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revision 1.9 by cnh, Wed Oct 24 15:21:27 2001 UTC revision 1.11 by adcroft, Tue Nov 13 18:15:26 2001 UTC
# Line 8  method. This amounts to a grid-point met Line 8  method. This amounts to a grid-point met
8  centered finite difference) in the fluid interior but allows  centered finite difference) in the fluid interior but allows
9  boundaries to intersect a regular grid allowing a more accurate  boundaries to intersect a regular grid allowing a more accurate
10  representation of the position of the boundary. We treat the  representation of the position of the boundary. We treat the
11  horizontal and veritical directions as seperable and differently.  horizontal and vertical directions as separable and differently.
12    
13  \input{part2/notation}  \input{part2/notation}
14    
# Line 37  the lengths $\Delta x_i$: Line 37  the lengths $\Delta x_i$:
37  \begin{displaymath}  \begin{displaymath}
38  \Delta x \partial_t \theta + \delta_i ( F ) = 0  \Delta x \partial_t \theta + \delta_i ( F ) = 0
39  \end{displaymath}  \end{displaymath}
40  is exact if $\theta(x)$ is peice-wise constant over the interval  is exact if $\theta(x)$ is piece-wise constant over the interval
41  $\Delta x_i$ or more generally if $\theta_i$ is defined as the average  $\Delta x_i$ or more generally if $\theta_i$ is defined as the average
42  over the interval $\Delta x_i$.  over the interval $\Delta x_i$.
43    
# Line 57  recovers the same ODE's resulting from f Line 57  recovers the same ODE's resulting from f
57  interior of a fluid. Differences arise at boundaries where a boundary  interior of a fluid. Differences arise at boundaries where a boundary
58  is not positioned on a regular or smoothly varying grid. This method  is not positioned on a regular or smoothly varying grid. This method
59  is used to represent the topography using lopped cell, see  is used to represent the topography using lopped cell, see
60  \cite{Adcroft98}. Subtle difference also appear in more than one  \cite{adcroft:97}. Subtle difference also appear in more than one
61  dimension away from boundaries. This happens because the each  dimension away from boundaries. This happens because the each
62  direction is discretized independantly in the finite difference method  direction is discretized independently in the finite difference method
63  while the integrating over finite volume implicitly treats all  while the integrating over finite volume implicitly treats all
64  directions simultaneously. Illustration of this is given in  directions simultaneously. Illustration of this is given in
65  \cite{Adcroft02}.  \cite{ac:02}.
66    
67  \subsection{C grid staggering of variables}  \subsection{C grid staggering of variables}
68    
# Line 79  equations. } Line 79  equations. }
79  The basic algorithm employed for stepping forward the momentum  The basic algorithm employed for stepping forward the momentum
80  equations is based on retaining non-divergence of the flow at all  equations is based on retaining non-divergence of the flow at all
81  times. This is most naturally done if the components of flow are  times. This is most naturally done if the components of flow are
82  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}.
83    
84  Fig. \ref{fig:cgrid3d} shows the components of flow ($u$,$v$,$w$)  Fig. \ref{fig:cgrid3d} shows the components of flow ($u$,$v$,$w$)
85  staggered in space such that the zonal component falls on the  staggered in space such that the zonal component falls on the
86  interface between continiuty cells in the zonal direction. Similarly  interface between continuity cells in the zonal direction. Similarly
87  for the meridional and vertical directions.  The continiuty cell is  for the meridional and vertical directions.  The continuity cell is
88  synonymous with tracer cells (they are one and the same).  synonymous with tracer cells (they are one and the same).
