--- manual/s_algorithm/text/spatial-discrete.tex	2001/10/25 18:36:53	1.10
+++ manual/s_algorithm/text/spatial-discrete.tex	2006/04/05 01:16:27	1.19
@@ -1,7 +1,10 @@
-% $Header: /home/ubuntu/mnt/e9_copy/manual/s_algorithm/text/spatial-discrete.tex,v 1.10 2001/10/25 18:36:53 cnh Exp $
+% $Header: /home/ubuntu/mnt/e9_copy/manual/s_algorithm/text/spatial-discrete.tex,v 1.19 2006/04/05 01:16:27 jmc Exp $
 % $Name:  $
 
 \section{Spatial discretization of the dynamical equations}
+\begin{rawhtml}
+<!-- CMIREDIR:spatial_discretization_of_dyn_eq: -->
+\end{rawhtml}
 
 Spatial discretization is carried out using the finite volume
 method. This amounts to a grid-point method (namely second-order
@@ -14,6 +17,11 @@
 
 
 \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
@@ -57,12 +65,12 @@
 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 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}
 
@@ -79,7 +87,7 @@
 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
@@ -142,8 +150,8 @@
 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 parallelized
-or not; see section \ref{sect:tiles} for more details. Although the
-tiles may be patched together in an unstructured manner
+or not; see section \ref{sect:domain_decomposition} 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 structured grid of quadrilateral cells. The horizontal coordinate
 system is orthogonal curvilinear meaning we can not necessarily treat
@@ -361,15 +369,23 @@
 
 The above grid (Fig.~\ref{fig:vgrid}a) is known as the cell centered
 approach because the tracer points are at cell centers; the cell
-centers are mid-way between the cell interfaces. An alternative, the
-vertex or interface centered approach, is shown in
+centers are mid-way between the cell interfaces. 
+This discretization is selected when the thickness of the
+levels are provided ({\bf delR}, parameter file {\em data}, 
+namelist {\em PARM04})
+An alternative, the vertex or interface centered approach, is shown in
 Fig.~\ref{fig:vgrid}b. Here, the interior interfaces are positioned
 mid-way between the tracer nodes (no longer cell centers). This
 approach is formally more accurate for evaluation of hydrostatic
 pressure and vertical advection but historically the cell centered
 approach has been used. An alternative form of subroutine {\em
 INI\_VERTICAL\_GRID} is used to select the interface centered approach
-but no run time option is currently available.
+This form requires to specify $Nr+1$ vertical distances {\bf delRc} 
+(parameter file {\em data}, namelist {\em PARM04}, e.g. 
+{\em verification/ideal\_2D\_oce/input/data})
+corresponding to surface to center, $Nr-1$ center to center, and center to 
+bottom distances.
+%but no run time option is currently available.
 
 \fbox{ \begin{minipage}{4.75in}
 {\em S/R INI\_VERTICAL\_GRID} ({\em
@@ -387,6 +403,9 @@
 
 
 \subsection{Topography: partially filled cells}
+\begin{rawhtml}
+<!-- CMIREDIR:topo_partial_cells: -->
+\end{rawhtml}
 
 \begin{figure}
 \begin{center}
@@ -400,7 +419,7 @@
 \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
@@ -457,6 +476,10 @@
 
 
 \section{Continuity and horizontal pressure gradient terms}
+\begin{rawhtml}
+<!-- CMIREDIR:continuity_and_horizontal_pressure: -->
+\end{rawhtml}
+
 
 The core algorithm is based on the ``C grid'' discretization of the
 continuity equation which can be summarized as:
@@ -471,7 +494,7 @@
 \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
@@ -495,6 +518,9 @@
 evaporation and only enters the top-level of the {\em ocean} model.
 
 \section{Hydrostatic balance}
+\begin{rawhtml}
+<!-- CMIREDIR:hydrostatic_balance: -->
+\end{rawhtml}
 
 The vertical momentum equation has the hydrostatic or
 quasi-hydrostatic balance on the right hand side. This discretization
@@ -524,7 +550,7 @@
 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 conversion
-terms is discussed at length in \cite{Adcroft01}.
+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
