Quasilinear theory of the diocotron instability for nonrelativistic non-neutral electron flow in planar geometry

Abstract

A macroscopic cold-fluid model is used to investigate the quasilinear stabilization of the diocotron instability for sheared, nonrelativistic electron flow. Planar diode geometry is assumed, with cathode and anode located at x=0 and x=d, respectively. The non-neutral plasma is immersed in a strong applied magnetic field B0êz, and the electrons are treated as a massless (m→0) guiding-center fluid with flow velocity Vb=−(c/B0)∇φ×êz, where ∂/∂z=0 is assumed, and the fields are electrostatic with E=−∇φ. All quantities are assumed to be periodic in the y direction with periodicity length L. The nonlinear continuity-Poisson equations are used to obtain coupled quasilinear kinetic equations describing the self-consistent evolution of the average density 〈nb〉(x,t) and spectral energy density ℰk(x,t) associated with the y-electric field perturbations. Here, the average flow velocity in the y direction is VE(x,t)=(c/B0)(∂/∂x)〈φ〉(x,t), where average quantities are defined by 〈ψ〉(x,t)=∫L0 (dy/L)ψ(x,y,t). Several general features of the quasilinear evolution of the system are discussed, including a derivation of exact conservation constraints. Typically, if the initial profile 〈nb〉(x, t=0) corresponds to instability with γk(0)>0, the perturbations amplify, and the density profile 〈nb〉(x,t) readjusts in such a way as to reduce the growth rate γk(t) and stabilize the instability. As a specific example, the quasilinear evolution of the diocotron instability is considered for 〈nb〉(x,0) corresponding to a gentle density bump superimposed on a rectangular density profile in contact with the cathode.