--- manual/s_phys_pkgs/text/exch2.tex	2004/03/12 20:58:19	1.9
+++ manual/s_phys_pkgs/text/exch2.tex	2004/03/17 19:49:22	1.13
@@ -1,4 +1,4 @@
-% $Header: /home/ubuntu/mnt/e9_copy/manual/s_phys_pkgs/text/exch2.tex,v 1.9 2004/03/12 20:58:19 afe Exp $
+% $Header: /home/ubuntu/mnt/e9_copy/manual/s_phys_pkgs/text/exch2.tex,v 1.13 2004/03/17 19:49:22 afe Exp $
 % $Name:  $
 
 %%  * Introduction
@@ -10,61 +10,79 @@
 %%    o automatically inserted at \section{Reference} 
 
 
-\section{Extended Cubed Sphere Exchange}
+\section{exch2: Extended Cubed Sphere \mbox{Topology}}
 \label{sec:exch2}
 
 
 \subsection{Introduction}
 
-The \texttt{exch2} package is an extension to the original cubed
-sphere topological configuration that allows more flexible domain
+The \texttt{exch2} package extends the original cubed
+sphere topology configuration to allow more flexible domain
 decomposition and parallelization.  Cube faces (also called
 subdomains) may be divided into any number of tiles that divide evenly
 into the grid point dimensions of the subdomain.  Furthermore, the
-individual tiles may be run on separate processors in different
+individual tiles can run on separate processors in different
 combinations, and whether exchanges between particular tiles occur
 between different processors is determined at runtime.  This
-flexibility provides for manual load balancing across a relatively
-arbitrary 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}
 and assigned in
 \filelink{pkg/exch2/w2\_e2setup.F}{pkg-exch2-w2_e2setup.F}. The
-validity of the cube topology depends on the \texttt{SIZE.h} file as
-detailed below.  Both files are generated by Matlab scripts in ??
-check these in already! and should not be edited.  The default files
-provided in the release configure a cubed sphere arrangement of six
-tiles, one per subdomain, each with 32$\times$32 grid points, all
-running on a single processor.  Pregenerated examples of these files 
-with alternate topologies are provided in ??.
+validity of the cube topology depends on the \file{SIZE.h} file as
+detailed below.  The default files provided in the release configure a
+cubed sphere topology of six tiles, one per subdomain, each with
+32$\times$32 grid points, all 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}
+for details on creating alternate topologies.  Pregenerated examples
+of these files with alternate topologies are provided under
+\file{utils/exch2/code-mods} along with the appropriate \file{SIZE.h}
+file for single-processor execution.
 
 \subsection{Invoking exch2}
 
-To use exch2 with the cubed sphere, the following conditions must be met:
+To use exch2 with the cubed sphere, the following conditions must be
+met: \\
 
-- the exch2 package is included when \texttt{genmake2} is run.  The
-  easiest way to do this is to add the line \texttt{exch2} to the
-  \texttt{profile.conf} file -- see Section \ref{sect:buildingCode}
-  for general details. \\
-
-- an example of \texttt{W2\_EXCH2\_TOPOLOGY.h} and
-  \texttt{w2\_e2setup.F} must reside in a directory containing code
-  linked when \texttt{genmake2} runs.  The safest place to put these
-  is the directory indicated in the \texttt{-mods=DIR} command line
-  modifier (typically \texttt{../code}), or the build directory.  The
-  default versions of these files reside in \texttt{pkg/exch2}, but
-  they should be left untouched to avoid breaking configurations other
-  than the one you intend to modify.\\
-
-- files containing grid parameters, named
-  \texttt{tile}xxx\texttt{.mitgrid} where xxx is \texttt{001} through
-  \texttt{006}, 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$32 cube
-  sides and are non-trivial to generate -- please contact MITgcm
-  support if you want to generate files for other configurations.
-  This is lame. ?? \\
+$\bullet$ The exch2 package is included when \file{genmake2} is run.
+  The easiest way to do this is to add the line \code{exch2} to the
+  \file{profile.conf} file -- see Section
+  \ref{sect:buildingCode} \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
+  linked when \file{genmake2} runs.  The safest place to put these
+  is the directory indicated in the \code{-mods=DIR} command line
+  modifier (typically \file{../code}), or the build directory.  The
+  default versions of these files reside in \file{pkg/exch2} and are
+  linked automatically if no other versions exist elsewhere in the
+  link 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$32 cube sides and are non-trivial to
+  generate -- please contact MITgcm support if you want to generate
+  files for other configurations. \\
+
+$\bullet$ As always when compiling MITgcm, the file \file{SIZE.h} must
+  be placed where \file{genmake2} will find it.  In particular for
+  exch2, 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
+  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
 used for guidance:
@@ -80,131 +98,318 @@
 
