11.3.1 Detailed anatomy

The inner workings of the memory window will now be exposed so it will become clear how the window is used for multi-tasking. The memory window is a chunk of memory dimensioned to hold a few latitude rows of 3-D prognostic data plus some workspace. A detailed schematic of the minimum sized MW appropriate for second order numerics (i.e.. numerics which requires access to nearest neighboring cells) is given in Figure 11.2a. Arrays are dimensioned within the MW to hold two prognostic variables (velocity and tracer) plus a workspace area. Each prognostic variable is an array subscripted by 5 indices^11.7. The first three i,k,j are used to denote longitude, depth, and latitude. The fourth denotes prognostic component (e.g. for velocity, a 1 pertains to the zonal component of velocity and a 2 pertains to the meridional component. For tracers, a 1 pertains to temperature and a 2 pertains to salinity) and the fifth index which denotes time level (e.g. typically, this index takes on values of $\{-1,0,1\}$ to represent discrete time levels $\{\tau-1$, $\tau$, or $\tau+1$}.). The latitude rows for which 3-D prognostic equations are solved (updated to time level $\tau+1$) are indicated in red. The remaining latitude rows indicated in blue are used as buffer rows for boundary conditions on the computed row.

A work area is also indicated as part of the memory window. The storage required for this area varies depending on which options are enabled and may be comparable to the storage required for prognostic arrays. In general, space is needed to hold diffusive and advective fluxes of prognostic quantities defined on the faces of each cell. Arrays for these fluxes are also subscripted by indices  i,k,j but without a fourth or fifth index since they are recalculated for each prognostic variable to conserve memory. Additionally, some options require diffusive coefficients with spatial dependence. In this case, three dimensional arrays are used for diffusive coefficients which are also defined on cell faces.

A simplified schematic of the MW is given on the right side of Figure 11.2a by collapsing all indices except for the local latitude index ``j''. Two important characteristics are indicated: the MW contains jmw=3 latitude rows and prognostic equations are integrated to update prognostic variables to time level $\tau+1$starting on local row j=2 and ending on local row j=2. Rows on which prognostic equations are solved are indicated by an ```x'' and buffer rows which hold data one row away are indicated by a ``circle'' in the simplified schematic. An example of a larger MW is given in Figure 11.2b. In this example, the window holds jmw=6rows and prognostic equations are solved for local rows j=2 through j=5. In both windows, local rows given by j=1 and j=jmw are needed as boundaries for the second order numerics. Note that the MW is always symmetrical in that there are as many boundary rows to the north of the computed rows as to the south. Why aren't prognostic equations solved on rows j=1 and j=jmw? Because indexing into arrays required by the 2nd order numerics would reach outside the MW (i.e. j<1 or j>jmw).

In general, the number of rows for which prognostic equations are solved within a MW is given by the size of the MW minus the number of northern and southern boundary or buffer rows ``jbuf''. Equations for intermediate quantities such as density or fluxes on cell faces are solved on these extra rows. However, details about which quantities are solved and where they are defined on the grid system within the MW are not needed here and are covered more thoroughly in Section 22.2.4. The important point for now is that the number of computed rows ``ncrows'' where prognostic equations are solved within the MW is given by

ncrows = jmw - 2*jbuf (11.1)

and prognostic equations are solved on every row within the MW from  j=js_calc through j=je_calc where

$$js\_calc$$ = 1+jbuf (11.2)

$$je\_calc$$ = jmw-jbuf (11.3)

In the case of second order numerics, jbuf=1, the minimum sized window is jmw=3, and there is only one row on which prognostic equations are solved: $js\_calc = je\_calc = 2$. For a larger window where jmw > 3, prognostic equations are solved from MW rows $js\_calc=2$through $je\_calc=jmw-1$.

For third and fourth order numerics, nearest neighbors of nearest neighbor cells are required which means a larger minimum sized MW is needed. An example is given in Figure 11.3a which is the minimum size required when executing with third or fourth order numerics. In this case, jbuf=2, the window holds jmw=5 rows, and prognostic equations are solved from rows $js\_calc=3$ through $je\_calc=3$. As with the 2nd order MW, equations for intermediate quantities are solved on the buffer rows as long as the indexing does not reach outside of the MW.