Cascades and Dissipative Anomalies in Nearly Collisionless Plasma Turbulence

We develop first-principles theory of kinetic plasma turbulence governed by the Vlasov-Maxwell-Landau equations in the limit of vanishing collision rates. Following an exact renormalization-group approach pioneered by Onsager, we demonstrate the existence of a "collisionless range" of scales (lengths and velocities) in 1-particle phase space where the ideal Vlasov-Maxwell equations are satisfied in a "coarse-grained sense". Entropy conservation may nevertheless be violated in that range by a "dissipative anomaly" due to nonlinear entropy cascade. We derive "4/5th-law" type expressions for the entropy flux, which allow us to characterize the singularities (structure-function scaling exponents) required for its non-vanishing. Conservation laws of mass, momentum and energy are not afflicted with anomalous transfers in the collisionless limit. In a subsequent limit of small gyroradii, however, anomalous contributions to inertial-range energy balance may appear due both to cascade of bulk energy and to turbulent redistribution of internal energy in phase space. In that same limit the "generalized Ohm's law" derived from the particle momentum balances reduces to an "ideal Ohm's law", but only in a coarse-grained sense that does not imply magnetic flux-freezing and that permits magnetic reconnection at all inertial-range scales. We compare our results with prior theory based on the gyrokinetic (high gyro-frequency) limit, with numerical simulations, and with spacecraft measurements of the solar wind and terrestrial magnetosphere.Comment: Several additions have been made that were requested by the referees of the PRX submission. In particular, discussion previously relegated to Supplemental Materials are now included in the main text as appendice

A Turbulent Constitutive Law for the Two-Dimensional Inverse Energy Cascade

We develop a fundamental approach to a turbulent constitutive law for the 2D inverse cascade, based upon a convergent multi-scale gradient (MSG) expansion. To first order in gradients we find that the turbulent stress generated by small-scale eddies is proportional not to strain but instead to `skew-strain,' i.e. the strain tensor rotated by $45^\circ.$ The skew-strain from a given scale of motion makes no contribution to energy flux across eddies at that scale, so that the inverse cascade cannot be strongly scale-local. We show that this conclusion extends a result of Kraichnan for spectral transfer and is due to absence of vortex-stretching in 2D. This `weakly local' mechanism of inverse cascade requires a relative rotation between the principal directions of strain at different scales and we argue for this using both the dynamical equations of motion and also a heuristic model of `thinning' of small-scale vortices by an imposed large-scale strain. Carrying out our expansion to second-order in gradients, we find two additional terms in the stress that can contribute to energy cascade. The first is a Newtonian stress with an `eddy-viscosity' due to differential strain-rotation, and the second is a tensile stress exerted along vorticity contour-lines. The latter was anticipated by Kraichnan for a very special model situation of small-scale vortex wave-packets in a uniform strain field. We prove a proportionality in 2D between the mean rates of differential strain-rotation and of vorticity-gradient stretching, analogous to a similar relation of Betchov for 3D. According to this result the second-order stresses will also contribute to inverse cascade when, as is plausible, vorticity contour-lines lengthen on average by turbulent advection.Comment: 24 pages, 1 figur