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revision 1.18 by afe, Thu May 6 15:21:01 2004 UTC revision 1.25 by edhill, Tue Aug 9 21:52:09 2005 UTC
# Line 10  Line 10 
10  %%    o automatically inserted at \section{Reference}  %%    o automatically inserted at \section{Reference}
11    
12    
13  \section{exch2: Extended Cubed Sphere \mbox{Topology}}  \subsection{exch2: Extended Cubed Sphere \mbox{Topology}}
14  \label{sec:exch2}  \label{sec:exch2}
15    
16    
17  \subsection{Introduction}  \subsubsection{Introduction}
18    
19  The \texttt{exch2} package extends the original cubed sphere topology  The \texttt{exch2} package extends the original cubed sphere topology
20  configuration to allow more flexible domain decomposition and  configuration to allow more flexible domain decomposition and
# Line 23  into any number of tiles that divide eve Line 23  into any number of tiles that divide eve
23  dimensions of the subdomain.  Furthermore, the tiles can run on  dimensions of the subdomain.  Furthermore, the tiles can run on
24  separate processors individually or in groups, which provides for  separate processors individually or in groups, which provides for
25  manual compile-time load balancing across a relatively arbitrary  manual compile-time load balancing across a relatively arbitrary
26  number of processors. \\  number of processors.
27    
28  The exchange parameters are declared in  The exchange parameters are declared in
29  \filelink{pkg/exch2/W2\_EXCH2\_TOPOLOGY.h}{pkg-exch2-W2_EXCH2_TOPOLOGY.h}  \filelink{pkg/exch2/W2\_EXCH2\_TOPOLOGY.h}{pkg-exch2-W2_EXCH2_TOPOLOGY.h}
# Line 41  of these files with alternate topologies Line 41  of these files with alternate topologies
41  \file{utils/exch2/code-mods} along with the appropriate \file{SIZE.h}  \file{utils/exch2/code-mods} along with the appropriate \file{SIZE.h}
42  file for single-processor execution.  file for single-processor execution.
43    
44  \subsection{Invoking exch2}  \subsubsection{Invoking exch2}
45    
46  To use exch2 with the cubed sphere, the following conditions must be  To use exch2 with the cubed sphere, the following conditions must be
47  met: \\  met:
48    
49  $\bullet$ The exch2 package is included when \file{genmake2} is run.  \begin{itemize}
50    The easiest way to do this is to add the line \code{exch2} to the  \item The exch2 package is included when \file{genmake2} is run.  The
51    \file{profile.conf} file -- see Section    easiest way to do this is to add the line \code{exch2} to the
52    \ref{sect:buildingCode} \sectiontitle{Building the code} for general    \file{packages.conf} file -- see Section \ref{sect:buildingCode}
53    details. \\    \sectiontitle{Building the code} for general
54      details.
55    
56  $\bullet$ An example of \file{W2\_EXCH2\_TOPOLOGY.h} and  \item An example of \file{W2\_EXCH2\_TOPOLOGY.h} and
57    \file{w2\_e2setup.F} must reside in a directory containing files    \file{w2\_e2setup.F} must reside in a directory containing files
58    symbolically linked when \file{genmake2} runs.  The safest place to    symbolically linked by the \file{genmake2} script.  The safest place
59    put these is the directory indicated in the \code{-mods=DIR} command    to put these is the directory indicated in the \code{-mods=DIR}
60    line modifier (typically \file{../code}), or the build directory.    command line modifier (typically \file{../code}), or the build
61    The default versions of these files reside in \file{pkg/exch2} and    directory.  The default versions of these files reside in
62    are linked automatically if no other versions exist elsewhere in the    \file{pkg/exch2} and are linked automatically if no other versions
63    build path, but they should be left untouched to avoid breaking    exist elsewhere in the build path, but they should be left untouched
64    configurations other than the one you intend to modify.\\    to avoid breaking configurations other than the one you intend to
65      modify.
