Instability of nonlinear Trivelpiece-Gould waves I: Wave degeneracies

Arguments based on energy conservation are used to evaluate the fluid theory of stability of nonlinear traveling waves (pump waves) in an ideal plasma system. Instabilities growing on the pump wave are associated with wave degeneracies. The relative signs of the energies of degenerate waves, as seen in the frame of the pump wave, determine whether their amplitudes grow exponentially or merely oscillate through resonant energy exchange. This energy analysis is carried out in detail for Trivelpiece-Gould (TG) waves and is compared to numerical calculations. It is verified that nonlinear TG waves are stable with respect to 3 wave processes, but weaker 4 wave, 5 wave, and higher order wave processes cause instability over narrow wavenumber bands. A modulational instability is also identified.