Nonlinear Beat Wave Decay of Diocotron Modes

We describe theory and experiments on nonlinear beat wave decay of diocotron modes on a nonneutral plasma column (or Kelvin waves on a vortex patch). Specifically, a diocotron "pump wave" varying as A exp[i (l₃ θ-ω₃t)] decays into two "daughter waves": a diocotron mode with exponentially growing amplitude a₂ , mode number l₂ < l₃, and frequency ω₂; and a nonlinear beat wave with mode number l₁ and frequency ω₁. Nonlinear wave-wave coupling requires l₁=l₃-l₂ and ω₁≅ω₃-ω₂. The new theory simplifies and extends a previous weak-turbulence theory for the exponential growth rate of this instability¹, by instead using an eigenmode expansion to describe the beat wave as a wavepacket of continuum (van Kampen) modes. The new theory predicts the growth rate, the nonlinear frequency shift, (both proportional to A²), and the functional form of the beat wave, with amplitude proportional to A a₂. Experiments observe beat wave decay on electron plasma columns for a range of mode numbers up to l₃=5 and l₂=4, with results in qualitative agreement with the theory for low amplitudes, including the l₂=1 case for which measured growth rates are negligible, as expected theoretically. At larger pump amplitude A (with radial diocotron oscillations greater than 10% of the plasma radius) the experimental growth rate is substantially reduced.

¹ N. Mattor, B. T. Chang and T. B. Mitchell, Phys. Rev. Lett. 96, 045003 (2006).