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revision 1.5 by adcroft, Thu Aug 9 20:45:27 2001 UTC revision 1.9 by cnh, Wed Oct 24 15:21:27 2001 UTC
# Line 17  horizontal and veritical directions as s Line 17  horizontal and veritical directions as s
17    
18  The finite volume method is used to discretize the equations in  The finite volume method is used to discretize the equations in
19  space. The expression ``finite volume'' actually has two meanings; one  space. The expression ``finite volume'' actually has two meanings; one
20  is the method of cut or instecting boundaries (shaved or lopped cells  is the method of embedded or intersecting boundaries (shaved or lopped
21  in our terminology) and the other is non-linear interpolation methods  cells in our terminology) and the other is non-linear interpolation
22  that can deal with non-smooth solutions such as shocks (i.e. flux  methods that can deal with non-smooth solutions such as shocks
23  limiters for advection). Both make use of the integral form of the  (i.e. flux limiters for advection). Both make use of the integral form
24  conservation laws to which the {\it weak solution} is a solution on  of the conservation laws to which the {\it weak solution} is a
25  each finite volume of (sub-domain). The weak solution can be  solution on each finite volume of (sub-domain). The weak solution can
26  constructed outof piece-wise constant elements or be  be constructed out of piece-wise constant elements or be
27  differentiable. The differentiable equations can not be satisfied by  differentiable. The differentiable equations can not be satisfied by
28  piece-wise constant functions.  piece-wise constant functions.
29    
# Line 67  directions simultaneously. Illustration Line 67  directions simultaneously. Illustration
67  \subsection{C grid staggering of variables}  \subsection{C grid staggering of variables}
68    
69  \begin{figure}  \begin{figure}
70  \centerline{ \resizebox{!}{2in}{ \includegraphics{part2/cgrid3d.eps}} }  \begin{center}
71    \resizebox{!}{2in}{ \includegraphics{part2/cgrid3d.eps}}
72    \end{center}
73  \caption{Three dimensional staggering of velocity components. This  \caption{Three dimensional staggering of velocity components. This
74  facilitates the natural discretization of the continuity and tracer  facilitates the natural discretization of the continuity and tracer
75  equations. }  equations. }
# Line 113  grid data: ({\em model/inc/GRID.h}) Line 115  grid data: ({\em model/inc/GRID.h})
115    
116    
117  \subsection{Horizontal grid}  \subsection{Horizontal grid}
118    \label{sec:spatial_discrete_horizontal_grid}
119    
120  \begin{figure}  \begin{figure}
121  \centerline{ \begin{tabular}{cc}  \begin{center}
122    \begin{tabular}{cc}
123    \raisebox{1.5in}{a)}\resizebox{!}{2in}{ \includegraphics{part2/hgrid-Ac.eps}}    \raisebox{1.5in}{a)}\resizebox{!}{2in}{ \includegraphics{part2/hgrid-Ac.eps}}
124  & \raisebox{1.5in}{b)}\resizebox{!}{2in}{ \includegraphics{part2/hgrid-Az.eps}}  & \raisebox{1.5in}{b)}\resizebox{!}{2in}{ \includegraphics{part2/hgrid-Az.eps}}
125  \\  \\
126    \raisebox{1.5in}{c)}\resizebox{!}{2in}{ \includegraphics{part2/hgrid-Au.eps}}    \raisebox{1.5in}{c)}\resizebox{!}{2in}{ \includegraphics{part2/hgrid-Au.eps}}
127  & \raisebox{1.5in}{d)}\resizebox{!}{2in}{ \includegraphics{part2/hgrid-Av.eps}}  & \raisebox{1.5in}{d)}\resizebox{!}{2in}{ \includegraphics{part2/hgrid-Av.eps}}
128  \end{tabular} }  \end{tabular}
129    \end{center}
130  \caption{  \caption{
131  Staggering of horizontal grid descriptors (lengths and areas). The  Staggering of horizontal grid descriptors (lengths and areas). The
132  grid lines indicate the tracer cell boundaries and are the reference  grid lines indicate the tracer cell boundaries and are the reference
# Line 313  other grids, the horizontal grid descrip Line 318  other grids, the horizontal grid descrip
318  \subsection{Vertical grid}  \subsection{Vertical grid}
319    
320  \begin{figure}  \begin{figure}
321  \centerline{ \begin{tabular}{cc}  \begin{center}
322      \begin{tabular}{cc}
323    \raisebox{4in}{a)} \resizebox{!}{4in}{    \raisebox{4in}{a)} \resizebox{!}{4in}{
