Tangential discontinuities in untidy magnetic topologies

Abstract

The equilibrium requirement ∇×B=αB for the Maxwell stresses in a magnetic field B has the curious property that the torsion coefficient α is constant along every individual field line (B⋅∇α=0) but is completely unrelated from one field line to the next. The magnetic fields to be seen in nature sometimes have untidy internal field topologies because of the interlacing of their field lines by convection of their footpoints. The field lines may be wrapped and swirled in arbitrary topological patterns at different locations along the field, with the result that the local magnetic circulation, represented by the invariant α, must be supplemented by the appearance of surfaces of tangential discontinuity. These incipient discontinuities introduce rapid resistive dissipation of the magnetic free energy, which may contribute to heating the x-ray corona of the Sun, etc.