Eulerian variational formulations and momentum conservation laws for kinetic plasma systems

H. Sugama,T.-H. Watanabe,M. Nunami,S. Satake

doi:10.1063/1.5031155

Abstract

The Eulerian variational principle for the Vlasov-Poisson-Ampère 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. 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 Ampère'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 to the Vlasov and drift kinetic equations.

Highlights

A large number of numerical simulations have been performed to investigate neoclassical and turbulent transport in toroidal plasmas.[1,2,3] As a modern theoretical technique for deriving basic kinetic model equations of such simulations, the variational principle[4,5,6,7] is used because the derived equations possess favorable conservation properties for long-time simulations to pursue evolutions of plasma profiles resulting from transport processes
It is shown for the Vlasov-Poisson-Ampère system that, in the presence of the magnetic field, the canonical momentum conservation law derived from the space translational symmetry contains the asymmetric pressure tensor
When self-consistent electromagnetic fields are treated as the solutions of the equations given simultaneously with the drift kinetic equations from the variational principle, the explicit dependence on the spatial coordinates is removed from the action integral, and the momentum conservation law is derived for the total system consisting of the charged particles and fields [see Sec

Summary

INTRODUCTION

A large number of numerical simulations have been performed to investigate neoclassical and turbulent transport in toroidal plasmas.[1,2,3] As a modern theoretical technique for deriving basic kinetic model equations of such simulations, the variational principle[4,5,6,7] is used because the derived equations possess favorable conservation properties for long-time simulations to pursue evolutions of plasma profiles resulting from transport processes. When self-consistent electromagnetic fields are treated as the solutions of the equations given simultaneously with the drift kinetic equations from the variational principle, the explicit dependence on the spatial coordinates is removed from the action integral, and the momentum conservation law is derived for the total system consisting of the charged particles and fields [see Sec. IV]. The Vlasov-Poisson-Ampère system[27] is considered as an example of kinetic systems, for which the Eulerian variational principle is presented It is shown for this system how to obtain the momentum conservation law from the invariance of the action integral under general coordinate transformations

Eulerian formulation of the variational principle in general coordinates

Quasineutrality and Ampère’s law combined with drift kinetic equations

CONCLUSIONS