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-revision 1.13 by afe,
Wed Mar 17 19:49:22 2004 UTC
+revision 1.27 by jmc,
Sat May  2 02:13:18 2009 UTC
 Line 10
  %%    o automatically inserted at \section{Reference}
- \section{exch2: Extended Cubed Sphere \mbox{Topology}}
+ \subsection{exch2: Extended Cubed Sphere \mbox{Topology}}
  \label{sec:exch2}
- \subsection{Introduction}
+ \subsubsection{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 43 
 of these files with alternate topologies
+Line 41 
 of these files with alternate topologies
  \file{utils/exch2/code-mods} along with the appropriate \file{SIZE.h}
  file for single-processor execution.
- \subsection{Invoking exch2}
+ \subsubsection{Invoking exch2}
  To use exch2 with the cubed sphere, the following conditions must be
- met: \\
+ met:
- $\bullet$ The exch2 package is included when \file{genmake2} is run.
+ \begin{itemize}
-   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
-   \file{profile.conf} file -- see Section
+   easiest way to do this is to add the line \code{exch2} to the
-   \ref{sect:buildingCode} \sectiontitle{Building the code} for general
+   \file{packages.conf} file -- see Section \ref{sect:buildingCode}
-   details. \\
+   \sectiontitle{Building the code} for general
+   details.
- $\bullet$ An example of \file{W2\_EXCH2\_TOPOLOGY.h} and
-   \file{w2\_e2setup.F} must reside in a directory containing code
+ \item An example of \file{W2\_EXCH2\_TOPOLOGY.h} and
-   linked when \file{genmake2} runs.  The safest place to put these
+   \file{w2\_e2setup.F} must reside in a directory containing files
-   is the directory indicated in the \code{-mods=DIR} command line
+   symbolically linked by the \file{genmake2} script.  The safest place
-   modifier (typically \file{../code}), or the build directory.  The
+   to put these is the directory indicated in the \code{-mods=DIR}
-   default versions of these files reside in \file{pkg/exch2} and are
+   command line modifier (typically \file{../code}), or the build
-   linked automatically if no other versions exist elsewhere in the
+   directory.  The default versions of these files reside in
-   link path, but they should be left untouched to avoid breaking
+   \file{pkg/exch2} and are linked automatically if no other versions
-   configurations other than the one you intend to modify.\\
+   exist elsewhere in the build path, but they should be left untouched
+   to avoid breaking configurations other than the one you intend to
- $\bullet$ Files containing grid parameters, named
+   modify.
-   \file{tile00$n$.mitgrid} where $n$=\code{(1:6)} (one per subdomain),
-   must be in the working directory when the MITgcm executable is run.
+ \item Files containing grid parameters, named \file{tile00$n$.mitgrid}
-   These files are provided in the example experiments for cubed sphere
+   where $n$=\code{(1:6)} (one per subdomain), must be in the working
-   configurations with 32$\times$32 cube sides and are non-trivial to
+   directory when the MITgcm executable is run.  These files are
-   generate -- please contact MITgcm support if you want to generate
+   provided in the example experiments for cubed sphere configurations
-   files for other configurations. \\
+   with 32$\times$32 cube sides -- 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
-   be placed where \file{genmake2} will find it.  In particular for
+ \item As always when compiling MITgcm, the file \file{SIZE.h} must be
-   exch2, the domain decomposition specified in \file{SIZE.h} must
+   placed where \file{genmake2} will find it.  In particular for exch2,
-   correspond with the particular configuration's topology specified in
+   the domain decomposition specified in \file{SIZE.h} must correspond
+   with the particular configuration's topology specified in
    \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
-   general background on the subject relevant to MITgcm is presented in
+     Multiprocessing}; a more general background on the subject
-   Section \ref{sect:specifying_a_decomposition}
+   relevant to MITgcm is presented in Section
-   \sectiontitle{Specifying a decomposition}.\\
+   \ref{sect:specifying_a_decomposition}
+   \sectiontitle{Specifying a decomposition}.
+ \end{itemize}
- 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 98 
 verification/hs94.cs-32x32x5
+Line 99 
 verification/hs94.cs-32x32x5
- \subsection{Generating Topology Files for exch2}
+ \subsubsection{Generating Topology Files for exch2}
  \label{sec:topogen}
  Alternate cubed sphere topologies may be created using the Matlab
-Line 108 
 m-file
+Line 109 
 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:18tile},
- subroutines of \file{driver.m} and should not be run ``bare'' except
+ and \ref{fig:48tile} 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:48tile} shows one possible topology for a twenty-four-tile
- cube, and figure \ref{fig:12tile} shows one for twelve tiles. \\
+ cube, and figure \ref{fig:6tile} shows one for six tiles. \\
  \begin{figure}
  \begin{center}
-  \resizebox{4in}{!}{
+  \resizebox{6in}{!}{
-   \includegraphics{part6/s24t_16x16.ps}
+ % \includegraphics{part6/s24t_16x16.ps}
+   \includegraphics{part6/adjust_cs.ps}
   }
  \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$32 subdomains, each of which is divided
- (\code{tnx=16, tny=16}) for a total of twentyfour tiles.
