Equilibrium of magnetic fields with arbitrary interweaving of the lines of force II. Discontinuities in the field

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Abstract
It is shown that surfaces ∑α of discontinuity in the torsion α in a force-free field (whose existence was established in the previous paper) have special properties. There is a discontinuity in the magnetic field (i.e. a current sheet) associated with ∑α if the extension of ∑α along the field is terminated anywhere within the fluid, or if two or more ∑α intersect. If the ∑α do not terminate, they extend arbitrarily far along the field from the location of the arbitrary change in winding pattern that produces them. In that case there would be as many ∑α as changes in winding pattern, each ∑α oriented arbitrarily with respect to the others. Intersections would be unavoidable, and discontinuities (current sheets) inevitable companions of the ∑α. Hence sequences of arbitrary patterns of winding and wrapping of the lines of force generally produce current sheets. The current sheets are the principal sources of dissipation of the field in a highly conducting fluid, because the interior of each current sheet is subject to the dynamical nonequilibrium that produces neutral point reconnection. We have suggested that all astrophysical magnetic fields wound into complex topologies (by convective motions of the gas in which the field is rooted) produce the internal dissipation of current sheets, no matter how high the electrical conductivity. The stellar X-ray corona appears to be the most striking consequence of this effect. There are, however, a number of outstanding theoretical questions that remain unanswered, e.g. the precise rate of dissipation of the field and whether current sheets are, or are not, localized in the neighborhood of the arbitrary pattern changes that cause the ∑α.