--- manual/s_phys_pkgs/text/exch2.tex 2004/02/11 20:48:14 1.7 +++ manual/s_phys_pkgs/text/exch2.tex 2004/10/12 17:27:17 1.21 @@ -1,4 +1,4 @@ -% $Header: /home/ubuntu/mnt/e9_copy/manual/s_phys_pkgs/text/exch2.tex,v 1.7 2004/02/11 20:48:14 afe Exp $ +% $Header: /home/ubuntu/mnt/e9_copy/manual/s_phys_pkgs/text/exch2.tex,v 1.21 2004/10/12 17:27:17 edhill Exp $ % $Name: $ %% * Introduction @@ -10,174 +10,527 @@ %% o automatically inserted at \section{Reference} -\section{exch2: Extended Cubed Sphere Exchange} +\section{exch2: Extended Cubed Sphere \mbox{Topology}} \label{sec:exch2} \subsection{Introduction} -The exch2 package is an extension to the original cubed sphere exchanges -to allow more flexible domain decomposition and parallelization. Cube faces -(subdomains) may be divided into whatever 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. - -The exchange parameters are declared in {\em W2\_EXCH2\_TOPOLOGY.h} and -assigned in {\em w2\_e2setup.F}, both in the -{\em pkg/exch2} directory. The validity of the cube topology depends -on the {\em SIZE.h} file as detailed below. Both files are generated by -Matlab scripts and -should not be edited. The default files provided in the release set up -a cube sphere arrangement of six tiles, one per subdomain, each with 32x32 grid -points, running on a single processor. +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. The default files provided in the release configure a +cubed sphere topology of six tiles, one per subdomain, each with +32$\times$32 grid points, with all tiles running on a single processor. Both +files are generated by Matlab scripts in +\file{utils/exch2/matlab-topology-generator}; see Section +\ref{sec:topogen} \sectiontitle{Generating Topology Files for exch2} +for details on creating alternate topologies. Pregenerated examples +of these files with alternate topologies are provided under +\file{utils/exch2/code-mods} along with the appropriate \file{SIZE.h} +file for single-processor execution. + +\subsection{Invoking exch2} + +To use exch2 with the cubed sphere, the following conditions must be +met: + +\begin{itemize} +\item The exch2 package is included when \file{genmake2} is run. The + easiest way to do this is to add the line \code{exch2} to the + \file{profile.conf} file -- see Section \ref{sect:buildingCode} + \sectiontitle{Building the code} for general details. + +\item An example of \file{W2\_EXCH2\_TOPOLOGY.h} and + \file{w2\_e2setup.F} must reside in a directory containing files + symbolically linked by the \file{genmake2} script. The safest place + to put these is the directory indicated in the \code{-mods=DIR} + command line modifier (typically \file{../code}), or the build + directory. The default versions of these files reside in + \file{pkg/exch2} and are linked automatically if no other versions + exist elsewhere in the build path, but they should be left untouched + to avoid breaking configurations other than the one you intend to + modify. + +\item Files containing grid parameters, named \file{tile00$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 -- please contact + \begin{rawhtml} + + \end{rawhtml} +\begin{verbatim} +MITgcm-support@mitgcm.org +\end{verbatim} + \begin{rawhtml} \end{rawhtml} + if you want to generate files for other configurations. + +\item 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}. +\end{itemize} + + + +At the time of this writing the following examples use exch2 and may +be used for guidance: + +\begin{verbatim} +verification/adjust_nlfs.cs-32x32x1 +verification/adjustment.cs-32x32x1 +verification/aim.5l_cs +verification/global_ocean.cs32x15 +verification/hs94.cs-32x32x5 +\end{verbatim} + + + + +\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 -- 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. \\ + +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 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 subdomains with differing resolutions.\\ + +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 +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$32 subdomains, each of which is divided +into four tiles of width \code{tnx=16} and height \code{tny=16} for a +total of twenty-four tiles. The colored borders of the subdomains +represent the parameters \code{nr} (red), \code{nb} (blue), and +\code{ng} (green). } \label{fig:24tile} +\end{figure} + +\begin{figure} +\begin{center} + \resizebox{4in}{!