--- manual/s_algorithm/text/spatial-discrete.tex	2001/09/25 20:13:42	1.6
+++ manual/s_algorithm/text/spatial-discrete.tex	2004/03/23 16:47:04	1.14
@@ -1,4 +1,4 @@
-% $Header: /home/ubuntu/mnt/e9_copy/manual/s_algorithm/text/spatial-discrete.tex,v 1.6 2001/09/25 20:13:42 adcroft Exp $
+% $Header: /home/ubuntu/mnt/e9_copy/manual/s_algorithm/text/spatial-discrete.tex,v 1.14 2004/03/23 16:47:04 afe Exp $
 % $Name:  $
 
 \section{Spatial discretization of the dynamical equations}
@@ -8,22 +8,27 @@
 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 differently.
+horizontal and vertical directions as separable and differently.
 
 \input{part2/notation}
 
 
 \subsection{The finite volume method: finite volumes versus finite difference}
+\begin{rawhtml}
+<!-- CMIREDIR:finite_volume: -->
+\end{rawhtml}
+
+
 
 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.
 
@@ -37,7 +42,7 @@
 \begin{displaymath}
 \Delta x \partial_t \theta + \delta_i ( F ) = 0
 \end{displaymath}
-is exact if $\theta(x)$ is peice-wise constant over the interval
+is exact if $\theta(x)$ is piece-wise constant over the interval
 $\Delta x_i$ or more generally if $\theta_i$ is defined as the average
 over the interval $\Delta x_i$.
 
@@ -57,17 +62,19 @@
 interior of a fluid. Differences arise at boundaries where a boundary
 is not positioned on a regular or smoothly varying grid. This method
 is used to represent the topography using lopped cell, see
-\cite{Adcroft98}. Subtle difference also appear in more than one
+\cite{adcroft:97}. Subtle difference also appear in more than one
 dimension away from boundaries. This happens because the each
-direction is discretized independantly in the finite difference method
+direction is discretized independently in the finite difference method
 while the integrating over finite volume implicitly treats all
 directions simultaneously. Illustration of this is given in
-\cite{Adcroft02}.
+\cite{ac:02}.
 
 \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. }
@@ -77,12 +84,12 @@
 The basic algorithm employed for stepping forward the momentum
 equations is based on retaining non-divergence of the flow at all
 times. This is most naturally done if the components of flow are
-staggered in space in the form of an Arakawa C grid \cite{Arakawa70}.
+staggered in space in the form of an Arakawa C grid \cite{arakawa:77}.
 
 Fig. \ref{fig:cgrid3d} shows the components of flow ($u$,$v$,$w$)
 staggered in space such that the zonal component falls on the
-interface between continiuty cells in the zonal direction. Similarly
-for the meridional and vertical directions.  The continiuty cell is
+interface between continuity cells in the zonal direction. Similarly
+for the meridional and vertical directions.  The continuity cell is
 synonymous with tracer cells (they are one and the same).
 
 
@@ -113,15 +120,18 @@
 
 
 \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
@@ -136,20 +146,20 @@
 
 The model domain is decomposed into tiles and within each tile a
 quasi-regular grid is used. A tile is the basic unit of domain
-decomposition for parallelization but may be used whether parallized
+decomposition for parallelization but may be used whether parallelized
 or not; see section \ref{sect:tiles} for more details. Although the
 tiles may be patched together in an unstructured manner
 (i.e. irregular or non-tessilating pattern), the interior of tiles is
-a structered grid of quadrilateral cells. The horizontal coordinate
+a structured grid of quadrilateral cells. The horizontal coordinate
 system is orthogonal curvilinear meaning we can not necessarily treat
-the two horizontal directions as seperable. Instead, each cell in the
+the two horizontal directions as separable. Instead, each cell in the
 horizontal grid is described by the length of it's sides and it's
 area.
 
 The grid information is quite general and describes any of the
 available coordinates systems, cartesian, spherical-polar or
 curvilinear. All that is necessary to distinguish between the
-coordinate systems is to initialize the grid data (discriptors)
+coordinate systems is to initialize the grid data (descriptors)
 appropriately.
 
