EL PENSAR DE LA ENFERMERA

Five distinct topological states were found in total, all of which appear on the case 1 bi- furcation diagram (figure4.2). Corresponding examples of four of these topologies were observed byBeveridge, Priest, and Brown(2004) in the planar photosphere case; one is new. They are differentiated from one another by the numbers and positioning of their topological features, such as null points and separatrices. For each topology I present an example with extrapolated spine and separatrix fieldlines, an idealised representation of the topological structure, and also a diagram called a domain graph that shows which pairs of sources are connected by flux domains. The domain graph (Longcope and Klap- per, 2002) is useful because it provides an instant insight into the connectivity of the sources present – two sources are linked on the graph if they are connected by a flux domain.

Firstly, thedetached state involves two prone photospheric nulls, as shown in figure4.3. In fact there is one null of each sign; the spines of the positive (negative) null connect to the two positive (negative) sources, and all of the fieldlines in its separatrix surface go to one of the negative (positive) sources, forming a dome enclosing just one of its own spine sources. The two domes do not intersect one another, so there are three flux domains and no separators. In the notation used on the bifurcation diagram, the topology in figure4.3

is called ‘detached (P1–N1)’. The two sources in brackets after the name of the state show which flux domain is unbounded, i.e. whose fieldlines are not confined within a separatrix dome but extend out to infinity. There is not a source at infinity, because the flux in all the topologies studied in this chapter is balanced, but in each topology there is one flux domain whose fieldines fill the entire space outside a certain radius out from the photosphere. Depending on which flux domain is unbounded, there are four possible variants of the detached state.

Thenestedstate (shown in figure 4.4) is very similar to the detached state just seen; the difference between them lies in the arrangement of the separatrix surfaces. One remains unchanged (the smaller one in the diagram), enclosing one of its own spine sources. But the other has now changed which of its own spine sources it encloses, and also it com- pletely covers the other separatrix dome. The two domes do not intersect, but are stacked up on top of each other. Again there are three flux domains, but a different one extends to infinity. In the notation used on the bifurcation diagram, the topology in figure4.4is called ‘nested (P2–N1)’. Depending on which flux domain is unbounded, there are four possible variants of the nested state.

4.4 Results for “2+2” scenario 82

(a) Example with extrapolated fieldlines

(b) Schematic of topological structure

(c) Domain graph

Figure 4.3: The detached state

(a) Example with extrapolated fieldlines

(b) Schematic of topological structure

(c) Domain graph

4.4 Results for “2+2” scenario 83

(a) Example with extrapolated fieldlines

(b) Schematic of topological structure. The black dashed line represents a separator in

the corona.

(c) Domain graph

Figure 4.5: The intersecting state

in figure4.5. The separatrix of the positive (negative) null again forms a dome enclosing one of its own spine sources, but this time the dome touches both negative (positive) sources and the null between them. So the two domes intersect one another and there is a separator running between the two null points, giving a total of four flux domains in the topology. The separator is shown in the schematic as a black dashed line.

In the notation used on the bifurcation diagram, the topology in figure4.5 is called ‘in- tersecting (P1–N1)’. Any of the four flux domains can be unbounded, so at least four possible variants of the intersecting state exist. In addition, the intersecting state has the special property that its mirror image is not identically the same topology. The intersect- ing state is unique amongst all the “2+2” topologies in this respect. Its mirror image is another distinct variant with opposite handedness. In total this means that there are eight possible variants of the intersecting state. An example of this handedness phenomenon can be seen on the bifurcation diagram (figure4.2). The two states covering the largest areas on the diagram are both labelled “intersecting (P1–N1)”, but in fact they are not the same variant but mirror images of each other.

Next, thedual intersectingstate (figure4.6), which is a new state not previously known to occur in the planar photospheric case. Its main distinguishing feature is the presence of two spatially distinct separators joining the same two null points. Such a phenomenon

4.4 Results for “2+2” scenario 84

(a) Example with extrapolated fieldlines

(b) Schematic of topo- logical structure

(c) Domain graph

Figure 4.6: The dual intersecting state

has never before been observed in simple cases with only four sources; however,Close, Parnell, and Priest (2005) found multiple separators joining pairs of nulls in topologies with higher numbers of flux sources. In the four-source case, the two separators arise from the fact that the same two separatrix domes intersect one another twice (hence the name). A source configuration close to being symmetrical is necessary to produce this state, but it exists for a range of parameter values and is therefore undoubtedly topologically stable. If we go through all the fieldlines in the fan plane of one of the nulls, in order from photosphere up into the corona and back down to the photosphere on the other side, we find the following connectivity: source of opposite sign, null of opposite sign, second source of opposite sign, back to the same null, then back to the first source. The other null shows identical behaviour. There are five flux domains in this topology. One is purely coronal, and engirdled by the loop formed by the two separators; the dashed line on the domain graph indicates this domain. Another two of the flux domains actually link the same two sources but are physically separated from one another. Dashed lines in the schematic represent coronal fieldlines. In the notation used on the bifurcation diagram, the topology in figure4.6is called ‘dual intersecting (P1–N1)’. Depending on which flux domain is unbounded, there are four possible variants of the dual intersecting state. The final state found is thecoronal null state, shown in figure 4.7. There are three null

4.4 Results for “2+2” scenario 85

(a) Example with extrapolated fieldlines

(b) Schematic of topological structure

(c) Domain graph

Figure 4.7: The coronal null state

points in this state, two of the same sign in the photosphere and one of the opposite sign in the corona. In the case when the coronal null is negative (i.e. the case shown in the diagram), the two photospheric nulls are positive, and their spines each connect to both positive sources, forming a loop in the photosphere around one of the negative sources. Similarly, each of their separatrices touches both negative sources, forming a dome around one of the positive sources. The separatrix of the coronal null also forms a dome, touching the photosphere along the loop of spines, and the intersection of these two domes gives rise to separators between the coronal null and each of the photospheric ones. There are four flux domains present in total, and two separators. In the notation used on the bifurcation diagram, the topology in figure4.7 is called ‘coronal null (P1– N1)’. Depending on which flux domain is unbounded, there are four possible variants of the coronal null state.