| 76 |
continuity equation over the entire depth of the fluid, |
continuity equation over the entire depth of the fluid, |
| 77 |
from $R_{min}(x,y)$ up to $R_o(x,y)$ |
from $R_{min}(x,y)$ up to $R_o(x,y)$ |
| 78 |
(Linear free surface): |
(Linear free surface): |
| 79 |
\begin{displaymath} |
\begin{eqnarray} |
| 80 |
\epsilon_{fs} \partial_t \eta = |
\epsilon_{fs} \partial_t \eta = |
| 81 |
\left. \dot{r} \right|_{r=r_{surf}} + \epsilon_{fw} (P-E) = |
\left. \dot{r} \right|_{r=r_{surf}} + \epsilon_{fw} (P-E) = |
| 82 |
- {\bf \nabla}_r \cdot \int_{R_{min}}^{R_o} \vec{\bf v} dr |
- {\bf \nabla}_r \cdot \int_{R_{min}}^{R_o} \vec{\bf v} dr |
| 83 |
+ \epsilon_{fw} (P-E) |
+ \epsilon_{fw} (P-E) |
| 84 |
\end{displaymath} |
\label{eq-cont-2D} |
| 85 |
|
\end{eqnarray} |
| 86 |
|
|
| 87 |
Where $\epsilon_{fs}$,$\epsilon_{fw}$ are two flags to |
Where $\epsilon_{fs}$,$\epsilon_{fw}$ are two flags to |
| 88 |
distinguish between a free-surface equation ($\epsilon_{fs}=1$) |
distinguish between a free-surface equation ($\epsilon_{fs}=1$) |
| 326 |
|
|
| 327 |
Substituting \ref{eq-rtd-hmom} into \ref{eq-rtd-cont}, assuming |
Substituting \ref{eq-rtd-hmom} into \ref{eq-rtd-cont}, assuming |
| 328 |
$\epsilon_{nh} = 0$ yields a Helmholtz equation for ${\eta}^{n+1}$: |
$\epsilon_{nh} = 0$ yields a Helmholtz equation for ${\eta}^{n+1}$: |
| 329 |
$$ |
\begin{eqnarray} |
| 330 |
\epsilon_{fs} {\eta}^{n+1} - |
\epsilon_{fs} {\eta}^{n+1} - |
| 331 |
{\bf \nabla}_r \cdot \Delta t^2 (R_o-R_{min}) |
{\bf \nabla}_r \cdot \Delta t^2 (R_o-R_{min}) |
| 332 |
{\bf \nabla}_r B_o {\eta}^{n+1} |
{\bf \nabla}_r B_o {\eta}^{n+1} |
| 333 |
= {\eta}^* |
= {\eta}^* |
| 334 |
\label{solve_2d} |
\label{solve_2d} |
| 335 |
$$ |
\end{eqnarray} |
| 336 |
where |
where |
| 337 |
$$ |
\begin{eqnarray} |
| 338 |
{\eta}^* = \epsilon_{fs} \: {\eta}^{n} - |
{\eta}^* = \epsilon_{fs} \: {\eta}^{n} - |
| 339 |
\Delta t {\bf \nabla}_r \cdot \int_{R_{min}}^{R_o} \vec{\bf v}^* dr |
\Delta t {\bf \nabla}_r \cdot \int_{R_{min}}^{R_o} \vec{\bf v}^* dr |
| 340 |
\: + \: \epsilon_{fw} \Delta_t (P-E)^{n} |
\: + \: \epsilon_{fw} \Delta_t (P-E)^{n} |
| 341 |
$$ |
\label{solve_2d_rhs} |
| 342 |
|
\end{eqnarray} |
| 343 |
|
|
| 344 |
Once ${\eta}^{n+1}$ has been found substituting into \ref{eq-rtd-hmom} |
Once ${\eta}^{n+1}$ has been found substituting into \ref{eq-rtd-hmom} |
| 345 |
would yield $\vec{\bf v}^{n+1}$ if the model is hydrostatic |
would yield $\vec{\bf v}^{n+1}$ if the model is hydrostatic |
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