--- manual/s_algorithm/text/spatial-discrete.tex	2001/08/09 20:45:27	1.5
+++ manual/s_algorithm/text/spatial-discrete.tex	2001/09/28 14:09:55	1.8
@@ -1,4 +1,4 @@
-% $Header: /home/ubuntu/mnt/e9_copy/manual/s_algorithm/text/spatial-discrete.tex,v 1.5 2001/08/09 20:45:27 adcroft Exp $
+% $Header: /home/ubuntu/mnt/e9_copy/manual/s_algorithm/text/spatial-discrete.tex,v 1.8 2001/09/28 14:09:55 adcroft Exp $
 % $Name:  $
 
 \section{Spatial discretization of the dynamical equations}
@@ -17,13 +17,13 @@
 
 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
-in our terminology) and the other is non-linear interpolation methods
-that can deal with non-smooth solutions such as shocks (i.e. flux
-limiters for advection). Both make use of the integral form of the
-conservation laws to which the {\it weak solution} is a solution on
-each finite volume of (sub-domain). The weak solution can be
-constructed outof piece-wise constant elements or be
+is the method of embedded or intersecting boundaries (shaved or lopped
+cells in our terminology) and the other is non-linear interpolation
+methods that can deal with non-smooth solutions such as shocks
+(i.e. flux limiters for advection). Both make use of the integral form
+of the conservation laws to which the {\it weak solution} is a
+solution on each finite volume of (sub-domain). The weak solution can
+be constructed out of piece-wise constant elements or be
 differentiable. The differentiable equations can not be satisfied by
 piece-wise constant functions.
 
@@ -67,7 +67,9 @@
 \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. }
@@ -115,13 +117,15 @@
 \subsection{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
@@ -313,11 +317,13 @@
 \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
@@ -382,9 +388,9 @@
 \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
@@ -454,9 +460,9 @@
 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 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' \\
-\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}' \\
+\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' \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}' \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}
@@ -479,8 +485,9 @@
 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(P-E)_{r=0}
+{\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(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
@@ -500,12 +507,14 @@
 \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