 In the following, we refer to the orientation of quantities on the
@@ -288,7 +298,7 @@
 spacing can be set to uniform via scalars {\bf dXspacing} and {\bf
 dYspacing} in namelist {\em PARM04} or to variable resolution by the
 vectors {\bf DELX} and {\bf DELY}. Units are normally
-meters. Non-dimensional coordinates can be used by interpretting the
+meters. Non-dimensional coordinates can be used by interpreting the
 gravitational constant as the Rayleigh number.
 
 \subsubsection{Spherical-polar coordinates}
@@ -313,11 +323,13 @@
 \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
@@ -340,7 +352,7 @@
 INI\_VERTICAL\_GRID} and specified via the vector {\bf DELR} in
 namelist {\em PARM04}. The units of ``r'' are either meters or Pascals
 depending on the isomorphism being used which in turn is dependent
-only on the choise of equation of state.
+only on the choice of equation of state.
 
 There are alternative namelist vectors {\bf DELZ} and {\bf DELP} which
 dictate whether z- or
@@ -382,9 +394,9 @@
 \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
@@ -393,13 +405,13 @@
 \label{fig:hfacs}
 \end{figure}
 
-\cite{Adcroft97} presented two alternatives to the step-wise finite
+\cite{adcroft:97} presented two alternatives to the step-wise finite
 difference representation of topography. The method is known to the
 engineering community as {\em intersecting boundary method}. It
 involves allowing the boundary to intersect a grid of cells thereby
 modifying the shape of those cells intersected. We suggested allowing
-the topgoraphy to take on a peice-wise linear representation (shaved
-cells) or a simpler piecewise constant representaion (partial step).
+the topography to take on a piece-wise linear representation (shaved
+cells) or a simpler piecewise constant representation (partial step).
 Both show dramatic improvements in solution compared to the
 traditional full step representation, the piece-wise linear being the
 best. However, the storage requirements are excessive so the simpler
@@ -413,7 +425,7 @@
 \marginpar{$h_s$: {\bf hFacS}}
 The physical thickness of a tracer cell is given by $h_c(i,j,k) \Delta
 r_f(k)$ and the physical thickness of the open side is given by
-$h_w(i,j,k) \Delta r_f(k)$. Three 3-D discriptors $h_c$, $h_w$ and
+$h_w(i,j,k) \Delta r_f(k)$. Three 3-D descriptors $h_c$, $h_w$ and
 $h_s$ are used to describe the geometry: {\bf hFacC}, {\bf hFacW} and
 {\bf hFacS} respectively. These are calculated in subroutine {\em
 INI\_MASKS\_ETC} along with there reciprocals {\bf RECIP\_hFacC}, {\bf
@@ -464,7 +476,7 @@
 \end{eqnarray}
 where the continuity equation has been most naturally discretized by
 staggering the three components of velocity as shown in
-Fig.~\ref{fig-cgrid3d}. The grid lengths $\Delta x_c$ and $\Delta y_c$
+Fig.~\ref{fig:cgrid3d}. The grid lengths $\Delta x_c$ and $\Delta y_c$
 are the lengths between tracer points (cell centers). The grid lengths
 $\Delta x_g$, $\Delta y_g$ are the grid lengths between cell
 corners. $\Delta r_f$ and $\Delta r_c$ are the distance (in units of
@@ -477,7 +489,7 @@
 \marginpar{$h_s$: {\bf hFacS}}
 
 The last equation, the discrete continuity equation, can be summed in
-the vertical to yeild the free-surface equation:
+the vertical to yield 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
@@ -496,7 +508,7 @@
 from the pressure gradient terms when forming the kinetic energy
 equation.
 
-In the ocean, using z-ccordinates, the hydrostatic balance terms are
+In the ocean, using z-coordinates, the hydrostatic balance terms are
 discretized:
 \begin{equation}
 \epsilon_{nh} \partial_t w
@@ -516,8 +528,8 @@
 
 The difference in approach between ocean and atmosphere occurs because
 of the direct use of the ideal gas equation in forming the potential
-energy conversion term $\alpha \omega$. The form of these consversion
-terms is discussed at length in \cite{Adcroft01}.
+energy conversion term $\alpha \omega$. The form of these conversion
+terms is discussed at length in \cite{adcroft:02}.
 
 Because of the different representation of hydrostatic balance between
 ocean and atmosphere there is no elegant way to represent both systems
