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10  %%    o automatically inserted at \section{Reference}  %%    o automatically inserted at \section{Reference}
11    
12    
13  \section{exch2: Extended Cubed Sphere Exchange}  \section{exch2: Extended Cubed Sphere \mbox{Topology}}
14  \label{sec:exch2}  \label{sec:exch2}
15    
16    
17  \subsection{Introduction}  \subsection{Introduction}
18    
19  The exch2 package is an extension to the original cubed sphere exchanges  The \texttt{exch2} package extends the original cubed
20  to allow more flexible domain decomposition and parallelization.  Cube faces  sphere topology configuration to allow more flexible domain
21  (subdomains) may be divided into whatever number of tiles that divide evenly  decomposition and parallelization.  Cube faces (also called
22  into the grid point dimensions of the subdomain.  Furthermore, the individual  subdomains) may be divided into any number of tiles that divide evenly
23  tiles may be run on separate processors in different combinations,  into the grid point dimensions of the subdomain.  Furthermore, the
24  and whether exchanges between particular tiles occur between different  individual tiles may be run on separate processors in different
25  processors is determined at runtime.  combinations, and whether exchanges between particular tiles occur
26    between different processors is determined at runtime.  This
27  The exchange parameters are declared in {\em W2\_EXCH2\_TOPOLOGY.h} and  flexibility provides for manual compile-time load balancing across a
28  assigned in {\em w2\_e2setup.F}, both in the  relatively arbitrary number of processors. \\
29  {\em pkg/exch2} directory.  The validity of the cube topology depends  
30  on the {\em SIZE.h} file as detailed below.  Both files are generated by  The exchange parameters are declared in
31  Matlab scripts and  \filelink{pkg/exch2/W2\_EXCH2\_TOPOLOGY.h}{pkg-exch2-W2_EXCH2_TOPOLOGY.h}
32  should not be edited.  The default files provided in the release set up  and assigned in
33  a cube sphere arrangement of six tiles, one per subdomain, each with 32x32 grid  \filelink{pkg/exch2/w2\_e2setup.F}{pkg-exch2-w2_e2setup.F}. The
34  points, running on a single processor.    validity of the cube topology depends on the \file{SIZE.h} file as
35    detailed below.  The default files provided in the release configure a
36    cubed sphere topology of six tiles, one per subdomain, each with
37    32$\times$32 grid points, all running on a single processor.  Both
38    files are generated by Matlab scripts in
39    \file{utils/exch2/matlab-topology-generator}; see Section
40    \ref{sec:topogen} \sectiontitle{Generating Topology Files for exch2}
41    for details on creating alternate topologies.  Pregenerated examples
42    of these files with alternate topologies are provided under
43    \file{utils/exch2/code-mods} along with the appropriate \file{SIZE.h}
44    file for single-processor execution.
45    
46    \subsection{Invoking exch2}
47    
48    To use exch2 with the cubed sphere, the following conditions must be
49    met: \\
50    
51    $\bullet$ The exch2 package is included when \file{genmake2} is run.
52      The easiest way to do this is to add the line \code{exch2} to the
53      \file{profile.conf} file -- see Section
54      \ref{sect:buildingCode} \sectiontitle{Building the code} for general
55      details. \\
56    
57    $\bullet$ An example of \file{W2\_EXCH2\_TOPOLOGY.h} and
58      \file{w2\_e2setup.F} must reside in a directory containing code
59      linked when \file{genmake2} runs.  The safest place to put these
60      is the directory indicated in the \code{-mods=DIR} command line
61      modifier (typically \file{../code}), or the build directory.  The
62      default versions of these files reside in \file{pkg/exch2} and are
63      linked automatically if no other versions exist elsewhere in the
64      link path, but they should be left untouched to avoid breaking
65      configurations other than the one you intend to modify.\\
66    
67    $\bullet$ Files containing grid parameters, named
68      \file{tile00$n$.mitgrid} where $n$=[1,6] (one per subdomain), must
69      be in the working directory when the MITgcm executable is run.