66  $\bullet$ Files containing grid parameters, named  
67    \file{tile00$n$.mitgrid} where $n$=\code{(1:6)} (one per subdomain),  \item Files containing grid parameters, named \file{tile00$n$.mitgrid}
68    must be in the working directory when the MITgcm executable is run.    where $n$=\code{(1:6)} (one per subdomain), must be in the working
69    These files are provided in the example experiments for cubed sphere    directory when the MITgcm executable is run.  These files are
70    configurations with 32$\times$32 cube sides    provided in the example experiments for cubed sphere configurations
71    -- please contact MITgcm support if you want to generate    with 32$\times$32 cube sides -- please contact MITgcm support if you
72    files for other configurations. \\    want to generate files for other configurations.
73    
74  $\bullet$ As always when compiling MITgcm, the file \file{SIZE.h} must  \item As always when compiling MITgcm, the file \file{SIZE.h} must be
75    be placed where \file{genmake2} will find it.  In particular for    placed where \file{genmake2} will find it.  In particular for exch2,
76    exch2, the domain decomposition specified in \file{SIZE.h} must    the domain decomposition specified in \file{SIZE.h} must correspond
77    correspond with the particular configuration's topology specified in    with the particular configuration's topology specified in
78    \file{W2\_EXCH2\_TOPOLOGY.h} and \file{w2\_e2setup.F}.  Domain    \file{W2\_EXCH2\_TOPOLOGY.h} and \file{w2\_e2setup.F}.  Domain
79    decomposition issues particular to exch2 are addressed in Section    decomposition issues particular to exch2 are addressed in Section
80    \ref{sec:topogen} \sectiontitle{Generating Topology Files for exch2}    \ref{sec:topogen} \sectiontitle{Generating Topology Files for exch2}
81    and \ref{sec:exch2mpi} \sectiontitle{exch2, SIZE.h, and MPI}; a more    and \ref{sec:exch2mpi} \sectiontitle{exch2, SIZE.h, and
82    general background on the subject relevant to MITgcm is presented in      Multiprocessing}; a more general background on the subject
83    Section \ref{sect:specifying_a_decomposition}    relevant to MITgcm is presented in Section
84    \sectiontitle{Specifying a decomposition}.\\    \ref{sect:specifying_a_decomposition}
85      \sectiontitle{Specifying a decomposition}.
86    \end{itemize}
87    
88  At the time of this writing the following examples use exch2 and may  At the time of this writing the following examples use exch2 and may
89  be used for guidance:  be used for guidance:
# Line 96  verification/hs94.cs-32x32x5 Line 99  verification/hs94.cs-32x32x5
99    
100    
101    
102  \subsection{Generating Topology Files for exch2}  \subsubsection{Generating Topology Files for exch2}
103  \label{sec:topogen}  \label{sec:topogen}
104    
105  Alternate cubed sphere topologies may be created using the Matlab  Alternate cubed sphere topologies may be created using the Matlab
# Line 107  from the Matlab prompt (there are no par Line 110  from the Matlab prompt (there are no par
110  exch2 topology files \file{W2\_EXCH2\_TOPOLOGY.h} and  exch2 topology files \file{W2\_EXCH2\_TOPOLOGY.h} and
111  \file{w2\_e2setup.F} in the working directory and displays a figure of  \file{w2\_e2setup.F} in the working directory and displays a figure of
112  the topology via Matlab -- figures \ref{fig:6tile}, \ref{fig:12tile},  the topology via Matlab -- figures \ref{fig:6tile}, \ref{fig:12tile},
113  and \ref{fig:24tile} are examples.  The other m-files in the directory are  and \ref{fig:24tile} are examples of the generated diagrams.  The other
114  subroutines of \file{driver.m} and should not be run ``bare'' except  m-files in the directory are
115    subroutines called from \file{driver.m} and should not be run ``bare'' except
116  for development purposes. \\  for development purposes. \\
117    
118  The parameters that determine the dimensions and topology of the  The parameters that determine the dimensions and topology of the
119  generated configuration are \code{nr}, \code{nb}, \code{ng},  generated configuration are \code{nr}, \code{nb}, \code{ng},
120  \code{tnx} and \code{tny}, and all are assigned early in the script. \\  \code{tnx} and \code{tny}, and all are assigned early in the script. \\
121    
122  The first three determine the size of the subdomains and  The first three determine the height and width of the subdomains and