89    
90    
# Line 141  is bordered by the lengths $\Delta x_f$ Line 141  is bordered by the lengths $\Delta x_f$
141    
142  The model domain is decomposed into tiles and within each tile a  The model domain is decomposed into tiles and within each tile a
143  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
144  decomposition for parallelization but may be used whether parallized  decomposition for parallelization but may be used whether parallelized
145  or not; see section \ref{sect:tiles} for more details. Although the  or not; see section \ref{sect:tiles} for more details. Although the
146  tiles may be patched together in an unstructured manner  tiles may be patched together in an unstructured manner
147  (i.e. irregular or non-tessilating pattern), the interior of tiles is  (i.e. irregular or non-tessilating pattern), the interior of tiles is
148  a structered grid of quadrilateral cells. The horizontal coordinate  a structured grid of quadrilateral cells. The horizontal coordinate
149  system is orthogonal curvilinear meaning we can not necessarily treat  system is orthogonal curvilinear meaning we can not necessarily treat
150  the two horizontal directions as seperable. Instead, each cell in the  the two horizontal directions as separable. Instead, each cell in the
151  horizontal grid is described by the length of it's sides and it's  horizontal grid is described by the length of it's sides and it's
152  area.  area.
153    
154  The grid information is quite general and describes any of the  The grid information is quite general and describes any of the
155  available coordinates systems, cartesian, spherical-polar or  available coordinates systems, cartesian, spherical-polar or
156  curvilinear. All that is necessary to distinguish between the  curvilinear. All that is necessary to distinguish between the
157  coordinate systems is to initialize the grid data (discriptors)  coordinate systems is to initialize the grid data (descriptors)
158  appropriately.  appropriately.
159    
160  In the following, we refer to the orientation of quantities on the  In the following, we refer to the orientation of quantities on the
# Line 293  using\-Cartes\-ianGrid} in namelist {\em Line 293  using\-Cartes\-ianGrid} in namelist {\em
293  spacing can be set to uniform via scalars {\bf dXspacing} and {\bf  spacing can be set to uniform via scalars {\bf dXspacing} and {\bf
294  dYspacing} in namelist {\em PARM04} or to variable resolution by the  dYspacing} in namelist {\em PARM04} or to variable resolution by the
295  vectors {\bf DELX} and {\bf DELY}. Units are normally  vectors {\bf DELX} and {\bf DELY}. Units are normally
296  meters. Non-dimensional coordinates can be used by interpretting the  meters. Non-dimensional coordinates can be used by interpreting the
297  gravitational constant as the Rayleigh number.  gravitational constant as the Rayleigh number.
298    
299  \subsubsection{Spherical-polar coordinates}  \subsubsection{Spherical-polar coordinates}
# Line 347  The vertical grid is calculated in subro Line 347  The vertical grid is calculated in subro
347  INI\_VERTICAL\_GRID} and specified via the vector {\bf DELR} in  INI\_VERTICAL\_GRID} and specified via the vector {\bf DELR} in
348  namelist {\em PARM04}. The units of ``r'' are either meters or Pascals  namelist {\em PARM04}. The units of ``r'' are either meters or Pascals
349  depending on the isomorphism being used which in turn is dependent  depending on the isomorphism being used which in turn is dependent
350  only on the choise of equation of state.  only on the choice of equation of state.
351    
352  There are alternative namelist vectors {\bf DELZ} and {\bf DELP} which  There are alternative namelist vectors {\bf DELZ} and {\bf DELP} which
353  dictate whether z- or  dictate whether z- or
# Line 400  thickness of the open side is given by $ Line 400  thickness of the open side is given by $
400  \label{fig:hfacs}  \label{fig:hfacs}
401  \end{figure}  \end{figure}
402    
403  \cite{Adcroft97} presented two alternatives to the step-wise finite  \cite{adcroft:97} presented two alternatives to the step-wise finite
404  difference representation of topography. The method is known to the  difference representation of topography. The method is known to the
405  engineering community as {\em intersecting boundary method}. It  engineering community as {\em intersecting boundary method}. It
406  involves allowing the boundary to intersect a grid of cells thereby  involves allowing the boundary to intersect a grid of cells thereby
407  modifying the shape of those cells intersected. We suggested allowing  modifying the shape of those cells intersected. We suggested allowing
408  the topgoraphy to take on a peice-wise linear representation (shaved  the topography to take on a piece-wise linear representation (shaved
409  cells) or a simpler piecewise constant representaion (partial step).  cells) or a simpler piecewise constant representation (partial step).
