A Resonant Far Fields and Amplitude Oscillations (Part IV. Applications to the Vlasov Plasma)

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The present paper considers a one dimensional monochromatic Langmuir wave of small but finite amplitude. In §1, long-time asymptotic behaviors of the wave due to resonant effects are briefly discussed as the resonant far fields of the Vlasov equation. In §2, following the idea of the reductive perturbation method the Vlasov-Poisson equations are investigated by Lagrangian method assuming appropriate orderings to give a set of equations of one slow time. The expansion parameter ε is the ratio of the decay constant of the Landau damping γ_L to the plasma frequency, ω_p, i. e., ε〓γ_L / ω_p. The resultant set of equations contains a parameter q〓γ_L / ω_B, with ω_B, the bounce frequency of electron. When q≫1 (but q≪ε^<-1>), the wave Landau-damps, in which case trapped electrons are not essential to the time evolution of the wave amplitude E. When q<1, a density perturbation of trapped electrons balances with a collective slow variation of E, and a system of equations which governs the time evolution of E is obtained and integrated numerically. An initial evolution of this system is solved analytically and a nonlinear corrections to the Landau damping is obtained. The results of numerical investigations of the system for all time given in §3 are as follows. For q<0.77(butq≫ω), in the time dependence of the amplitude a plateau is formed after oscillations. For q≈0.77 the amplitude approaches a plateau without oscillation but for q>0.77 the wave is damped (not necessarily Landau damped). A time evolution of the number of "trapped electrons"and also of the distribution of resonant electrons are obtained. For q≪1 even when E asymptotically approaches a constant a corresponding velocity distribution of resonant electrons is not necessarily uniform in the trapped region. The time dependence of E for various q is compared with experimental results and is shown to be in good agreement with them. In §4, for stronger fields or weaker damping (i. e., when q become O (ε)) an amplitude oscillation of new type is predicted, for which an initial perturbed density of resonant electrons is responsible. The most remarkable feature of this oscillation is that the amplitude does not damp initially but grows, then oscillate about an amplitude larger than the initial one. In §5, the theory is reformulated in the Eulerian representation so as to be compared with that of Al'tshul' and Karpman ; differences between these theories are emphasized.
理論物理学刊行会の論文
1975-01-31

A Resonant Far Fields and Amplitude Oscillations (Part IV. Applications to the Vlasov Plasma)

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