Universal fluctuation spectrum of Vlasov-Poisson turbulence

The thermal fluctuation spectrum of the electric field arising due to particle noise in a quiescent Vlasov-Poisson plasma was derived in the 1960s. Here, we derive the universal fluctuation spectrum of the electric field, at Debye and sub-Debye scales, for a turbulent Vlasov-Poisson plasma. This spectrum arises from 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 $C_2$, the quadratic Casimir invariant of the particle distribution function. $C_2$ cascades to small scales in position and velocity space via linear and nonlinear phase mixing, in such a way that the time scales of the two processes are critically balanced at every scale. We construct a scaling theory of the fluctuation spectrum of $C_2$ and of the electric field in wavenumber space. The electric-field spectrum is sufficiently steep for the nonlinear mixing to be controlled by the largest-scale electric fields, and so the $C_2$ cascade resembles the Batchelor cascade of a passive scalar. Our theory is supported by simulations of a forced 1D-1V plasma. We predict that the cascade is terminated at the wavenumber where the turbulent electric-field spectrum gives way to the thermal noise spectrum. The time scale for this small-scale cutoff to be reached is the dynamical time of phase-space mixing times a logarithmic factor in the plasma parameter - this is the first concrete demonstration of this property of Vlasov-Poisson turbulence, akin to how fluid turbulence dissipates energy at a rate independent (or nearly independent) of molecular diffusion. In the presence of the sub-Debye phase-space cascade - a scenario that may be ubiquitous - standard collisional plasma theory ceases to be valid. This calls for the development of new collision operators suited to such turbulent environments.