| 11 |
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| 12 |
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| 13 |
\subsubsection{pressure/geo-potential and free surface} |
\subsubsection{pressure/geo-potential and free surface} |
| 14 |
|
\label{sect:phi-freesurface} |
| 15 |
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| 16 |
For the atmosphere, since $\phi = \phi_{topo} - \int^p_{p_s} \alpha dp$, |
For the atmosphere, since $\phi = \phi_{topo} - \int^p_{p_s} \alpha dp$, |
| 17 |
subtracting the reference state defined in section |
subtracting the reference state defined in section |
| 308 |
\vec{\bf v}^{*} & = & \vec{\bf v}^{n-1/2} + \Delta t \frac{dh^{n-1}}{dh^{n}} \left( |
\vec{\bf v}^{*} & = & \vec{\bf v}^{n-1/2} + \Delta t \frac{dh^{n-1}}{dh^{n}} \left( |
| 309 |
\vec{\bf G}_{\vec{\bf v}}^{(n)} + F_{\vec{\bf v}}^{n}/dh^{n-1} \right) |
\vec{\bf G}_{\vec{\bf v}}^{(n)} + F_{\vec{\bf v}}^{n}/dh^{n-1} \right) |
| 310 |
- \Delta t \nabla \phi_{hyd}^{n} |
- \Delta t \nabla \phi_{hyd}^{n} |
| 311 |
\label{eq:vstar-nlfs} \\ |
\label{eq:vstar-nlfs} |
| 312 |
\mathrm{update}\hspace{-4mm} & & \hspace{-4mm}\mathrm{ |
\end{eqnarray} |
| 313 |
model~geometry~:~{\bf hFac}}(dh^n)\nonumber \\ |
\hspace{3cm}$\longrightarrow$~~{\it update model~geometry~:~}${\bf hFac}(dh^n)$\\ |
| 314 |
|
\begin{eqnarray} |
| 315 |
\eta^{n+1/2} \hspace{-2mm} & = & |
\eta^{n+1/2} \hspace{-2mm} & = & |
| 316 |
\eta^{n-1/2} + \Delta t P^{n+1/2} - \Delta t |
\eta^{n-1/2} + \Delta t P^{n+1/2} - \Delta t |
| 317 |
\nabla \cdot \int \vec{\bf v}^{n+1/2} dh^{n} \nonumber \\ |
\nabla \cdot \int \vec{\bf v}^{n+1/2} dh^{n} \nonumber \\ |
| 420 |
part of the r* coordinate motion, as in a $\sigma_{z},\sigma_{p}$ |
part of the r* coordinate motion, as in a $\sigma_{z},\sigma_{p}$ |
| 421 |
model, but the fixed part related to topography is treated |
model, but the fixed part related to topography is treated |
| 422 |
as in a height or pressure coordinate model. |
as in a height or pressure coordinate model. |
| 423 |
A complete description is given in \cite{adcroft:04}. |
A complete description is given in \cite{adcroft:04a}. |
| 424 |
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| 425 |
The time-stepping implementation of the $r^*$ coordinate is |
The time-stepping implementation of the $r^*$ coordinate is |
| 426 |
identical to the non-linear free-surface in $r$ coordinate, |
identical to the non-linear free-surface in $r$ coordinate, |
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