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revision 1.10 by cnh, Thu Oct 25 18:36:53 2001 UTC revision 1.19 by jmc, Wed Apr 5 01:16:27 2006 UTC
# Line 2  Line 2 
2  % $Name$  % $Name$
3    
4  \section{Spatial discretization of the dynamical equations}  \section{Spatial discretization of the dynamical equations}
5    \begin{rawhtml}
6    <!-- CMIREDIR:spatial_discretization_of_dyn_eq: -->
7    \end{rawhtml}
8    
9  Spatial discretization is carried out using the finite volume  Spatial discretization is carried out using the finite volume
10  method. This amounts to a grid-point method (namely second-order  method. This amounts to a grid-point method (namely second-order
# Line 14  horizontal and vertical directions as se Line 17  horizontal and vertical directions as se
17    
18    
19  \subsection{The finite volume method: finite volumes versus finite difference}  \subsection{The finite volume method: finite volumes versus finite difference}
20    \begin{rawhtml}
21    <!-- CMIREDIR:finite_volume: -->
22    \end{rawhtml}
23    
24    
25    
26  The finite volume method is used to discretize the equations in  The finite volume method is used to discretize the equations in
27  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 65  recovers the same ODE's resulting from f
65  interior of a fluid. Differences arise at boundaries where a boundary  interior of a fluid. Differences arise at boundaries where a boundary
66  is not positioned on a regular or smoothly varying grid. This method  is not positioned on a regular or smoothly varying grid. This method
67  is used to represent the topography using lopped cell, see  is used to represent the topography using lopped cell, see
68  \cite{Adcroft98}. Subtle difference also appear in more than one  \cite{adcroft:97}. Subtle difference also appear in more than one
69  dimension away from boundaries. This happens because the each  dimension away from boundaries. This happens because the each
70  direction is discretized independently in the finite difference method  direction is discretized independently in the finite difference method
71  while the integrating over finite volume implicitly treats all  while the integrating over finite volume implicitly treats all
72  directions simultaneously. Illustration of this is given in  directions simultaneously. Illustration of this is given in
73  \cite{Adcroft02}.  \cite{ac:02}.
74    
75  \subsection{C grid staggering of variables}  \subsection{C grid staggering of variables}
76    
# Line 79  equations. } Line 87  equations. }
87  The basic algorithm employed for stepping forward the momentum  The basic algorithm employed for stepping forward the momentum
88  equations is based on retaining non-divergence of the flow at all  equations is based on retaining non-divergence of the flow at all
89  times. This is most naturally done if the components of flow are  times. This is most naturally done if the components of flow are
90  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}.
91    
92  Fig. \ref{fig:cgrid3d} shows the components of flow ($u$,$v$,$w$)  Fig. \ref{fig:cgrid3d} shows the components of flow ($u$,$v$,$w$)
93  staggered in space such that the zonal component falls on the  staggered in space such that the zonal component falls on the
# Line 142  is bordered by the lengths $\Delta x_f$ Line 150  is bordered by the lengths $\Delta x_f$
150  The model domain is decomposed into tiles and within each tile a  The model domain is decomposed into tiles and within each tile a
151  quasi-regular grid is used. A tile is the basic unit of domain  quasi-regular grid is used. A tile is the basic unit of domain
152  decomposition for parallelization but may be used whether parallelized  decomposition for parallelization but may be used whether parallelized
153  or not; see section \ref{sect:tiles} for more details. Although the  or not; see section \ref{sect:domain_decomposition} for more details.
154  tiles may be patched together in an unstructured manner  Although the tiles may be patched together in an unstructured manner
155  (i.e. irregular or non-tessilating pattern), the interior of tiles is  (i.e. irregular or non-tessilating pattern), the interior of tiles is
156  a structured grid of quadrilateral cells. The horizontal coordinate  a structured grid of quadrilateral cells. The horizontal coordinate
157  system is orthogonal curvilinear meaning we can not necessarily treat  system is orthogonal curvilinear meaning we can not necessarily treat
# Line 361  vertical grid descriptors are stored in Line 369  vertical grid descriptors are stored in
369    
370  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
371  approach because the tracer points are at cell centers; the cell  approach because the tracer points are at cell centers; the cell
372  centers are mid-way between the cell interfaces. An alternative, the  centers are mid-way between the cell interfaces.
