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

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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 Pϑ, i.e., fe0(H, Pϑ) = (n0R0/2πm) δ (H+eφ̄0−mV02/2) δ (Pϑ−P0), where n0, R0, φ̄0, V0, and P0 are constants. No a priori restriction is made that the density is low (ωp02≪ωc2), and equilibrium and stability behavior is studied for both fast (P0≳0) and slow (P0E layer when ωp02≈ωc2. The electrostatic stability analysis includes both negative‐mass (orbit variations with Pϑ) 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=ni0/ne0 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 ωp0.