• 6.18.5.1 Scalars
• 6.18.5.2 Arrays indexed to tile number
• 6.18.5.3 Arrays Indexed to Tile Number and Neighbor

6.18.5 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 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.

6.18.5.1 Scalars

The number of tiles in a particular topology is set with the parameter NTILES, and the maximum number of neighbors of any tiles by 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 driver.m.

The scalar parameters exch2_domain_nxt and exch2_domain_nyt express the number of tiles in the and global indices. For example, the default setup of six tiles (Fig. 6.6) has exch2_domain_nxt=6 and exch2_domain_nyt=1. A topology of twenty-four square tiles, four per subdomain (as in figure 6.4), will have exch2_domain_nxt=12 and 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 bi=(1:NTILES) in the axis, and the axis variable bj is assumed to equal 1 throughout the package.

6.18.5.2 Arrays indexed to tile number

The following arrays are of length NTILES and are indexed to the tile number, which is indicated in the diagrams with the notation t. The indices are omitted in the descriptions.

The arrays exch2_tnx and exch2_tny express the and dimensions of each tile. At present for each tile exch2_tnx=sNx and exch2_tny=sNy, as assigned in SIZE.h and described in Section 6.18.4 exch2, SIZE.h, and Multiprocessing. Future releases of MITgcm may allow varying tile sizes.

The arrays exch2_tbasex and 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. 6.6) each index in these arrays is set to 0 since a tile occupies its entire subdomain. The twenty-four-tile case discussed above will have values of 0 or 16, depending on the quadrant of the tile within the subdomain. The elements of the arrays exch2_txglobalo and exch2_txglobalo are similar to exch2_tbasex and exch2_tbasey, but locate the tile edges within the global address space, similar to that used by global output and input files.

The array exch2_myFace contains the number of the subdomain of each tile, in a range (1:6) in the case of the standard cube topology and indicated by f in figures 6.5 and 6.4. exch2_nNeighbours contains a count of the neighboring tiles each tile has, and sets the bounds for looping over neighboring tiles. exch2_tProc holds the process rank of each tile, and is used in interprocess communication.

The arrays exch2_isWedge, exch2_isEedge, exch2_isSedge, and exch2_isNedge are set to 1 if the indexed tile lies on the edge of its 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 requires special exchange and numerical handling for the singularities at the eight corners of the cube.

6.18.5.3 Arrays Indexed to Tile Number and Neighbor

The following arrays have vectors of length MAX_NEIGHBOURS and NTILES and describe the orientations between the the tiles.

The array exch2_neighbourId(a,T) holds the tile number Tn for each of the tile number T's neighboring tiles a. The neighbor tiles are indexed (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 exch2_opposingSend_record(a,T) array holds the index b of the element in exch2_neighbourId(b,Tn) that holds the tile number T, given Tn=exch2_neighborId(a,T). In other words,

   exch2_neighbourId( exch2_opposingSend_record(a,T),
                      exch2_neighbourId(a,T) ) = T

The arrays exch2_pi and 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 exch2_pi(t,N,T) and exch2_pj(t,N,T) are the neighbor ID N and the tile number T as explained above, plus a vector of length 2 containing transformation factors 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, exch2_pi(1,N,T) holds the mapping of the axis index of tile T to the axis of tile T's neighbor N, and exch2_pi(2,N,T) holds the mapping of T's index to the neighbor N's index.

One of the two elements of exch2_pi or exch2_pj for a given tile T and neighbor N will be 0, reflecting the fact that the two axes are orthogonal. The other element will be 1 or -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 (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 (0,1). Those with the opposite index direction will have (0,-1) in order to reverse the ordering.

The arrays exch2_oi, exch2_oj, exch2_oi_f, and 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 N to a local tile T. Consider T=1 in the six-tile topology (Fig. 6.6), where

       exch2_oi(1,1)=33
       exch2_oi(2,1)=0
       exch2_oi(3,1)=32
       exch2_oi(4,1)=-32

The simplest case is exch2_oi(2,1), the southern neighbor, which is Tn=6. The axes of T and Tn have the same orientation and their axes have the same origin, and so an exchange between the two requires no changes to the index. For the western neighbor (Tn=5), code_oi(3,1)=32 since the x=0 vector on T corresponds to the y=32 vector on Tn. The eastern edge of T shows the reverse case (exch2_oi(4,1)=-32)), where x=32 on T exchanges with x=0 on Tn=2.

The most interesting case, where exch2_oi(1,1)=33 and Tn=3, involves a reversal of indices. As in every case, the offset exch2_oi is added to the original index of T multiplied by the transformation factor exch2_pi(t,N,T). Here exch2_pi(1,1,1)=0 since the axis of T is orthogonal to the axis of Tn. exch2_pi(2,1,1)=-1 since the axis of T corresponds to the axis of Tn, but the index is reversed. The result is that the index of the northern edge of T, which runs (1:32), is transformed to (-1:-32). exch2_oi(1,1) is then added to this range to get back (32:1) - the index of the axis of Tn relative to 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, exch2_itlo_c, exch2_ithi_c, exch2_jtlo_c and exch2_jthi_c hold the location and index bounds of the edge segment of the neighbor tile N's subdomain that gets exchanged with the local tile T. To take the example of tile T=2 in the twelve-tile topology (Fig. 6.5):

       exch2_itlo_c(4,2)=17
       exch2_ithi_c(4,2)=17
       exch2_jtlo_c(4,2)=0
       exch2_jthi_c(4,2)=33

Here N=4, indicating the western neighbor, which is Tn=1. Tn resides on the same subdomain as T, so the tiles have the same orientation and the same and axes. The axis is orthogonal to the western edge and the tile is 16 points wide, so exch2_itlo_c and exch2_ithi_c indicate the column beyond 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 axis bounds exch2_jtlo_c to exch2_jthi_c cover the height of (1:32), plus 1 in either direction to cover part of the halo.

For the north edge of the same tile T=2 where N=1 and the neighbor tile is Tn=5:

       exch2_itlo_c(1,2)=0
       exch2_ithi_c(1,2)=0
       exch2_jtlo_c(1,2)=0
       exch2_jthi_c(1,2)=17

T's northern edge is parallel to the axis, but since Tn's axis corresponds to T's axis, T's northern edge exchanges with Tn's western edge. The western edge of the tiles corresponds to the lower bound of the axis, so exch2_itlo_c and exch2_ithi_c are 0, in the western halo region of Tn. The range of exch2_jtlo_c and exch2_jthi_c correspond to the width of T's northern edge, expanded by one into the halo.

Copyright © 2002 Massachusetts Institute of Technology