Eur. Phys. J. D (2014) 68: 296
https://doi.org/10.1140/epjd/e2014-50157-8

Regular Article

Irreversible energy flow in forced Vlasov dynamics^*

¹ Max Planck Institute for Plasma Physics, EURATOM Association, 17491 Greifswald, Germany
² Oxford Centre for Collaborative Applied Mathematics, University of Oxford, Oxford OX2 6GG, UK

^a e-mail: gplunk@ipp.mpg.de

Received: 28 February 2014
Received in final form: 1 July 2014
Published online: 14 October 2014

Abstract

The recent paper of Plunk [G.G. Plunk, Phys. Plasmas 20, 032304 (2013)] considered the forced linear Vlasov equation as a model for the quasi-steady state of a single stable plasma wavenumber interacting with a bath of turbulent fluctuations. This approach gives some insight into possible energy flows without solving for nonlinear dynamics. The central result of the present work is that the forced linear Vlasov equation exhibits asymptotically zero (irreversible) dissipation to all orders under a detuning of the forcing frequency and the characteristic frequency associated with particle streaming. We first prove this by direct calculation, tracking energy flow in terms of certain exact conservation laws of the linear (collisionless) Vlasov equation. Then we analyze the steady-state solutions in detail using a weakly collisional Hermite-moment formulation, and compare with numerical solution. This leads to a detailed description of the Hermite energy spectrum, and a proof of no dissipation at all orders, complementing the collisionless Vlasov result.