3.4 Topological models of anchored fields

Much of the literature on the topology of anchored magnetic fields concerns the properties of the footpoint mapping. Most of this literature cleaves into two categories according to the assumed form of the photospheric normal field and the level of detail with which the mapping is represented. Some models, hereafter called pointwise mapping models, consider the detailed structure of the point-to-point footpoint mapping $X(x)$ . The alternative, called Magnetic Charge Topology (MCT), reduce the mapping to its connectivity between distinct photospheric sources: regions of unipolar photospheric flux surrounded by a strictly field-free “sea” ( $Bn = 0$ ) ¹² . Some models make a further simplification by replacing each region with a point magnetic charge - the leading order in its multipole expansion. In contrast to this intermittent distribution, pointwise mapping models generally assume the normal photospheric field is a generic, non-intermittent function on the surface, vanishing only along curves known as polarity inversion lines (PILs). It is useful to organize the existing literature into these categories, because their different approaches lead to different concepts of topology and different topological elements; for example they use mutually inconsistent definitions of separatrices. While both seek to describe, in their own way, a common underlying reality, they do so using subtly different conceptual frameworks. The situation is further muddled by their use of the same terms, such as separatrix, to denote different things. One rather popular model, which we call the submerged poles model, appears to defy this categorization. In fact, this one model is used by some authors as an MCT model and by others as a pointwise mapping model. We find that formulating rigorous frameworks for each of category, MCT in Section 4 and pointwise mapping models in Section 5, leads to a relatively clear presentation of most existing literature. We refer briefly to the submerged poles model with the MCT models, but defer its full presentation to a separate section, Section 6, since it draws elements from both categories.

The need for two model categories arises partly from the diversity of data. The quiet Sun, for example, is revealed by line-of-sight magnetograms (see top panel of Figure 7 for an example) to have an intermittent photospheric field consisting of small unipolar regions, called magnetic elements, separated by distances far exceeding their own diameters (see Zwaan, 1987 , for a review of the hierarchy of photospheric magnetic fields). This suggests that the magnetic charge topology model would be a good approximation for this portion of the solar atmosphere. Almost all modeling of the quiet Sun magnetic field uses some version of an MCT model.

Figure 7:

Two small sections of a full-disk magnetogram made on 2000 March 17 by the SOI/MDI instrument on the SOHO spacecraft (Scherrer et al., 1995

). The grey-scale shows the component of the magnetic field along the line of sight: Black is negative (away from the detector), white is positive, and grey is zero. Axes are labeled in arc seconds from the center of the solar disk. The top panel is a small ( $214 Mm$ on a side) region of the quiet Sun. Black and white specks are unipolar magnetic elements, each roughly $3 × 1018 Mx$ , with maximum field strengths $|Blos|~ 150 G$ ; the grey-scale extends from $-150 G$ to $+150 G$ . Bottom is a small, young active region (NOAA 8910) plotted on a grey-scale extending from $- 1000 G$ to $+1000 G$ .

Detailed chromospheric models suggest that the magnetic field expands above these isolated features until it merges at a merging height or canopy to form a volume-filling coronal field (Kopp and Kuperus, 1968 ; Gabriel, 1976 ). More careful studies of the apparently field-free sea surrounding the elements (Livingston and Harvey, 1971 ; Lin and Rimmele, 1999 ) reveals that it contains an even finer inter-mixture of smaller positive and negative flux elements. To date this further complication in the quiet Sun field has been modeled by MCT models with smaller charges (i.e. points with less magnetic charge, see Schrijver and Title, 2003 ). It is not clear if future efforts will turn to pointwise mapping models or be forced to discard topology altogether due to the higher collisionality, shorter time scales and lower Alfvén speeds in this very complex layer of the solar atmosphere.

Active region photospheric fields (see bottom panel of Figure 7 ), on the other hand, are less clearly separated into distinct unipolar structures and are consequently less amenable to MCT models. The alternative is to assume the photospheric magnetic field is a non-intermittent function vanishing only along the PILs. Models of this type consider the point-to-point mapping function from positive to negative regions. This is almost always the model used when analyzing time-dependent numerical simulations of active region evolution. The observed active region field does, nevertheless, appear to be organized into distinct positive and negative regions. Therefore, the literature includes both MCT and pointwise mapping model analyses of active regions.