+ into eight tiles of width \code{tnx=16} and height \code{tny=8} for a
- } \label{fig:24tile}
+ total of forty-eight tiles. The colored borders of the subdomains
+ represent the parameters \code{nr} (red), \code{ng} (green), and
+ \code{nb} (blue).
+ This tiling is used in the example
+ verification/adjustment.cs-32x32x1/
+ with the option (blanklist.txt) to remove the land-only 4 tiles
+ (11,12,13,14) which are filled in red on the plot.
+ } \label{fig:48tile}
  \end{figure}
  \begin{figure}
  \begin{center}
-  \resizebox{4in}{!}{
+  \resizebox{6in}{!}{
-   \includegraphics{part6/s12t_16x32.ps}
+ % \includegraphics{part6/s12t_16x32.ps}
+   \includegraphics{part6/polarcap.ps}
   }
  \end{center}
- \caption{Plot of a cubed sphere topology with a 32$\times$192 domain
+ \caption{Plot of a non-square cubed sphere topology with
- divided into six 32$\times$32 subdomains of two tiles each
+subdomains of different size (nr=90,ng=360,nb=90),
-  (\code{tnx=16, tny=32}).
+ divided into one to four tiles each
- } \label{fig:12tile}
+  (\code{tnx=90, tny=90}), resulting in a total of 18 tiles.
+ } \label{fig:18tile}
  \end{figure}
  \begin{figure}
  \begin{center}
   \resizebox{4in}{!}{
+ % \includegraphics{part6/s6t_32x32.ps}
    \includegraphics{part6/s6t_32x32.ps}
   }
  \end{center}
-Line 181 
 by their tile number in the topology, se
+Line 194 
 by their tile number in the topology, se
- \subsection{exch2, SIZE.h, and multiprocessing}
+ \subsubsection{exch2, SIZE.h, and Multiprocessing}
  \label{sec:exch2mpi}
  Once the topology configuration files are created, the Fortran
  \code{PARAMETER}s in \file{SIZE.h} must be configured to match.
  Section \ref{sect:specifying_a_decomposition} \sectiontitle{Specifying
- a decomposition} provides a general description of domain
+   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
- imposes as well as describes how to enable parallel execution with
+ and describes how to enable parallel execution with MPI.
- MPI. \\
  As in the general case, the parameters \varlink{sNx}{sNx} and
  \varlink{sNy}{sNy} define the size of the individual tiles, and so
  must be assigned the same respective values as \code{tnx} and
- \code{tny} in \file{driver.m}.\\
+ \code{tny} in \file{driver.m}.
  The halo width parameters \varlink{OLx}{OLx} and \varlink{OLy}{OLy}
  have no special bearing on exch2 and may be assigned as in the general
  case. The same holds for \varlink{Nr}{Nr}, the number of vertical
- levels in the model.\\
+ 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. \\
+ regenerating the topology, \code{\varlink{nPy}{nPy}=1} as well.