70      These files are provided in the example experiments for cubed sphere
71      configurations with 32$\times$32 cube sides and are non-trivial to
72      generate -- please contact MITgcm support if you want to generate
73      files for other configurations. \\
74    
75    $\bullet$ As always when compiling MITgcm, the file \file{SIZE.h} must
76      be placed where \file{genmake2} will find it.  In particular for the
77      exch2, the domain decomposition specified in \file{SIZE.h} must
78      correspond with the particular configuration's topology specified in
79      \file{W2\_EXCH2\_TOPOLOGY.h} and \file{w2\_e2setup.F}.  Domain
80      decomposition issues particular to exch2 are addressed in Section
81      \ref{sec:topogen} \sectiontitle{Generating Topology Files for exch2}
82      and \ref{sec:exch2mpi} \sectiontitle{exch2, SIZE.h, and MPI}; a more
83      general background on the subject relevant to MITgcm is presented in
84      Section \ref{sect:specifying_a_decomposition}
85      \sectiontitle{Specifying a decomposition}.\\
86    
87  \subsection{Key Variables}  As of the time of writing the following examples use exch2 and may be
88    used for guidance:
89    
90  The descriptions of the variables are divided up into scalars,  \begin{verbatim}
91  one-dimensional arrays indexed to the tile number, and two and three  verification/adjust_nlfs.cs-32x32x1
92  dimensional  verification/adjustment.cs-32x32x1
93  arrays indexed to tile number and neighboring tile.  This division  verification/aim.5l_cs
94  actually reflects  the functionality of these variables: the scalars  verification/global_ocean.cs32x15
95  are common to every part of the topology, the tile-indexed arrays to  verification/hs94.cs-32x32x5
96  individual tiles, and the arrays indexed to tile and neighbor to  \end{verbatim}
 relationships between tiles and their neighbors.  
97    
 \subsubsection{Scalars}  
98    
 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.  
99    
 \subsubsection{Arrays Indexed to Tile Number}  
100    
101  The following arrays are of size {\em NTILES}, are indexed to the tile number,  \subsection{Generating Topology Files for exch2}
102  and the indices are omitted in their descriptions.  \label{sec:topogen}
103    
104  The arrays {\em exch2\_tnx} and {\em exch2\_tny}  Alternate cubed sphere topologies may be created using the Matlab
105  express the x and y dimensions of each tile.  At present for each tile  scripts in \file{utils/exch2/matlab-topology-generator}. Running the
106  {\em exch2\_tnx = sNx}  m-file
107  and {\em exch2\_tny = sNy}, as assigned in {\em SIZE.h}.  Future releases of  \filelink{driver.m}{utils-exch2-matlab-topology-generator_driver.m}
108  MITgcm are to allow varying tile sizes.  from the Matlab prompt (there are no parameters to pass) generates
109    exch2 topology files \file{W2\_EXCH2\_TOPOLOGY.h} and
110  The location of the tiles' Cartesian origin within a subdomain are determined  \file{w2\_e2setup.F} in the working directory and displays a figure of
111  by the arrays {\em exch2\_tbasex} and {\em exch2\_tbasey}.  These variables  the topology via Matlab.  The other m-files in the directory are
112  are used to relate the location of the edges of the tiles to each other.  As  subroutines of \file{driver.m} and should not be run ``bare'' except
113  an example, in the default six-tile topology (the degenerate case)  for development purposes. \\
114  each index in these arrays are  
115  set to 0.  The twenty-four, 32x32 cube face case discussed above will have  The parameters that determine the dimensions and topology of the
116  values of 0 or 16, depending on the quadrant the tile falls within the  generated configuration are \code{nr}, \code{nb}, \code{ng},
117  subdomain.  {\em exch2\_myFace} contains the number of the  \code{tnx} and \code{tny}, and all are assigned early in the script. \\
118  cubeface/subdomain of each tile, numbered 1-6 in the case of the standard  
119  cube topology.    The first three determine the size of the subdomains and
120    hence the size of the overall domain.  Each one determines the number
121  The arrays {\em exch2\_txglobalo} and {\em exch2\_txglobalo} are similar to  of grid points, and therefore the resolution, along the subdomain
122  {\em exch2\_tbasex} and {\em exch2\_tbasey}, but locate the tiles within  sides in a ``great circle'' around each axis of the cube.  At the time
123  the global address space, similar to that used by global files.    of this writing MITgcm requires these three parameters to be equal,
124    but they provide for future releases  to accomodate different
125  The arrays {\em exch2\_isWedge}, {\em exch2\_isEedge}, {\em exch2\_isSedge},  resolutions around the axes to allow (for example) greater resolution
126  and {\em exch2\_isNedge} are set to 1 if the indexed tile lies on the edge  around the equator.\\
127  of a subdomain, 0 if not.  The values are used within the topology generator  
128  to determine the orientation of neighboring tiles and to indicate whether  The parameters \code{tnx} and \code{tny} determine the dimensions of
129  a tile lies on the corner of a subdomain.  The latter case indicates  the tiles into which the subdomains are decomposed, and must evenly
130  special exchange and numerical handling for the singularities at the eight  divide the integer assigned to \code{nr}, \code{nb} and \code{ng}.
