| 454 |
The core algorithm is based on the ``C grid'' discretization of the |
The core algorithm is based on the ``C grid'' discretization of the |
| 455 |
continuity equation which can be summarized as: |
continuity equation which can be summarized as: |
| 456 |
\begin{eqnarray} |
\begin{eqnarray} |
| 457 |
\partial_t u + \frac{1}{\Delta x_c} \delta_i \left. \frac{ \partial \Phi}{\partial r}\right|_{s} \eta + \frac{\epsilon_{nh}}{\Delta x_c} \delta_i \Phi_{nh}' & = & G_u - \frac{1}{\Delta x_c} \delta_i \Phi_h' \\ |
\partial_t u + \frac{1}{\Delta x_c} \delta_i \left. \frac{ \partial \Phi}{\partial r}\right|_{s} \eta + \frac{\epsilon_{nh}}{\Delta x_c} \delta_i \Phi_{nh}' & = & G_u - \frac{1}{\Delta x_c} \delta_i \Phi_h' \label{eq:discrete-momu} \\ |
| 458 |
\partial_t v + \frac{1}{\Delta y_c} \delta_j \left. \frac{ \partial \Phi}{\partial r}\right|_{s} \eta + \frac{\epsilon_{nh}}{\Delta y_c} \delta_j \Phi_{nh}' & = & G_v - \frac{1}{\Delta y_c} \delta_j \Phi_h' \\ |
\partial_t v + \frac{1}{\Delta y_c} \delta_j \left. \frac{ \partial \Phi}{\partial r}\right|_{s} \eta + \frac{\epsilon_{nh}}{\Delta y_c} \delta_j \Phi_{nh}' & = & G_v - \frac{1}{\Delta y_c} \delta_j \Phi_h' \label{eq:discrete-momv} \\ |
| 459 |
\epsilon_{nh} \left( \partial_t w + \frac{1}{\Delta r_c} \delta_k \Phi_{nh}' \right) & = & \epsilon_{nh} G_w + \overline{b}^k - \frac{1}{\Delta r_c} \delta_k \Phi_{h}' \\ |
\epsilon_{nh} \left( \partial_t w + \frac{1}{\Delta r_c} \delta_k \Phi_{nh}' \right) & = & \epsilon_{nh} G_w + \overline{b}^k - \frac{1}{\Delta r_c} \delta_k \Phi_{h}' \label{eq:discrete-momw} \\ |
| 460 |
\delta_i \Delta y_g \Delta r_f h_w u + |
\delta_i \Delta y_g \Delta r_f h_w u + |
| 461 |
\delta_j \Delta x_g \Delta r_f h_s v + |
\delta_j \Delta x_g \Delta r_f h_s v + |
| 462 |
\delta_k {\cal A}_c w & = & {\cal A}_c \delta_k (P-E)_{r=0} |
\delta_k {\cal A}_c w & = & {\cal A}_c \delta_k (P-E)_{r=0} |
| 479 |
The last equation, the discrete continuity equation, can be summed in |
The last equation, the discrete continuity equation, can be summed in |
| 480 |
the vertical to yeild the free-surface equation: |
the vertical to yeild the free-surface equation: |
| 481 |
\begin{equation} |
\begin{equation} |
| 482 |
{\cal A}_c \partial_t \eta + \delta_i \sum_k \Delta y_g \Delta r_f h_w u + \delta_j \sum_k \Delta x_g \Delta r_f h_s v = |
{\cal A}_c \partial_t \eta + \delta_i \sum_k \Delta y_g \Delta r_f h_w |
| 483 |
{\cal A}_c(P-E)_{r=0} |
u + \delta_j \sum_k \Delta x_g \Delta r_f h_s v = {\cal |
| 484 |
|
A}_c(P-E)_{r=0} \label{eq:discrete-freesurface} |
| 485 |
\end{equation} |
\end{equation} |
| 486 |
The source term $P-E$ on the rhs of continuity accounts for the local |
The source term $P-E$ on the rhs of continuity accounts for the local |
| 487 |
addition of volume due to excess precipitation and run-off over |
addition of volume due to excess precipitation and run-off over |
| 501 |
\begin{equation} |
\begin{equation} |
| 502 |
\epsilon_{nh} \partial_t w |
\epsilon_{nh} \partial_t w |
| 503 |
+ g \overline{\rho'}^k + \frac{1}{\Delta z} \delta_k \Phi_h' = \ldots |
+ g \overline{\rho'}^k + \frac{1}{\Delta z} \delta_k \Phi_h' = \ldots |
| 504 |
|
\label{eq:discrete_hydro_ocean} |
| 505 |
\end{equation} |
\end{equation} |
| 506 |
|
|
| 507 |
In the atmosphere, using p-coordinates, hydrostatic balance is |
In the atmosphere, using p-coordinates, hydrostatic balance is |
| 508 |
discretized: |
discretized: |
| 509 |
\begin{equation} |
\begin{equation} |
| 510 |
\overline{\theta'}^k + \frac{1}{\Delta \Pi} \delta_k \Phi_h' = 0 |
\overline{\theta'}^k + \frac{1}{\Delta \Pi} \delta_k \Phi_h' = 0 |
| 511 |
|
\label{eq:discrete_hydro_atmos} |
| 512 |
\end{equation} |
\end{equation} |
| 513 |
where $\Delta \Pi$ is the difference in Exner function between the |
where $\Delta \Pi$ is the difference in Exner function between the |
| 514 |
pressure points. The non-hydrostatic equations are not available in |
pressure points. The non-hydrostatic equations are not available in |
Parent Directory
| 
Revision Log
| 
Revision Graph
| 
Patch