--- manual/s_phys_pkgs/text/exch2.tex	2004/03/15 22:39:28	1.11
+++ manual/s_phys_pkgs/text/exch2.tex	2005/07/18 20:45:27	1.24
@@ -1,4 +1,4 @@
-% $Header: /home/ubuntu/mnt/e9_copy/manual/s_phys_pkgs/text/exch2.tex,v 1.11 2004/03/15 22:39:28 afe Exp $
+% $Header: /home/ubuntu/mnt/e9_copy/manual/s_phys_pkgs/text/exch2.tex,v 1.24 2005/07/18 20:45:27 molod Exp $
 % $Name:  $
 
 %%  * Introduction
@@ -10,40 +10,38 @@
 %%    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 is an extension to the original cubed
-sphere topological configuration that allows 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
-combinations, and whether exchanges between particular tiles occur
-between different processors is determined at runtime.  This
-flexibility provides for manual compile-time load balancing across a
-relatively arbitrary number of processors. \\
+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 tiles can run on
+separate processors individually or in groups, which 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 \file{SIZE.h} file as
-detailed below.  Both files are generated by Matlab scripts in
+detailed below.  The default files provided in the release configure a
+cubed sphere topology of six tiles, one per subdomain, each with
+32$\times$32 grid points, with all tiles running on a single processor.  Both
+files are generated by Matlab scripts in
 \file{utils/exch2/matlab-topology-generator}; see Section
 \ref{sec:topogen} \sectiontitle{Generating Topology Files for exch2}
-for details on creating alternate topologies.  The default files
-provided in the release configure a cubed sphere topology of six
-tiles, one per subdomain, each with 32$\times$32 grid points, all
-running on a single processor.  Pregenerated examples of these files
-with alternate topologies are provided under
+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}
+\subsubsection{Invoking exch2}
 
 To use exch2 with the cubed sphere, the following conditions must be
 met: \\
@@ -51,42 +49,41 @@
 $\bullet$ 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
+  \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
+  \file{w2\_e2setup.F} must reside in a directory containing files
+  symbolically linked by the \file{genmake2} script.  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
+  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{tile???.mitgrid} where \file{???} is \file{001} through
-  \file{006} (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 the 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}; a more general background on the subject
-  relvant to MITgcm is presented in Section
-  \ref{sect:specifying_a_decomposition}\sectiontitle{Specifying a
-  decomposition}.\\
+  \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$32 cube sides 
+  -- please contact MITgcm support if you want to generate
+  files for other configurations. \\
+
+$\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 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
-used for guidance:
+At the time of this writing the following examples use exch2 and may
+be used for guidance:
 
 \begin{verbatim}
 verification/adjust_nlfs.cs-32x32x1
@@ -99,36 +96,39 @@
 
 
 
-\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
 scripts in \file{utils/exch2/matlab-topology-generator}. Running the
-m-file \file{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 except for
-development purposes. \\
+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 -- figures \ref{fig:6tile}, \ref{fig:12tile}, 
+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.
+\code{tnx} and \code{tny}, and all are assigned early in the script. \\
 
-The first three determine the size of the subdomains (cube faces) 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 each 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 of MITgcm to accomodate different
-resolutions around the axes to allow (for example) greater resolution
-around the equator.\\
+but they provide for future releases  to accomodate different
+resolutions around the axes to allow subdomains with differing resolutions.\\
 
-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 twenty-four 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}
@@ -137,9 +137,13 @@
   \includegraphics{part6/s24t_16x16.ps}
  }
 \end{center} 
-\caption{Plot of cubed sphere topology with a 32$\times$32 grid and
-twenty-four tiles (\code{tnx=16, tny=16})
-} \label{fig:24tile}
+
+\caption{Plot of a cubed sphere topology with a 32$\times$192 domain
+divided into six 32$\times$32 subdomains, each of which is divided
+into four tiles of width \code{tnx=16} and height \code{tny=16} for a 
+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}
@@ -148,172 +152,382 @@
   \includegraphics{part6/s12t_16x32.ps}
  }
 \end{center} 
-\caption{Plot of cubed sphere topology with a 32$\times$32 grid and
-twelve tiles (\code{tnx=16, tny=32})
+\caption{Plot of a cubed sphere topology with a 32$\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 kind otuning is useful in
-ocean modeling for omitting tiles that fall entirely on land.  The
-tiles omitted are specified in the file \file{blanklist.txt} by
-their tile number in the topology, separated by a newline. \\
+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. \\
 
