |
is the deviation from the vertically averaged velocity at timestep |
$(u')^{n}$ is the deviation from the vertically averaged velocity at |
| 393 |
$n$ one grid point inward from the boundary. |
timestep $n$ on the boundary. $(u')^{n}$ is computed in the previous |
| 394 |
|
time step $n$ from the intermediate velocity $u^*$ prior to the |
| 395 |
|
correction step (see section \ref{sec:time_stepping}, e.g., |
| 396 |
|
eq.\,(\ref{eq:ustar-backward-free-surface})). |
| 397 |
|
% and~(\ref{eq:vstar-backward-free-surface})). |
| 398 |
|
(This velocity is not |
| 399 |
|
available at the beginning of the next time step $n+1$, when |
| 400 |
|
S/R~OBCS\_CALC/OBCS\_CALC\_STEVENS are called, therefore it needs to |
| 401 |
|
be saved in S/R~DYNAMICS by calling S/R~OBCS\_SAVE\_UV\_N and also |
| 402 |
|
stored in a separate restart files |
| 403 |
|
\verb+pickup_stevens[N/S/E/W].${iteration}.data+) |
| 404 |
|
% Define CPP-flag OBCS\_STEVENS\_USE\_INTERIOR\_VELOCITY to use the |
| 405 |
|
% velocity one grid point inward from the boundary. |
| 406 |
\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 |
| 407 |
value for tracer $\chi$ is restored to the prescribed values: |
value for tracer $\chi$ is restored to the prescribed values: |
| 408 |
\[\chi^{n+1} = \chi^{n} + \frac{\Delta{t}}{\tau_\chi} (\chi_{ob} - |
\[\chi^{n+1} = \chi^{n} + \frac{\Delta{t}}{\tau_\chi} (\chi_{ob} - |
| 411 |
to the advection by $u^{n+1}$. |
to the advection by $u^{n+1}$. |
| 412 |
\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 |
| 413 |
$\chi^{n+1}$ on the boundary at timestep $n+1$ is estimated from |
$\chi^{n+1}$ on the boundary at timestep $n+1$ is estimated from |
| 414 |
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 |
| 415 |
a phase velocity estimated as |
a phase velocity estimated as |
| 416 |
$\frac{1}{2}\frac{\partial\chi}{\partial{t}}/\frac{\partial\chi}{\partial{x}}$. The |
$\frac{1}{2}\frac{\partial\chi}{\partial{t}}/\frac{\partial\chi}{\partial{x}}$. The |
| 417 |
numerical scheme is (as an example for an eastern boundary): |
numerical scheme is (as an example for an eastern boundary): |
| 418 |
\[\chi_{i,j,k}^{n+1} = \chi_{i,j,k}^{n} + \Delta{t} |
\[\chi_{i_{b},j,k}^{n+1} = \chi_{i_{b},j,k}^{n} + \Delta{t} |
| 419 |
(u^{n+1}+c)_{i_{b},j,k}\frac{\chi_{i_{b},j,k}^{n} |
(u^{n+1}+c)_{i_{b},j,k}\frac{\chi_{i_{b},j,k}^{n} |
| 420 |
- \chi_{i_{b}-1,j,k}^{n}}{\Delta{x}_{i_{b},j}^{C}}\mbox{, if }u_{i_{b},j,k}^{n+1}>0, |
- \chi_{i_{b}-1,j,k}^{n}}{\Delta{x}_{i_{b},j}^{C}}\mbox{, if }u_{i_{b},j,k}^{n+1}>0, |
| 421 |
\] where $i_{b}$ is the boundary index. |
\] where $i_{b}$ is the boundary index.\\ |
|
|
|
| 422 |
For test purposes, the phase velocity contribution or the entire |
For test purposes, the phase velocity contribution or the entire |
| 423 |
advection can be turned off by setting the corresponding parameters |
advection can be turned off by setting the corresponding parameters |
| 424 |
\texttt{useStevensPhaseVel} and \texttt{useStevensAdvection} to |
\texttt{useStevensPhaseVel} and \texttt{useStevensAdvection} to |
| 425 |
\texttt{.FALSE.}.\end{itemize} See \citet{stevens:90} for details. |
\texttt{.FALSE.}. |
| 426 |
|
\end{itemize} |
| 427 |
|
See \citet{stevens:90} for details. With this boundary condition |
| 428 |
|
specifying the exact net transport across the open boundary is simple, |
| 429 |
|
so that balancing the flow with (S/R~OBCS\_BALANCE\_FLOW, see next |
| 430 |
|
paragraph) is usually not necessary. |
| 431 |
|
|
| 432 |
\paragraph{OBCS\_BALANCE\_FLOW:} ~ \\ |
\paragraph{OBCS\_BALANCE\_FLOW:} ~ \\ |
| 433 |
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