To illustrate the method let us consider two different models of flaring active region 2776 on November 5, 1980, as shown in Figure 18. Gorbachev and Somov (1989) proposed a model with four charges all placed at (0.1 units), while Démoulin et al. (1994) produced a more accurate representation using 18 charges at depths ranging from to . Gorbachev and Somov (1989) selected their charge distribution in order to produce a vertical field which resembled the basic appearance of AR 2776. Démoulin et al. (1994) employed an automated algorithm to define their parameters; the algorithm minimized the squared difference in vertical photospheric field between the model, Equation (23), and a vector magnetogram obtained at Marshal Space Flight Center (MSFC). Figure 18 illustrates how the use of more sources permits a more complex photospheric field and thereby permits a more faithful representation of observation.
The next step in a submerged poles model is to associate every photospheric footpoint with the pole to which it maps by the sub-photospheric model field. This process, at least in principle, partitions the photosphere into regions which serve the same function as in MCT models. The coronal field may now be divided into domains according to the regions at each footpoint, exactly as in MCT models. Separatrices are defined as the boundaries between such domains, and separators as the intersections of these separatrices.
This partitioning scheme resembles that of MCT models in that fan surfaces from null points produce separatrices. In this case the null points are frequently sub-photospheric, and map up to the photosphere to produce region boundaries. Figure 19 shows how a portion of one null’s fan surface crosses the photosphere to produce a separatrix in the corona. The first crossing produces a photospheric boundary between regions approximately resembling the null’s spine sources, in this case N17 and N14 (not shown). The region boundary forms the footpoints of a separatrix surface (solid curves) extending into the corona. The opposite footpoints, shown as dashed curves, complete the trace of this particular separatrix.
Submerged poles models differ from MCT models in that they have a non-intermittent photospheric field with a PIL and therefore can have bald patches (Seehafer, 1986). The skeleton of a submerged poles model must therefore include BP separatrices as well as the fan surfaces.
Submerged poles models serve an important role as a conceptual bridge between MCT models and pointwise mapping models. A set of submerged point charges, at depths , may be converted to a point-source MCT model by simply taking each depth continuously to zero. Through this process the sequence of non-intermittent of photospheric fields will continuously approach the singular, intermittent MCT field. If the submerged sources are coplanar (all depths are equal) then this is equivalent to moving the photospheric surface downward until it coincides with the charge-surface. In this case the actual magnetic field never changes, but certain features defined using the photosphere, such as PILs and BPs, do change. Separatrices from fan surfaces are the same in both models, while BPs will vanish in the MCT limit. Figure 20 shows the footprint of the MCT which results from taking the 4-charge co-planar model of Gorbachev and Somov (1989) to the photosphere. The correspondence is illustrated by comparing the footprint in Figure 20 to the separatrix trace from the submerged poles model in Figure 18.
Submerged dipoles were introduced by Démoulin et al. (1992) as an alternative to point charges. Dipoles with moments pointing either vertically upward or vertically downward produce positive and negative flux concentrations, respectively. These models promise improved representation of the photospheric field because their field is more vertical at the concentrations periphery, and there will automatically be a surrounding layer of opposing (Démoulin et al., 1992). When using potential fields it is often hard to see significant the differences in the photospheric fields produced by dipoles and monopoles (see Démoulin et al., 1994, for example). An added complication which arises from dipoles is that a given dipole has terminations of both senses (i.e. the field goes both into and out of a dipole). This opens up numerous new, and often perplexing, possibilities for domains connecting like-signed poles or even connecting a pole to itself. With the new connections come new separatrices (Démoulin et al., 1992).
Submerged sources can generate constant- force free fields as well as potential fields. Démoulin and Priest (1992) proposed a submerged poles model using force-free dipoles. In spherical coordinates centered on it, a single dipole with moment has the axi-symmetric field
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