Force-Free Magnetic Fields with Constant Alpha

Abstract

According to Taylor’s conjecture (1986), the force-free field with constant α is a final state of a magnetic configuration after reconnection and subsequent relaxation.This conjecture is based on the assumption of existence and completeness of the set of eigenfunctions of the operator ∇×, i.e., the set of force-free solutions with constant α. It turns out, however, that the existence of solutions is very sensitive to the geometry of the plasma configuration. Actually, only for a boundary with high degree of symmetry the solutions are found. If the shape of the boundary possesses only axial symmetry, the force-free fields are presumably axisymmetric as well. Therefore, the set of eigenfunctions is not complete in that case. For a body with a “bad” shape force-free fields with constant α may not exist at all. As a result, the question of the final state of a magnetic configuration inside the body remains open. As pointed out by Moffatt (1985), there is an analogy between magnetostatic equilibria and steady Euler flows. According to this analogy, constant-alpha force-free fields are equivalent to Beltrami flows. Additionally, only this latter class of fields may be stochastic, so that magnetic surfaces do not exist (Arnold, 1973). According to Moffatt (1985), the relaxation of an initially ergodic field (in some domain) under topological constraints has to end up as a constant-alpha force-free field. In addition to Moffatt’s conclusion, we would suggest that the relaxation of the field under topological constraints is accompanied by a change of the shape towards a more symmetric geometry of the domain.