Vlasov equilibrium and nonlocal stability properties of an inhomogeneous plasma column

Abstract

A fully kinetic, nonlocal, matrix dispersion equation is derived for electrostatic perturbations about a spatially nonuniform cylindrical plasma equilibrium. The analysis is carried out for the class of radially confined rigid-rotor equilibria described by f0j(x,v) = (n̂jmj/2πTj) F (H⊥/Tj− ωjPϑ/Tj,vz), where Pϑ is the canonical angular momentum, vz is the axial velocity, H⊥ is the perpendicular energy, and n̂j, Tj, and ωj are constants. Assuming equilibrium charge neutrality and negligible spatial variation in the axial magnetic field B0êz, it is shown that the particle trajectories (in the equilibrium electric and magnetic fields) and the orbit integrals required in the stability analysis can be evaluated in closed form. Expanding the perturbed electrostatic potential in terms of the vacuum eigenfunctions {Jl(λnr) } for the conducting cylinder leads to a matrix dispersion equation of the form det[δn′,n+ Σjχjn′,n(ω)]=0, where the susceptibility χjn′,n(ω) is expressed as a phase-space integral over f0j(x,v) and known functions of ω, r λn, etc. The limiting case of strongly magnetized electrons and unmagnetized ions is considered together with a preliminary application to the lower-hyprid-drift instability.