4.2 Skeletons

A complete topological description of a three-dimensional MCT field is provided by its skeleton (Priest et al., 1997 ), comprising all of its null points, spines, fans and separators. The footprint described above is the photospheric slice of the full skeleton. Longcope and Klapper (2002

) present a systematic method for analyzing the skeleton of a potential field by “scanning” each null point: tracing its spines and then tracing the fan lines originating in each direction from the local fan plane. Since all MCT separatrices originate in null points this process yields the complete skeleton of the field.

The field’s domain graph (Longcope, 2001 ) provides a schematic summary of the field’s connectivity. The right panel of Figure 9 shows the domain graph of the field from six sources, while the left panel is the footprint of the field’s skeleton.

The skeletons of the simplest non-trivial system, the potential field arising from three photospheric point sources, were completely cataloged and characterized by Brown and Priest (1999a ). A different and more systematic method for cataloging the skeletons was introduced by Pontin et al. (2003 ) and applied to these same three-source configurations. Considering all possible locations, magnitudes and signs of three sources there are eight different skeletons. Since the sum of photospheric fluxes from the three sources does not necessarily vanish, there will in general be a fourth balancing source at infinity. The sign of the balancing source (opposite to the sign of the sum) will match the signs of $MS$ of the photospheric sources, where $MS$ can be zero, one or two. Two of the eight skeletons correspond to $MS = 0$ , three to $MS = 1$ , and three to $MS = 2$ . The cases $MS = 0$ and $MS = 2$ have trivial domain graphs with one source (possibly infinity) connecting separately to each of the other three. The case $MS = 1$ corresponds to a version of Sweet’s configuration where one of the four sources has been removed to infinity. Depending on relative strengths there may be three or four flux domains.

Four photospheric sources offer many more possibilities, which have not yet been so methodically cataloged. Baum and Bratenahl (1980 ) pioneered the field by constructing the skeleton of a potential field from Sweet’s configuration (two positive and negative sources, all of equal magnitude). Seehafer (1986 ) considered all possible configurations of these four equiflux sources and ruled out coronal null points except in very special arrangments such as perfect rectangles. Gorbachev et al. (1988 ) performed a comprehensive analysis of more general configurations in which the sources have arbitrary magnitudes so long as the two positives cancelled the two negatives. They showed that only two generic domain graphs were possible, shown in Figure 10 . The first, case A, has four domains and a separator, while the second, case B, has only three, and no separator. The separator in case A is a single curve composed of either one or two null-null lines encircling one domain (Figure 8 shows one possibility). The photospheric field always includes exactly two prone nulls; some case A configurations also include a coronal null linked by separators to each prone null. Subsequent investigations (Brown and Priest, 2001 ; Beveridge et al., 2002 ) have characterized the broader realm of four-source configurations including cases with net flux in the photosphere (including infinity, these are actually five-source systems).

Figure 10:

The two possible connectivities of Sweet’s configuration. Footprints are shown on the left and corresponding domain graphs on the right. The top and bottom rows are cases A and B, respectively, of Gorbachev et al. (1988

). Case A (topologically equivalent to Figure 8

) has four domains, and the fan surfaces from the two nulls interect along a separator (not shown), so both fans are broken fans. For example, the fan traces from $B1$ ( $\~/$ ) connect to $N 3$ (downward) and to $N 4$ (upward, although the complete fan trace is not shown). In case B (bottom row) the fans from null points $B1$ and $A2$ are unbroken enclosing domains $P 1$ - $N 3$ and $P2$ - $N 4$ , respectively. For example, both fan traces (dashed lines) from null $B1$ connect to $N 4$ . A potential field will switch from case B to case A through a global separator bifurcation as sources $P 2$ and $N 3$ approach one another.

Figure 11:

An example of a global spine-fan bifurcation. The two panels show a portion of the footprint of the field before (left) and after (right) the bifurcation. The bifurcation occurs as the spine (solid curve) of null point $B1$ sweeps across the fan (dashed curve) of null point $B2$ . As a consequence the spine source of $B1$ switches from $P 1$ to $P 6$ , and the fan trace of $B2$ sweeps from $N 5$ to $N 4$ . At the instant of bifurcation (not shown) the spine from $B1$ is part of the fan of $B2$ .

The topology of a field is a robust property which will not change, in general, as the field is continuously deformed or changed. When a change of topology does occur during some continuous process, say non-ideal time evolution or hypothetical parameter variation, it is a singular event known as a bifurcation. Bifurcations are designated either local or global following the nomenclature from ordinary differential equations (Guckenheimer and Holmes, 1983

). Local bifurcations, discussed further in Section 7, create or destroy null points without changing the domain structure of the field. Global bifurcations change the domain structure by changing global elements of its skeleton - fans, spines and separators - without affecting the null points themselves. The most important of these are the global separator bifurcation and the spine-fan bifurcation (Brown and Priest, 1999a

), each of which is an analog of a two-dimensional heteroclinic saddle bifurcation (Guckenheimer and Holmes, 1983

In a global separator bifurcation the fans of two opposing nulls encounter one another creating a pair of separators at their intersections. Gorbachev et al. (1988 ) describe such a bifurcation between two prone nulls which converts a three-domain case B field to Sweet’s four-domain field, case A (see Figure 10 ). When both are prone photospheric nulls one separator is in the mirror corona, while in all other cases both are coronal separators. In the former case, fan traces from each of the opposing null points will appear to sweep past one another, as shown in Figure 10 . This process will create one or two new separators in the corona and must create an equal number of new domains in order to preserve the inter-relation between these two skeletal elements (this relationship is quantified below).

The other common bifurcation, a global spine-fan bifurcation (Brown and Priest, 1999a ), occurs when the spine of one null passes through the fan of a like-signed null point. At the instant of bifurcation the spine of the first null ends at the second null; this is a structurally unstable configuration as any bifurcation must be (Hornig and Schindler, 1996 ). The final effect of a global fan spine bifurcation is to swap the spine sources of one null, and create and destroy separators linking to the other null (see Figure 11 ). This will result in complicated changes to the skeleton and thereby to the domain graph. Maclean et al. (2005 ) present a detailed analysis of the bifurcation and present a systematic prescription for predicting the changes to the field’s skeleton.

These two global bifurcations account for most of the topological transitions which fields undergo under continuous change. Any coronal field, be it potential or not, equilibrium or dynamic, will change its connectivity either through a global separator bifurcation or a global spine-fan bifurcation. Events involving the creation of new connections, as in the breakout model of coronal mass ejections (Antiochos, 1998 ), must occur through a global bifurcation. In a topological analysis of breakout in three dimensions, Maclean et al. (2005 ) show that it is most often the result of a global spine-fan bifurcation, although some are due to global separator bifurcation.