--- manual/s_algorithm/text/spatial-discrete.tex 2004/10/13 18:50:54 1.16 +++ manual/s_algorithm/text/spatial-discrete.tex 2017/06/14 21:05:01 1.25 @@ -1,7 +1,10 @@ -% $Header: /home/ubuntu/mnt/e9_copy/manual/s_algorithm/text/spatial-discrete.tex,v 1.16 2004/10/13 18:50:54 jmc Exp $ +% $Header: /home/ubuntu/mnt/e9_copy/manual/s_algorithm/text/spatial-discrete.tex,v 1.25 2017/06/14 21:05:01 jmc Exp $ % $Name: $ \section{Spatial discretization of the dynamical equations} +\begin{rawhtml} + +\end{rawhtml} Spatial discretization is carried out using the finite volume method. This amounts to a grid-point method (namely second-order @@ -10,8 +13,6 @@ representation of the position of the boundary. We treat the horizontal and vertical directions as separable and differently. -\input{part2/notation} - \subsection{The finite volume method: finite volumes versus finite difference} \begin{rawhtml} @@ -73,7 +74,7 @@ \begin{figure} \begin{center} -\resizebox{!}{2in}{ \includegraphics{part2/cgrid3d.eps}} +\resizebox{!}{2in}{ \includegraphics{s_algorithm/figs/cgrid3d.eps}} \end{center} \caption{Three dimensional staggering of velocity components. This facilitates the natural discretization of the continuity and tracer @@ -125,11 +126,11 @@ \begin{figure} \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}{a)}\resizebox{!}{2in}{ \includegraphics{s_algorithm/figs/hgrid-Ac.eps}} +& \raisebox{1.5in}{b)}\resizebox{!}{2in}{ \includegraphics{s_algorithm/figs/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}} + \raisebox{1.5in}{c)}\resizebox{!}{2in}{ \includegraphics{s_algorithm/figs/hgrid-Au.eps}} +& \raisebox{1.5in}{d)}\resizebox{!}{2in}{ \includegraphics{s_algorithm/figs/hgrid-Av.eps}} \end{tabular} \end{center} \caption{ @@ -138,8 +139,8 @@ grid for all panels. a) The area of a tracer cell, $A_c$, is bordered by the lengths $\Delta x_g$ and $\Delta y_g$. b) The area of a vorticity cell, $A_\zeta$, is bordered by the lengths $\Delta x_c$ and -$\Delta y_c$. c) The area of a u cell, $A_c$, is bordered by the -lengths $\Delta x_v$ and $\Delta y_f$. d) The area of a v cell, $A_c$, +$\Delta y_c$. c) The area of a u cell, $A_w$, is bordered by the +lengths $\Delta x_v$ and $\Delta y_f$. d) The area of a v cell, $A_s$, is bordered by the lengths $\Delta x_f$ and $\Delta y_u$.} \label{fig:hgrid} \end{figure} @@ -147,8 +148,8 @@ 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{sec: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 @@ -166,7 +167,7 @@ computational grid using geographic terminology such as points of the compass. \marginpar{Caution!} -This is purely for convenience but should note be confused +This is purely for convenience but should not be confused with the actual geographic orientation of model quantities. Fig.~\ref{fig:hgrid}a shows the tracer cell (synonymous with the @@ -185,7 +186,7 @@ Fig.~\ref{fig:hgrid}b shows the vorticity cell. The length of the southern edge, $\Delta x_c$, western edge, $\Delta y_c$ and surface area, $A_\zeta$, presented in the vertical are stored in arrays {\bf -DXg}, {\bf DYg} and {\bf rAz}. +DXc}, {\bf DYc} and {\bf rAz}. \marginpar{$A_\zeta$: {\bf rAz}} \marginpar{$\Delta x_c$: {\bf DXc}} \marginpar{$\Delta y_c$: {\bf DYc}} @@ -193,7 +194,7 @@ cell centers and the ``$\zeta$'' suffix associates points with the vorticity points. The quantities are staggered in space and the indexing is such that {\bf DXc(i,j)} is positioned to the north of -{\bf rAc(i,j)} and {\bf DYc(i,j)} positioned to the east. +{\bf rAz(i,j)} and {\bf DYc(i,j)} positioned to the east. Fig.~\ref{fig:hgrid}c shows the ``u'' or western (w) cell. The length of the southern edge, $\Delta x_v$, eastern edge, $\Delta y_f$ and @@ -326,8 +327,8 @@ \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}} + \includegraphics{s_algorithm/figs/vgrid-cellcentered.eps}} & \raisebox{4in}{b)} + \resizebox{!}{4in}{ \includegraphics{s_algorithm/figs/vgrid-accurate.eps}} \end{tabular} \end{center} \caption{Two versions of the vertical grid. a) The cell centered @@ -367,7 +368,7 @@ 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. -This discretisation is selected when the thickness of the +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 @@ -406,7 +407,7 @@ \begin{figure} \begin{center} -\resizebox{4.5in}{!}{\includegraphics{part2/vgrid-xz.eps}} +\resizebox{4.5in}{!}{\includegraphics{s_algorithm/figs/vgrid-xz.eps}} \end{center} \caption{ A schematic of the x-r plane showing the location of the @@ -473,6 +474,10 @@ \section{Continuity and horizontal pressure gradient terms} +\begin{rawhtml} + +\end{rawhtml} + The core algorithm is based on the ``C grid'' discretization of the continuity equation which can be summarized as: @@ -511,6 +516,9 @@ evaporation and only enters the top-level of the {\em ocean} model. \section{Hydrostatic balance} +\begin{rawhtml} + +\end{rawhtml} The vertical momentum equation has the hydrostatic or quasi-hydrostatic balance on the right hand side. This discretization