/[MITgcm]/manual/s_phys_pkgs/text/exch2.tex

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-revision 1.14 by afe,
Wed Mar 17 21:44:02 2004 UTC
+revision 1.23 by afe,
Fri Jun 10 19:34:09 2005 UTC
 Line 16
  \subsection{Introduction}
- The \texttt{exch2} package extends the original cubed
+ The \texttt{exch2} package extends the original cubed sphere topology
- sphere topology configuration to allow more flexible domain
+ configuration to allow more flexible domain decomposition and
- decomposition and parallelization.  Cube faces (also called
+ parallelization.  Cube faces (also called subdomains) may be divided
- subdomains) may be divided into any number of tiles that divide evenly
+ into any number of tiles that divide evenly into the grid point
- into the grid point dimensions of the subdomain.  Furthermore, the
+ dimensions of the subdomain.  Furthermore, the tiles can run on
- individual tiles can run on separate processors in different
+ separate processors individually or in groups, which provides for
- combinations, and whether exchanges between particular tiles occur
+ manual compile-time load balancing across a relatively arbitrary
- between different processors is determined at runtime.  This
+ number of processors. \\
- flexibility provides for manual compile-time load balancing across a
- relatively arbitrary number of processors. \\
  The exchange parameters are declared in
  \filelink{pkg/exch2/W2\_EXCH2\_TOPOLOGY.h}{pkg-exch2-W2_EXCH2_TOPOLOGY.h}
-Line 34 
 and assigned in
+Line 32 
 and assigned in
  validity of the cube topology depends on the \file{SIZE.h} file as
  detailed below.  The default files provided in the release configure a
  cubed sphere topology of six tiles, one per subdomain, each with
-$\times$32 grid points, all running on a single processor.  Both
+$\times$32 grid points, with all tiles running on a single processor.  Both
  files are generated by Matlab scripts in
  \file{utils/exch2/matlab-topology-generator}; see Section
  \ref{sec:topogen} \sectiontitle{Generating Topology Files for exch2}
-Line 55 
 $\bullet$ The exch2 package is included
+Line 53 
 $\bullet$ The exch2 package is included
    details. \\
  $\bullet$ An example of \file{W2\_EXCH2\_TOPOLOGY.h} and
-   \file{w2\_e2setup.F} must reside in a directory containing code
+   \file{w2\_e2setup.F} must reside in a directory containing files
-   linked when \file{genmake2} runs.  The safest place to put these
+   symbolically linked by the \file{genmake2} script.  The safest place to
-   is the directory indicated in the \code{-mods=DIR} command line
+   put these is the directory indicated in the \code{-mods=DIR} command
-   modifier (typically \file{../code}), or the build directory.  The
+   line modifier (typically \file{../code}), or the build directory.
-   default versions of these files reside in \file{pkg/exch2} and are
+   The default versions of these files reside in \file{pkg/exch2} and
-   linked automatically if no other versions exist elsewhere in the
+   are linked automatically if no other versions exist elsewhere in the
-   link path, but they should be left untouched to avoid breaking
+   build path, but they should be left untouched to avoid breaking
    configurations other than the one you intend to modify.\\
  $\bullet$ Files containing grid parameters, named
    \file{tile00$n$.mitgrid} where $n$=\code{(1:6)} (one per subdomain),
    must be in the working directory when the MITgcm executable is run.
