16 - Magnetic Relaxation under Topological Constraints

Summary

The theory of magnetic relaxation in a perfectly conducting but viscous fluid is presented as the counterpart of the dynamo process: both processes must be present in a statistically steady state in which the mean level of magnetic energy is constant. It is shown that the frozen-field property implies magnetic helicity conservation, which in turn implies a lower bound for the magnetic energy and so the existence of a steady state with arbitrarily prescribed magnetic field topology. Alternative relaxation procedures are described. Two-dimensional relaxation in an incompressible fluid is described, and the invariant magnetic signature function is introduced. The evolution of current sheets from saddle points of the initial field is demonstrated, with Y-type singularities at the ends of the current sheets. The relaxation of knotted structures in 3D is also discussed, and the minimum energy for a given knotted flux-tube is expressed in terms of its invariant flux and volume. Current sheets in general develop during such relaxation. Stability criteria for arbitrary magnetostatic states are obtained. Relaxation to steady solutions of the equations of ideal magnetohydrodynamics, with prescribed magnetic helicity and cross-helicity, is similarly described.