Infinite contraction in force-free magnetic field evolution in cylindrical geometry

Abstract

This paper provides a new formulation of the resistive force-free evolution of cylindrically symmetric magnetic fields subject to purely radial motions. It is shown analytically that the evolution bounded by a perfect conductor ceases to exist after a finite time if the initial field has total axial flux of opposite sign to the field on the axis of symmetry. A numerical solution indicates that the evolution ceases to exist owing to the unlimited contraction of the field profile producing a line of infinite current density. The asymptotic form of this ‘blow-up’ is identified as the particular self-similar contraction for which the field direction is exactly reversed in the limit of large radius. Possible applications to solar flares and the reversed-field pinch are discussed.