Stability of parallel flow in a parallel magnetic field at small magnetic Reynolds numbers

Abstract

The hydromagnetic stability of a basic two-dimensional parallel flow of an incompressible conducting fluid in a uniform magnetic field parallel to the flow is considered. By use of the generalization of the Orr–Sommerfeld equation for an electrically conducting fluid, it is shown that any given small wave disturbance can be stabilized by a sufficiently strong magnetic field if the Reynolds number is finite and the magnetic Reynolds number small.Stability of velocity profiles with a point of inflexion at small magnetic Reynolds number and infinite Reynolds number is considered in detail. Perturbation methods are developed to find stability characteristics in two cases, when the magnetic field is weak, and when the disturbance is a long wave. These methods are applied to the jet and the half-jet, which are both found to be unstable to long-wave disturbances, however strong the magnetic field. Nonetheless, these two flows can be stabilized for any given harmonic disturbance of finite wavelength. The analysis of the jet reveals the surprising result that the magnetic field makes inviscid long-wave disturbances more unstable.