 
 
-\subsection{Generating Topology Files}
+\subsection{Generating Topology Files for exch2}
+\label{sec:topogen}
+
+Alternate cubed sphere topologies may be created using the Matlab
+scripts in \file{utils/exch2/matlab-topology-generator}. Running the
+m-file
+\filelink{driver.m}{utils-exch2-matlab-topology-generator_driver.m}
+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
+subroutines of \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
+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
+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
+around the equator.\\
+
+The parameters \code{tnx} and \code{tny} determine the dimensions 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
+cube, and figure \ref{fig:12tile} shows one for twelve tiles. \\
+
+\begin{figure}
+\begin{center}
+ \resizebox{4in}{!}{
+  \includegraphics{part6/s24t_16x16.ps}
+ }
+\end{center} 
+
+\caption{Plot of a cubed sphere topology with a 32$\times$192 domain
+divided into six 32$\times$32 subdomains, each of which is divided into four tiles 
+(\code{tnx=16, tny=16}) for a total of twentyfour tiles.
+} \label{fig:24tile}
+\end{figure}
+
+\begin{figure}
+\begin{center}
+ \resizebox{4in}{!}{
+  \includegraphics{part6/s12t_16x32.ps}
+ }
+\end{center} 
+\caption{Plot of a cubed sphere topology with a 32$\times$192 domain
+divided into six 32$\times$32 subdomains of two tiles each
+ (\code{tnx=16, tny=32}).
+} \label{fig:12tile}
+\end{figure}
+
+\begin{figure}
+\begin{center}
+ \resizebox{4in}{!}{
+  \includegraphics{part6/s6t_32x32.ps}
+ }
+\end{center} 
+\caption{Plot of a cubed sphere topology with a 32$\times$192 domain
+divided into six 32$\times$32 subdomains with one tile each
+(\code{tnx=32, tny=32}).  This is the default configuration.
+  }
+\label{fig:6tile}
+\end{figure}
+
+
+Tiles can be selected from the topology to be omitted from being
+allocated memory and processors.  This tuning is useful in ocean
+modeling for omitting tiles that fall entirely on land.  The tiles
+omitted are specified in the file
+\filelink{blanklist.txt}{utils-exch2-matlab-topology-generator_blanklist.txt}
+by their tile number in the topology, separated by a newline. \\
+
+
+
+
+\subsection{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
+decomposition within MITgcm and its relation to \file{SIZE.h}. The
+current section specifies certain constraints the exch2 package
+imposes as well as describes how to enable parallel execution with
+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}.\\
+
+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.\\
+
+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
+\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. \\
+
+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
+number of tiles in the topology minus those listed in
+\file{blanklist.txt} must equal \code{nSx*nPx}. \\
+
+The following is an example of \file{SIZE.h} for the twelve-tile
+configuration illustrated in figure \ref{fig:12tile} running on 
+one processor: \\
+
+\begin{verbatim}
+      PARAMETER (
+     &           sNx =  16,
+     &           sNy =  32,
+     &           OLx =   2,
+     &           OLy =   2,
+     &           nSx =  12,
+     &           nSy =   1,
+     &           nPx =   1,
+     &           nPy =   1,
+     &           Nx  = sNx*nSx*nPx,
+     &           Ny  = sNy*nSy*nPy,
+     &           Nr  =   5)
+\end{verbatim}
+
+The following is an example for the twentyfour-tile topology in figure
+\ref{fig:24tile} running on six processors:
+
+\begin{verbatim}
+      PARAMETER (
+     &           sNx =  16,
+     &           sNy =  16,
+     &           OLx =   2,
+     &           OLy =   2,
+     &           nSx =   4,
+     &           nSy =   1,
+     &           nPx =   6,
+     &           nPy =   1,
+     &           Nx  = sNx*nSx*nPx,
+     &           Ny  = sNy*nSy*nPy,
+     &           Nr  =   5)
+\end{verbatim}
+
+
+
+
 
 \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
 dimensional arrays indexed to tile number and neighboring tile.  This
-division actually reflects the functionality of these variables: the
+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 to tile and
-neighbor to relationships between tiles and their neighbors.
+arrays to individual tiles, and the arrays indexed by tile and
+neighbor to relationships between tiles and their neighbors. \\
 
