Numerical investigation of kinetic instabilities in BGK equilibria under collisional effects

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.

💡 Research Summary

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This paper presents a comprehensive numerical investigation of the stability and long‑term evolution of a one‑dimensional Bernstein‑Greene‑Kruskal (BGK) equilibrium in the presence of short‑range collisional effects. The authors first generate a turbulent, periodic BGK state by solving the Vlasov‑Poisson system for initially thermal electrons while applying an external electric field that excites high‑amplitude electron acoustic waves (EAWs). When the external field is switched off, resonant electrons become trapped, forming a stationary two‑hole phase‑space configuration.

To probe the stability of this configuration, a very low‑amplitude sinusoidal density perturbation (amplitude A = 10⁻¹², wavelength equal to the simulation box length) is introduced at t = 0. The perturbation triggers an electrostatic instability that drives the two vortices to merge into a single coherent hole located near the wave phase velocity v ≈ 1 (in units of the electron thermal speed). The authors perform a series of high‑resolution simulations (Nx = 8192, Nv = 8001, Hermite truncation M = 800) using a third‑order upwind finite‑volume scheme for the Vlasov advection and a Dougherty operator to model binary collisions. The collision frequency ν is varied over several orders of magnitude (ν = 0, 10⁻⁵, 10⁻⁴, …) to assess its impact on both the growth rate γ of the instability and the onset time t₀.

Key findings are:

1. Growth Rate Independence from Collisions – By measuring the exponential growth of the electric field energy, the authors find that γ is essentially constant across all ν values examined (γ ≈ 0.018 ωₚ). This agrees with Landau’s theory for weakly inhomogeneous equilibria, which predicts that the growth rate depends only on the slope of the distribution function ∂f/∂v evaluated at the resonant velocity, not on collisional damping.

2. Delay of Instability Onset by Collisions – The time at which the vortex merging occurs (t₀) shifts to later times as ν increases. In the collisionless case (ν = 0) merging takes place around t ≈ 3750 ωₚ⁻¹, whereas for ν = 10⁻⁵ the same event is delayed to t ≈ 4500 ωₚ⁻¹. The authors interpret this as a consequence of collisional diffusion in velocity space, which smooths the fine‑scale structures that are essential for the trapped‑particle driven instability.

3. Hermite Spectral Analysis of the Saturated State – After the instability saturates (t ≈ 10⁴ ωₚ⁻¹), the electron distribution function is decomposed into Hermite modes. In the collisionless runs the Hermite spectrum follows a power‑law |fₘ|² ∝ m⁻⁴⁄³ over an intermediate range of mode numbers, indicating a constant enstrophy (or free‑energy) flux from large to small velocity‑space scales, as predicted by kinetic turbulence theories (Schekochihin et al., Servidio et al.). When collisions are strong enough, a clear cutoff appears at high m, and the spectrum decays exponentially, reflecting the suppression of fine velocity‑space structures by the collisional operator.

4. Numerical Robustness – Convergence tests varying Nx, Nv, and M confirm that the observed growth rates and onset times are not artifacts of grid resolution. Energy conservation is maintained within ΔE/E < 10⁻⁴ throughout all runs.

The study provides several important insights for plasma physics. First, it demonstrates that BGK equilibria can become unstable even when only a single hole is present, extending earlier work that focused on multi‑hole configurations. Second, it clarifies the distinct roles of the distribution‑function slope (governing γ) and collisional diffusion (governing t₀) in the dynamics of trapped‑particle instabilities. Third, the Hermite‑space analysis offers a quantitative diagnostic of kinetic turbulence, showing how an enstrophy cascade operates in a weakly collisional plasma and how collisions truncate the cascade.

These results are directly relevant to space‑plasma environments (e.g., auroral electron holes, shock‑driven EAWs) and laboratory settings (e.g., Penning traps, laser‑produced plasmas) where BGK‑type structures coexist with finite collisionality. The methodology—high‑order Vlasov solvers combined with Hermite diagnostics—provides a framework for future studies that could incorporate multi‑dimensional effects, magnetic fields, or more realistic collision operators, and could be benchmarked against spacecraft observations or laboratory measurements.