A solvable model of Vlasov-kinetic plasma turbulence in Fourier-Hermite phase space

TL;DR

This paper introduces an exactly solvable kinetic model of plasma turbulence in Fourier-Hermite space, revealing that stochastic electric fields suppress Landau damping and alter energy transfer dynamics.

Contribution

It presents a novel solvable model of Vlasov-kinetic plasma turbulence that captures the competition between phase mixing and nonlinear effects in phase space.

Findings

Suppression of free-energy flux from low to high Hermite moments.

Derivation of the full Fourier-Hermite spectrum with specific asymptotics.

Identification of the stochastic electric field's role in scaling and dissipation processes.

Abstract

A class of simple kinetic systems is considered, described by the 1D Vlasov-Landau equation with Poisson or Boltzmann electrostatic response and an energy source. Assuming a stochastic electric field, a solvable model is constructed for the phase-space turbulence of the particle distribution. The model is a kinetic analog of the Kraichnan-Batchelor model of chaotic advection. The solution of the model is found in Fourier-Hermite space and shows that the free-energy flux from low to high Hermite moments is suppressed, with phase mixing cancelled on average by anti-phase-mixing (stochastic plasma echo). This implies that Landau damping is an ineffective route to dissipation (i.e., to thermalisation of electric energy via velocity space). The full Fourier-Hermite spectrum is derived. Its asymptotics are $m^{-3/2}$ at low wave numbers and high Hermite moments ($m$) and $m^{-1/2}k^{-2}$ at low Hermite moments and high wave numbers ($k$). These conclusions hold at wave numbers below a certain cut off (analog of Kolmogorov scale), which increases with the amplitude of the stochastic electric field and scales as inverse square of the collision rate. The energy distribution and flows in phase space are a simple and, therefore, useful example of competition between phase mixing and nonlinear dynamics in kinetic turbulence, reminiscent of more realistic but more complicated multi-dimensional systems that have not so far been amenable to complete analytical solution.