131  corners of the cube.  {\em exch2\_isNedge} contains a count of how many  The result is a rectangular tiling of the subdomain.  Figure
132  neighboring tiles each tile has, and is used for setting bounds for looping  \ref{fig:24tile} shows one possible topology for a twenty-four tile
133  over neighboring tiles.  {\em exch2\_tProc} holds the process rank of each tile,  cube, and figure \ref{fig:12tile} shows one for twelve tiles. \\
134  and is used in interprocess communication.  
135    \begin{figure}
136    \begin{center}
137     \resizebox{4in}{!}{
138      \includegraphics{part6/s24t_16x16.ps}
139     }
140    \end{center}
141    
142    \caption{Plot of cubed sphere topology with a 32$\times$192 domain
143    divided into six 32$\times$32 subdomains, each of which is divided into four tiles
144    (\code{tnx=16, tny=16}) for a total of twenty-four tiles.
145    } \label{fig:24tile}
146    \end{figure}
147    
148    \begin{figure}
149    \begin{center}
150     \resizebox{4in}{!}{
151      \includegraphics{part6/s12t_16x32.ps}
152     }
153    \end{center}
154    \caption{Plot of cubed sphere topology with a 32$\times$192 domain
155    divided into six 32$\times$32 subdomains of two tiles each
156     (\code{tnx=16, tny=32}).
157    } \label{fig:12tile}
158    \end{figure}
159    
160    Tiles can be selected from the topology to be omitted from being
161    allocated memory and processors.  This tuning is useful in ocean
162    modeling for omitting tiles that fall entirely on land.  The tiles
163    omitted are specified in the file
164    \filelink{blanklist.txt}{utils-exch2-matlab-topology-generator_blanklist.txt}
165    by their tile number in the topology, separated by a newline. \\
166    
167    
168    
169    
170    \subsection{exch2, SIZE.h, and multiprocessing}
171    \label{sec:exch2mpi}
172    
173    Once the topology configuration files are created, the Fortran
174    parameters in \file{SIZE.h} must be configured to match.  Section
175    \ref{sect:specifying_a_decomposition} \sectiontitle{Specifying a
176    decomposition} provides a general description of domain decomposition
177    within MITgcm and its relation to \file{SIZE.h}. The current section
178    specifies certain constraints the exch2 package imposes as well as
179    describes how to enable parallel execution with MPI. \\
180    
181    As in the general case, the parameters \varlink{sNx}{sNx} and
182    \varlink{sNy}{sNy} define the size of the individual tiles, and so
183    must be assigned the same respective values as \code{tnx} and
184    \code{tny} in \file{driver.m}.\\
185    
186    The halo width parameters \varlink{OLx}{OLx} and \varlink{OLy}{OLy}
187    have no special bearing on exch2 and may be assigned as in the general
188    case. The same holds for \varlink{Nr}{Nr}, the number of vertical
189    levels in the model.\\
190    
