Topological bifurcations in three-dimensional magnetic fields

Most of the dynamical processes that take place in the Sun's corona (its ou ter atmosphere) are dominated by the magnetic field. The sources of the cor onal field are magnetic fragments scattered over the solar surface and most ly clustered around the edges of large convection cells called supergranule s. These sources are not static but continually move about over the surface , coalescing, fragmenting and cancelling with one another. The resulting co ronal magnetic field has an incredibly complex topology. In order to begin to understand this complexity it is important to consider, as building bloc ks, the field generated by a small number of discrete sources. Priest and c o-workers started this task by studying some of the different topological s tates of a three-source system together with some of the types of bifurcati on between states. They considered the case where the sources are collinear and the special non-collinear case with a positive source at the origin an d two negative sources of equal strength equidistant from the positive sour ce. The present work extends their analysis by considering a general unbala nced three-source system and classifying the eight stable topological state s that arise and their location in parameter space: six of the states occur when two of the sources have polarity opposite to the third and the remain ing two states occur when all three sources have the same polarity. In addi tion, the bifurcations from one topological state to another, both local an d global, are analysed. Particular study is made of a local separator bifur cation (in which two linear nulls and a separator linking them are created or destroyed); a global spine bifurcation (at which the spine of one null l ies in the field of the other); and a global separator bifurcation (at whic h a topologically stable separator is created or destroyed).