}{ + \includegraphics{part6/s12t_16x32.ps} + } +\end{center} +\caption{Plot of a cubed sphere topology with a 32$\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 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. \\ + +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} 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 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. +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 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 -{\em NTILES}, and the maximum number of neighbors of any tiles by -{\em 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. - -The scalar parameters {\em exch2\_domain\_nxt} and -{\em exch2\_domain\_nyt} express the number of tiles in the x and y global -indices. For example, the default setup of six tiles has -{\em exch2\_domain\_nxt=6} and {\em exch2\_domain\_nyt=1}. A topology of -twenty-four square (in gridpoints) tiles, four (2x2) per subdomain, will -have {\em exch2\_domain\_nxt=12} and {\em 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 {\em 1,bi} (in the -x axis) and y-axis variable {\em bj} is generally ignored within the package. - -\subsubsection{Arrays Indexed to Tile Number} - -The following arrays are of size {\em NTILES}, are indexed to the tile number, -and the indices are omitted in their descriptions. - -The arrays {\em exch2\_tnx} and {\em exch2\_tny} -express the x and y dimensions of each tile. At present for each tile -{\em exch2\_tnx = sNx} -and {\em exch2\_tny = sNy}, as assigned in {\em 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 {\em exch2\_tbasex} and {\em 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. {\em 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 {\em exch2\_txglobalo} and {\em exch2\_txglobalo} are similar to -{\em exch2\_tbasex} and {\em exch2\_tbasey}, but locate the tiles within -the global address space, similar to that used by global files. - -The arrays {\em exch2\_isWedge}, {\em exch2\_isEedge}, {\em exch2\_isSedge}, -and {\em 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. {\em exch2\_isNedge} contains a count of how many -neighboring tiles each tile has, and is used for setting bounds for looping -over neighboring tiles. {\em exch2\_tProc} holds the process rank of each tile, -and is used in interprocess communication. +\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 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 (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. \\ + +\subsubsection{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 +\code{tn}. 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 +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 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{fn}} in +figures \ref{fig:12tile} and \ref{fig:24tile}. The +\varlink{exch2\_nNeighbours}{exch2_nNeighbours} variable contains a +count of the neighboring tiles each tile has, and sets the bounds for +looping over neighboring tiles. And +\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 \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. \\ + \subsubsection{Arrays Indexed to Tile Number and Neighbor} -The following arrays are all of size {\em MAX\_NEIGHBOURS}x{\em NTILES} and -describe the orientations between the the tiles. +The following arrays have vectors of length \code{MAX\_NEIGHBOURS} and +\code{NTILES} and describe the orientations between the the tiles. \\ -The array {\em exch2\_neighbourId(a,T)} holds the tile number $T_{n}$ for each tile -{\em T}'s neighbor tile {\em a}. The neighbor tiles are indexed {\em 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? - -{\em exch2\_opposingSend\_record(a,T)} holds -the index c in {\em exch2\_neighbourId(b,$T_{n}$)} that holds the tile number T. -In other words, - -\begin{verbatim} -exch2_neighbourId( exch2_opposingSend_record(a,T), - exch2_neighbourId(a,T) ) = T -\end{verbatim} - -% {\em exch2\_neighbourId(exch2\_opposingSend\_record(a,T),exch2\_neighbourId(a,T))=T}. -% alternate version - -This is to provide a backreference from the neighbor tiles. - -The arrays {\em exch2\_pi }, {\em exch2\_pj }, {\em exch2\_oi }, -{\em exch2\_oj }, {\em exch2\_oi\_f }, and {\em exch2\_oj\_f } specify -the transformations in exchanges between the neighboring tiles. The dimensions -of {\em exch2\_pi(t,N,T) } and {\em exch2\_pj(t,N,T) } are