191    The parameters \varlink{nSx}{nSx}, \varlink{nSy}{nSy},
192    \varlink{nPx}{nPx}, and \varlink{nPy}{nPy} relate to the number of
193    tiles and how they are distributed on processors.  When using exch2,
194    the tiles are stored in single dimension, and so
195    \code{\varlink{nSy}{nSy}=1} in all cases.  Since the tiles as
196    configured by exch2 cannot be split up accross processors without
197    regenerating the topology, \code{\varlink{nPy}{nPy}=1} as well. \\
198    
199    The number of tiles MITgcm allocates and how they are distributed
200    between processors depends on \varlink{nPx}{nPx} and
201    \varlink{nSx}{nSx}.  \varlink{nSx}{nSx} is the number of tiles per
202    processor and \varlink{nPx}{nPx} the number of processors.  The total
203    number of tiles in the topology minus those listed in
204    \file{blanklist.txt} must equal \code{nSx*nPx}. \\
205    
206    The following is an example of \file{SIZE.h} for the twelve-tile
207    configuration illustrated in figure \ref{fig:12tile} running on
208    one processor: \\
209    
210  \subsubsection{Arrays Indexed to Tile Number and Neighbor}  \begin{verbatim}
211          PARAMETER (
212  The following arrays are all of size {\em MAX\_NEIGHBOURS}x{\em NTILES} and       &           sNx =  16,
213  describe the orientations between the the tiles.       &           sNy =  32,
214         &           OLx =   2,
215         &           OLy =   2,
216         &           nSx =  12,
217         &           nSy =   1,
218         &           nPx =   1,
219         &           nPy =   1,
220         &           Nx  = sNx*nSx*nPx,
221         &           Ny  = sNy*nSy*nPy,
222         &           Nr  =   5)
223    \end{verbatim}
224    
225  The array {\em exch2\_neighbourId(a,T)} holds the tile number $T_{n}$ for each tile  The following is an example for the twentyfour-tile topology in figure
226  {\em T}'s neighbor tile {\em a}, and {\em exch2\_opposingSend\_record(a,T)} holds  \ref{fig:24tile} running on six processors:
 the index c in {\em exch2\_neighbourId(b,$T_{n}$)} that holds the tile number T.  
 In other words,  
227    
228  \begin{verbatim}    \begin{verbatim}
229  exch2_neighbourId( exch2_opposingSend_record(a,T), exch2_neighbourId(a,T) ) = T        PARAMETER (
230         &           sNx =  16,
231         &           sNy =  16,
232         &           OLx =   2,
233         &           OLy =   2,
234         &           nSx =   4,
235         &           nSy =   1,
236         &           nPx =   6,
237         &           nPy =   1,
238         &           Nx  = sNx*nSx*nPx,
239         &           Ny  = sNy*nSy*nPy,
240         &           Nr  =   5)
241  \end{verbatim}  \end{verbatim}
242    
 % {\em exch2\_neighbourId(exch2\_opposingSend\_record(a,T),exch2\_neighbourId(a,T))=T}.  
 % alternate version  
243    
 This is to provide a backreference from the neighbor tiles.  
