Role of Landau Damping in Crossed-Field Electron Beams and Inviscid Shear Flow

Under certain circumstances the problem of the stability of low-density crossed-field electron beams is precisely analogous to the problem of the stability of incompressible inviscid shear flows. The electron density, the drift velocity, and the potential correspond to the fluid vorticity, velocity, and the stream function, respectively. Neutrally stable normal modes of oscillation that are found when certain suitable step-function unperturbed electron density (vorticity) profiles are assumed seem to disappear when small gradients are present. In the context of a Laplace transform solution of the initial value problem, the vanishing normal modes can be found by analytic continuation of the solution onto another sheet of the cut complex ω plane. The “recovered” modes are found to be damped by a fluid-wave resonant interaction which closely resembles the Landau damping of plasma oscillations in warm plasmas by particle-wave resonance. An interesting feature of the neutral waves is that they have a negative energy and so are destabilized by the removal of energy from the system (for example, by slightly dissipative walls). The sign and magnitude of the Landau damping term is evaluated for a simple case and compared (for the electronic case) with the growth provoked by slightly lossy walls.