Nonlinear stability theorem for rotating coherent structures in a non-neutral plasma column

Abstract

A nonlinear stability theorem is developed for arbitrary-amplitude, two-dimensional coherent structures, nR(θ − ωr t) and φR(θ − ωr t), in a strongly magnetized, low-density (ω2pe/ω2ce ≪ 1) non-neutral plasma column confined radially by a uniform axial magnetic field B0êz. Here, a grounded, perfectly conducting, cylindrical wall is located at radius r = rw, and ωr = const is the angular rotation velocity of the coherent structures about the cylinder axis (r=0). A cold-fluid guiding-center model based on the continuity-Poisson equations is used to describe the nonlinear evolution of the electron density ne(r,θ,t) = nR + δne and electrostatic potential φ(r,θ,t) = φR + δφ. Making use of global (spatially averaged) nonlinear conservation constraints, it is shown that ∂nR(ψR)/∂ψR ≤ 0 is a sufficient condition for nonlinear stability of the rotating equilibrium state (φR,nR) to arbitrary-amplitude perturbations δne and δφ. Here, ψR(r,θ − ωr t) = −eφR(r,θ − ωr t) + ωr(eB0/2c)r 2 is an effective streamfunction, and the stability theorem is valid no matter how complex the radial and azimuthal structure of φR(r,θ − ωr t) and nR(r,θ − ωr t).