Helmholtz Instability of a Plasma

A plasma having infinite electrical conductivity and no viscosity is assumed to be in contact with a uniform magnetic field along a plane boundary which is parallel to the field. The behavior of small perturbations of this boundary when the plasma is flowing at velocity $v_0$ perpendicular to the magnetic field is calculated by linearized theory. Perturbations which only move lines of force parallel to themselves are unstable; for small $\frac{v_0}{c}$ the motion is incompressible and the rate of growth of the perturbation can be obtained from the incompressible hydrodynamic expression by replacing the mass density of each fluid in the hydrodynamic case by the sum of twice the magnetic energy density divided by $c^2$ and the mass density of each magnetohydrodynamic fluid. The magnetic field is to be considered as a "fluid" having only magnetic mass. It is shown that this analogy holds even in the nonlinear equations for two-dimensional incompressible flow. Perturbations which only bend lines of force are stable, while those which both move lines parallel to themselves and bend them are stable if the bending wavelength is short enough.