    These files are provided in the example experiments for cubed sphere
-   configurations with 32$\times$32 cube sides and are non-trivial to
+   configurations with 32$\times$32 cube sides
-   generate -- please contact MITgcm support if you want to generate
+   -- please contact MITgcm support if you want to generate
    files for other configurations. \\
  $\bullet$ As always when compiling MITgcm, the file \file{SIZE.h} must
-Line 79 
 $\bullet$ As always when compiling MITgc
+Line 77 
 $\bullet$ As always when compiling MITgc
    \file{W2\_EXCH2\_TOPOLOGY.h} and \file{w2\_e2setup.F}.  Domain
    decomposition issues particular to exch2 are addressed in Section
    \ref{sec:topogen} \sectiontitle{Generating Topology Files for exch2}
-   and \ref{sec:exch2mpi} \sectiontitle{exch2, SIZE.h, and MPI}; a more
+   and \ref{sec:exch2mpi} \sectiontitle{exch2, SIZE.h, and Multiprocessing}; a more
    general background on the subject relevant to MITgcm is presented in
    Section \ref{sect:specifying_a_decomposition}
    \sectiontitle{Specifying a decomposition}.\\
- As of the time of writing the following examples use exch2 and may be
+ At the time of this writing the following examples use exch2 and may
- used for guidance:
+ be used for guidance:
  \begin{verbatim}
  verification/adjust_nlfs.cs-32x32x1
-Line 108 
 m-file
+Line 106 
 m-file
  from the Matlab prompt (there are no parameters to pass) generates
  exch2 topology files \file{W2\_EXCH2\_TOPOLOGY.h} and
  \file{w2\_e2setup.F} in the working directory and displays a figure of
- the topology via Matlab.  The other m-files in the directory are
+ the topology via Matlab -- figures \ref{fig:6tile}, \ref{fig:12tile},
- subroutines of \file{driver.m} and should not be run ``bare'' except
+ and \ref{fig:24tile} are examples of the generated diagrams.  The other
+ m-files in the directory are
+ subroutines called from \file{driver.m} and should not be run ``bare'' except
  for development purposes. \\
  The parameters that determine the dimensions and topology of the
  generated configuration are \code{nr}, \code{nb}, \code{ng},
  \code{tnx} and \code{tny}, and all are assigned early in the script. \\
- The first three determine the size of the subdomains and
+ The first three determine the height and width of the subdomains and
  hence the size of the overall domain.  Each one determines the number
  of grid points, and therefore the resolution, along the subdomain
- sides in a ``great circle'' around an axis of the cube.  At the time
+ sides in a ``great circle'' around each the three spatial axes of the cube.  At the time
  of this writing MITgcm requires these three parameters to be equal,
  but they provide for future releases  to accomodate different
- resolutions around the axes to allow (for example) greater resolution
+ resolutions around the axes to allow subdomains with differing resolutions.\\
- around the equator.\\
- The parameters \code{tnx} and \code{tny} determine the dimensions of
+ The parameters \code{tnx} and \code{tny} determine the width and height of
  the tiles into which the subdomains are decomposed, and must evenly
  divide the integer assigned to \code{nr}, \code{nb} and \code{ng}.
  The result is a rectangular tiling of the subdomain.  Figure
- \ref{fig:24tile} shows one possible topology for a twentyfour-tile
+ \ref{fig:24tile} shows one possible topology for a twenty-four-tile
  cube, and figure \ref{fig:12tile} shows one for twelve tiles. \\
  \begin{figure}
-Line 140 
 cube, and figure \ref{fig:12tile} shows
+Line 139 
 cube, and figure \ref{fig:12tile} shows
  \end{center}
  \caption{Plot of a cubed sphere topology with a 32$\times$192 domain
- divided into six 32$\times$32 subdomains, each of which is divided into four tiles
+ divided into six 32$\times$32 subdomains, each of which is divided
- (\code{tnx=16, tny=16}) for a total of twentyfour tiles.
