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revision 1.10 by cnh, Thu Oct 25 18:36:53 2001 UTC revision 1.16 by jmc, Wed Oct 13 18:50:54 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 361  vertical grid descriptors are stored in Line 366  vertical grid descriptors are stored in
366    
367  The above grid (Fig.~\ref{fig:vgrid}a) is known as the cell centered  The above grid (Fig.~\ref{fig:vgrid}a) is known as the cell centered
368  approach because the tracer points are at cell centers; the cell  approach because the tracer points are at cell centers; the cell
369  centers are mid-way between the cell interfaces. An alternative, the  centers are mid-way between the cell interfaces.
370  vertex or interface centered approach, is shown in  This discretisation is selected when the thickness of the
371    levels are provided ({\bf delR}, parameter file {\em data},
372    namelist {\em PARM04})
373    An alternative, the vertex or interface centered approach, is shown in
374  Fig.~\ref{fig:vgrid}b. Here, the interior interfaces are positioned  Fig.~\ref{fig:vgrid}b. Here, the interior interfaces are positioned
375  mid-way between the tracer nodes (no longer cell centers). This  mid-way between the tracer nodes (no longer cell centers). This
376  approach is formally more accurate for evaluation of hydrostatic  approach is formally more accurate for evaluation of hydrostatic
377  pressure and vertical advection but historically the cell centered  pressure and vertical advection but historically the cell centered
378  approach has been used. An alternative form of subroutine {\em  approach has been used. An alternative form of subroutine {\em
379  INI\_VERTICAL\_GRID} is used to select the interface centered approach  INI\_VERTICAL\_GRID} is used to select the interface centered approach
380  but no run time option is currently available.  This form requires to specify $Nr+1$ vertical distances {\bf delRc}
381    (parameter file {\em data}, namelist {\em PARM04}, e.g.
382    {\em verification/ideal\_2D\_oce/input/data})
383    corresponding to surface to center, $Nr-1$ center to center, and center to
384    bottom distances.
385    %but no run time option is currently available.
386    
387  \fbox{ \begin{minipage}{4.75in}  \fbox{ \begin{minipage}{4.75in}
388  {\em S/R INI\_VERTICAL\_GRID} ({\em  {\em S/R INI\_VERTICAL\_GRID} ({\em
# Line 387  $\Delta r_c^{-1}$: {\bf RECIP\_DRc} ({\e Line 400  $\Delta r_c^{-1}$: {\bf RECIP\_DRc} ({\e
400    
401    
402  \subsection{Topography: partially filled cells}  \subsection{Topography: partially filled cells}
403    \begin{rawhtml}
404    <!-- CMIREDIR:topo_partial_cells: -->
405    \end{rawhtml}
406    
407  \begin{figure}  \begin{figure}
408  \begin{center}  \begin{center}
# Line 400  thickness of the open side is given by $ Line 416  thickness of the open side is given by $
416  \label{fig:hfacs}  \label{fig:hfacs}
417  \end{figure}  \end{figure}
418    
419  \cite{Adcroft97} presented two alternatives to the step-wise finite  \cite{adcroft:97} presented two alternatives to the step-wise finite
420  difference representation of topography. The method is known to the  difference representation of topography. The method is known to the
421  engineering community as {\em intersecting boundary method}. It  engineering community as {\em intersecting boundary method}. It
422  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 487  continuity equation which can be summari
487  \end{eqnarray}  \end{eqnarray}
488  where the continuity equation has been most naturally discretized by  where the continuity equation has been most naturally discretized by
489  staggering the three components of velocity as shown in  staggering the three components of velocity as shown in
490  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 $\Delta y_c$
491  are the lengths between tracer points (cell centers). The grid lengths  are the lengths between tracer points (cell centers). The grid lengths
492  $\Delta x_g$, $\Delta y_g$ are the grid lengths between cell  $\Delta x_g$, $\Delta y_g$ are the grid lengths between cell
493  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 540  the atmosphere.
540  The difference in approach between ocean and atmosphere occurs because  The difference in approach between ocean and atmosphere occurs because
541  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
542  energy conversion term $\alpha \omega$. The form of these conversion  energy conversion term $\alpha \omega$. The form of these conversion
543  terms is discussed at length in \cite{Adcroft01}.  terms is discussed at length in \cite{adcroft:02}.
544    
545  Because of the different representation of hydrostatic balance between  Because of the different representation of hydrostatic balance between
546  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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