Numerical Simulation of the Bump-on-Tail Instability

Abstract

Wave-particle interaction is among the most important and extensively studied problems in plasma physics. Langmuir waves and their Landau damping or growth are fundamental examples of wave-particle interaction. The bump-on-tail instability is an example of wave growth and is one of the most fundamental and basic instabilities in plasma physics. When the bump in the tail of the distribution function presents a positive slope, a wave perturbation whose phase velocity lies along the positive slope of the distribution function becomes unstable. The bump-on-tail instability has been generally studied analytically and numerically under various approximations, either assuming a cold beam, or the presence of a single wave, or assuming conditions where the beam density is weak so that the unstable wave representing the collective oscillations of the bulk particles exhibits a small growth and can be considered as essentially of slowly varying amplitude in an envelope approximation (see for instance Umeda et al., 2003, Doveil et al. 2001, and references therein). Some early numerical simulations have studied the growth, saturation and stabilization mechanism for the beam-plasma instability (Dawson and Shanny, 1968, Denavit and Kruer, 1971, Joyce et al., 1971, Nuhrenberg, 1971). Using Eulerian codes for the solution of the Vlasov-Poisson system (Cheng and Knorr, 1976, Gagne and Shoucri, 1977), it has been possible to present a better picture of the nonlinear evolution of the bump-on-tail instability (Shoucri, 1979), where it has been shown that for a single wave perturbation the initial bump in the tail of the distribution is distorted during the instability, and evolves to an asymptotic state having another bump in the tail of the spatially averaged distribution function, with a minimum of zero slope at the phase velocity of the initially unstable wave (in this way the large amplitude wave can oscillate at constant amplitude without growth or damping). The phase-space in this case shows in the asymptotic state a Bernstein-GreeneKruskal (BGK) vortex structure traveling at the phase-velocity of the wave (Bernstein et al., 1957, Bertrand et al., 1988, Buchanan and Dorning, 1995). These results are also confirmed in several simulations (see for instance Nakamura and Yabe, 1999, Crouseilles et al., 2009). Since the early work of Berk and Roberts, 1967, the existence of steady-state phase-space holes in plasmas has been discussed in several publications. A discussion on the formation and dynamics of coherent structures involving phase-space holes in plasmas has been presented for instance in the recent works of Schamel, 2000, Eliasson and Shukla, 2006. There are of course situations where a single wave theory and a weak beam density do not apply. In the present Chapter, we present a study for the long-time evolution of the Vlasov-