Numerical investigation of kinetic instabilities in BGK equilibria under collisional effects

TL;DR

This study investigates the instability of a one-dimensional BGK mode in a kinetic-electron plasma using high-resolution 1D-1V Vlasov-Poisson simulations with a Dougherty collision operator. An external field excites electron acoustic waves that generate a two-hole BGK equilibrium, which becomes unstable under large-scale density noise, leading to vortex merging into a single hole. The growth rate of the instability is found to be largely independent of weak collisionality, whileCollisions delay the trigger time and affect the long-time saturation analyzed via Hermite-space spectra. The Hermite analysis reveals a persistent enstrophy cascade in velocity space for collisionless cases and demonstrates collisional dissipation and eventual spectral cutoff at higher collisionality, in agreement with theoretical predictions. Overall, the work links phase-space structure dynamics to enstrophy-cascade concepts and provides quantitative benchmarks for how collisional effects modify kinetic turbulence in BGK-like equilibria.

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.

Figures (10)

• Figure 1: Contour plots of the phase-space electron distribution function at $t=0$ [panel (a)], $t=3750$ [panel (b)], $t=4500$ [panel (c)], and $t=10500$ [panel (d)] for $g=0$ (collisionless case).
• Figure 2: Contour plots of the phase-space electron distribution function at $t=0$ [panel (a)], $t=3750$ [panel (b)], $t=4500$ [panel (c)], and $t=10500$ [panel (d)] for $g=10^{-5}$ (weakly collisional case).
• Figure 3: Time evolution of the amplitude of the electric potential Fourier components $|\widehat{\phi}_1|$ and $|\widehat{\phi}_2|$ in the collisional and collisionless cases. Panel (a): mode 1 (blue curve) and mode 2 (orange curve) for $g=0$, compared with the typical trend $e^{\gamma t}$ (black dashed line). Panel (b): $| \widehat{\phi}_1 |$ evaluated for $g=0$ (blue curve) and $g=10^{-5}$ (orange curve), compared with the best fits $e^{\gamma t}$ (black dotted lines). Panel (c): comparison between $| \widehat{\phi}_1 |$ (blue lines) and $| \widehat{\phi}_2|$ (orange lines) in the merging phase for for $g=0$ (solid lines) and $g=10^{-5}$ (dashed lines). The crossing time $\tau_{cross}$ is highlighted by black dashed lines.
• Figure 4: Panel (a): delay time $\tau_{del}$ (blue curve) and crossing time $\tau_{cross}$ (orange curve) as functions of the collisional parameter $g$. Panel (b): growth rate $\gamma$ (blue curve) and saturation amplitude $\phi_{sat}$ (orange curve) as functions of $g$.
• Figure 5: Contour plots of Hermite transform during the merging phase and at the end of the simulation for $g=0$ [panels (a)-(b)], $g=10^{-6}$ [panels (c)-(d)], and $g=10^{-5}$ [panels (e)-(f)].