TEORÍA DEL CUIDADO

When the topology changes from one of the states in the previous section to another, we say that it has undergone a bifurcation. All bifurcations in potential fields are fully

4.4 Results for “2+2” scenario 86

reversible,i.e.if the parameters are changed back to their original values then any bifur- cations which have happened will be reversed and the original topology will reappear. As can be seen on the case 1 bifurcation diagram (figure 4.2), four types of bifurcation occur in the case of two sources of each sign. One of these is a local bifurcation, two are of global type, and one a quasi-bifurcation, so that at least one of each known class is present. Let us now understand in more detail how each of these bifurcations takes place. Thelocal double-separator bifurcation (Brown and Priest, 1999b), being of local type, must create or destroy null points. It occurs in the topology when magnetic sources are moved in relation to each other in certain ways, for example in the four-source case if a negative source moves all the way round a positive source, a local double-separator bifurcation must occur. It is an extension of a classical pitchfork-type bifurcation, in which one stable fixed point (corresponding to a null point here) in a system of ordinary differential equations bifurcates to become three stable fixed points, or vice versa. A sketch of how it takes place can be found in figure4.8. The lefthand image shows an initial generic first-order null point sitting on the photosphere before the bifurcation. At the point of bifurcation, the null changes its character and becomes a third-order null. Afterwards, as shown in the righthand image, the null splits into three: one (of opposite sign to the original) still in the photosphere, plus one in the corona and one in the mirror corona, both of the same sign as the original null. Equation 1.24 is thus satisfied. This entire process can also run in reverse. Note that this bifurcation cannot change the structure of the flux domains present in a topology. Of course, the null created in the mirror corona is meaningless in terms of the real coronal magnetic field, it is simply an artefact that must exist in the model to keep the model self-consistent, but it does not correspond to any feature in the real sub-photospheric field, where the model no longer applies.

In the model, the local double-separator bifurcation is responsible for the change between the intersecting and coronal null states. The intersecting state involves a separator linking two photospheric nulls of opposite sign. When the bifurcation happens, one of the photo- spheric nulls changes sign, and this causes a new coronal null to appear, moving up into the corona along the arc of the pre-existing separator.

The next type of bifurcation present in the model is the global separator bifurcation. A global bifurcation such as the global separator bifurcation (Brown and Priest, 1999b) will, unlike the local double-separator bifurcation, change the flux domain structure of a topology. It is a variant of a classic heteroclinic bifurcation. Heteroclinic bifurcations involve the creation or destruction of heteroclinic orbits, lines that join two equilibrium points; these orbits correspond to the topological separator fieldlines. An example of this type of bifurcation is given in figure4.9. In the lefthand image, before the bifurcation,

4.4 Results for “2+2” scenario 87

Figure 4.8: The local double-separator bifurcation. The lefthand image is before the bifurcation takes place, and the righthand one after. Dashed lines indicate fieldlines below the photosphere.

Figure 4.9: The global separator bifurcation.

there are two simple separatrix domes. At the point of bifurcation these domes touch along the photosphere on one side, so that an unstable photospheric separator connects the two null points, as shown in the middle image. After the bifurcation, as shown in the righthand image, the two domes intersect one another, creating a stable separator in the corona. The effect of this is to create a new flux domain, joining the upper positive source to the lower negative one. The number of nulls is unaffected. This bifurcation can also take place in reverse, destroying a separator and a flux domain rather than creating them. There are several examples of the global separator bifurcation in the model. It takes the topology between the detached and intersecting, nested and intersecting, and intersecting and dual intersecting states. In the first two cases, this is simply a question of the two previously isolated separatrix domes moving together and intersecting one another to form a separator. In the last case, the separatrix domes must intersect one another again, in a different spatial location, so that a new separator is created between the same two

4.4 Results for “2+2” scenario 88

Figure 4.10: The global spine-spine bifurcation.

null points.

The third type of bifurcation under consideration here is theglobal spine-spine bifurca- tion. This bifurcation can be classified as global as it involves passing through a state involving an unstable separator. This is the case even though, in the case of the “2+2” model studied here, the flux domain structure is identical before and after the bifurcation, as the bifurcating spines are not carrying any separatrices. As shown in figure4.10, the bifurcation involves the spines of two oppositely-signed null points passing through one another, so that the final topology is essentially a mirror image of the initial one. An unstable separator will be formed between the two nulls at the point of bifurcation. Such a bifurcation occurs near the top of the bifurcation diagram, taking one version of the nested state to its mirror image and back again. On later bifurcation diagrams, we will see that the global spine-spine bifurcation can also flip a detached state to its own mirror image.

The final bifurcation occurring in the current case is theglobal separatrix quasi-bifurcation

(Beveridge, Priest, and Brown,2002), shown in figure4.11. It is not analogous to any of the classical bifurcations, and indeed does not show any finite bifurcation behaviour; it occurs when fieldlines in the separatrix surface of a null point extend out to infinity then move back into a different arrangement. Figure4.11shows how a separatrix surface can enlarge itself, until in the central image the fieldlines reach out to form an infinite surface. When they collapse back down, the separatrix dome that they form encloses a different set of sources to the set it enclosed before the bifurcation. No flux domains are created or de- stroyed (which is why this cannot be considered a global bifurcation); what has changed is the sources enclosed by the separatrix domes.

4.4 Results for “2+2” scenario 89

Figure 4.11: The global separatrix quasi-bifurcation, shown taking place on a sphere to emphasise how the separatrix moves across the back of the sphere during the bifurcation.

In the model, the global separatrix quasi-bifurcation acts to move the topology between different versions of the intersecting state, and also from detached to nested and back. The detached and nested states are in fact indistinguishable except by noting which flux domain is the unbounded one; only the sources enclosed by the separatrix domes have changed.