Nonlinear Trivelpiece-Gould waves: Frequency, functional form, and stability

Abstract

This paper considers the frequency, spatial form, and stability of nonlinear Trivelpiece-Gould (TG) waves on a cylindrical plasma column of length L and radius rp, treating both traveling waves and standing waves, and focussing on the regime of experimental interest in which L/rp≫1. In this regime, TG waves are weakly dispersive, allowing strong mode-coupling between Fourier harmonics. The mode coupling implies that linear theory for such waves is a poor approximation even at fairly small amplitude, and nonlinear theories that include a small number of harmonics, such as three-wave parametric resonance theory, also fail to fully capture the stability properties of the system. It is found that nonlinear standing waves suffer jumps in their functional form as their amplitude is varied continuously. The jumps are caused by nonlinear resonances between the standing wave and nearly linear waves whose frequencies and wave numbers are harmonics of the standing wave. Also, the standing waves are found to be unstable to a multi-wave version of three-wave parametric resonance, with an amplitude required for instability onset that is much larger than expected from three wave theory. It is found that traveling waves are linearly stable for all amplitudes that could be studied, in contradiction to three-wave theory.