Numerical investigation of kinetic instabilities in BGK equilibria under collisional effects

TL;DR

This study uses high-precision numerical simulations to explore how collisional effects influence the development and saturation of kinetic instabilities in BGK equilibria, revealing the impact of collisions on instability onset and spectral behavior.

Contribution

It provides new insights into the role of collisions in BGK mode instabilities, including how they affect trigger time, growth rate, and spectral saturation, extending understanding of plasma kinetic behavior.

Findings

Growth rate depends on the slope of the distribution function in the resonant region.

Collisional effects can delay the onset of instability.

Hermite spectral analysis reveals different saturation behaviors with and without collisions.

Abstract

An unstable one-dimensional Bernstein-Greene-Kruskal (BGK) mode has been studied through high-precision numerical simulations. The initial turbulent, periodic equilibrium state is obtained by solving a Vlasov-Poisson system for initially thermalized electrons, with the addition of an external electric field able to trigger undamped, high-amplitude electron acoustic waves (EAWs). Once the external field is turned off, resonant particles are trapped in a stationary two-hole phase-space configuration. This equilibrium scenario is perturbed by some large-scale density noise, leading to an electrostatic instability with the merging of vortices into a final one-hole state. Numerical runs investigate several features of this regime, focusing on the dependence of the instability trigger time and growth rate on the rate of short-range collisions and grid resolution. According to Landau theory for weakly inhomogeneous equilibria, we observe that the growth rate of the instability depends only on the slope of the distribution function in the resonant region. Conversely, the onset time of the instability is affected by the collisional rate, which is able to postpone the onset of the instability. Moreover, by extending the simulations to a long-time scale, we investigate the saturation stage of the instability, which can be analyzed through the Hermite spectral analysis. In collisionless simulations where grid effects are negligible, the Hermite spectrum follows a power law typical of a constant enstrophy flux scenario. Otherwise, if collisional effects become significant, a cutoff is observed at high Hermite modes, leading to a decaying trend.