244    
245    
 //  
246    
247    \subsection{Key Variables}
248    
249    The descriptions of the variables are divided up into scalars,
250    one-dimensional arrays indexed to the tile number, and two and three
251    dimensional arrays indexed to tile number and neighboring tile.  This
252    division reflects the functionality of these variables: The
253    scalars are common to every part of the topology, the tile-indexed
254    arrays to individual tiles, and the arrays indexed by tile and
255    neighbor to relationships between tiles and their neighbors. \\
256    
257    \subsubsection{Scalars}
258    
259    The number of tiles in a particular topology is set with the parameter
260    \code{NTILES}, and the maximum number of neighbors of any tiles by
261    \code{MAX\_NEIGHBOURS}.  These parameters are used for defining the
262    size of the various one and two dimensional arrays that store tile
263    parameters indexed to the tile number and are assigned in the files
264    generated by \file{driver.m}.\\
265    
266    The scalar parameters \varlink{exch2\_domain\_nxt}{exch2_domain_nxt}
267    and \varlink{exch2\_domain\_nyt}{exch2_domain_nyt} express the number
268    of tiles in the $x$ and $y$ global indices.  For example, the default
269    setup of six tiles has \code{exch2\_domain\_nxt=6} and
270    \code{exch2\_domain\_nyt=1}.  A topology of twenty-four square tiles,
271    four per subdomain (as in figure \ref{fig:24tile}), will have
272    \code{exch2\_domain\_nxt=12} and \code{exch2\_domain\_nyt=2}.  Note
273    that these parameters express the tile layout to allow global data
274    files that are tile-layout-neutral and have no bearing on the internal
275    storage of the arrays.  The tiles are internally stored in a range
276    from [1,\varlink{bi}{bi}] the $x$ axis and $y$ axis variable
277    \varlink{bj}{bj} is generally ignored within the package. \\
278    
279    \subsubsection{Arrays Indexed to Tile Number}
280    
281    The following arrays are of size \code{NTILES}, are indexed to the
282    tile number, and the indices are omitted in their descriptions. \\
283    
284    The arrays \varlink{exch2\_tnx}{exch2_tnx} and
285    \varlink{exch2\_tny}{exch2_tny} express the $x$ and $y$ dimensions of
286    each tile.  At present for each tile \texttt{exch2\_tnx=sNx} and
287    \texttt{exch2\_tny=sNy}, as assigned in \file{SIZE.h} and described in
288    section \ref{sec:exch2mpi} \sectiontitle{exch2, SIZE.h, and
289    multiprocessing}.  Future releases of MITgcm are to allow varying tile
290    sizes. \\
291    
292    The location of the tiles' Cartesian origin within a subdomain are
293    determined by the arrays \varlink{exch2\_tbasex}{exch2_tbasex} and
294    \varlink{exch2\_tbasey}{exch2_tbasey}.  These variables are used to
295    relate the location of the edges of different tiles to each other.  As
296    an example, in the default six-tile topology ??  each index in these
297    arrays are set to \code{0}.  The twentyfour-tile case discussed above
298    will have values of \code{0} or \code{16}, depending on the quadrant
299    the tile falls within the subdomain.  The array
300    \varlink{exch2\_myFace}{exch2_myFace} contains the number of the
301    subdomain of each tile, numbered \code{(1:6)} in the case of the
302    standard cube topology and indicated by \textbf{\textsf{f}}$n$ in
303    figures \ref{fig:12tile}) and \ref{fig:24tile}). \\
304    
305    The elements of the arrays \varlink{exch2\_txglobalo}{exch2_txglobalo}
306    and \varlink{exch2\_txglobalo}{exch2_txglobalo} are similar to
307    \varlink{exch2\_tbasex}{exch2_tbasex} and
308    \varlink{exch2\_tbasey}{exch2_tbasey}, but locate the tiles within the
309    global address space, similar to that used by global files. \\
310    
311    The arrays \varlink{exch2\_isWedge}{exch2_isWedge},
312    \varlink{exch2\_isEedge}{exch2_isEedge},
313    \varlink{exch2\_isSedge}{exch2_isSedge}, and
314    \varlink{exch2\_isNedge}{exch2_isNedge} are set to \code{1} if the
315    indexed tile lies on the edge of a subdomain, \code{0} if not.  The
316    values are used within the topology generator to determine the
317    orientation of neighboring tiles, and to indicate whether a tile lies
318    on the corner of a subdomain.  The latter case requires special
319    exchange and numerical handling for the singularities at the eight
320    corners of the cube.  \varlink{exch2\_nNeighbours}{exch2_nNeighbours}
321    contains a count of how many neighboring tiles each tile has, and is
322    used for setting bounds for looping over neighboring tiles.