123  hence the size of the overall domain.  Each one determines the number  hence the size of the overall domain.  Each one determines the number
124  of grid points, and therefore the resolution, along the subdomain  of grid points, and therefore the resolution, along the subdomain
125  sides in a ``great circle'' around each the three spatial axes of the cube.  At the time  sides in a ``great circle'' around each the three spatial axes of the cube.  At the time
126  of this writing MITgcm requires these three parameters to be equal,  of this writing MITgcm requires these three parameters to be equal,
127  but they provide for future releases  to accomodate different  but they provide for future releases  to accomodate different
128  resolutions around the axes to allow (for example) greater resolution  resolutions around the axes to allow subdomains with differing resolutions.\\
 around the equator.\\  
129    
130  The parameters \code{tnx} and \code{tny} determine the width and height of  The parameters \code{tnx} and \code{tny} determine the width and height of
131  the tiles into which the subdomains are decomposed, and must evenly  the tiles into which the subdomains are decomposed, and must evenly
# Line 182  by their tile number in the topology, se Line 185  by their tile number in the topology, se
185    
186    
187    
188  \subsection{exch2, SIZE.h, and multiprocessing}  \subsubsection{exch2, SIZE.h, and Multiprocessing}
189  \label{sec:exch2mpi}  \label{sec:exch2mpi}
190    
191  Once the topology configuration files are created, the Fortran  Once the topology configuration files are created, the Fortran
192  \code{PARAMETER}s in \file{SIZE.h} must be configured to match.  \code{PARAMETER}s in \file{SIZE.h} must be configured to match.
193  Section \ref{sect:specifying_a_decomposition} \sectiontitle{Specifying  Section \ref{sect:specifying_a_decomposition} \sectiontitle{Specifying
194  a decomposition} provides a general description of domain    a decomposition} provides a general description of domain
195  decomposition within MITgcm and its relation to \file{SIZE.h}. The  decomposition within MITgcm and its relation to \file{SIZE.h}. The
196  current section specifies certain constraints the exch2 package  current section specifies constraints that the exch2 package imposes
197  imposes as well as describes how to enable parallel execution with  and describes how to enable parallel execution with MPI.
 MPI. \\  
198    
199  As in the general case, the parameters \varlink{sNx}{sNx} and  As in the general case, the parameters \varlink{sNx}{sNx} and
200  \varlink{sNy}{sNy} define the size of the individual tiles, and so  \varlink{sNy}{sNy} define the size of the individual tiles, and so
201  must be assigned the same respective values as \code{tnx} and  must be assigned the same respective values as \code{tnx} and
202  \code{tny} in \file{driver.m}.\\  \code{tny} in \file{driver.m}.
203    
204  The halo width parameters \varlink{OLx}{OLx} and \varlink{OLy}{OLy}  The halo width parameters \varlink{OLx}{OLx} and \varlink{OLy}{OLy}
205  have no special bearing on exch2 and may be assigned as in the general  have no special bearing on exch2 and may be assigned as in the general
206  case. The same holds for \varlink{Nr}{Nr}, the number of vertical  case. The same holds for \varlink{Nr}{Nr}, the number of vertical
207  levels in the model.\\  levels in the model.
208    
209  The parameters \varlink{nSx}{nSx}, \varlink{nSy}{nSy},  The parameters \varlink{nSx}{nSx}, \varlink{nSy}{nSy},
210  \varlink{nPx}{nPx}, and \varlink{nPy}{nPy} relate to the number of  \varlink{nPx}{nPx}, and \varlink{nPy}{nPy} relate to the number of
211  tiles and how they are distributed on processors.  When using exch2,  tiles and how they are distributed on processors.  When using exch2,
212  the tiles are stored in a single dimension, and so  the tiles are stored in the $x$ dimension, and so
213  \code{\varlink{nSy}{nSy}=1} in all cases.  Since the tiles as  \code{\varlink{nSy}{nSy}=1} in all cases.  Since the tiles as
214  configured by exch2 cannot be split up accross processors without  configured by exch2 cannot be split up accross processors without
215  regenerating the topology, \code{\varlink{nPy}{nPy}=1} as well. \\  regenerating the topology, \code{\varlink{nPy}{nPy}=1} as well.
