Topological bifurcations in three-dimensional magnetic fields

Most of the dynamical processes that take place in the Sun's corona (its outer atmosphere) are dominated by the magnetic field. The sources of the coronal field are magnetic fragments scattered over the solar surface and mostly clustered around the edges of large convection cells called supergranules. These sources are not static but continually move about over the surface, coalescing, fragmenting and cancelling with one another. The resulting coronal magnetic field has an incredibly complex topology. In order to begin to understand this complexity it is important to consider, as building blocks, the field generated by a small number of discrete sources. Priest and co-workers started this task by studying some of the different topological states of a three-source system together with some of the types of bifurcation between states. They considered the case where the sources are collinear and the special non-collinear case with a positive source at the origin and two negative sources of equal strength equidistant from the positive source. The present work extends their analysis by considering a general unbalanced three-source system and classifying the eight stable topological states 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 remaining two states occur when all three sources have the same polarity. In addition, the bifurcations from one topological state to another, both local and global, are analysed. Particular study is made of a local separator bifurcation (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 lies in the field of the other); and a global separator bifurcation (at which a topologically stable separator is created or destroyed).