| 80 |
\>\> CALC\_GRAD\_PHI\_SURF \` $\nabla \eta^{n+1}$ \\ |
\>\> CALC\_GRAD\_PHI\_SURF \` $\nabla \eta^{n+1}$ \\ |
| 81 |
\>\> CORRECTION\_STEP \` $u^{n+1}$,$v^{n+1}$ (\ref{eq:un+1-rigid-lid},\ref{eq:vn+1-rigid-lid}) |
\>\> CORRECTION\_STEP \` $u^{n+1}$,$v^{n+1}$ (\ref{eq:un+1-rigid-lid},\ref{eq:vn+1-rigid-lid}) |
| 82 |
\end{tabbing} \end{minipage} } \end{center} |
\end{tabbing} \end{minipage} } \end{center} |
| 83 |
\caption{Calling tree for the pressure method algorihtm} |
\caption{Calling tree for the pressure method algorihtm} |
| 84 |
\label{fig:call-tree-pressure-method} |
\label{fig:call-tree-pressure-method} |
| 85 |
\end{figure} |
\end{figure} |
| 86 |
|
|
| 554 |
{\em PARM01} of {\em data}. |
{\em PARM01} of {\em data}. |
| 555 |
|
|
| 556 |
The only difficulty with this approach is apparent in equation |
The only difficulty with this approach is apparent in equation |
| 557 |
$\ref{eq:Gt-n-staggered}$ and illustrated by the dotted arrow |
\ref{eq:Gt-n-staggered} and illustrated by the dotted arrow |
| 558 |
connecting $u,v^n$ with $G_\theta^{n-1/2}$. The flow used to advect |
connecting $u,v^n$ with $G_\theta^{n-1/2}$. The flow used to advect |
| 559 |
tracers around is not naturally located in time. This could be avoided |
tracers around is not naturally located in time. This could be avoided |
| 560 |
by applying the Adams-Bashforth extrapolation to the tracer field |
by applying the Adams-Bashforth extrapolation to the tracer field |
| 611 |
|
|
| 612 |
|
|
| 613 |
Once ${\eta}^{n+1}$ has been found, substituting into |
Once ${\eta}^{n+1}$ has been found, substituting into |
| 614 |
\ref{eq-tDsC-Hmom} yields $\vec{\bf v}^{n+1}$ if the model is |
\ref{eq:discrete-time-u,eq:discrete-time-v} yields $\vec{\bf v}^{n+1}$ if the model is |
| 615 |
hydrostatic ($\epsilon_{nh}=0$): |
hydrostatic ($\epsilon_{nh}=0$): |
| 616 |
$$ |
$$ |
| 617 |
\vec{\bf v}^{n+1} = \vec{\bf v}^{*} |
\vec{\bf v}^{n+1} = \vec{\bf v}^{*} |
| 621 |
This is known as the correction step. However, when the model is |
This is known as the correction step. However, when the model is |
| 622 |
non-hydrostatic ($\epsilon_{nh}=1$) we need an additional step and an |
non-hydrostatic ($\epsilon_{nh}=1$) we need an additional step and an |
| 623 |
additional equation for $\phi'_{nh}$. This is obtained by substituting |
additional equation for $\phi'_{nh}$. This is obtained by substituting |
| 624 |
\ref{eq-tDsC-Hmom} and \ref{eq-tDsC-Vmom} into |
\ref{eq:discrete-time-u}, \ref{eq:discrete-time-v} and \ref{eq:discrete-time-w} |
| 625 |
\ref{eq-tDsC-cont}: |
into continuity: |
| 626 |
\begin{equation} |
\begin{equation} |
| 627 |
\left[ {\bf \nabla}_h^2 + \partial_{rr} \right] {\phi'_{nh}}^{n+1} |
\left[ {\bf \nabla}_h^2 + \partial_{rr} \right] {\phi'_{nh}}^{n+1} |
| 628 |
= \frac{1}{\Delta t} \left( |
= \frac{1}{\Delta t} \left( |
| 702 |
{\it implicSurfPress}, {\it implicDiv2DFlow}. They are read from |
{\it implicSurfPress}, {\it implicDiv2DFlow}. They are read from |
| 703 |
the main data file "{\it data}" and are set by default to 1,1. |
the main data file "{\it data}" and are set by default to 1,1. |
| 704 |
|
|
| 705 |
Equations \ref{eq-tDsC-Hmom} and \ref{eq-tDsC-eta} are modified as follows: |
Equations \ref{eq:ustar-backward-free-surface} -- |
| 706 |
|
\ref{eq:vn+1-backward-free-surface} are modified as follows: |
| 707 |
$$ |
$$ |
| 708 |
\frac{ \vec{\bf v}^{n+1} }{ \Delta t } |
\frac{ \vec{\bf v}^{n+1} }{ \Delta t } |
| 709 |
+ {\bf \nabla}_h b_s [ \beta {\eta}^{n+1} + (1-\beta) {\eta}^{n} ] |
+ {\bf \nabla}_h b_s [ \beta {\eta}^{n+1} + (1-\beta) {\eta}^{n} ] |
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