410  Both show dramatic improvements in solution compared to the  Both show dramatic improvements in solution compared to the
411  traditional full step representation, the piece-wise linear being the  traditional full step representation, the piece-wise linear being the
412  best. However, the storage requirements are excessive so the simpler  best. However, the storage requirements are excessive so the simpler
# Line 420  the thickness of a level is determined a Line 420  the thickness of a level is determined a
420  \marginpar{$h_s$: {\bf hFacS}}  \marginpar{$h_s$: {\bf hFacS}}
421  The physical thickness of a tracer cell is given by $h_c(i,j,k) \Delta  The physical thickness of a tracer cell is given by $h_c(i,j,k) \Delta
422  r_f(k)$ and the physical thickness of the open side is given by  r_f(k)$ and the physical thickness of the open side is given by
423  $h_w(i,j,k) \Delta r_f(k)$. Three 3-D discriptors $h_c$, $h_w$ and  $h_w(i,j,k) \Delta r_f(k)$. Three 3-D descriptors $h_c$, $h_w$ and
424  $h_s$ are used to describe the geometry: {\bf hFacC}, {\bf hFacW} and  $h_s$ are used to describe the geometry: {\bf hFacC}, {\bf hFacW} and
425  {\bf hFacS} respectively. These are calculated in subroutine {\em  {\bf hFacS} respectively. These are calculated in subroutine {\em
426  INI\_MASKS\_ETC} along with there reciprocals {\bf RECIP\_hFacC}, {\bf  INI\_MASKS\_ETC} along with there reciprocals {\bf RECIP\_hFacC}, {\bf
# Line 484  A}_c$.  The factors $h_w$ and $h_s$ are Line 484  A}_c$.  The factors $h_w$ and $h_s$ are
484  \marginpar{$h_s$: {\bf hFacS}}  \marginpar{$h_s$: {\bf hFacS}}
485    
486  The last equation, the discrete continuity equation, can be summed in  The last equation, the discrete continuity equation, can be summed in
487  the vertical to yeild the free-surface equation:  the vertical to yield the free-surface equation:
488  \begin{equation}  \begin{equation}
489  {\cal A}_c \partial_t \eta + \delta_i \sum_k \Delta y_g \Delta r_f h_w  {\cal A}_c \partial_t \eta + \delta_i \sum_k \Delta y_g \Delta r_f h_w
490  u + \delta_j \sum_k \Delta x_g \Delta r_f h_s v = {\cal  u + \delta_j \sum_k \Delta x_g \Delta r_f h_s v = {\cal
# Line 503  derived from the buoyancy equation exact Line 503  derived from the buoyancy equation exact
503  from the pressure gradient terms when forming the kinetic energy  from the pressure gradient terms when forming the kinetic energy
504  equation.  equation.
505    
506  In the ocean, using z-ccordinates, the hydrostatic balance terms are  In the ocean, using z-coordinates, the hydrostatic balance terms are
507  discretized:  discretized:
508  \begin{equation}  \begin{equation}
509  \epsilon_{nh} \partial_t w  \epsilon_{nh} \partial_t w
# Line 523  the atmosphere. Line 523  the atmosphere.
523    
524  The difference in approach between ocean and atmosphere occurs because  The difference in approach between ocean and atmosphere occurs because
525  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
526  energy conversion term $\alpha \omega$. The form of these consversion  energy conversion term $\alpha \omega$. The form of these conversion
527  terms is discussed at length in \cite{Adcroft01}.  terms is discussed at length in \cite{adcroft:02}.
528    
529  Because of the different representation of hydrostatic balance between  Because of the different representation of hydrostatic balance between
530  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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