323    \varlink{exch2\_tProc}{exch2_tProc} holds the process rank of each
324    tile, and is used in interprocess communication.  \\
325    
326    \subsubsection{Arrays Indexed to Tile Number and Neighbor}
327    
328    The following arrays are all of size
329    \code{MAX\_NEIGHBOURS}$\times$\code{NTILES} and describe the
330    orientations between the the tiles. \\
331    
332    The array \code{exch2\_neighbourId(a,T)} holds the tile number
333    \code{Tn} for each of the tile number \code{T}'s neighboring tiles
334    \code{a}.  The neighbor tiles are indexed \code{(1:MAX\_NEIGHBOURS)}
335    in the order right to left on the north then south edges, and then top
336    to bottom on the east and west edges.  Maybe throw in a fig here, eh?
337    \\
338    
339    The \code{exch2\_opposingSend\_record(a,T)} array holds the index
340    \code{b} in \texttt{exch2\_neighbourId(b,Tn)} that holds the tile
341    number \code{T}.  In other words,
342  \begin{verbatim}  \begin{verbatim}
343       exch2_neighbourId( exch2_opposingSend_record(a,T),
344                          exch2_neighbourId(a,T) ) = T
345    \end{verbatim}
346    This provides a back-reference from the neighbor tiles. \\
347    
348    The arrays \varlink{exch2\_pi}{exch2_pi},
349    \varlink{exch2\_pj}{exch2_pj}, \varlink{exch2\_oi}{exch2_oi},
350    \varlink{exch2\_oj}{exch2_oj}, \varlink{exch2\_oi\_f}{exch2_oi_f}, and
351    \varlink{exch2\_oj\_f}{exch2_oj_f} specify the transformations in
352    exchanges between the neighboring tiles.  The dimensions of
353    \code{exch2\_pi(t,N,T)} and \code{exch2\_pj(t,N,T)} are the neighbor
354    ID \code{N} and the tile number \code{T} as explained above, plus a
355    vector of length 2 containing transformation factors \code{t}.  The
356    first element of the transformation vector indicates the factor
357    \code{t} by which variables representing the same vector component of
358    a tile \code{T} will be multiplied in exchanges with neighbor
359    \code{N}, and the second element indicates the transform to the
360    variable in the other direction.  As an example,
361    \code{exch2\_pi(1,N,T)} holds the transform of the $i$ component of a
362    vector variable in tile \code{T} to the $i$ component of tile
363    \code{T}'s neighbor \code{N}, and \code{exch2\_pi(2,N,T)} hold the
364    component of neighbor \code{N}'s $j$ component. \\
365    
366    Under the current cube topology, one of the two elements of
367    \code{exch2\_pi} or \code{exch2\_pj} for a given tile \code{T} and
368    neighbor \code{N} will be \code{0}, reflecting the fact that the two
369    vector components are orthogonal.  The other element will be 1 or -1,
370    depending on whether the components are indexed in the same or
371    opposite directions.  For example, the transform vector of the arrays
372    for all tile neighbors on the same subdomain will be \code{(1,0)},
373    since all tiles on the same subdomain are oriented identically.  A
374    vector direction that corresponds to the orthogonal dimension with the
375    same index direction in a particular tile-neighbor orientation will
376    have \code{(0,1)}, whereas those in the opposite index direction will
377    have \code{(0,-1)}.  This needs some diagrams.
378    
379    
380    {\footnotesize
381    \begin{verbatim}
382  C      exch2_pi          :: X index row of target to source permutation  C      exch2_pi          :: X index row of target to source permutation
383  C                        :: matrix for each neighbour entry.              C                        :: matrix for each neighbour entry.            
384  C      exch2_pj          :: Y index row of target to source permutation  C      exch2_pj          :: Y index row of target to source permutation
# Line 143  C      exch2_oj_f        :: Y index elem Line 396  C      exch2_oj_f        :: Y index elem
396  C                        :: offset vector for face quantities            C                        :: offset vector for face quantities          
397  C                        :: of each neighbor entry.                      C                        :: of each neighbor entry.                    
398  \end{verbatim}  \end{verbatim}
399    }
400    
401    
402    
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