324    \includegraphics{part2/vgrid-cellcentered.eps}} & \raisebox{4in}{b)}    \includegraphics{part2/vgrid-cellcentered.eps}} & \raisebox{4in}{b)}
325    \resizebox{!}{4in}{ \includegraphics{part2/vgrid-accurate.eps}}    \resizebox{!}{4in}{ \includegraphics{part2/vgrid-accurate.eps}}
326  \end{tabular} }  \end{tabular}
327    \end{center}
328  \caption{Two versions of the vertical grid. a) The cell centered  \caption{Two versions of the vertical grid. a) The cell centered
329  approach where the interface depths are specified and the tracer  approach where the interface depths are specified and the tracer
330  points centered in between the interfaces. b) The interface centered  points centered in between the interfaces. b) The interface centered
# Line 382  $\Delta r_c^{-1}$: {\bf RECIP\_DRc} ({\e Line 389  $\Delta r_c^{-1}$: {\bf RECIP\_DRc} ({\e
389  \subsection{Topography: partially filled cells}  \subsection{Topography: partially filled cells}
390    
391  \begin{figure}  \begin{figure}
392  \centerline{  \begin{center}
393  \resizebox{4.5in}{!}{\includegraphics{part2/vgrid-xz.eps}}  \resizebox{4.5in}{!}{\includegraphics{part2/vgrid-xz.eps}}
394  }  \end{center}
395  \caption{  \caption{
396  A schematic of the x-r plane showing the location of the  A schematic of the x-r plane showing the location of the
397  non-dimensional fractions $h_c$ and $h_w$. The physical thickness of a  non-dimensional fractions $h_c$ and $h_w$. The physical thickness of a
# Line 454  $h_s^{-1}$: {\bf RECIP\_hFacS} ({\em GRI Line 461  $h_s^{-1}$: {\bf RECIP\_hFacS} ({\em GRI
461  The core algorithm is based on the ``C grid'' discretization of the  The core algorithm is based on the ``C grid'' discretization of the
462  continuity equation which can be summarized as:  continuity equation which can be summarized as:
463  \begin{eqnarray}  \begin{eqnarray}
464  \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} \\
465  \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} \\
466  \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} \\
467  \delta_i \Delta y_g \Delta r_f h_w u +  \delta_i \Delta y_g \Delta r_f h_w u +
468  \delta_j \Delta x_g \Delta r_f h_s v +  \delta_j \Delta x_g \Delta r_f h_s v +
469  \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}
# Line 479  A}_c$.  The factors $h_w$ and $h_s$ are Line 486  A}_c$.  The factors $h_w$ and $h_s$ are
486  The last equation, the discrete continuity equation, can be summed in  The last equation, the discrete continuity equation, can be summed in
487  the vertical to yeild the free-surface equation:  the vertical to yeild the free-surface equation:
488  \begin{equation}  \begin{equation}
489  {\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
490  {\cal A}_c(P-E)_{r=0}  u + \delta_j \sum_k \Delta x_g \Delta r_f h_s v = {\cal
491    A}_c(P-E)_{r=0} \label{eq:discrete-freesurface}
492  \end{equation}  \end{equation}
493  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
494  addition of volume due to excess precipitation and run-off over  addition of volume due to excess precipitation and run-off over
# Line 500  discretized: Line 508  discretized:
508  \begin{equation}  \begin{equation}
509  \epsilon_{nh} \partial_t w  \epsilon_{nh} \partial_t w
510  + 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
511    \label{eq:discrete_hydro_ocean}
512  \end{equation}  \end{equation}
513    
514  In the atmosphere, using p-coordinates, hydrostatic balance is  In the atmosphere, using p-coordinates, hydrostatic balance is
515  discretized:  discretized:
516  \begin{equation}  \begin{equation}
517  \overline{\theta'}^k + \frac{1}{\Delta \Pi} \delta_k \Phi_h' = 0  \overline{\theta'}^k + \frac{1}{\Delta \Pi} \delta_k \Phi_h' = 0
518    \label{eq:discrete_hydro_atmos}
519  \end{equation}  \end{equation}
520  where $\Delta \Pi$ is the difference in Exner function between the  where $\Delta \Pi$ is the difference in Exner function between the
521  pressure points. The non-hydrostatic equations are not available in  pressure points. The non-hydrostatic equations are not available in
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