the neighbor ID -{ \em N } and the tile number {\em 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, {\em exch2\_pi(1,N,T) } holds the -transform of the i-component of a vector variable in tile {\em T } to the i-component of -tile {\em T }'s neighbor {\em N }, and {\em exch2\_pi(2,N,T) } hold the component -of neighbor {\em N }'s j-component. - -Under the current cube topology, one of the two elements of {\em exch2\_pi } or {\em exch2\_pj } -for a given tile {\em T } and neighbor {\em 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 {\em [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 {\em [0 , 1] }, whereas those in the opposite index direction will have -{\em [0 , -1] }. - - - - -// +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} +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 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 \begin{verbatim} + exch2_oi(1,1)=33 + exch2_oi(2,1)=0 + exch2_oi(3,1)=32 + exch2_oi(4,1)=-32 +\end{verbatim} + +The simplest case is \code{exch2\_oi(2,1)}, the southern neighbor, +which is \code{Tn=6}. The axes of \code{T} and \code{Tn} have the +same orientation and their $x$ axes have the same origin, and so an +exchange between the two requires no changes to the $x$ index. For +the western neighbor (\code{Tn=5}), \code{code\_oi(3,1)=32} since the +\code{x=0} vector on \code{T} corresponds to the \code{y=32} vector on +\code{Tn}. The eastern edge of \code{T} shows the reverse case +(\code{exch2\_oi(4,1)=-32)}), where \code{x=32} on \code{T} exchanges +with \code{x=0} on \code{Tn=2}. \\ + + The most interesting case, where \code{exch2\_oi(1,1)=33} and +\code{Tn=3}, involves a reversal of indices. As in every case, the +offset \code{exch2\_oi} is added to the original $x$ index of \code{T} +multiplied by the transformation factor \code{exch2\_pi(t,N,T)}. Here +\code{exch2\_pi(1,1,1)=0} since the $x$ axis of \code{T} is orthogonal +to the $x$ axis of \code{Tn}. \code{exch2\_pi(2,1,1)=-1} since the +$x$ axis of \code{T} corresponds to the $y$ axis of \code{Tn}, but the +index is reversed. The result is that the index of the northern edge +of \code{T}, which runs \code{(1:32)}, is transformed to +\code{(-1:-32)}. \code{exch2\_oi(1,1)} is then added to this range to +get back \code{(32:1)} -- the index of the $y$ axis of \code{Tn} +relative to \code{T}. This transformation may seem overly convoluted +for the six-tile case, but it is necessary to provide a general +solution for various topologies. \\ + +Finally, \varlink{exch2\_itlo\_c}{exch2_itlo_c}, +\varlink{exch2\_ithi\_c}{exch2_ithi_c}, +\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}): \\ -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. +\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. \\ \subsection{Key Routines} +Most of the subroutines particular to exch2 handle the exchanges +themselves and are of the same format as those described in +\ref{sect:cube_sphere_communication} \sectiontitle{Cube sphere +communication}. Like the original routines, they are written as +templates which the local Makefile converts from \code{RX} into +\code{RL} and \code{RS} forms. \\ + +The interfaces with the core model subroutines are +\code{EXCH\_UV\_XY\_RX}, \code{EXCH\_UV\_XYZ\_RX} and +\code{EXCH\_XY\_RX}. They override the standard exchange routines +when \code{genmake2} is run with \code{exch2} option. They in turn +call the local exch2 subroutines \code{EXCH2\_UV\_XY\_RX} and +\code{EXCH2\_UV\_XYZ\_RX} for two and three-dimensional vector +quantities, and \code{EXCH2\_XY\_RX} and \code{EXCH2\_XYZ\_RX} for two +and three-dimensional scalar quantities. These subroutines set the +dimensions of the area to be exchanged, call \code{EXCH2\_RX1\_CUBE} +for scalars and \code{EXCH2\_RX2\_CUBE} for vectors, and then handle +the singularities at the cube corners. \\ + +The separate scalar and vector forms of \code{EXCH2\_RX1\_CUBE} and +\code{EXCH2\_RX2\_CUBE} reflect that the vector-handling subroutine +needs to pass both the $u$ and $v$ components of the physical vectors. +This swapping arises from the topological folding discussed above, where the +$x$ and $y$ axes get swapped in some cases, and is not an +issue with the scalar case. These subroutines call +\code{EXCH2\_SEND\_RX1} and \code{EXCH2\_SEND\_RX2}, which do most of +the work using the variables discussed above. \\ - -\subsection{References}