216    
217  The number of tiles MITgcm allocates and how they are distributed  The number of tiles MITgcm allocates and how they are distributed
218  between processors depends on \varlink{nPx}{nPx} and  between processors depends on \varlink{nPx}{nPx} and
219  \varlink{nSx}{nSx}.  \varlink{nSx}{nSx} is the number of tiles per  \varlink{nSx}{nSx}.  \varlink{nSx}{nSx} is the number of tiles per
220  processor and \varlink{nPx}{nPx} the number of processors.  The total  processor and \varlink{nPx}{nPx} is the number of processors.  The
221  number of tiles in the topology minus those listed in  total number of tiles in the topology minus those listed in
222  \file{blanklist.txt} must equal \code{nSx*nPx}. \\  \file{blanklist.txt} must equal \code{nSx*nPx}.  Note that in order to
223    obtain maximum usage from a given number of processors in some cases,
224    this restriction might entail sharing a processor with a tile that
225    would otherwise be excluded because it is topographically outside of
226    the domain and therefore in \file{blanklist.txt}.  For example,
227    suppose you have five processors and a domain decomposition of
228    thirty-six tiles that allows you to exclude seven tiles.  To evenly
229    distribute the remaining twenty-nine tiles among five processors, you
230    would have to run one ``dummy'' tile to make an even six tiles per
231    processor.  Such dummy tiles are \emph{not} listed in
232    \file{blanklist.txt}.
233    
234  The following is an example of \file{SIZE.h} for the twelve-tile  The following is an example of \file{SIZE.h} for the twelve-tile
235  configuration illustrated in figure \ref{fig:12tile} running on  configuration illustrated in figure \ref{fig:12tile} running on one
236  one processor: \\  processor:
237    
238  \begin{verbatim}  \begin{verbatim}
239        PARAMETER (        PARAMETER (
# Line 257  figure \ref{fig:24tile} running on six p Line 269  figure \ref{fig:24tile} running on six p
269  \end{verbatim}  \end{verbatim}
270    
271    
272    \subsubsection{Key Variables}
   
   
 \subsection{Key Variables}  
273    
274  The descriptions of the variables are divided up into scalars,  The descriptions of the variables are divided up into scalars,
275  one-dimensional arrays indexed to the tile number, and two and  one-dimensional arrays indexed to the tile number, and two and
# Line 270  scalars are common to every part of the Line 279  scalars are common to every part of the
279  arrays to individual tiles, and the arrays indexed by tile and  arrays to individual tiles, and the arrays indexed by tile and
280  neighbor to relationships between tiles and their neighbors. \\  neighbor to relationships between tiles and their neighbors. \\
281    
282  \subsubsection{Scalars}  Scalars:
283    
284  The number of tiles in a particular topology is set with the parameter  The number of tiles in a particular topology is set with the parameter
285  \code{NTILES}, and the maximum number of neighbors of any tiles by  \code{NTILES}, and the maximum number of neighbors of any tiles by
# Line 287  setup of six tiles (Fig. \ref{fig:6tile} Line 296  setup of six tiles (Fig. \ref{fig:6tile}
296  topology of twenty-four square tiles, four per subdomain (as in figure  topology of twenty-four square tiles, four per subdomain (as in figure
297  \ref{fig:24tile}), will have \code{exch2\_domain\_nxt=12} and  \ref{fig:24tile}), will have \code{exch2\_domain\_nxt=12} and
298  \code{exch2\_domain\_nyt=2}.  Note that these parameters express the  \code{exch2\_domain\_nyt=2}.  Note that these parameters express the
299  tile layout to allow global data files that are tile-layout-neutral  tile layout in order to allow global data files that are tile-layout-neutral.
