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

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-revision 1.4 by adcroft,
Thu Aug  9 19:48:39 2001 UTC
+revision 1.9 by cnh,
Wed Oct 24 15:21:27 2001 UTC
 Line 8 
 method. This amounts to a grid-point met
  centered finite difference) in the fluid interior but allows
  boundaries to intersect a regular grid allowing a more accurate
  representation of the position of the boundary. We treat the
- horizontal and veritical directions as seperable and thus slightly
+ horizontal and veritical directions as seperable and differently.
- differently.
- Initialization of grid data is controlled by subroutine {\em
+ \input{part2/notation}
- INI\_GRID} which in calls {\em INI\_VERTICAL\_GRID} to initialize the
- vertical grid, and then either of {\em INI\_CARTESIAN\_GRID}, {\em
- INI\_SPHERICAL\_POLAR\_GRID} or {\em INI\_CURV\-ILINEAR\_GRID} to
- initialize the horizontal grid for cartesian, spherical-polar or
- curvilinear coordinates respectively.
- The reciprocals of all grid quantities are pre-calculated and this is
- done in subroutine {\em INI\_MASKS\_ETC} which is called later by
- subroutine {\em INITIALIZE\_FIXED}.
- All grid descriptors are global arrays and stored in common blocks in
- {\em GRID.h} and a generally declared as {\em \_RS}.
- \fbox{ \begin{minipage}{4.75in}
- {\em S/R INI\_GRID} ({\em model/src/ini\_grid.F})
- {\em S/R INI\_MASKS\_ETC} ({\em model/src/ini\_masks\_etc.F})
- grid data: ({\em model/inc/GRID.h})
- \end{minipage} }
  \subsection{The finite volume method: finite volumes versus finite difference}
  The finite volume method is used to discretize the equations in
  space. The expression ``finite volume'' actually has two meanings; one
- is the method of cut or instecting boundaries (shaved or lopped cells
+ is the method of embedded or intersecting boundaries (shaved or lopped
- in our terminology) and the other is non-linear interpolation methods
+ cells in our terminology) and the other is non-linear interpolation
- that can deal with non-smooth solutions such as shocks (i.e. flux
+ methods that can deal with non-smooth solutions such as shocks
- 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
- conservation laws to which the {\it weak solution} is a solution on
+ of the conservation laws to which the {\it weak solution} is a
- each finite volume of (sub-domain). The weak solution can be
+ solution on each finite volume of (sub-domain). The weak solution can
- constructed outof piece-wise constant elements or be
+ be constructed out of piece-wise constant elements or be
  differentiable. The differentiable equations can not be satisfied by
  piece-wise constant functions.
-Line 88 
 directions simultaneously. Illustration
+Line 67 
 directions simultaneously. Illustration
  \subsection{C grid staggering of variables}
  \begin{figure}
- \centerline{ \resizebox{!}{2in}{ \includegraphics{part2/cgrid3d.eps}} }
+ \begin{center}
+ \resizebox{!}{2in}{ \includegraphics{part2/cgrid3d.eps}}
+ \end{center}
  \caption{Three dimensional staggering of velocity components. This
  facilitates the natural discretization of the continuity and tracer
  equations. }
-Line 108 
 synonymous with tracer cells (they are o
+Line 89 
 synonymous with tracer cells (they are o
+ \subsection{Grid initialization and data}
+ Initialization of grid data is controlled by subroutine {\em
+ INI\_GRID} which in calls {\em INI\_VERTICAL\_GRID} to initialize the
+ vertical grid, and then either of {\em INI\_CARTESIAN\_GRID}, {\em
+ INI\_SPHERICAL\_POLAR\_GRID} or {\em INI\_CURV\-ILINEAR\_GRID} to
+ initialize the horizontal grid for cartesian, spherical-polar or
+ curvilinear coordinates respectively.
+ The reciprocals of all grid quantities are pre-calculated and this is
+ done in subroutine {\em INI\_MASKS\_ETC} which is called later by
+ subroutine {\em INITIALIZE\_FIXED}.
+ All grid descriptors are global arrays and stored in common blocks in
+ {\em GRID.h} and a generally declared as {\em \_RS}.