+ into four tiles of width \code{tnx=16} and height \code{tny=16} for a
- } \label{fig:24tile}
+ total of twenty-four tiles.  The colored borders of the subdomains
+ represent the parameters \code{nr} (red), \code{nb} (blue), and
+ \code{ng} (green).  } \label{fig:24tile}
  \end{figure}
  \begin{figure}
-Line 181 
 by their tile number in the topology, se
+Line 182 
 by their tile number in the topology, se
- \subsection{exch2, SIZE.h, and multiprocessing}
+ \subsection{exch2, SIZE.h, and Multiprocessing}
  \label{sec:exch2mpi}
  Once the topology configuration files are created, the Fortran
-Line 189 
 Once the topology configuration files ar
+Line 190 
 Once the topology configuration files ar
  Section \ref{sect:specifying_a_decomposition} \sectiontitle{Specifying
  a decomposition} provides a general description of domain
  decomposition within MITgcm and its relation to \file{SIZE.h}. The
- current section specifies certain constraints the exch2 package
+ current section specifies constraints that the exch2 package
- imposes as well as describes how to enable parallel execution with
+ imposes and describes how to enable parallel execution with
  MPI. \\
  As in the general case, the parameters \varlink{sNx}{sNx} and
-Line 206 
 levels in the model.\\
+Line 207 
 levels in the model.\\
  The parameters \varlink{nSx}{nSx}, \varlink{nSy}{nSy},
  \varlink{nPx}{nPx}, and \varlink{nPy}{nPy} relate to the number of
  tiles and how they are distributed on processors.  When using exch2,
- the tiles are stored in single dimension, and so
+ the tiles are stored in the $x$ dimension, and so
  \code{\varlink{nSy}{nSy}=1} in all cases.  Since the tiles as
  configured by exch2 cannot be split up accross processors without
  regenerating the topology, \code{\varlink{nPy}{nPy}=1} as well. \\
-Line 214 
 regenerating the topology, \code{\varlin
+Line 215 
 regenerating the topology, \code{\varlin
  The number of tiles MITgcm allocates and how they are distributed
  between processors depends on \varlink{nPx}{nPx} and
  \varlink{nSx}{nSx}.  \varlink{nSx}{nSx} is the number of tiles per
- processor and \varlink{nPx}{nPx} the number of processors.  The total
+ processor and \varlink{nPx}{nPx} is the number of processors.  The
- number of tiles in the topology minus those listed in
+ total number of tiles in the topology minus those listed in
- \file{blanklist.txt} must equal \code{nSx*nPx}. \\
+ \file{blanklist.txt} must equal \code{nSx*nPx}.  Note that in order to
+ obtain maximum usage from a given number of processors in some cases,
+ this restriction might entail sharing a processor with a tile that
+ would otherwise be excluded because it is topographically outside of
+ the domain and therefore in \file{blanklist.txt}.  For example,
+ suppose you have five processors and a domain decomposition of
+ thirty-six tiles that allows you to exclude seven tiles.  To evenly
+ distribute the remaining twenty-nine tiles among five processors, you
+ would have to run one ``dummy'' tile to make an even six tiles per
+ processor.  Such dummy tiles are \emph{not} listed in
+ \file{blanklist.txt}.\\
  The following is an example of \file{SIZE.h} for the twelve-tile
  configuration illustrated in figure \ref{fig:12tile} running on
-Line 237 
 one processor: \\
+Line 249 
 one processor: \\
       &           Nr  =   5)
  \end{verbatim}
- The following is an example for the twentyfour-tile topology in figure
+ The following is an example for the twenty-four-tile topology in
- \ref{fig:24tile} running on six processors:
+ figure \ref{fig:24tile} running on six processors:
  \begin{verbatim}
        PARAMETER (
-Line 262 
 The following is an example for the twen
+Line 274 
 The following is an example for the twen
  \subsection{Key Variables}
  The descriptions of the variables are divided up into scalars,
- one-dimensional arrays indexed to the tile number, and two and three
+ one-dimensional arrays indexed to the tile number, and two and
- dimensional arrays indexed to tile number and neighboring tile.  This
+ three-dimensional arrays indexed to tile number and neighboring tile.
