Eulerian variational formulations and momentum conservation laws for kinetic plasma systems

TL;DR

This paper develops Eulerian variational principles for kinetic plasma systems, deriving momentum conservation laws and symmetric pressure tensors through coordinate invariance, applicable to Vlasov-Poisson-Ampère and drift kinetic equations.

Contribution

It introduces a unified Eulerian variational framework for plasma kinetic systems that directly yields momentum conservation laws and pressure tensors, improving upon traditional methods.

Findings

Derived momentum conservation laws from coordinate invariance.

Presented Eulerian formulations for extended drift kinetic systems.

Analyzed momentum balances with collision and source terms.

Abstract

The Eulerian variational principle for the Vlasov-Poisson-Amp\`{e}re system of equations in a general coordinate system is presented. The invariance of the action integral under an arbitrary spatial coordinate transformation is used to obtain the momentum conservation law and the symmetric pressure in a more direct way than using the translational and rotational symmetries of the system. Next, the Eulerian variational principle is given for the collisionless drift kinetic equation, where particles' phase-space trajectories in given electromagnetic fields are described by Littlejohn's guiding center equations~[R. G. Littlejohn, J. Plasma Phys.\ {\bf 29}, 111 (1983)]. Then, it is shown that, in comparison with the conventional moment method, the invariance under a general spatial coordinate transformation yields a more convenient way to obtain the momentum balance as a three-dimensional vector equation in which the symmetric pressure tensor, the Lorentz force, and the magnetization current are properly expressed. Furthermore, the Eulerian formulation is presented for the extended drift kinetic system, for which, in addition to the drift kinetic equations for the distribution functions of all particle species, the quasineutrality condition and Amp\`{e}re's law to determine the self-consistent electromagnetic fields are given. Again, the momentum conservation law for the extended system is derived from the invariance under the general spatial coordinate transformation. Besides, the momentum balances are investigated for the cases where the collision and/or external source terms are added into the Vlasov and drift kinetic equations.