Phase-space turbulence in electrostatic plasmas

We propose a theory of Vlasov-Poisson kinetic plasma turbulence in which the cascaded invariant is not energy, but rather the generalized (negative) entropy C₂ ∝ ∬dxdv f². As particles ballistically stream and get accelerated by turbulent electric fields, the particle distribution function stretches and folds in both position and velocity space, and C₂ cascades from large (injection) to small (collisional) phase-space scales. The phase-space eddy turnover time is determined by the ‘critical balance’ between the time scales of linear phase mixing and nonlinear mixing by the electric field. We derive scalings for the wavenumber spectrum of C₂ in phase space. The only electric field spectrum dimensionally consistent with a constant-flux, critically-balanced cascade is one that is ∝ k^-(d+3), where d is the dimension. This spectrum is sufficiently steep that the phase-space mixing is dominated by the largest scales of the electric field, and so the C₂ cascade is analogous to Batchelor turbulence in fluid passive scalars. The mixing is efficient—collisional dissipation is activated on a time scale that scales logarithmically with the collision frequency. The effect of the δf fluctuations (small scales) on the equilibrium f₀ (large scales) is stochastic heating, viz., non-resonant energization of particles accelerated by the chaotic electric fields. We verify this theory using direct numerical simulations of a forced, 1D-1V plasma and find good agreement. We discuss the implications of our work on the turbulent dissipation and relaxation of particle distribution functions in nearly collisionless plasmas.