- division reflects the functionality of these variables: The
+ This division reflects the functionality of these variables: The
  scalars are common to every part of the topology, the tile-indexed
  arrays to individual tiles, and the arrays indexed by tile and
  neighbor to relationships between tiles and their neighbors. \\
-Line 281 
 generated by \file{driver.m}.\\
+Line 293 
 generated by \file{driver.m}.\\
  The scalar parameters \varlink{exch2\_domain\_nxt}{exch2_domain_nxt}
  and \varlink{exch2\_domain\_nyt}{exch2_domain_nyt} express the number
  of tiles in the $x$ and $y$ global indices.  For example, the default
- setup of six tiles (Fig. \ref{fig:6tile}) has \code{exch2\_domain\_nxt=6} and
+ setup of six tiles (Fig. \ref{fig:6tile}) has
- \code{exch2\_domain\_nyt=1}.  A topology of twenty-four square tiles,
+ \code{exch2\_domain\_nxt=6} and \code{exch2\_domain\_nyt=1}.  A
- four per subdomain (as in figure \ref{fig:24tile}), will have
+ topology of twenty-four square tiles, four per subdomain (as in figure
- \code{exch2\_domain\_nxt=12} and \code{exch2\_domain\_nyt=2}.  Note
+ \ref{fig:24tile}), will have \code{exch2\_domain\_nxt=12} and
- that these parameters express the tile layout to allow global data
+ \code{exch2\_domain\_nyt=2}.  Note that these parameters express the
- files that are tile-layout-neutral and have no bearing on the internal
+ tile layout in order to allow global data files that are tile-layout-neutral.
- storage of the arrays.  The tiles are internally stored in a range
+ They have no bearing on the internal storage of the arrays.  The tiles
- from \code{(1:\varlink{bi}{bi})} the $x$ axis, and $y$ axis variable
+ are stored internally in a range from \code{\varlink{bi}{bi}=(1:NTILES)} in the
- \varlink{bj}{bj} is generally ignored within the package. \\
+ $x$ axis, and the $y$ axis variable \varlink{bj}{bj} is assumed to
+ equal \code{1} throughout the package. \\
- \subsubsection{Arrays Indexed to Tile Number}
+ \subsubsection{Arrays indexed to tile number}
- The following arrays are of length \code{NTILES}, are indexed to the
- tile number, and the indices are omitted in their descriptions. \\
+ The following arrays are of length \code{NTILES} and are indexed to
+ the tile number, which is indicated in the diagrams with the notation
+ \textsf{t}$n$.  The indices are omitted in the descriptions. \\
  The arrays \varlink{exch2\_tnx}{exch2_tnx} and
  \varlink{exch2\_tny}{exch2_tny} express the $x$ and $y$ dimensions of
  each tile.  At present for each tile \texttt{exch2\_tnx=sNx} and
  \texttt{exch2\_tny=sNy}, as assigned in \file{SIZE.h} and described in
- section \ref{sec:exch2mpi} \sectiontitle{exch2, SIZE.h, and
+ Section \ref{sec:exch2mpi} \sectiontitle{exch2, SIZE.h, and
- multiprocessing}.  Future releases of MITgcm are to allow varying tile
+ Multiprocessing}.  Future releases of MITgcm may allow varying tile
  sizes. \\
- The location of the tiles' Cartesian origin within a subdomain are
+ The arrays \varlink{exch2\_tbasex}{exch2_tbasex} and
- determined by the arrays \varlink{exch2\_tbasex}{exch2_tbasex} and
+ \varlink{exch2\_tbasey}{exch2_tbasey} determine the tiles'
- \varlink{exch2\_tbasey}{exch2_tbasey}.  These variables are used to
+ Cartesian origin within a subdomain
- 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
  an example, in the default six-tile topology (Fig. \ref{fig:6tile})
  each index in these arrays is set to \code{0} since a tile occupies
- its entire subdomain.  The twentyfour-tile case discussed above will
+ its entire subdomain.  The twenty-four-tile case discussed above will
- 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
- tile falls within the subdomain.  The elements of the arrays
+ tile within the subdomain.  The elements of the arrays
  \varlink{exch2\_txglobalo}{exch2_txglobalo} and
  \varlink{exch2\_txglobalo}{exch2_txglobalo} are similar to
  \varlink{exch2\_tbasex}{exch2_tbasex} and
- \varlink{exch2\_tbasey}{exch2_tbasey}, but locate the tiles within the
+ \varlink{exch2\_tbasey}{exch2_tbasey}, but locate the tile edges within the
- global address space, similar to that used by global files. \\
+ global address space, similar to that used by global output and input
+ files. \\
  The array \varlink{exch2\_myFace}{exch2_myFace} contains the number of
  the subdomain of each tile, in a range \code{(1:6)} in the case of the
  standard cube topology and indicated by \textbf{\textsf{f}}$n$ in
  figures \ref{fig:12tile} and
  \ref{fig:24tile}. \varlink{exch2\_nNeighbours}{exch2_nNeighbours}
- contains a count of how many neighboring tiles each tile has, and is
+ contains a count of the neighboring tiles each tile has, and sets
- used for setting bounds for looping over neighboring tiles.
