4 Magnetic Charge Topology Models

All Magnetic Charge Topology (MCT) models share two basic assumptions. First, they assume the photospheric field can be partitioned into distinct unipolar regions. Second, they consider any two field lines with both their footpoints in the same regions to be topologically equivalent. The most natural partitioning occurs when each unipolar region is surrounded by a field-free “sea”, or is a point magnetic source located at $z = 0$ . MCT’s definition of topological equivalence is most natural with point charges since there is only one footpoint location for each region: the charge itself.

A related class of models use the potential field from a set of submerged ( $z < 0$ ) charges or dipoles to produce a smooth photospheric field (Seehafer, 1986 ; Gorbachev and Somov, 1988 ; Démoulin et al., 1992 ). Photospheric regions are then defined by the mapping from submerged poles and may be delineated by curves mapping from fans of submerged null points. These submerged pole models share many elements with MCT and many authors consider them to be the category’s prototype. They differ from the MCT models, as defined here, in several critical respects. The flux $Pa$ in photospheric region $a$ depends on how many field lines from its source “reach” the photosphere. This value will change as source locations evolve, so the model does not constrain the fluxes of its regions. This fact is also responsible for the seeming arbitrariness in the definition of their separatrices: The separatrix extends from a photospheric curve whose actual definition is not topological, but depends on the modeling of the photosphere. Finally, submerged poles models are often used to analyze properties of the point-for-point mapping $X(x)$ . This means they include bald patches and quasi-separatrix layers, which are not elements of MCT as it is defined here. We therefore defer the discussion of submerged poles models to a separate section, Section 6, after the discussion of pointwise mapping models. Hereafter we apply the term MCT only to models whose photospheric field consists of separated unipolar regions or point charges located at $z = 0$ .

In contrast to the intermittent photospheric field, the coronal field, $z > 0$ , is assumed to be continuous and volume-filling, vanishing at only isolated points¹³ . There is therefore a unique field line passing through every point in the corona, except the null points. Almost all field lines can be assigned to one of a countable number of equivalence classes according to source regions at each footpoint. Open field lines are considered to have a footpoint at infinity, which therefore counts as another source region. This divides the corona into sub-volumes, known as domains or cells. A separatrix, as defined in MCT models, is any surface between two field-line domains. A separatrix surface must consist of field lines, and by definition these must have at least one end which is not at a source; it must end at a magnetic null point. We have excluded those models, such as submerged poles models, where photospheric source regions might be separated by a curve with footpoints of its own; there are no footpoints in the field-free sea. Therefore, each separatrix in an MCT model is the fan surface of a null point¹⁴ . The fan surfaces of null points divide the coronal field into domains. Longcope and Klapper (2002 ) present a systematic method for constructing the separatrices and domains of an arbitrary potential magnetic field. Sweet (1958b ) proposed the first MCT model for a hypothetical flaring active region consisting of two positive and two negative sources interconnected by four domains of field lines, as shown in Figure 8 . Sweet’s configuration has been thoroughly studied by subsequent authors using coronae consisting of potential fields (Baum and Bratenahl, 1980 ; Seehafer, 1986 ; Gorbachev et al., 1988 ), linear force free equilibria (Hudson and Wheatland, 1999 ; Brown and Priest, 1999b ; Petrie and Lothian, 2003 ), time-dependent numerical solutions (Longcope and Magara, 2004 ), and approximate semi-analytic equilibria (Longcope, 1996 ). In most models of Sweet’s configuration there are two magnetic null points located in the photospheric plane, one positive and one negative (see Figure 8 ). The fan surface from the positive null separates those field lines originating in $P1$ from those originating in $P2$ . The negative fan divides the field lines ending at $N 3$ from those ending at $N 4$ . The intersection between the two separatrices forms a separator lying at the junction of all four domains at once.

Figure 8:

A version of Sweet’s original model of four interacting flux domains (cells) from four discrete photospheric sources. The top panel shows the locations of the 2 positive (white) and two negative (black) sources. The two magnetic null points, $B1$ and $A2$ are shown by triangles. Dashed and solid lines are fans and spines, respectively. On the bottom is a perspective view of one representative field line from each of the four flux domains: $P 1$ - $N 3$ and $P 2$ - $N 4$ (red) and $P 1$ - $N 4$ and $P 2$ - $N 3$ (green). The blue line is the field’s separator, running from the positive to negative null. Black lines are the spines from the two nulls.

A field line undergoes topological change in the MCT model when the source region at one of its footpoints changes. This occurs kinematically when two field lines, from different domains, approach the separator, temporarily join the spine-separator-spine combination, and then emerge in the other two domains (Greene, 1988 ; Lau and Finn, 1990 ). This occurs in Sweet’s configuration when, for example, field lines from domains $P 1$ - $N 4$ and $P 2$ - $N 3$ are converted, through some non-ideal process, into field lines in domains $P1$ - $N 3$ and $P 2$ - $N 4$ .

The rate of reconnection in Sweet’s configuration can be quantified as the time rate of change of the flux in domain $P 2$ - $N 3$ . According to Faraday’s law this changing flux is proportional to the voltage drop along the separator (Sweet, 1958b ; Longcope, 1996 ). Of course, since a perfect conductor is an equipotential, this requires some departure from the ideal induction equation. Longcope and Klapper (2002 ) show that the flux changes in each domain in an MCT field of arbitrary complexity is linearly related to a vector composed of voltage drops across each of its separators.

4.1 The photospheric field
4.2 Skeletons
4.3 Connectivity
4.4 Applications