Theory of Parametric Instabilities and Harmonic Generation in Tonks-Dattner Resonances

Onedimensional fluid theory of Tonks-Dattner resonances in inhomogeneous, bounded plasmas is extended to stronger exciting electric fields. The relation of parametric instabilities and harmonic generation with Tonks-Dattner resonances will be worked out in a more direct way than in previous work. Electric field thresholds and growth rates for parametric instabilities will be obtained from theory of homogeneous Mathieu equation. Stationary excitation at the driving frequency and its harmonics follows from a particular solution of an inhomogeneous Mathieu equation.