+ the bounds for looping over neighboring tiles.
  \varlink{exch2\_tProc}{exch2_tProc} holds the process rank of each
  tile, and is used in interprocess communication.  \\
-Line 334 
 The arrays \varlink{exch2\_isWedge}{exch
+Line 349 
 The arrays \varlink{exch2\_isWedge}{exch
  \varlink{exch2\_isEedge}{exch2_isEedge},
  \varlink{exch2\_isSedge}{exch2_isSedge}, and
  \varlink{exch2\_isNedge}{exch2_isNedge} are set to \code{1} if the
- indexed tile lies on the edge of a subdomain, \code{0} if not.  The
+ indexed tile lies on the edge of its subdomain, \code{0} if
- values are used within the topology generator to determine the
+ not.  The values are used within the topology generator to determine
- orientation of neighboring tiles, and to indicate whether a tile lies
+ the orientation of neighboring tiles, and to indicate whether a tile
- on the corner of a subdomain.  The latter case requires special
+ lies on the corner of a subdomain.  The latter case requires special
  exchange and numerical handling for the singularities at the eight
  corners of the cube. \\
  \subsubsection{Arrays Indexed to Tile Number and Neighbor}
- The following arrays are all of size
+ The following arrays have vectors of length \code{MAX\_NEIGHBOURS} and
- \code{MAX\_NEIGHBOURS}$\times$\code{NTILES} and describe the
+ \code{NTILES} and describe the orientations between the the tiles. \\
- orientations between the the tiles. \\
  The array \code{exch2\_neighbourId(a,T)} holds the tile number
  \code{Tn} for each of the tile number \code{T}'s neighboring tiles
- \code{a}.  The neighbor tiles are indexed \code{(1:MAX\_NEIGHBOURS)}
+ \code{a}.  The neighbor tiles are indexed
- in the order right to left on the north then south edges, and then top
+ \code{(1:exch2\_nNeighbours(T))} in the order right to left on the
- to bottom on the east and west edges.  Maybe throw in a fig here, eh?
+ north then south edges, and then top to bottom on the east then west
- \\
+ edges.  \\
- \sloppy
+  The \code{exch2\_opposingSend\_record(a,T)} array holds the
- The \code{exch2\_opposingSend\_record(a,T)} array holds the index
+ index \code{b} of the element in \texttt{exch2\_neighbourId(b,Tn)}
- \code{b} in \texttt{exch2\_neighbourId(b,Tn)} that holds the tile
+ that holds the tile number \code{T}, given
- number \code{T}.  In other words,
+ \code{Tn=exch2\_neighborId(a,T)}.  In other words,
  \begin{verbatim}
     exch2_neighbourId( exch2_opposingSend_record(a,T),
                        exch2_neighbourId(a,T) ) = T
-Line 366 
 number \code{T}.  In other words,
+Line 380 
 number \code{T}.  In other words,
  This provides a back-reference from the neighbor tiles. \\
  The arrays \varlink{exch2\_pi}{exch2_pi} and
- \varlink{exch2\_pj}{exch2_pj} specify the transformations of variables
+ \varlink{exch2\_pj}{exch2_pj} specify the transformations of indices
  in exchanges between the neighboring tiles.  These transformations are
- necessary in exchanges between subdomains because a physical vector
+ necessary in exchanges between subdomains because a horizontal dimension
- component in one direction may map to one in a different direction in
+ in one subdomain
- an adjacent subdomain, and may be have its indexing reversed. This
+ may map to other horizonal dimension in an adjacent subdomain, and
- swapping arises from the ``folding'' of two-dimensional arrays into a
+ may also have its indexing reversed. This swapping arises from the
- three-dimensional cube.
