Vlasov equilibrium and electrostatic stability properties of a nonrelativistic nonneutral E layer

Abstract

Vlasov equilibrium and stability properties of a nonrelativistic nonneutral E layer are investigated for the class of thin (a ≪R₀) E‐layer equilibria in which all electrons have the same value of the energy H and the same value of canonical angular momentum P_ϑ, i.e., f_e⁰(H, P_ϑ) = (n₀R₀/2πm) δ (H+eφ̄₀−mV₀²/2) δ (P_ϑ−P₀), where n₀, R₀, φ̄₀, V₀, and P₀ are constants. No a priori restriction is made that the density is low (ω_p0²≪ω_c²), and equilibrium and stability behavior is studied for both fast (P₀≳0) and slow (P₀E layer when ω_p0²≈ω_c². The electrostatic stability analysis includes both negative‐mass (orbit variations with P_ϑ) and diocotron (angular velocity shear) effects. The system is shown to be stable whenever the equilibrium flow is space‐charge neutralized, i.e., whenever f=1, where f=n_i⁰/n_e⁰ is the fractional charge neutralization. For f≪1, a necessary and sufficient condition for instability is given by (l−2) a/R₀≳f, where l is the azimuthal mode number of the perturbation. Moreover, depending on system parameters, the growth rate can be a substantial fraction of the electron plasma frequency ω_p0.