/[MITgcm]/manual/s_algorithm/text/spatial-discrete.tex
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revision 1.10 by cnh, Thu Oct 25 18:36:53 2001 UTC revision 1.13 by afe, Tue Mar 23 15:29:40 2004 UTC
# Line 14  horizontal and vertical directions as se Line 14  horizontal and vertical directions as se
14    
15    
16  \subsection{The finite volume method: finite volumes versus finite difference}  \subsection{The finite volume method: finite volumes versus finite difference}
17    \begin{rawhtml}
18    <!-- CMIREDIR:finite_volume -->
19    \end{rawhtml}
20    
21    
22    
23  The finite volume method is used to discretize the equations in  The finite volume method is used to discretize the equations in
24  space. The expression ``finite volume'' actually has two meanings; one  space. The expression ``finite volume'' actually has two meanings; one
# Line 57  recovers the same ODE's resulting from f Line 62  recovers the same ODE's resulting from f
62  interior of a fluid. Differences arise at boundaries where a boundary  interior of a fluid. Differences arise at boundaries where a boundary
63  is not positioned on a regular or smoothly varying grid. This method  is not positioned on a regular or smoothly varying grid. This method
64  is used to represent the topography using lopped cell, see  is used to represent the topography using lopped cell, see
65  \cite{Adcroft98}. Subtle difference also appear in more than one  \cite{adcroft:97}. Subtle difference also appear in more than one
66  dimension away from boundaries. This happens because the each  dimension away from boundaries. This happens because the each
67  direction is discretized independently in the finite difference method  direction is discretized independently in the finite difference method
68  while the integrating over finite volume implicitly treats all  while the integrating over finite volume implicitly treats all
69  directions simultaneously. Illustration of this is given in  directions simultaneously. Illustration of this is given in
70  \cite{Adcroft02}.  \cite{ac:02}.
71    
72  \subsection{C grid staggering of variables}  \subsection{C grid staggering of variables}
73    
# Line 79  equations. } Line 84  equations. }
84  The basic algorithm employed for stepping forward the momentum  The basic algorithm employed for stepping forward the momentum
85  equations is based on retaining non-divergence of the flow at all  equations is based on retaining non-divergence of the flow at all
86  times. This is most naturally done if the components of flow are  times. This is most naturally done if the components of flow are
87  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}.
88    
89  Fig. \ref{fig:cgrid3d} shows the components of flow ($u$,$v$,$w$)  Fig. \ref{fig:cgrid3d} shows the components of flow ($u$,$v$,$w$)
90  staggered in space such that the zonal component falls on the  staggered in space such that the zonal component falls on the
# Line 400  thickness of the open side is given by $ Line 405  thickness of the open side is given by $
405  \label{fig:hfacs}  \label{fig:hfacs}
406  \end{figure}  \end{figure}
407    
408  \cite{Adcroft97} presented two alternatives to the step-wise finite  \cite{adcroft:97} presented two alternatives to the step-wise finite
409  difference representation of topography. The method is known to the  difference representation of topography. The method is known to the
410  engineering community as {\em intersecting boundary method}. It  engineering community as {\em intersecting boundary method}. It
411  involves allowing the boundary to intersect a grid of cells thereby  involves allowing the boundary to intersect a grid of cells thereby
# Line 471  continuity equation which can be summari Line 476  continuity equation which can be summari
476  \end{eqnarray}  \end{eqnarray}
477  where the continuity equation has been most naturally discretized by  where the continuity equation has been most naturally discretized by
478  staggering the three components of velocity as shown in  staggering the three components of velocity as shown in
479  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 $\Delta y_c$
480  are the lengths between tracer points (cell centers). The grid lengths  are the lengths between tracer points (cell centers). The grid lengths
481  $\Delta x_g$, $\Delta y_g$ are the grid lengths between cell  $\Delta x_g$, $\Delta y_g$ are the grid lengths between cell
482  corners. $\Delta r_f$ and $\Delta r_c$ are the distance (in units of  corners. $\Delta r_f$ and $\Delta r_c$ are the distance (in units of
# Line 524  the atmosphere. Line 529  the atmosphere.
529  The difference in approach between ocean and atmosphere occurs because  The difference in approach between ocean and atmosphere occurs because
530  of the direct use of the ideal gas equation in forming the potential  of the direct use of the ideal gas equation in forming the potential
531  energy conversion term $\alpha \omega$. The form of these conversion  energy conversion term $\alpha \omega$. The form of these conversion
532  terms is discussed at length in \cite{Adcroft01}.  terms is discussed at length in \cite{adcroft:02}.
533    
534  Because of the different representation of hydrostatic balance between  Because of the different representation of hydrostatic balance between
535  ocean and atmosphere there is no elegant way to represent both systems  ocean and atmosphere there is no elegant way to represent both systems
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