+ ``folding'' of two-dimensional arrays into a three-dimensional
+ cube. \\
  The dimensions of \code{exch2\_pi(t,N,T)} and \code{exch2\_pj(t,N,T)}
  are the neighbor ID \code{N} and the tile number \code{T} as explained
- above, plus a vector of length 2 containing transformation factors
+ above, plus a vector of length \code{2} containing transformation
- \code{t}.  The first element of the transformation vector indicates
+ factors \code{t}.  The first element of the transformation vector
- the factor \code{t} by which variables representing the same
+ holds the factor to multiply the index in the same dimension, and the
- \emph{physical} vector component of a tile \code{T} will be multiplied
+ second element holds the the same for the orthogonal dimension.  To
- in exchanges with neighbor \code{N}, and the second element indicates
+ clarify, \code{exch2\_pi(1,N,T)} holds the mapping of the $x$ axis
- the transform to the physical vector in the other direction.  To
+ index of tile \code{T} to the $x$ axis of tile \code{T}'s neighbor
- clarify (hopefully), \code{exch2\_pi(1,N,T)} holds the transform of
+ \code{N}, and \code{exch2\_pi(2,N,T)} holds the mapping of \code{T}'s
- the $i$ component of a vector variable in tile \code{T} to the $i$
+ $x$ index to the neighbor \code{N}'s $y$ index. \\
- component of tile \code{T}'s neighbor \code{N}, and
- \code{exch2\_pi(2,N,T)} holds the transform of \code{T}'s $i$
- components to the neighbor \code{N}'s $j$ component. \\
- Under the current cube topology, one of the two elements of
+ One of the two elements of \code{exch2\_pi} or \code{exch2\_pj} for a
- \code{exch2\_pi} or \code{exch2\_pj} for a given tile \code{T} and
+ given tile \code{T} and neighbor \code{N} will be \code{0}, reflecting
- neighbor \code{N} will be \code{0}, reflecting the fact that the two
+ the fact that the two axes are orthogonal.  The other element will be
- vector components are orthogonal.  The other element will be \code{1}
+ \code{1} or \code{-1}, depending on whether the axes are indexed in
- or \code{-1}, depending on whether the components are indexed in the
+ the same or opposite directions.  For example, the transform vector of
- same or opposite directions.  For example, the transform vector of the
+ the arrays for all tile neighbors on the same subdomain will be
- arrays for all tile neighbors on the same subdomain will be
  \code{(1,0)}, since all tiles on the same subdomain are oriented
- identically.  A vector direction that corresponds to the orthogonal
+ identically.  An axis that corresponds to the orthogonal dimension
- dimension with the same index direction in a particular tile-neighbor
+ with the same index direction in a particular tile-neighbor
- orientation will have \code{(0,1)}, whereas those in the opposite
+ orientation will have \code{(0,1)}.  Those with the opposite index
- index direction will have \code{(0,-1)}. \\
+ direction will have \code{(0,-1)} in order to reverse the ordering. \\
  The arrays \varlink{exch2\_oi}{exch2_oi},