300  and have no bearing on the internal storage of the arrays.  The tiles  They have no bearing on the internal storage of the arrays.  The tiles
301  are stored internally in a range from \code{(1:\varlink{bi}{bi})} the  are stored internally in a range from \code{\varlink{bi}{bi}=(1:NTILES)} in the
302  $x$ axis, and the $y$ axis variable \varlink{bj}{bj} is assumed to  $x$ axis, and the $y$ axis variable \varlink{bj}{bj} is assumed to
303  equal \code{1} throughout the package. \\  equal \code{1} throughout the package. \\
304    
305  \subsubsection{Arrays Indexed to Tile Number}  Arrays indexed to tile number:
306    
307  The following arrays are of length \code{NTILES} and are indexed to  The following arrays are of length \code{NTILES} and are indexed to
308  the tile number, which is indicated in the diagrams with the notation  the tile number, which is indicated in the diagrams with the notation
# Line 303  The arrays \varlink{exch2\_tnx}{exch2_tn Line 312  The arrays \varlink{exch2\_tnx}{exch2_tn
312  \varlink{exch2\_tny}{exch2_tny} express the $x$ and $y$ dimensions of  \varlink{exch2\_tny}{exch2_tny} express the $x$ and $y$ dimensions of
313  each tile.  At present for each tile \texttt{exch2\_tnx=sNx} and  each tile.  At present for each tile \texttt{exch2\_tnx=sNx} and
314  \texttt{exch2\_tny=sNy}, as assigned in \file{SIZE.h} and described in  \texttt{exch2\_tny=sNy}, as assigned in \file{SIZE.h} and described in
315  section \ref{sec:exch2mpi} \sectiontitle{exch2, SIZE.h, and  Section \ref{sec:exch2mpi} \sectiontitle{exch2, SIZE.h, and
316  multiprocessing}.  Future releases of MITgcm may allow varying tile  Multiprocessing}.  Future releases of MITgcm may allow varying tile
317  sizes. \\  sizes. \\
318    
319  The location of the tiles' Cartesian origin within a subdomain are  The arrays \varlink{exch2\_tbasex}{exch2_tbasex} and
320  determined by the arrays \varlink{exch2\_tbasex}{exch2_tbasex} and  \varlink{exch2\_tbasey}{exch2_tbasey} determine the tiles'
321  \varlink{exch2\_tbasey}{exch2_tbasey}.  These variables are used to  Cartesian origin within a subdomain  
322  relate the location of the edges of different tiles to each other.  As  and locate the edges of different tiles relative to each other.  As
323  an example, in the default six-tile topology (Fig. \ref{fig:6tile})  an example, in the default six-tile topology (Fig. \ref{fig:6tile})
324  each index in these arrays is set to \code{0} since a tile occupies  each index in these arrays is set to \code{0} since a tile occupies
325  its entire subdomain.  The twenty-four-tile case discussed above will  its entire subdomain.  The twenty-four-tile case discussed above will
326  have values of \code{0} or \code{16}, depending on the quadrant the  have values of \code{0} or \code{16}, depending on the quadrant of the
327  tile falls within the subdomain.  The elements of the arrays  tile within the subdomain.  The elements of the arrays
328  \varlink{exch2\_txglobalo}{exch2_txglobalo} and  \varlink{exch2\_txglobalo}{exch2_txglobalo} and
329  \varlink{exch2\_txglobalo}{exch2_txglobalo} are similar to  \varlink{exch2\_txglobalo}{exch2_txglobalo} are similar to
330  \varlink{exch2\_tbasex}{exch2_tbasex} and  \varlink{exch2\_tbasex}{exch2_tbasex} and
331  \varlink{exch2\_tbasey}{exch2_tbasey}, but locate the tiles within the  \varlink{exch2\_tbasey}{exch2_tbasey}, but locate the tile edges within the
332  global address space, similar to that used by global output and input  global address space, similar to that used by global output and input
333  files. \\  files. \\
334    
# Line 328  the subdomain of each tile, in a range \ Line 337  the subdomain of each tile, in a range \
337  standard cube topology and indicated by \textbf{\textsf{f}}$n$ in  standard cube topology and indicated by \textbf{\textsf{f}}$n$ in
338  figures \ref{fig:12tile} and  figures \ref{fig:12tile} and
339  \ref{fig:24tile}. \varlink{exch2\_nNeighbours}{exch2_nNeighbours}  \ref{fig:24tile}. \varlink{exch2\_nNeighbours}{exch2_nNeighbours}
340  contains a count of the neighboring tiles each tile has, and is used  contains a count of the neighboring tiles each tile has, and sets
341  for setting bounds for looping over neighboring tiles.  the bounds for looping over neighboring tiles.
