Full second-order cold-fluid theory of the diocotron and magnetron resonances

This is the authors' abstract. We don't add key points for this paper.

The full second-order, cold-fluid theory of the diocotron and magnetron resonances for nonrelativistic, non-neutral electron flow in planar, crossed-field geometry is investigated using the method of multiple-scales perturbation theory. We find that, for both resonances, the zeroth-order density evolves in the slow time and spatial plane according to a diffusion equation. The quasilinear treatment of the diocotron instability [R. C. Davidson, Phys. Fluids 28, 1937 (1985)] is recovered from the full treatment near the wave-particle, or diocotron, resonance. We also conjecture that, under constant-voltage conditions, this diffusion will cause a small dc current to be set up from the anode to the cathode and that this second-order diffusion equation could possess equilibrium solutions as well. Such solutions would have negative density gradients between all unstable resonances (magnetron as well as diocotron), with plateau formation occurring at the spatial position of all these resonances. Although we cannot show stabilization, we can argue that the introduction of negative density gradients, which would occur as the density profile seeks to re-form into an equilibrium profile, should tend to reduce the linear-instability growth rates, thereby causing the system to tend toward stabilization.