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.

Fluidization of collisionless plasma turbulence

“…Here, we explore for the first time the possibility of stochastic echoes in a plasma with moving field lines (b = 0). As described above, the compressive fluctuations couple to the Alfvénic fluctuations in a nontrivial way, as the field lines along which the particles stream are being advected by the same flows as the particles' distribution function g. Although there is as yet no complete theory for this problem, our numerical results suggest that elements of the electrostatic theory [23,61] are germane. This generalization opens the way to understanding kinetic turbulence in heliospheric (solarwind) and similarly collisionless astrophysical plasmas, which are usually well into the electromagnetic regime.…”

“…The most familiar example of this phase unmixing is the textbook phenomenon of plasma echo [47,48]. The idea of a stream of stochastic echoes produced by a sequence of nonlinear interactions has been the object of several recent studies of electrostatic (b = 0) plasma turbulence [23,24,60,61]. Here, we explore for the first time the possibility of stochastic echoes in a plasma with moving field lines (b = 0).…”

“…For electrostatic kinetic turbulence, where b = 0, it is possible to decompose gm explicitly into phase-mixing and anti-phase-mixing components and thus calculate directly the "forward" and "backward" free-energy fluxes in Hermite space[23,61]. However, this decomposition involves the sign of k , which in our case would have to be calculated with respect to the perturbed, turbulent magnetic field.…”

Conservative discontinuous Galerkin scheme of a gyro-averaged Dougherty collision operator

“…using Gaussian quadrature) and use that as an initial condition, the system will not be static and will evolve some. A Maxwellian is equivalent to the (properly normalized) Gaussian-weighted zeroth Hermite basis [36]. This is also an eigenfunction of C[ f ], as we will show in section 5.2.…”

Thermal disequilibration of ions and electrons by collisionless plasma turbulence

“…We do not have a quantitative theory that would explain why Qnormali/Qnormale should saturate at the value that we observe numerically (which, based on a resolution study, appears to be converged). Presumably, this is decided by the details of the operation of ion Landau damping in a turbulent environment [a tricky subject (44–46)] and by the efficiency with which energy can be channeled from the MHD scales into the magnetic cascade below ρ* and the KAW cascade below ρnormali. In the absence of a definitive theory, Qnormali/Qnormale≈30 should be viewed as an “experimental” result.…”