+ \fbox{ \begin{minipage}{4.75in}
+ {\em S/R INI\_GRID} ({\em model/src/ini\_grid.F})
+ {\em S/R INI\_MASKS\_ETC} ({\em model/src/ini\_masks\_etc.F})
+ grid data: ({\em model/inc/GRID.h})
+ \end{minipage} }
  \subsection{Horizontal grid}
+ \label{sec:spatial_discrete_horizontal_grid}
  \begin{figure}
- \centerline{ \begin{tabular}{cc}
+ \begin{center}
+ \begin{tabular}{cc}
    \raisebox{1.5in}{a)}\resizebox{!}{2in}{ \includegraphics{part2/hgrid-Ac.eps}}
  & \raisebox{1.5in}{b)}\resizebox{!}{2in}{ \includegraphics{part2/hgrid-Az.eps}}
  \\
    \raisebox{1.5in}{c)}\resizebox{!}{2in}{ \includegraphics{part2/hgrid-Au.eps}}
  & \raisebox{1.5in}{d)}\resizebox{!}{2in}{ \includegraphics{part2/hgrid-Av.eps}}
- \end{tabular} }
+ \end{tabular}
+ \end{center}
  \caption{
  Staggering of horizontal grid descriptors (lengths and areas). The
  grid lines indicate the tracer cell boundaries and are the reference
-Line 310 
 other grids, the horizontal grid descrip
+Line 318 
 other grids, the horizontal grid descrip
  \subsection{Vertical grid}
  \begin{figure}
- \centerline{ \begin{tabular}{cc}
+ \begin{center}
+   \begin{tabular}{cc}
    \raisebox{4in}{a)} \resizebox{!}{4in}{
    \includegraphics{part2/vgrid-cellcentered.eps}} & \raisebox{4in}{b)}
    \resizebox{!}{4in}{ \includegraphics{part2/vgrid-accurate.eps}}
- \end{tabular} }
+ \end{tabular}
+ \end{center}
  \caption{Two versions of the vertical grid. a) The cell centered
  approach where the interface depths are specified and the tracer
  points centered in between the interfaces. b) The interface centered
-Line 379 
 $\Delta r_c^{-1}$: {\bf RECIP\_DRc} ({\e
+Line 389 
 $\Delta r_c^{-1}$: {\bf RECIP\_DRc} ({\e
  \subsection{Topography: partially filled cells}
  \begin{figure}
- \centerline{
+ \begin{center}
  \resizebox{4.5in}{!}{\includegraphics{part2/vgrid-xz.eps}}
- }
+ \end{center}
  \caption{
  A schematic of the x-r plane showing the location of the
  non-dimensional fractions $h_c$ and $h_w$. The physical thickness of a
-Line 446 
 $h_s^{-1}$: {\bf RECIP\_hFacS} ({\em GRI
+Line 456 
 $h_s^{-1}$: {\bf RECIP\_hFacS} ({\em GRI
  \end{minipage} }
- \subsection{Continuity and horizontal pressure gradient terms}
+ \section{Continuity and horizontal pressure gradient terms}
  The core algorithm is based on the ``C grid'' discretization of the
  continuity equation which can be summarized as:
  \begin{eqnarray}
- \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} \\
- \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} \\
- \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} \\
  \delta_i \Delta y_g \Delta r_f h_w u +
  \delta_j \Delta x_g \Delta r_f h_s v +
  \delta_k {\cal A}_c w & = & {\cal A}_c \delta_k (P-E)_{r=0}
-Line 476 
 A}_c$.  The factors $h_w$ and $h_s$ are
+Line 486 
 A}_c$.  The factors $h_w$ and $h_s$ are
  The last equation, the discrete continuity equation, can be summed in
  the vertical to yeild the free-surface equation:
  \begin{equation}
- {\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
- {\cal A}_c(P-E)_{r=0}
+ u + \delta_j \sum_k \Delta x_g \Delta r_f h_s v = {\cal
+ A}_c(P-E)_{r=0} \label{eq:discrete-freesurface}
  \end{equation}
  The source term $P-E$ on the rhs of continuity accounts for the local
  addition of volume due to excess precipitation and run-off over
  evaporation and only enters the top-level of the {\em ocean} model.
- \subsection{Hydrostatic balance}
+ \section{Hydrostatic balance}
  The vertical momentum equation has the hydrostatic or
  quasi-hydrostatic balance on the right hand side. This discretization
-Line 497 
 discretized:
+Line 508 
 discretized:
  \begin{equation}
  \epsilon_{nh} \partial_t w
  + g \overline{\rho'}^k + \frac{1}{\Delta z} \delta_k \Phi_h' = \ldots
+ \label{eq:discrete_hydro_ocean}
  \end{equation}
  In the atmosphere, using p-coordinates, hydrostatic balance is
  discretized:
  \begin{equation}
  \overline{\theta'}^k + \frac{1}{\Delta \Pi} \delta_k \Phi_h' = 0
+ \label{eq:discrete_hydro_atmos}
  \end{equation}
  where $\Delta \Pi$ is the difference in Exner function between the
  pressure points. The non-hydrostatic equations are not available in

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