Linear and nonlinear behaviour of two-stream instabilities in collisionless plasmas

Abstract

The transient, growth and nonlinear saturation stages in the evolution of the electrostatic two-stream instabilities as described by the Vlasov–Poisson system are reconsidered by numerically following the evolution of the total wave energy of the plasma oscillations excited from (numerical) noise. Except for peculiarities related to the necessarily finite (even though very small) magnitude of the perturbations in the numerical simulation, the existence and initial growth properties of the instabilities from the numerical results are found to be consistent with those from linear normal mode analysis and the Penrose criteria. However, contradictory to the traditional point of view, the growth of instability before saturation is not always linear. The initial stage of the growth can exhibit fine structures that can be attributed to the harmonics of the excited plasma oscillations, whose wavelengths are determined by the system size and the numerical noise. As expected, saturation of the unstable oscillations is due to electron trapping when they reach sufficiently large amplitudes.