Influence of profile shape on the diocotron instability in a non-neutral plasma column

Abstract

In this paper we examine theoretically the influence of density profile shape on the diocotron instability in a cylindrical, low-density (ωpe2≪ωce2) non-neutral electron plasma column confined radially by a uniform axial magnetic field B0êz. The analysis assumes electrostatic flute perturbations (∂/∂z=0) about an axisymmetric equilibrium density profile ne0(r), where r=(x2+y2)1/2 is the radial distance from the column axis. Two classes of density profiles with inverted population in radius r are considered. These are the following: (a) a step-function density profile with uniform density n̂e⋅Δ in the column interior 0⩽r<rb−, and uniform density n̂e in an outer annular region rb−<r<rb+; and (b) a continuously-varying density profile of the form ne0(r)=n̂e(Δ+r2/rb2)(1−r2/rb2)2 over the interval 0⩽r<rb. Here, n̂e, rb−, rb+ and rb are positive constants, and the dimensionless parameter Δ measures the degree of “hollowness” of the equilibrium density profile ne0(r). Detailed linear stability properties are calculated for a wide range of system parameters, including values of the “filling factor” Δ, radial location rw of the cylindrical conducting wall, azimuthal mode number l, etc. As a general remark, in both cases, it is found that small increases in Δ from the value Δ=0 (corresponding to the strongest diocotron instability) can have a large effect on the growth rate and detailed properties of the instability. In addition, for the step-function density profile, the instability tends to be algebraic in nature and have a large growth rate in the unstable region of parameter space, whereas for the continuously-varying density profile, the instability is typically much weaker and involves a narrow class of resonant particles at radius r=rs satisfying the resonance condition ωr−lωE(rs)=0. Here, ωr=Re ω is the real oscillation frequency, and ωE(r)=−cEr0(r)/rB0 is the equilibrium E0×B0êz rotation velocity of the plasma column.