A dynamic runaway effect associated with flux expulsion in magnetohydrodynamic channel flow

Abstract

Pressure-driven flow along a channel in the presence of an applied magnetic field which is periodic in the streamwise direction is considered. The configuration is such that the transverse component of field By, is non-zero on the centreline y = 0, but its streamwise average 〈By〉 is zero. In this situation, flux expulsion due to reconnection of field lines occurs when the pressure gradient is sufficiently large. This leads to a decrease in the Lorentz forces, hence to an acceleration of the flow, and hence to stronger flux expulsion. When viscous effects are weak (i.e. at high Hartmann number) this creates a runaway effect, which appears at a critical value of the pressure gradient. This critical value is determined in the inviscid limit, and numerical and analytical methods are used to explore the associated ‘cusp-catastrophe’ behaviour when effects of weak viscosity are taken into account.