Instability of counter-rotating Trivelpiece-Gould (TG) waves in a cylindrical electron plasma

The stability properties of a small amplitude l =+1 or l =-1 (here l is the azimuthal wavenumber) TG plasma wave in an electron plasma column with a nonuniform density n(r) are studied. In the laboratory frame, these azimuthally counter-propagating TG waves have high frequencies (MHz) with a very small separation gap (tens of kHz) determined by the E×B rotation frequency of the plasma. Using a phase-locked excitation pulse, it is possible to excite each of these two waves separately with a contrast ratio of more than 20:1 in initial amplitudes. The higher frequency l =+1 TG wave is a stable "pump wave" and no growth of daughter waves has been observed. However, excitation of the lower frequency l =-1 TG wave leads to a fast instability that results in exponential growth of the high frequency l =+1 TG daughter wave when interacting with the nonlinearly coupled l = 2 diocotron wave. The three-frequency detuning size sets the threshold for the minimum amplitude of the l =-1 TG pump wave at which instability occurs. The experimental results are in qualitative agreement with an emerging theory of a nonlinear 3 wave interaction involving the negative energy diocotron mode and the two counter-propagating TG modes.