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We discuss the formation of current singularities and reconnection in magnetic fields with complex 3D topology. First, we argue that since the photospheric field is observed to consist of a complicated mixture of positive and negative polarity regions, the coronal magnetic field must, in general, contain a large number of separatrix surfaces and null points. Using numerical simulations, we calculate the effect of photospheric stressing on such a field. As initial conditions in the numerical model, we assume a cylindrically-symmetric potential field consisting of a small dipole field imbedded in a background larger dipole; hence, there are three polarity regions on the photosphere. In the corona the field has a hemisperical separatrix surface with a null point at the apex of this surface.
The initial field is then stressed by footpoint motions at the photosphere that have the form of a vortical flow of finite width. Results are discussed for two different photospheric locations of this flow, one in which the flow is centered on the symmetry axis so that the system retains its cylindrical symmetry, and one in which the flow is offset from the symmetry axis. The results of these simulations are discussed, in particular, the nature of reconnection in a true 3D geometry. We find that in both cases current sheets form at the separatrix. We argue that this mechanism for current sheet formation may play a central role in coronal heating.
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