 \subsubsection{Scalars}
 
 The number of tiles in a particular topology is set with the parameter
-\texttt{NTILES}, and the maximum number of neighbors of any tiles by
-\texttt{MAX\_NEIGHBOURS}.  These parameters are used for defining the
+\code{NTILES}, and the maximum number of neighbors of any tiles by
+\code{MAX\_NEIGHBOURS}.  These parameters are used for defining the
 size of the various one and two dimensional arrays that store tile
-parameters indexed to the tile number.\\
+parameters indexed to the tile number and are assigned in the files
+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 has \texttt{exch2\_domain\_nxt=6} and
-\texttt{exch2\_domain\_nyt=1}.  A topology of twenty-four square (in
-gridpoints) tiles, four (2x2) per subdomain, will have
-\texttt{exch2\_domain\_nxt=12} and \texttt{exch2\_domain\_nyt=2}.
-Note that these parameters express the tile layout to allow global
-data files that are tile-layout-neutral and have no bearing on the
-internal storage of the arrays.  The tiles are internally stored in a
-range from \texttt{1,bi} (in the x axis) and y-axis variable
-\texttt{bj} is generally ignored within the package.
+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
+\code{exch2\_domain\_nyt=1}.  A topology of twenty-four square tiles,
+four per subdomain (as in figure \ref{fig:24tile}), will have
+\code{exch2\_domain\_nxt=12} and \code{exch2\_domain\_nyt=2}.  Note
+that these parameters express the tile layout to allow global data
+files that are tile-layout-neutral and have no bearing on the internal
+storage of the arrays.  The tiles are internally stored in a range
+from \code{(1:\varlink{bi}{bi})} the $x$ axis, and $y$ axis variable
+\varlink{bj}{bj} is generally ignored within the package. \\
 
 \subsubsection{Arrays Indexed to Tile Number}
 
-The following arrays are of size \texttt{NTILES}, are indexed to the
-tile number, and the indices are omitted in their descriptions.
+The following arrays are of length \code{NTILES}, are indexed to the
+tile number, and the indices are omitted in their 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 \texttt{SIZE.h}.  Future
-releases of MITgcm are to allow varying tile sizes.
+\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
+multiprocessing}.  Future releases of MITgcm are to allow varying tile
+sizes. \\
 
 The location of the tiles' Cartesian origin within a subdomain are
 determined by the arrays \varlink{exch2\_tbasex}{exch2_tbasex} and
 \varlink{exch2\_tbasey}{exch2_tbasey}.  These variables are used to
-relate the location of the edges of the tiles to each other.  As an
-example, in the default six-tile topology (the degenerate case) each
-index in these arrays are set to 0.  The twenty-four, 32x32 cube face
-case discussed above will have values of 0 or 16, depending on the
-quadrant the tile falls within the subdomain.  The array 
-\varlink{exch2\_myFace}{exch2_myFace} contains the number of the
-cubeface/subdomain of each tile, numbered 1-6 in the case of the
-standard cube topology.
-
-The arrays \varlink{exch2\_txglobalo}{exch2_txglobalo} and
+relate the location of the edges of different tiles 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
+have values of \code{0} or \code{16}, depending on the quadrant the
+tile falls 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
-global address space, similar to that used by global files.
+global address space, similar to that used by global 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
+used for setting bounds for looping over neighboring tiles.
+\varlink{exch2\_tProc}{exch2_tProc} holds the process rank of each
+tile, and is used in interprocess communication.  \\
+
 
 The arrays \varlink{exch2\_isWedge}{exch2_isWedge},
 \varlink{exch2\_isEedge}{exch2_isEedge},
 \varlink{exch2\_isSedge}{exch2_isSedge}, and
-\varlink{exch2\_isNedge}{exch2_isNedge} are set to 1 if the indexed
-tile lies on the edge of a subdomain, 0 if not.  The values are used
-within the topology generator to determine the orientation of
-neighboring tiles and to indicate whether a tile lies on the corner of
-a subdomain.  The latter case indicates special exchange and numerical
-handling for the singularities at the eight corners of the cube.
-\varlink{exch2\_nNeighbours}{exch2_nNeighbours} contains a count of
-how many neighboring tiles each tile has, and is used for setting
-bounds for looping over neighboring tiles.
-\varlink{exch2\_tProc}{exch2_tProc} holds the process rank of each
-tile, and is used in interprocess communication.
+\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
+values are used within the topology generator to determine the
+orientation of neighboring tiles, and to indicate whether a tile 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 \texttt{MAX\_NEIGHBOURS} $\times$
-\texttt{NTILES} and describe the orientations between the the tiles.
+\subsubsection{Arrays Indexed to Tile Number and Neighbor}
 