  \varlink{exch2\_oj}{exch2_oj}, \varlink{exch2\_oi\_f}{exch2_oi_f}, and
  \varlink{exch2\_oj\_f}{exch2_oj_f} are indexed to tile number and
  neighbor and specify the relative offset within the subdomain of the
- array index of a variable going from a neighboring tile $N$ to a local
+ array index of a variable going from a neighboring tile \code{N} to a
- tile $T$.  Consider the six-tile case (Fig. \ref{fig:6tile}), where
+ local tile \code{T}.  Consider \code{T=1} in the six-tile topology
- \code{exch2\_oi(1,1)=33}, \code{exch2\_oi(2,1)=0},
+ (Fig. \ref{fig:6tile}), where
- \code{exch2\_oi(3,1)=32}, and \code{exch2\_oi(4,1)=-32}.  Each of these
- indicates the offset in the $x$ direction \\
+ \begin{verbatim}
+        exch2_oi(1,1)=33
+        exch2_oi(2,1)=0
+        exch2_oi(3,1)=32
+        exch2_oi(4,1)=-32
+ \end{verbatim}
+ The simplest case is \code{exch2\_oi(2,1)}, the southern neighbor,
+ which is \code{Tn=6}.  The axes of \code{T} and \code{Tn} have the
+ same orientation and their $x$ axes have the same origin, and so an
+ exchange between the two requires no changes to the $x$ index.  For
+ the western neighbor (\code{Tn=5}), \code{code\_oi(3,1)=32} since the
+ \code{x=0} vector on \code{T} corresponds to the \code{y=32} vector on
+ \code{Tn}.  The eastern edge of \code{T} shows the reverse case
+ (\code{exch2\_oi(4,1)=-32)}), where \code{x=32} on \code{T} exchanges
+ with \code{x=0} on \code{Tn=2}. \\
+  The most interesting case, where \code{exch2\_oi(1,1)=33} and
+ \code{Tn=3}, involves a reversal of indices.  As in every case, the
+ offset \code{exch2\_oi} is added to the original $x$ index of \code{T}
+ multiplied by the transformation factor \code{exch2\_pi(t,N,T)}.  Here
+ \code{exch2\_pi(1,1,1)=0} since the $x$ axis of \code{T} is orthogonal
+ to the $x$ axis of \code{Tn}.  \code{exch2\_pi(2,1,1)=-1} since the
+ $x$ axis of \code{T} corresponds to the $y$ axis of \code{Tn}, but the
+ index is reversed.  The result is that the index of the northern edge
+ of \code{T}, which runs \code{(1:32)}, is transformed to
+ \code{(-1:-32)}. \code{exch2\_oi(1,1)} is then added to this range to
+ get back \code{(32:1)} -- the index of the $y$ axis of \code{Tn}
+ relative to \code{T}.  This transformation may seem overly convoluted
+ for the six-tile case, but it is necessary to provide a general
+ solution for various topologies. \\
  Finally, \varlink{exch2\_itlo\_c}{exch2_itlo_c},
  \varlink{exch2\_ithi\_c}{exch2_ithi_c},
-Line 427 
 of tile \code{T=2} in the twelve-tile to
+Line 470 
 of tile \code{T=2} in the twelve-tile to
         exch2_jthi_c(4,2)=33
  \end{verbatim}
- Here \code{N=4}, indicating the western neighbor, which is \code{Tn=1}.
+ Here \code{N=4}, indicating the western neighbor, which is
- \code{Tn=1} resides on the same subdomain as \code{T=2}, so the tiles
+ \code{Tn=1}.  \code{Tn} resides on the same subdomain as \code{T}, so
- have the same orientation and the same $x$ and $y$ axes.  The $i$
+ the tiles have the same orientation and the same $x$ and $y$ axes.
