| 8 |
centered finite difference) in the fluid interior but allows |
centered finite difference) in the fluid interior but allows |
| 9 |
boundaries to intersect a regular grid allowing a more accurate |
boundaries to intersect a regular grid allowing a more accurate |
| 10 |
representation of the position of the boundary. We treat the |
representation of the position of the boundary. We treat the |
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horizontal and veritical directions as seperable and thus slightly |
horizontal and veritical directions as seperable and differently. |
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differently. |
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| 12 |
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| 13 |
Initialization of grid data is controlled by subroutine {\em |
\input{part2/notation} |
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INI\_GRID} which in calls {\em INI\_VERTICAL\_GRID} to initialize the |
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vertical grid, and then either of {\em INI\_CARTESIAN\_GRID}, {\em |
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INI\_SPHERICAL\_POLAR\_GRID} or {\em INI\_CURV\-ILINEAR\_GRID} to |
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initialize the horizontal grid for cartesian, spherical-polar or |
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curvilinear coordinates respectively. |
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The reciprocals of all grid quantities are pre-calculated and this is |
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done in subroutine {\em INI\_MASKS\_ETC} which is called later by |
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subroutine {\em INITIALIZE\_FIXED}. |
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All grid descriptors are global arrays and stored in common blocks in |
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{\em GRID.h} and a generally declared as {\em \_RS}. |
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\fbox{ \begin{minipage}{4.75in} |
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{\em S/R INI\_GRID} ({\em model/src/ini\_grid.F}) |
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{\em S/R INI\_MASKS\_ETC} ({\em model/src/ini\_masks\_etc.F}) |
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grid data: ({\em model/inc/GRID.h}) |
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\end{minipage} } |
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| 14 |
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| 15 |
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| 16 |
\subsection{The finite volume method: finite volumes versus finite difference} |
\subsection{The finite volume method: finite volumes versus finite difference} |
| 87 |
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| 88 |
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| 89 |
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| 90 |
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\subsection{Grid initialization and data} |
| 91 |
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| 92 |
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Initialization of grid data is controlled by subroutine {\em |
| 93 |
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INI\_GRID} which in calls {\em INI\_VERTICAL\_GRID} to initialize the |
| 94 |
|
vertical grid, and then either of {\em INI\_CARTESIAN\_GRID}, {\em |
| 95 |
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INI\_SPHERICAL\_POLAR\_GRID} or {\em INI\_CURV\-ILINEAR\_GRID} to |
| 96 |
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initialize the horizontal grid for cartesian, spherical-polar or |
| 97 |
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curvilinear coordinates respectively. |
| 98 |
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|
| 99 |
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The reciprocals of all grid quantities are pre-calculated and this is |
| 100 |
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done in subroutine {\em INI\_MASKS\_ETC} which is called later by |
| 101 |
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subroutine {\em INITIALIZE\_FIXED}. |
| 102 |
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| 103 |
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All grid descriptors are global arrays and stored in common blocks in |
| 104 |
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{\em GRID.h} and a generally declared as {\em \_RS}. |
| 105 |
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| 106 |
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\fbox{ \begin{minipage}{4.75in} |
| 107 |
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{\em S/R INI\_GRID} ({\em model/src/ini\_grid.F}) |
| 108 |
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| 109 |
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{\em S/R INI\_MASKS\_ETC} ({\em model/src/ini\_masks\_etc.F}) |
| 110 |
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| 111 |
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grid data: ({\em model/inc/GRID.h}) |
| 112 |
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\end{minipage} } |
| 113 |
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| 114 |
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| 115 |
\subsection{Horizontal grid} |
\subsection{Horizontal grid} |
| 116 |
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| 449 |
\end{minipage} } |
\end{minipage} } |
| 450 |
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| 451 |
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| 452 |
\subsection{Continuity and horizontal pressure gradient terms} |
\section{Continuity and horizontal pressure gradient terms} |
| 453 |
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|
| 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 |
| 488 |
evaporation and only enters the top-level of the {\em ocean} model. |
evaporation and only enters the top-level of the {\em ocean} model. |
| 489 |
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| 490 |
\subsection{Hydrostatic balance} |
\section{Hydrostatic balance} |
| 491 |
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|
| 492 |
The vertical momentum equation has the hydrostatic or |
The vertical momentum equation has the hydrostatic or |
| 493 |
quasi-hydrostatic balance on the right hand side. This discretization |
quasi-hydrostatic balance on the right hand side. This discretization |
| 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 |
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\label{eq:discrete_hydro_ocean} |
| 505 |
\end{equation} |
\end{equation} |
| 506 |
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|
| 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 |
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\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 |
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