Liquid instability and energy transformation near a magnetic neutral line: a soluble non-linear hydromagnetic problem

Abstract

Hydromagnetic flow of a conducting fluid has a special character near a magnetic neutral line. This is investigated with reference to the two-dimensional motion of a cylinder of perfectly conducting liquid in a permanent magnetic field, of which the axis of the cylinder is a neutral line. Electric current is induced in the liquid by its irrotational motion in the magnetic field. The liquid is uniform, incompressible and frictionless. The surface is elliptically cylindrical, infinitely long, free and in vacuo. The motion is governed by the force exerted on the electric current by the magnetic field, permanent and induced. The stream lines are constant rectangular hyperbolas, in planes normal to the cylinder axis. The permanent magnetic field lines are orthogonal rectangular hyperbolas. The cylinder axis is a stagnation line and a magnetic neutral line. If the liquid is initially at rest, with circular cross-section, and no electric current, its state is unstable. A small motion imparted to it, of the kind indicated, will grow indefinitely, magnetic energy being converted into kinetic energy. The initial motion, however, need not be small. This non-linear hydromagnetic problem is completely soluble. The initial conditions may be chosen in more than one way. The bearing of the solution on the theories of solar flares and the aurora is briefly considered.