Nonlinear beat wave decay of Kelvin/diocotron waves on a two-dimensional vortex

Abstract

We describe theory and experiments investigating nonlinear beat wave decay of diocotron modes on a nonneutral plasma column (or Kelvin waves on a vortex). Specifically, a Kelvin/diocotron pump wave varying as Ap exp [i(lpθ−ωpt)] decays into two waves: a Kelvin/diocotron daughter wave with exponentially growing amplitude Ad(t), mode number ld<lp, and frequency ωd; and an exponentially growing “beat wave” with mode number lb and frequency ωb. Nonlinear wave–wave coupling requires lb=lp−ld and ωb=ωp−ωd. The new theory simplifies and extends a previous weak-turbulence theory for the exponential growth rate of this instability, by instead using an eigenmode expansion to describe the beat wave as a wavepacket of continuum (Case/van Kampen) modes. The new theory predicts the growth rate, the nonlinear frequency shift (both proportional to Ap2), and the functional form of the beat wave, with amplitude proportional to ApAd*(t). Experiments observe beat wave decay on electron plasma columns for a range of mode numbers up to lp=5 and ld = 4, with results in quantitative agreement with the theory, including the ld = 1 case for which measured growth rates are negligible, as expected theoretically.