  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,
- The following is an example of \file{SIZE.h} for the twelve-tile
+ this restriction might entail sharing a processor with a tile that
- configuration illustrated in figure \ref{fig:12tile} running on
+ would otherwise be excluded because it is topographically outside of
- one processor: \\
+ 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 six-tile
+ configuration illustrated in figure \ref{fig:6tile}
+ running on one processor:
  \begin{verbatim}
        PARAMETER (
-      &           sNx =  16,
+      &           sNx =  32,
       &           sNy =  32,
       &           OLx =   2,
       &           OLy =   2,
-      &           nSx =  12,
+      &           nSx =   6,
       &           nSy =   1,
       &           nPx =   1,
       &           nPy =   1,
-Line 237 
 one processor: \\
+Line 259 
 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 forty-eight-tile topology in
- \ref{fig:24tile} running on six processors:
+ figure \ref{fig:48tile} running on six processors:
  \begin{verbatim}
        PARAMETER (
       &           sNx =  16,
-      &           sNy =  16,
+      &           sNy =   8,
       &           OLx =   2,
       &           OLy =   2,
-      &           nSx =   4,
+      &           nSx =   8,
       &           nSy =   1,
       &           nPx =   6,
       &           nPy =   1,
-Line 256 
 The following is an example for the twen
+Line 278 
 The following is an example for the twen
  \end{verbatim}
+ \subsubsection{Key Variables}
- \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. \\
- \subsubsection{Scalars}
+ Scalars:
  The number of tiles in a particular topology is set with the parameter
  \code{NTILES}, and the maximum number of neighbors of any tiles by
-Line 281 
 generated by \file{driver.m}.\\
+Line 300 
 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 forty-eight tiles, eight per subdomain (as in figure
- \code{exch2\_domain\_nxt=12} and \code{exch2\_domain\_nyt=2}.  Note
+ \ref{fig:48tile}), will have \code{exch2\_domain\_nxt=12} and
- that these parameters express the tile layout to allow global data
+ \code{exch2\_domain\_nyt=4}.  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}
+ 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
+ figures \ref{fig:6tile} and
- \ref{fig:24tile}. \varlink{exch2\_nNeighbours}{exch2_nNeighbours}
+ \ref{fig:48tile}. \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 356 
 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}
+ 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 387 
 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. \\
- \varlink{exch2\_oi}{exch2_oi},
+ 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}
+ \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 \code{N} to a
+ local tile \code{T}.  Consider \code{T=1} in the six-tile topology
+ (Fig. \ref{fig:6tile}), where
+ \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. \\
- This needs some diagrams. \\
+ Finally, \varlink{exch2\_itlo\_c}{exch2_itlo_c},
+ \varlink{exch2\_ithi\_c}{exch2_ithi_c},
+ \varlink{exch2\_jtlo\_c}{exch2_jtlo_c} and
+ \varlink{exch2\_jthi\_c}{exch2_jthi_c} hold the location and index
+ bounds of the edge segment of the neighbor tile \code{N}'s subdomain
+ that gets exchanged with the local tile \code{T}.  To take the example
+ of tile \code{T=2} in the forty-eight-tile topology
+ (Fig. \ref{fig:48tile}): \\
- {\footnotesize
  \begin{verbatim}
- C      exch2_pi          :: X index row of target to source permutation
+        exch2_itlo_c(4,2)=17
- C                        :: matrix for each neighbour entry.
+        exch2_ithi_c(4,2)=17
- C      exch2_pj          :: Y index row of target to source permutation
+        exch2_jtlo_c(4,2)=0
- C                        :: matrix for each neighbour entry.
+        exch2_jthi_c(4,2)=33
- C      exch2_oi          :: X index element of target to source
- C                        :: offset vector for cell-centered quantities
- C                        :: of each neighbor entry.
- C      exch2_oj          :: Y index element of target to source
- C                        :: offset vector for cell-centered quantities
- C                        :: of each neighbor entry.
- C      exch2_oi_f        :: X index element of target to source
- C                        :: offset vector for face quantities
- C                        :: of each neighbor entry.
- C      exch2_oj_f        :: Y index element of target to source
- C                        :: offset vector for face quantities
- C                        :: of each neighbor entry.
  \end{verbatim}
- }
+ Here \code{N=4}, indicating the western neighbor, which is
+ \code{Tn=1}.  \code{Tn} resides on the same subdomain as \code{T}, so
+ the tiles have the same orientation and the same $x$ and $y$ axes.
- \subsection{Key Routines}
+ The $x$ axis is orthogonal to the western edge and the tile is 16
+ points wide, so \code{exch2\_itlo\_c} and \code{exch2\_ithi\_c}
+ indicate the column beyond \code{Tn}'s eastern edge, in that tile's
+ 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 of \code{(1:32)}, plus 1 in
+ 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}:
- \subsection{References}
+ \begin{verbatim}
+        exch2_itlo_c(1,2)=0
+        exch2_ithi_c(1,2)=0
+        exch2_jtlo_c(1,2)=0
+        exch2_jthi_c(1,2)=17
+ \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{T}'s
+ northern edge exchanges with \code{Tn}'s western edge.  The western
+ edge of the tiles corresponds to the lower bound of the $x$ axis, so
+ \code{exch2\_itlo\_c} and \code{exch2\_ithi\_c} are \code{0}, in the
+ western halo region of \code{Tn}. The range of
+ \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. \\
+ \subsubsection{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. \\
+ \subsubsection{Experiments and tutorials that use exch2}
+ \label{sec:pkg:exch2:experiments}
+ \begin{itemize}
+ \item{Held Suarez tutorial, in tutorial\_held\_suarez\_cs verification directory,
+ described in section \ref{sect:eg-hs} }
+ \end{itemize}

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