--- manual/s_algorithm/text/spatial-discrete.tex 2001/10/25 18:36:53 1.10 +++ 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 1.10 2001/10/25 18:36:53 cnh 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} @@ -14,6 +14,11 @@ \subsection{The finite volume method: finite volumes versus finite difference} +\begin{rawhtml} + +\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 +62,12 @@ 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 +84,7 @@ 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 @@ -400,7 +405,7 @@ \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 @@ -471,7 +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 @@ -524,7 +529,7 @@ 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