 
 
 
+\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
+decomposition within MITgcm and its relation to \file{SIZE.h}. The
+current section specifies constraints that the exch2 package
+imposes and 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 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. \\
+
+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} is the number of processors.  The
+total number of tiles in the topology minus those listed in
+\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 
+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 twenty-four-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}
+\subsubsection{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
+one-dimensional arrays indexed to the tile number, and two and
+three-dimensional arrays indexed to tile number and neighboring tile.
+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 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}
+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.
-
-\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.
+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 in order to allow global data files that are tile-layout-neutral.
+They have no bearing on the internal storage of the arrays.  The tiles
+are stored internally in a range from \code{\varlink{bi}{bi}=(1:NTILES)} in the
+$x$ axis, and the $y$ axis variable \varlink{bj}{bj} is assumed to 
+equal \code{1} throughout the package. \\
+
+Arrays indexed to tile number:
+
+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 \texttt{SIZE.h}.  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
+\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 may allow varying tile
+sizes. \\
+
+The arrays \varlink{exch2\_tbasex}{exch2_tbasex} and
+\varlink{exch2\_tbasey}{exch2_tbasey} determine the tiles' 
+Cartesian origin within a subdomain  
+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 twenty-four-tile case discussed above will
+have values of \code{0} or \code{16}, depending on the quadrant of the
+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
-global address space, similar to that used by global files.
+\varlink{exch2\_tbasey}{exch2_tbasey}, but locate the tile edges within the
+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 the neighboring tiles each tile has, and sets 
+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.  \\
+
 
 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.
-
-\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.
-
-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, 
+\varlink{exch2\_isNedge}{exch2_isNedge} are set to \code{1} if the
+indexed tile lies on the edge of its 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. \\
+
+
+Arrays Indexed to Tile Number and Neighbor:
+
+The following arrays have vectors of length \code{MAX\_NEIGHBOURS} and
+\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:exch2\_nNeighbours(T))} in the order right to left on the
+north then south edges, and then top to bottom on the east then west
+edges.  \\
+
+ The \code{exch2\_opposingSend\_record(a,T)} array holds the
+index \code{b} of the element in \texttt{exch2\_neighbourId(b,Tn)}
+that holds the tile number \code{T}, given
+\code{Tn=exch2\_neighborId(a,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},
-\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.
-
-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].
+The arrays \varlink{exch2\_pi}{exch2_pi} and
+\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 horizontal dimension 
+in one subdomain 
+may map to other horizonal dimension in an adjacent subdomain, and
+may also 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 \code{2} containing transformation
+factors \code{t}.  The first element of the transformation vector
+holds the factor to multiply the index in the same dimension, and the
+second element holds the the same for the orthogonal dimension.  To
+clarify, \code{exch2\_pi(1,N,T)} holds the mapping of the $x$ axis
+index of tile \code{T} to the $x$ axis of tile \code{T}'s neighbor
+\code{N}, and \code{exch2\_pi(2,N,T)} holds the mapping of \code{T}'s
+$x$ index to the neighbor \code{N}'s $y$ index. \\
+ 
+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 axes are orthogonal.  The other element will be
+\code{1} or \code{-1}, depending on whether the axes 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.  An axis that corresponds to the orthogonal dimension
+with the same index direction in a particular tile-neighbor
+orientation will have \code{(0,1)}.  Those with the opposite index
+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 \code{N} to a
+local tile \code{T}.  Consider \code{T=1} in the six-tile topology
+(Fig. \ref{fig:6tile}), where
 
-{\footnotesize
 \begin{verbatim}
-C      exch2_pi          :: X index row of target to source permutation 
-C                        :: matrix for each neighbour entry.            
-C      exch2_pj          :: Y index row of target to source permutation 
-C                        :: matrix for each neighbour entry.            
-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.                     
+       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. \\
 
 
-\subsection{Key Routines}
 
+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 twelve-tile topology
+(Fig. \ref{fig:12tile}): \\
 
+\begin{verbatim}
+       exch2_itlo_c(4,2)=17
+       exch2_ithi_c(4,2)=17
+       exch2_jtlo_c(4,2)=0
+       exch2_jthi_c(4,2)=33
+\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.
+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}:
+
+\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. \\
 
-\subsection{References}