342  \varlink{exch2\_tProc}{exch2_tProc} holds the process rank of each  \varlink{exch2\_tProc}{exch2_tProc} holds the process rank of each
343  tile, and is used in interprocess communication.  \\  tile, and is used in interprocess communication.  \\
344    
# Line 338  The arrays \varlink{exch2\_isWedge}{exch Line 347  The arrays \varlink{exch2\_isWedge}{exch
347  \varlink{exch2\_isEedge}{exch2_isEedge},  \varlink{exch2\_isEedge}{exch2_isEedge},
348  \varlink{exch2\_isSedge}{exch2_isSedge}, and  \varlink{exch2\_isSedge}{exch2_isSedge}, and
349  \varlink{exch2\_isNedge}{exch2_isNedge} are set to \code{1} if the  \varlink{exch2\_isNedge}{exch2_isNedge} are set to \code{1} if the
350  indexed tile lies on the respective edge of a subdomain, \code{0} if  indexed tile lies on the edge of its subdomain, \code{0} if
351  not.  The values are used within the topology generator to determine  not.  The values are used within the topology generator to determine
352  the orientation of neighboring tiles, and to indicate whether a tile  the orientation of neighboring tiles, and to indicate whether a tile
353  lies on the corner of a subdomain.  The latter case requires special  lies on the corner of a subdomain.  The latter case requires special
# Line 346  exchange and numerical handling for the Line 355  exchange and numerical handling for the
355  corners of the cube. \\  corners of the cube. \\
356    
357    
358  \subsubsection{Arrays Indexed to Tile Number and Neighbor}  Arrays Indexed to Tile Number and Neighbor:
359    
360  The following arrays have vectors of length \code{MAX\_NEIGHBOURS} and  The following arrays have vectors of length \code{MAX\_NEIGHBOURS} and
361  \code{NTILES} and describe the orientations between the the tiles. \\  \code{NTILES} and describe the orientations between the the tiles. \\
# Line 371  This provides a back-reference from the Line 380  This provides a back-reference from the
380  The arrays \varlink{exch2\_pi}{exch2_pi} and  The arrays \varlink{exch2\_pi}{exch2_pi} and
381  \varlink{exch2\_pj}{exch2_pj} specify the transformations of indices  \varlink{exch2\_pj}{exch2_pj} specify the transformations of indices
382  in exchanges between the neighboring tiles.  These transformations are  in exchanges between the neighboring tiles.  These transformations are
383  necessary in exchanges between subdomains because the array index in  necessary in exchanges between subdomains because a horizontal dimension
384  one dimension may map to the other index in an adjacent subdomain, and  in one subdomain
385  may be have its indexing reversed. This swapping arises from the  may map to other horizonal dimension in an adjacent subdomain, and
386    may also have its indexing reversed. This swapping arises from the
387  ``folding'' of two-dimensional arrays into a three-dimensional  ``folding'' of two-dimensional arrays into a three-dimensional
388  cube. \\  cube. \\
389    
# Line 381  The dimensions of \code{exch2\_pi(t,N,T) Line 391  The dimensions of \code{exch2\_pi(t,N,T)
391  are the neighbor ID \code{N} and the tile number \code{T} as explained  are the neighbor ID \code{N} and the tile number \code{T} as explained
392  above, plus a vector of length \code{2} containing transformation  above, plus a vector of length \code{2} containing transformation
393  factors \code{t}.  The first element of the transformation vector  factors \code{t}.  The first element of the transformation vector
394  holds the factor to multiply the index in the same axis, and the  holds the factor to multiply the index in the same dimension, and the
395  second element holds the the same for the orthogonal index.  To  second element holds the the same for the orthogonal dimension.  To
396  clarify, \code{exch2\_pi(1,N,T)} holds the mapping of the $x$ axis  clarify, \code{exch2\_pi(1,N,T)} holds the mapping of the $x$ axis
397  index of tile \code{T} to the $x$ axis of tile \code{T}'s neighbor  index of tile \code{T} to the $x$ axis of tile \code{T}'s neighbor
398  \code{N}, and \code{exch2\_pi(2,N,T)} holds the mapping of \code{T}'s  \code{N}, and \code{exch2\_pi(2,N,T)} holds the mapping of \code{T}'s
