6.1 The method

One drawback of point charge MCT models is that they produce a photospheric magnetic field which is singular at each charge. This is not surprising in a multipole expansion which should not, after all, be used in the immediate vicinity of the sources. It means, however, that the field may not be directly compared to magnetograms. A solution introduced by Seehafer (1986

), and then by Gorbachev and Somov (1988

), is to place point charges below the photospheric surface. The charges themselves, and the field in the region $z < 0$ , are artifacts of the modeling and not intended to represent the true sub-photospheric field. These original submerged poles models used a potential field from a set of point charges. They produce a vertical photospheric field

which depends on the depths $di$ , charges $qi$ and horizontal placements $(xi,yi)$ of the $M$ sources. By adjusting these parameters it is possible to approximate the vertical field observed by a magnetogram. The salient features of such representations are that each charge generates a smooth, circular flux concentration of radius comparable to its depth. The combination of poles produces a non-intermittent photospheric field with smooth PILs separating regions of opposing polarity.

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 $d = 4.4 Mm$ (0.1 units), while Démoulin et al. (1994 ) produced a more accurate representation using 18 charges at depths ranging from $d9 = 3.99 Mm$ to $d14 = 102 Mm$ . 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.

Figure 18:

Two submerged poles models of AR 2776, November 5, 1980. The grey scale shows the vertical magnetic field at the photosphere from the collection of submerged poles. The top panel shows the 4-charge co-planar model by Gorbachev and Somov (1989

). The bottom panel shows the 18-pole model by Démoulin et al. (1994

). The axes are labeled in units used in that paper. The projected location of each charge is indicated. The PIL is a thin broken line, and several separatrix traces are shown as solid and dashed dark curves. The solid curve is the first intersection of the fan surface from the submerged null (triangle), the dashed curve is its second crossing. Thin solid lines show the sub-photospheric spines of the selected null points.

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 $N 17$ and $N 14$ (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.

Figure 19:

One of the separatrices in the 18-source model of Démoulin et al. (1994

). A portion of the fan surface of a submerged negative null point (triangle), and its spine curves (solid), one extending to $N 17$ and the other leaving the box. The box is a section of the region below the photosphere, showing charges at their respective depths as $+$ s and $×$ s. Dashed lines show field lines from the null’s fan surface extending upward to the photosphere. A dark solid curve indicates where they cross $z = 0$ , and the thin solid lines are the same field lines above the surface. A dark dashed line shows where the separatrix descends again below the photosphere, mapping to sources $P 8$ , $P15$ and $P 16$ . These same photospheric curves appear in the lower panel of Figure 18

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 $di$ , 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 $di$ 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 .

Figure 20:

The footprint of the 4-charge co-planar model of Gorbachev and Somov (1989

) after source-depths are taken to zero. Symbols are the same as MCT footprints. Dotted lines are the field’s two separators. The top panel of Figure 18

is the central portion of this footprint. This is identical to Figure 4 in Gorbachev and Somov (1989

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 $Bz$ (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- $a$ 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 $m = m^z$ has the axi-symmetric field

$B = \~/ f × \~/ f + af \~/ f, (24)$

where the flux function (see Section 2.1) is given by

This matches the potential dipole in the center, $ar « 1$ , but falls off less rapidly at large distances. A full field is produced by superposing contributions form every dipole. The weak fall-off is somewhat unphysical, so the model should be restricted to distances $<~ 1/|a|$ of all poles. Within that region the field will differ from a potential field principally by its overall twist, including shear at the PIL.