Forced Vlasov-Poisson turbulence

A theory is proposed, and confirmed by numerical simulations, of Vlasov-Poisson kinetic plasma turbulence at Debye and sub-Debye scales. This theory describes what is likely to be the “final cascade”—a universal regime to be encountered at the extreme small-scale end of any turbulent cascade in a nearly collisionless plasma. The cascaded invariant is the generalized (negative) entropy C₂, which is the quadratic Casimir invariant of the distribution function. C₂ cascades to small scales in both position and velocity space via both linear and nonlinear phase mixing, in such a way that the time scales associated with the two processes are “critically balanced” at every scale. We derive the scalings of the wavenumber spectra of C₂ and the electric field. The electric-field spectrum is sufficiently steep for the mixing to be controlled by the largest scales of the electric field, and so the C₂ cascade resembles the Batchelor cascade of a passive scalar field, albeit in the kinetic phase space. Our theory’s conclusions are corroborated by direct numerical simulations of a forced 1D-1V plasma. The phase-space cascade enables the irreversibility of particle heating to be achieved on collisionless time scales. The mean distribution function arising from this heating process is shown to be derivable from quasilinear considerations and is not a Maxwellian.