Formation of electron holes in the long-time evolution of the bump-on-tail instability

Abstract

AbstractAn Eulerian Vlasov code is applied for the numerical solution of the one-dimensional Vlasov–Poisson system of equations for electrons, and with ions forming an immobile background. We study the non-linear evolution of the bump-on-tail instability in the case when the system length L is greater than the wavelength λ of the unstable mode, with a beam density of 10% of the total density, nb = 0.1. We follow the growth and the saturation of an initially unstable wave perturbation, and the formation of a traveling Bernstein–Greene–Kruskal (BGK) mode, which evolves out of the instability. This first stage is followed by sidebands growing from round-off errors which develop and disrupt the BGK equilibrium. In the excited spectrum, mode coupling is mediated by the oscillating resonant particles and results in the electric energy of the system flowing to the longest wavelengths (inverse cascade), and reaching in the asymptotic state a steady state with constant amplitude oscillation modulated by the persistent oscillation of the trapped particles. Coherent phase-space electron holes are formed, which are localized phase-space regions of reduced density on trapped electron orbits, where the electron density is lower than the surrounding plasma electron density. The distribution function evolves to a shape with stationary inflection points of zero slope, at the phase velocities of the excited waves. The longest wavelengths show oscillations at frequencies below the plasma frequency, with phase velocities higher than that of the injected beam, which can accelerate electrons to energies in excess of the initial beam energy. The present work makes a connection between the formation of electron holes, the existence of inflection points of zero slopes in the electron distribution function at the phase velocities of the dominant waves, and at frequencies below the plasma frequency. A fine resolution grid is used in the Eulerian Vlasov code in the phase space and time to allow an accurate calculation of the time history of the system and of the dynamic and oscillation of the trapped particles in the low-density regions of the phase space, and of those particles at the separatrix regions of the vortex structures which evolve periodically between trapping and untrapping states and which can only be accurately studied using a fine-resolution phase-space grid.