4.4 Applications

The literature includes only a few cases where the connectivity of an active region is analyzed using strictly MCT methods, here again distinguished from submerged poles models. Longcope and Silva (1998

) partition the active region complex 6993/6994 (1998 January 7) into 13 regions, each of which is represented by a point charge. The potential field from this field has 12 nulls, all photospheric and prone, 24 photospheric domains and 4 additional coronal domains. Equation (19

) predicts $X = 15$ separators, of which Longcope and Silva (1998 ) report 12 (the remaining 3 can, however, be found from their reported configuration). Longcope and Klapper (2002 ) exhibit an example of a small, diffuse active region (1993 June 5) modeled using 20 sources plus a balancing source at infinity. The complete skeleton of the potential field includes 19 nulls (all prone and photospheric), 28 separators and 33 domains, 16 of which are purely coronal.

Figure 12:

The footprint of a potential field modeling the lower left quadrant of the quiet Sun field shown in Figure 7

Most applications of pure MCT models have been to the quiet Sun corona. Even a small patch of the quiet Sun will have a connectivity far more complex than the three-source and four-source prototypes. Figure 12

shows the footprint of the potential field generated from the lower left quadrant of the quiet Sun magnetogram in Figure 7

. Aside from verifying general relationships among topological elements, it is not clear that anything would be learned from an intensive study of a single example of the quiet Sun. Consequently, studies have tended to characterize the quiet Sun topology statistically.

Schrijver and Title (2002 ) modeled the quiet Sun by randomly distributing 288 point sources over a square region and assigning them fluxes from a random distribution with zero mean (the net flux in the region was forced to vanish). Using a potential field anchored in these sources they found the connectivity of each one by tracing selected field lines from it. To reduce the effects of edges, connectivity was determined only for sources within the central one-ninth of the square. From 200 realizations of this type they concluded that elements connect, on average, to 8.0 opposing elements, although a given element might connect to as many as 32 or as few as one. Of the 8.0 domains linking a given source, roughly half (3.8) are photospheric; the remainder are coronal domains. The majority of flux is found in short connections to the nearest 2-6 neighbors, however, there are some very long connections to distant sources. Schrijver and Title (2002 ) then compare $171 ºA$ and $195 ºA$ EUV coronal images made by TRACE (Handy et al., 1999 ) with magnetograms from SOHO/MDI (Scherrer et al., 1995 ), finding evidence for only the shortest connections predicted by the potential field model.

Close et al. (2003 ) studied two overlapping regions in a high resolution magnetogram of the quiet Sun from June 13, 1998. They extrapolated potential fields from each $264 Mm × 264 Mm$ region, and characterized the connectivities between source regions (not point charges) within the central one-ninth. The first region analyzed contained 375 sources (defined by $|Bz |> 20 G$ ) within the central $88 Mm × 88 Mm$ , with approximate flux balance. Each source connected on average to 5 others, although one particular source connected to 65. The second region contained 414 sources in its central region, with a 1:2 mix of positive:negative flux. The majority (negative) sources averaged 6.7 connections while the minority averaged 3.7. In both the balanced and unbalanced cases a source’s single largest connection accounted, on average, for two-thirds of its net flux (69% and 65% in the balanced and unbalanced regions, respectively).

Beveridge et al. (2003 ) used a Monte Carlo approach to model a long coronal loop whose footpoints comprised numerous small elements. They modeled each footpoint by randomly distributing point sources with one sign of flux but with a distribution of magnitudes. The footprint of such a region had an average of 0.1 upright nulls and 1.1 prone null point for each source. Mapping the footprints between the two ends they found 18 separatrix intersections, meaning 18 separators, for each prone null. Applying these statistical findings to Equation (19 ) implies that each source connects to an average of 20 opposing sources at the other end of the loop. The tendency of the separatrices to cluster together led to bundles of many dozen separators. Consequently, many of the 20 connections to an average source would have very little flux in them; so little flux that they would never occur in any random selection of field lines such as those used by Schrijver and Title (2002 ) or Close et al. (2003 ).