/[MITgcm]/manual/s_algorithm/text/spatial-discrete.tex
ViewVC logotype

Diff of /manual/s_algorithm/text/spatial-discrete.tex

Parent Directory Parent Directory | Revision Log Revision Log | View Revision Graph Revision Graph | View Patch Patch

revision 1.5 by adcroft, Thu Aug 9 20:45:27 2001 UTC revision 1.7 by adcroft, Wed Sep 26 21:59:33 2001 UTC
# 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 115  grid data: ({\em model/inc/GRID.h}) Line 117  grid data: ({\em model/inc/GRID.h})
117  \subsection{Horizontal grid}  \subsection{Horizontal grid}
118    
119  \begin{figure}  \begin{figure}
120  \centerline{ \begin{tabular}{cc}  \begin{center}
121    \begin{tabular}{cc}
122    \raisebox{1.5in}{a)}\resizebox{!}{2in}{ \includegraphics{part2/hgrid-Ac.eps}}    \raisebox{1.5in}{a)}\resizebox{!}{2in}{ \includegraphics{part2/hgrid-Ac.eps}}
123  & \raisebox{1.5in}{b)}\resizebox{!}{2in}{ \includegraphics{part2/hgrid-Az.eps}}  & \raisebox{1.5in}{b)}\resizebox{!}{2in}{ \includegraphics{part2/hgrid-Az.eps}}
124  \\  \\
125    \raisebox{1.5in}{c)}\resizebox{!}{2in}{ \includegraphics{part2/hgrid-Au.eps}}    \raisebox{1.5in}{c)}\resizebox{!}{2in}{ \includegraphics{part2/hgrid-Au.eps}}
126  & \raisebox{1.5in}{d)}\resizebox{!}{2in}{ \includegraphics{part2/hgrid-Av.eps}}  & \raisebox{1.5in}{d)}\resizebox{!}{2in}{ \includegraphics{part2/hgrid-Av.eps}}
127  \end{tabular} }  \end{tabular}
128    \end{center}
129  \caption{  \caption{
130  Staggering of horizontal grid descriptors (lengths and areas). The  Staggering of horizontal grid descriptors (lengths and areas). The
131  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 317  other grids, the horizontal grid descrip
317  \subsection{Vertical grid}  \subsection{Vertical grid}
318    
319  \begin{figure}  \begin{figure}
320  \centerline{ \begin{tabular}{cc}  \begin{center}
321      \begin{tabular}{cc}
322    \raisebox{4in}{a)} \resizebox{!}{4in}{    \raisebox{4in}{a)} \resizebox{!}{4in}{
323    \includegraphics{part2/vgrid-cellcentered.eps}} & \raisebox{4in}{b)}    \includegraphics{part2/vgrid-cellcentered.eps}} & \raisebox{4in}{b)}
324    \resizebox{!}{4in}{ \includegraphics{part2/vgrid-accurate.eps}}    \resizebox{!}{4in}{ \includegraphics{part2/vgrid-accurate.eps}}
325  \end{tabular} }  \end{tabular}
326    \end{center}
327  \caption{Two versions of the vertical grid. a) The cell centered  \caption{Two versions of the vertical grid. a) The cell centered
328  approach where the interface depths are specified and the tracer  approach where the interface depths are specified and the tracer
329  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 388  $\Delta r_c^{-1}$: {\bf RECIP\_DRc} ({\e
388  \subsection{Topography: partially filled cells}  \subsection{Topography: partially filled cells}
389    
390  \begin{figure}  \begin{figure}
391  \centerline{  \begin{center}
392  \resizebox{4.5in}{!}{\includegraphics{part2/vgrid-xz.eps}}  \resizebox{4.5in}{!}{\includegraphics{part2/vgrid-xz.eps}}
393  }  \end{center}
394  \caption{  \caption{
395  A schematic of the x-r plane showing the location of the  A schematic of the x-r plane showing the location of the
396  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 460  $h_s^{-1}$: {\bf RECIP\_hFacS} ({\em GRI
460  The core algorithm is based on the ``C grid'' discretization of the  The core algorithm is based on the ``C grid'' discretization of the
461  continuity equation which can be summarized as:  continuity equation which can be summarized as:
462  \begin{eqnarray}  \begin{eqnarray}
463  \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} \\
464  \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} \\
465  \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} \\
466  \delta_i \Delta y_g \Delta r_f h_w u +  \delta_i \Delta y_g \Delta r_f h_w u +
467  \delta_j \Delta x_g \Delta r_f h_s v +  \delta_j \Delta x_g \Delta r_f h_s v +
468  \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 485  A}_c$.  The factors $h_w$ and $h_s$ are
485  The last equation, the discrete continuity equation, can be summed in  The last equation, the discrete continuity equation, can be summed in
486  the vertical to yeild the free-surface equation:  the vertical to yeild the free-surface equation:
487  \begin{equation}  \begin{equation}
488  {\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
489  {\cal A}_c(P-E)_{r=0}  u + \delta_j \sum_k \Delta x_g \Delta r_f h_s v = {\cal
490    A}_c(P-E)_{r=0} \label{eq:discrete-freesurface}
491  \end{equation}  \end{equation}
492  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
493  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 507  discretized:
507  \begin{equation}  \begin{equation}
508  \epsilon_{nh} \partial_t w  \epsilon_{nh} \partial_t w
509  + 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
510    \label{eq:discrete_hydro_ocean}
511  \end{equation}  \end{equation}
512    
513  In the atmosphere, using p-coordinates, hydrostatic balance is  In the atmosphere, using p-coordinates, hydrostatic balance is
514  discretized:  discretized:
515  \begin{equation}  \begin{equation}
516  \overline{\theta'}^k + \frac{1}{\Delta \Pi} \delta_k \Phi_h' = 0  \overline{\theta'}^k + \frac{1}{\Delta \Pi} \delta_k \Phi_h' = 0
517    \label{eq:discrete_hydro_atmos}
518  \end{equation}  \end{equation}
519  where $\Delta \Pi$ is the difference in Exner function between the  where $\Delta \Pi$ is the difference in Exner function between the
520  pressure points. The non-hydrostatic equations are not available in  pressure points. The non-hydrostatic equations are not available in
Legend:
Removed from v.1.5  
changed lines
  Added in v.1.7
  ViewVC Help
Powered by ViewVC 1.1.22