373  vertex or interface centered approach, is shown in  This discretization is selected when the thickness of the
374    levels are provided ({\bf delR}, parameter file {\em data},
375    namelist {\em PARM04})
376    An alternative, the vertex or interface centered approach, is shown in
377  Fig.~\ref{fig:vgrid}b. Here, the interior interfaces are positioned  Fig.~\ref{fig:vgrid}b. Here, the interior interfaces are positioned
378  mid-way between the tracer nodes (no longer cell centers). This  mid-way between the tracer nodes (no longer cell centers). This
379  approach is formally more accurate for evaluation of hydrostatic  approach is formally more accurate for evaluation of hydrostatic
380  pressure and vertical advection but historically the cell centered  pressure and vertical advection but historically the cell centered
381  approach has been used. An alternative form of subroutine {\em  approach has been used. An alternative form of subroutine {\em
382  INI\_VERTICAL\_GRID} is used to select the interface centered approach  INI\_VERTICAL\_GRID} is used to select the interface centered approach
383  but no run time option is currently available.  This form requires to specify $Nr+1$ vertical distances {\bf delRc}
384    (parameter file {\em data}, namelist {\em PARM04}, e.g.
385    {\em verification/ideal\_2D\_oce/input/data})
386    corresponding to surface to center, $Nr-1$ center to center, and center to
387    bottom distances.
388    %but no run time option is currently available.
389    
390  \fbox{ \begin{minipage}{4.75in}  \fbox{ \begin{minipage}{4.75in}
391  {\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 403  $\Delta r_c^{-1}$: {\bf RECIP\_DRc} ({\e
403    
404    
405  \subsection{Topography: partially filled cells}  \subsection{Topography: partially filled cells}
406    \begin{rawhtml}
407    <!-- CMIREDIR:topo_partial_cells: -->
408    \end{rawhtml}
409    
410  \begin{figure}  \begin{figure}
411  \begin{center}  \begin{center}
# Line 400  thickness of the open side is given by $ Line 419  thickness of the open side is given by $
419  \label{fig:hfacs}  \label{fig:hfacs}
420  \end{figure}  \end{figure}
421    
422  \cite{Adcroft97} presented two alternatives to the step-wise finite  \cite{adcroft:97} presented two alternatives to the step-wise finite
423  difference representation of topography. The method is known to the  difference representation of topography. The method is known to the
424  engineering community as {\em intersecting boundary method}. It  engineering community as {\em intersecting boundary method}. It
425  involves allowing the boundary to intersect a grid of cells thereby  involves allowing the boundary to intersect a grid of cells thereby
# Line 457  $h_s^{-1}$: {\bf RECIP\_hFacS} ({\em GRI Line 476  $h_s^{-1}$: {\bf RECIP\_hFacS} ({\em GRI
476    
477    
478  \section{Continuity and horizontal pressure gradient terms}  \section{Continuity and horizontal pressure gradient terms}
479    \begin{rawhtml}
480    <!-- CMIREDIR:continuity_and_horizontal_pressure: -->
481    \end{rawhtml}
482    
483    
484  The core algorithm is based on the ``C grid'' discretization of the  The core algorithm is based on the ``C grid'' discretization of the
485  continuity equation which can be summarized as:  continuity equation which can be summarized as:
# Line 471  continuity equation which can be summari Line 494  continuity equation which can be summari
494  \end{eqnarray}  \end{eqnarray}
495  where the continuity equation has been most naturally discretized by  where the continuity equation has been most naturally discretized by
496  staggering the three components of velocity as shown in  staggering the three components of velocity as shown in
497  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 $\Delta y_c$
498  are the lengths between tracer points (cell centers). The grid lengths  are the lengths between tracer points (cell centers). The grid lengths
499  $\Delta x_g$, $\Delta y_g$ are the grid lengths between cell  $\Delta x_g$, $\Delta y_g$ are the grid lengths between cell
500  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 495  addition of volume due to excess precipi Line 518  addition of volume due to excess precipi
518  evaporation and only enters the top-level of the {\em ocean} model.  evaporation and only enters the top-level of the {\em ocean} model.
519    
520  \section{Hydrostatic balance}  \section{Hydrostatic balance}
521    \begin{rawhtml}
522    <!-- CMIREDIR:hydrostatic_balance: -->
523    \end{rawhtml}
524    
525  The vertical momentum equation has the hydrostatic or  The vertical momentum equation has the hydrostatic or
526  quasi-hydrostatic balance on the right hand side. This discretization  quasi-hydrostatic balance on the right hand side. This discretization
# Line 524  the atmosphere. Line 550  the atmosphere.
550  The difference in approach between ocean and atmosphere occurs because  The difference in approach between ocean and atmosphere occurs because
551  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
552  energy conversion term $\alpha \omega$. The form of these conversion  energy conversion term $\alpha \omega$. The form of these conversion
553  terms is discussed at length in \cite{Adcroft01}.  terms is discussed at length in \cite{adcroft:02}.
554    
555  Because of the different representation of hydrostatic balance between  Because of the different representation of hydrostatic balance between
556  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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