Field Line Topology and Rapid Reconnection

Abstract

Consider a magnetic field extending through an infinitely conducting fluid between end plates z = 0, z = L, with arbitrary interlacing of the field lines throughout. The field is fixed in both endplates and is allowed to relax to an equilibrium described by the force-free equilibrium equation ∇ ×B = αB. The divergence of this field equation yields ∇ ⋅α = 0, requiring that the torsion coefficient α be constant along each individual field line, and showing that the field lines represent a family of real characteristics of the equilibrium field equation. So the field line topology plays a direct role in determining the nature of the equilibrium field. For an arbitrarily prescribed interlacing field line topology a continuous field generally cannot provide an α that is constant along field lines. Yet with the field fixed at both ends it is obvious that every topology has an equilibrium. So there must be a mathematical solution to the field equation for each and every topology. This dilemma is resolved by the fact that the field lines represent a family of real characteristics, so that surfaces of discontinuity (current sheets) can form along the flux surfaces. In almost all field line topologies, then, the continuous field is cut up by surfaces of tangential discontinuity between regions of continuous field.