Analysis of the Hermite spectrum in plasma turbulence

The properties of the Hermite spectrum associated with the linear drift-kinetic equation—as used in studies of gyrokinetic turbulence—are examined. A rigorous uniform asymptotic expression is derived for the steady-state spectrum with a Lenard-Bernstein collision operator. It is found that the spectrum is partitioned into three regions whose boundaries are determined by the ratio of the collision frequency ν to the parallel transit frequency kvth. In the regime of small Hermite index, n, with n ≲ (ν/kvth)2/3, collisions play no role, and the free energy decays like n−1/2 due to phase mixing. In the previously unexplored large-n regime, n≥(ν/kvth)2, collisions are dominant, and the decay of the free energy spectrum is extremely steep, falling off like (n/e)−n. Most of the free energy is dissipated in the intermediate regime, (ν/kvth)2/3 ≲ n≪(ν/kvth)2, where the asymptotic spectrum is in close agreement with the exponentially decaying “continuization” estimate. Our analysis shows that collisions act as a singular perturbation, giving rise to the intermediate regime, where collisions are significantly altering the spectrum well inside the general large-n asymptotic region.