The skeleton of complex magnetic fields due to many sources consists of the null points and a network of spine curves and separatrix fan surfaces.
Fig 6.1. Topological skeletons in 2D and 3D.
We have therefore been considering the topological properties of the key building blocks of complex fields, namely the field created by two, three and four sources. For example, three sources in the solar surface can produce eight structurally stable types of topology for the coronal field (Priest, Bungey and Titov, 1997, Geophys. Astrophys. Fluid Dyn. 84, 127; Brown and Priest, 1997, submitted).
Changes of topology from one of the above types to another occur either by local bifurcations, in which null points are created or destroyed, or by global bifurcations involving heteroclinic connections between null points.
For example, the above figure shows how, as you move three sources around, the topology changes as follows: in (a) we start with two separate separatrices and two nulls N1 and N2; in (b) a second-order null appears; in (c) a local bifurcation has created two new nulls N3 and N4 and the separatrices are touching; in (d) N1, N2 and N4 have coalesced via another local bifurcation to leave behind one null and the other one (N3) with one separatrix enclosed in the other.
We have also classified the local bifurcations of axisymmetric systems and have begun to investigate fully three-dimensional bifurcations of nulls (Priest, Lonie and Titov, 1996, J. Plasma Phys. 56, 507).
For example, in the above figure, we start with a flux tube in (a), which due to a saddle-node-Hopf bifurcation develops a pair of spiral nulls and an isolated closed field line (representing a ring of nulls in vertical planes). Then in (b) we break the symmetry by means of a non-axisymmetric perturbation; this destroys the heteroclinic surface joining the two spiral nulls and creates a region of chaotic field lines as shown in a Poincare section.
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