- component is orthogonal to the western edge and the tile is 16 points
+ The $x$ axis is orthogonal to the western edge and the tile is 16
- wide, so \code{exch2\_itlo\_c} and \code{exch2\_ithi\_c} indicate the
+ points wide, so \code{exch2\_itlo\_c} and \code{exch2\_ithi\_c}
- column beyond \code{Tn=1}'s eastern edge, in that tile's halo
+ indicate the column beyond \code{Tn}'s eastern edge, in that tile's
- region. Since the border of the tiles extends through the entire
+ halo region. Since the border of the tiles extends through the entire
  height of the subdomain, the $y$ axis bounds \code{exch2\_jtlo\_c} to
- \code{exch2\_jthi\_c} cover the height, plus 1 in either direction to
+ \code{exch2\_jthi\_c} cover the height of \code{(1:32)}, plus 1 in
- cover part of the halo. \\
+ either direction to cover part of the halo. \\
  For the north edge of the same tile \code{T=2} where \code{N=1} and
  the neighbor tile is \code{Tn=5}:
-Line 449 
 the neighbor tile is \code{Tn=5}:
+Line 492 
 the neighbor tile is \code{Tn=5}:
  \end{verbatim}
  \code{T}'s northern edge is parallel to the $x$ axis, but since
- \code{Tn}'s $y$ axis corresponds to \code{T}'s $x$ axis,
+ \code{Tn}'s $y$ axis corresponds to \code{T}'s $x$ axis, \code{T}'s
- \code{T}'s northern edge exchanges with \code{Tn}'s western edge.
+ northern edge exchanges with \code{Tn}'s western edge.  The western
- The western edge of the tiles corresponds to the lower bound of the
+ edge of the tiles corresponds to the lower bound of the $x$ axis, so
- $x$ axis, so \code{exch2\_itlo\_c} \code{exch2\_ithi\_c} are \code{0}. The
+ \code{exch2\_itlo\_c} and \code{exch2\_ithi\_c} are \code{0}, in the
- range of \code{exch2\_jtlo\_c} and \code{exch2\_jthi\_c} correspond to the
+ western halo region of \code{Tn}. The range of
- width of \code{T}'s northern edge, plus the halo. \\
+ \code{exch2\_jtlo\_c} and \code{exch2\_jthi\_c} correspond to the
+ width of \code{T}'s northern edge, expanded by one into the halo. \\
- This needs some diagrams. \\
  \subsection{Key Routines}
+ Most of the subroutines particular to exch2 handle the exchanges
+ themselves and are of the same format as those described in
+ \ref{sect:cube_sphere_communication} \sectiontitle{Cube sphere
+ communication}.  Like the original routines, they are written as
+ templates which the local Makefile converts from \code{RX} into
+ \code{RL} and \code{RS} forms. \\
+ The interfaces with the core model subroutines are
+ \code{EXCH\_UV\_XY\_RX}, \code{EXCH\_UV\_XYZ\_RX} and
+ \code{EXCH\_XY\_RX}.  They override the standard exchange routines
+ when \code{genmake2} is run with \code{exch2} option.  They in turn
+ call the local exch2 subroutines \code{EXCH2\_UV\_XY\_RX} and
+ \code{EXCH2\_UV\_XYZ\_RX} for two and three-dimensional vector
+ quantities, and \code{EXCH2\_XY\_RX} and \code{EXCH2\_XYZ\_RX} for two
+ and three-dimensional scalar quantities.  These subroutines set the
+ dimensions of the area to be exchanged, call \code{EXCH2\_RX1\_CUBE}
+ for scalars and \code{EXCH2\_RX2\_CUBE} for vectors, and then handle
+ the singularities at the cube corners. \\
+ The separate scalar and vector forms of \code{EXCH2\_RX1\_CUBE} and
+ \code{EXCH2\_RX2\_CUBE} reflect that the vector-handling subroutine
+ needs to pass both the $u$ and $v$ components of the physical vectors.
+ This swapping arises from the topological folding discussed above, where the
+ $x$ and $y$ axes get swapped in some cases, and is not an
+ issue with the scalar case. These subroutines call
+ \code{EXCH2\_SEND\_RX1} and \code{EXCH2\_SEND\_RX2}, which do most of
+ the work using the variables discussed above. \\
- \subsection{References}

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