# Line 397  the arrays for all tile neighbors on the Line 407  the arrays for all tile neighbors on the
407  \code{(1,0)}, since all tiles on the same subdomain are oriented  \code{(1,0)}, since all tiles on the same subdomain are oriented
408  identically.  An axis that corresponds to the orthogonal dimension  identically.  An axis that corresponds to the orthogonal dimension
409  with the same index direction in a particular tile-neighbor  with the same index direction in a particular tile-neighbor
410  orientation will have \code{(0,1)}.  Those in the opposite index  orientation will have \code{(0,1)}.  Those with the opposite index
411  direction will have \code{(0,-1)} in order to reverse the ordering. \\  direction will have \code{(0,-1)} in order to reverse the ordering. \\
412    
413  The arrays \varlink{exch2\_oi}{exch2_oi},  The arrays \varlink{exch2\_oi}{exch2_oi},
# Line 483  the neighbor tile is \code{Tn=5}: Line 493  the neighbor tile is \code{Tn=5}:
493  \code{Tn}'s $y$ axis corresponds to \code{T}'s $x$ axis, \code{T}'s  \code{Tn}'s $y$ axis corresponds to \code{T}'s $x$ axis, \code{T}'s
494  northern edge exchanges with \code{Tn}'s western edge.  The western  northern edge exchanges with \code{Tn}'s western edge.  The western
495  edge of the tiles corresponds to the lower bound of the $x$ axis, so  edge of the tiles corresponds to the lower bound of the $x$ axis, so
496  \code{exch2\_itlo\_c} \code{exch2\_ithi\_c} are \code{0}. The range of  \code{exch2\_itlo\_c} and \code{exch2\_ithi\_c} are \code{0}, in the
497    western halo region of \code{Tn}. The range of
498  \code{exch2\_jtlo\_c} and \code{exch2\_jthi\_c} correspond to the  \code{exch2\_jtlo\_c} and \code{exch2\_jthi\_c} correspond to the
499  width of \code{T}'s northern edge, plus the halo. \\  width of \code{T}'s northern edge, expanded by one into the halo. \\
500    
501    
502  \subsection{Key Routines}  \subsubsection{Key Routines}
503    
504  Most of the subroutines particular to exch2 handle the exchanges  Most of the subroutines particular to exch2 handle the exchanges
505  themselves and are of the same format as those described in  themselves and are of the same format as those described in
506  \ref{sect:cube_sphere_communication} \sectiontitle{Cube sphere  \ref{sect:cube_sphere_communication} \sectiontitle{Cube sphere
507  communication}.  Like the original routines, they are written as  communication}.  Like the original routines, they are written as
508  templates which the local Makefile converts from RX into RL and RS  templates which the local Makefile converts from \code{RX} into
509  forms. \\  \code{RL} and \code{RS} forms. \\
510    
511  The interfaces with the core model subroutines are  The interfaces with the core model subroutines are
512  \code{EXCH\_UV\_XY\_RX}, \code{EXCH\_UV\_XYZ\_RX} and  \code{EXCH\_UV\_XY\_RX}, \code{EXCH\_UV\_XYZ\_RX} and
# Line 510  for scalars and \code{EXCH2\_RX2\_CUBE} Line 521  for scalars and \code{EXCH2\_RX2\_CUBE}
521  the singularities at the cube corners. \\  the singularities at the cube corners. \\
522    
523  The separate scalar and vector forms of \code{EXCH2\_RX1\_CUBE} and  The separate scalar and vector forms of \code{EXCH2\_RX1\_CUBE} and
524  \code{EXCH2\_RX2\_CUBE} reflect that the vector-handling subrouine  \code{EXCH2\_RX2\_CUBE} reflect that the vector-handling subroutine
525  needs to pass both the $u$ and $v$ components of the phsical vectors.  needs to pass both the $u$ and $v$ components of the physical vectors.
526  This arises from the topological folding discussed above, where the  This swapping arises from the topological folding discussed above, where the
527  $x$ and $y$ axes get swapped in some cases.  This swapping is not an  $x$ and $y$ axes get swapped in some cases, and is not an
528  issue with the scalar version. These subroutines call  issue with the scalar case. These subroutines call
529  \code{EXCH2\_SEND\_RX1} and \code{EXCH2\_SEND\_RX2}, which do most of  \code{EXCH2\_SEND\_RX1} and \code{EXCH2\_SEND\_RX2}, which do most of
530  the work using the variables discussed above. \\  the work using the variables discussed above. \\
531    
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