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

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.

35 pages, minor edits, final version accepted by JPP

References in corpus (6)

• On magnetic field generation in Kolmogorov turbulence
• Gyrokinetic turbulence: a nonlinear route to dissipation through phase space
• Magnetosperic Multiscale (MMS) observation of plasma velocity-space cascade: Hermite representation and theory
• Diagnosing collisionless energy transfer using field-particle correlations: gyrokinetic turbulence
• Fourier-Hermite spectral representation for the Vlasov-Poisson system in the weakly collisional limit
• Kinetic intermittency in magnetized plasma turbulence

Cited by in corpus (16)

• Reconciling cosmic-ray transport theory with phenomenological models motivated by Milky-Way data
• Kinetic plasma turbulence: recent insights and open questions from 3D3V simulations
• On stochastic heating and its phase-space signatures in low-$\beta$ kinetic turbulence
• Ion versus electron heating in compressively driven astrophysical gyrokinetic turbulence
• Conservative discontinuous Galerkin scheme of a gyro-averaged Dougherty collision operator
• Collisionless relaxation of a Lynden-Bell plasma
• Electromagnetic instabilities and plasma turbulence driven by electron-temperature gradient
• Phase-space entropy cascade and irreversibility of stochastic heating in nearly collisionless plasma turbulence
• Space-filter techniques for quasi-neutral hybrid-kinetic models
• A quantum algorithm for the linear Vlasov equation with collisions
• Gravitational Turbulence: the Small-Scale Limit of the Cold-Dark-Matter Power Spectrum
• Four-dimensional equations for the study of electromagnetic plasma turbulence in a drift kinetic limit
• Dimensional measures of generalized entropy
• Scale invariance and critical balance in electrostatic drift-kinetic turbulence
• Machine-learning Closure for Vlasov-Poisson Dynamics in Fourier-Hermite Space
• Turbulent heating in collisionless low-beta plasmas: imbalance, Landau damping, and electron-ion energy partition