Nonlinear Oscillations in a Weakly Ionized Plasma

Abstract

A perturbation theory is developed for a nonlinear longitudinal monochromatic wave in a weakly ionized plasma. The effect of collisions between electrons and neutrals is treated by means of a relaxation term in the Boltzmann-Vlasov equation. With the distribution function developed to all orders in the amplitude E0 of the oscillating electric field (assumed exponentially damped sinusoidal in form), a constant damping decrement λ is indicated for two cases: Either (i) λ must be negligible or (ii) the collision frequency vR of the resonance electrons must be negligible in the calculation of the distribution function of the resonance electrons in terms of the electric field. With the nonlinear correction developed to lowest order, the Landau damping decrement is reduced from its linear value γk0 by the factor 1−⅛(Ω4/vR4) for case (i), and by the factor 1−116[Ω4/(|γk0|+v̄)4] for case (ii). Here v̄ is the collision frequency appropriately averaged for direct collisional damping, Ω=(eE0k0/m)1/2 is the frequency of oscillation of the resonance electrons, k0 is the wavenumber, and e and m are the electronic charge and mass, respectively.