Instability of a plane conducting free surface submitted to an alternating magnetic field

Abstract

This paper considers the stability of a horizontal liquid-metal free surface in the presence of a horizontal alternating magnetic field. A weak formulation is used to derive a generalized Mathieu–Hill equation for the evolution of surface perturbations. Previous studies which rely on time-averaging the electromagnetic force over one field cycle have predicted a generally weak instability, but we find much larger growth rates near the resonances, where the surface wave frequency is an integral multiple of the field frequency. The method can be extended to include viscous and ohmic damping; the former has little effect, while the latter damps all waves except those whose frequency is close to the field frequency. Growth rates can be closely approximated by simple algebraic formulae, as can the critical magnetic field strength for the onset of instability.