-The array \texttt{exch2\_neighbourId(a,T)} holds the tile number for
-each of the $n$ neighboring tiles.  The neighbor tiles are indexed
-\texttt{(1,MAX\_NEIGHBOURS} in the order right to left on the north
-then south edges, and then top to bottom on the east and west edges.
-Maybe throw in a fig here, eh?
-
-The \texttt{exch2\_opposingSend\_record(a,T)} array holds the index c
-in \texttt{exch2\_neighbourId(b,$T_{n}$)} that holds the tile number T.
-In other words, 
+The following arrays are all of size
+\code{MAX\_NEIGHBOURS}$\times$\code{NTILES} and describe the
+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)}
+in the order right to left on the north then south edges, and then top
+to bottom on the east and west edges.  Maybe throw in a fig here, eh?
+\\
+
+\sloppy
+The \code{exch2\_opposingSend\_record(a,T)} array holds the index
+\code{b} in \texttt{exch2\_neighbourId(b,Tn)} that holds the tile
+number \code{T}.  In other words,
 \begin{verbatim}
    exch2_neighbourId( exch2_opposingSend_record(a,T),
                       exch2_neighbourId(a,T) ) = T
 \end{verbatim}
-and this provides a back-reference from the neighbor tiles.
+This provides a back-reference from the neighbor tiles. \\
 
-The arrays \varlink{exch2\_pi}{exch2_pi},
-\varlink{exch2\_pj}{exch2_pj}, \varlink{exch2\_oi}{exch2_oi},
+The arrays \varlink{exch2\_pi}{exch2_pi} and
+\varlink{exch2\_pj}{exch2_pj} specify the transformations of variables
+in exchanges between the neighboring tiles.  These transformations are
+necessary in exchanges between subdomains because a physical vector
+component in one direction may map to one in a different direction in
+an adjacent subdomain, and may be have its indexing reversed. This
+swapping arises from the ``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
+\code{t}.  The first element of the transformation vector indicates
+the factor \code{t} by which variables representing the same
+\emph{physical} vector component of a tile \code{T} will be multiplied
+in exchanges with neighbor \code{N}, and the second element indicates
+the transform to the physical vector in the other direction.  To
+clarify (hopefully), \code{exch2\_pi(1,N,T)} holds the transform of
+the $i$ component of a vector variable in tile \code{T} to the $i$
+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
+\code{exch2\_pi} or \code{exch2\_pj} for a given tile \code{T} and
+neighbor \code{N} will be \code{0}, reflecting the fact that the two
+vector components are orthogonal.  The other element will be \code{1}
+or \code{-1}, depending on whether the components are indexed in the
+same or opposite directions.  For example, the transform vector of the
+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
+dimension with the same index direction in a particular tile-neighbor
+orientation will have \code{(0,1)}, whereas those in the opposite
+index direction will have \code{(0,-1)}. \\
+
+
+\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} specify the transformations in
-exchanges between the neighboring tiles.  The dimensions of
-\texttt{exch2\_pi(t,N,T)} and \texttt{exch2\_pj(t,N,T)} are the
-neighbor ID \textit{N} and the tile number \textit{T} as explained
-above, plus the transformation vector {\em t }, of length two.  The
-first element of the transformation vector indicates the factor by
-which variables representing the same vector component of a tile will
-be multiplied, and the second element indicates the transform to the
-variable in the other direction.  As an example,
-\texttt{exch2\_pi(1,N,T)} holds the transform of the i-component of a
-vector variable in tile \texttt{T} to the i-component of tile
-\texttt{T}'s neighbor \texttt{N}, and \texttt{exch2\_pi(2,N,T)} hold
-the component of neighbor \texttt{N}'s j-component.
+\varlink{exch2\_oj\_f}{exch2_oj_f}
 
-Under the current cube topology, one of the two elements of
-\texttt{exch2\_pi} or \texttt{exch2\_pj} for a given tile \texttt{T}
-and neighbor \texttt{N} will be 0, reflecting the fact that the vector
-components are orthogonal.  The other element will be 1 or -1,
-depending on whether the components are indexed in the same or
-opposite directions.  For example, the transform dimension of the
-arrays for all tile neighbors on the same subdomain will be [1,0],
-since all tiles on the same subdomain are oriented identically.
-Vectors that correspond to the orthogonal dimension with the same
-index direction will have [0,1], whereas those in the opposite index
-direction will have [0,-1].
+
+
+
+This needs some diagrams. \\
 
 
 {\footnotesize
