| 263 |
\begin{eqnarray} |
\begin{eqnarray} |
| 264 |
u^{*} & = & u^{n} + \Delta t G_u^{(n+1/2)} \label{eq:ustar-backward-free-surface} \\ |
u^{*} & = & u^{n} + \Delta t G_u^{(n+1/2)} \label{eq:ustar-backward-free-surface} \\ |
| 265 |
v^{*} & = & v^{n} + \Delta t G_v^{(n+1/2)} \label{eq:vstar-backward-free-surface} \\ |
v^{*} & = & v^{n} + \Delta t G_v^{(n+1/2)} \label{eq:vstar-backward-free-surface} \\ |
| 266 |
\eta^* & = & \epsilon_{fs} \left( \eta^{n} + \Delta t (P-E) \right)- \Delta t |
\eta^* & = & \epsilon_{fs} \left( \eta^{n} + \Delta t (P-E) \right) |
| 267 |
\partial_x H \widehat{u^{*}} |
- \Delta t \left( \partial_x H \widehat{u^{*}} |
| 268 |
+ \partial_y H \widehat{v^{*}} |
+ \partial_y H \widehat{v^{*}} \right) |
| 269 |
\\ |
\\ |
| 270 |
\partial_x g H \partial_x \eta^{n+1} |
\partial_x g H \partial_x \eta^{n+1} |
| 271 |
& + & \partial_y g H \partial_y \eta^{n+1} |
& + & \partial_y g H \partial_y \eta^{n+1} |
| 425 |
\end{eqnarray} |
\end{eqnarray} |
| 426 |
where ${\cal L}_\tau^{-1}$ is the inverse of the operator |
where ${\cal L}_\tau^{-1}$ is the inverse of the operator |
| 427 |
\begin{equation} |
\begin{equation} |
| 428 |
{\cal L} = \left[ 1 + \Delta t \partial_r \kappa_v \partial_r \right] |
{\cal L}_\tau = \left[ 1 + \Delta t \partial_r \kappa_v \partial_r \right] |
| 429 |
\end{equation} |
\end{equation} |
| 430 |
Equation \ref{eq:taustar-implicit} looks exactly as \ref{eq:taustar} |
Equation \ref{eq:taustar-implicit} looks exactly as \ref{eq:taustar} |
| 431 |
while \ref{eq:tau-n+1-implicit} involves an operator or matrix |
while \ref{eq:tau-n+1-implicit} involves an operator or matrix |
| 613 |
allowing the use of the most recent velocities to compute |
allowing the use of the most recent velocities to compute |
| 614 |
the advection terms. Once the thermodynamics fields are |
the advection terms. Once the thermodynamics fields are |
| 615 |
updated, the hydrostatic pressure is computed |
updated, the hydrostatic pressure is computed |
| 616 |
to step forwrad the dynamics. |
to step forward the dynamics. |
| 617 |
Note that the pressure gradient must also be taken out of the |
Note that the pressure gradient must also be taken out of the |
| 618 |
Adams-Bashforth extrapolation. Also, retaining the integer time-levels, |
Adams-Bashforth extrapolation. Also, retaining the integer time-levels, |
| 619 |
$n$ and $n+1$, does not give a user the sense of where variables are |
$n$ and $n+1$, does not give a user the sense of where variables are |
| 718 |
The non-hydrostatic formulation re-introduces the full vertical |
The non-hydrostatic formulation re-introduces the full vertical |
| 719 |
momentum equation and requires the solution of a 3-D elliptic |
momentum equation and requires the solution of a 3-D elliptic |
| 720 |
equations for non-hydrostatic pressure perturbation. We still |
equations for non-hydrostatic pressure perturbation. We still |
| 721 |
intergrate vertically for the hydrostatic pressure and solve a 2-D |
integrate vertically for the hydrostatic pressure and solve a 2-D |
| 722 |
elliptic equation for the surface pressure/elevation for this reduces |
elliptic equation for the surface pressure/elevation for this reduces |
| 723 |
the amount of work needed to solve for the non-hydrostatic pressure. |
the amount of work needed to solve for the non-hydrostatic pressure. |
| 724 |
|
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