An introductory guide to fluid models with anisotropic temperatures. Part 2. Kinetic theory, Padé approximants and Landau fluid closures

Abstract

In Part 2 of our guide to collisionless fluid models, we concentrate on Landau fluid closures. These closures were pioneered by Hammett and Perkins and allow for the rigorous incorporation of collisionless Landau damping into a fluid framework. It is Landau damping that sharply separates traditional fluid models and collisionless kinetic theory, and is the main reason why the usual fluid models do not converge to the kinetic description, even in the long-wavelength low-frequency limit. We start with a brief introduction to kinetic theory, where we discuss in detail the plasma dispersion function Z(ζ), and the associated plasma response function R(ζ) = 1 + ζZ(ζ) = −Z ′ (ζ)/2. We then consider a 1D (electrostatic) geometry and make a significant effort to map all possible Landau fluid closures that can be constructed at the 4th-order moment level. These closures for parallel moments have general validity from the largest astrophysical scales down to the Debye length, and we verify their validity by considering examples of the (proton and electron) Landau damping of the ion-acoustic mode, and the electron Landau damping of the Langmuir mode. We proceed by considering 1D closures at higher-order moments than the 4th-order, and as was concluded in Part 1, this is not possible without Landau fluid closures. We show that it is possible to reproduce linear Landau damping in the fluid framework to any desired precision, thus showing the convergence of the fluid and collisionless kinetic descriptions. We then consider a 3D (electromagnetic) geometry in the gyrotropic (long-wavelength low-frequency) limit and map all closures that are available at the 4th-order moment level. In the Appendix A, we provide comprehensive tables with Padé approximants of R(ζ) up to the 8th-pole order, with many given in an analytic form.

A Hamiltonian gyrofluid model based on a quasi-static closure

“…This latter dispersion relation involves the plasma response function R(ζ s ) of the particles of species s, which is related to the corresponding plasma dispersion function Z(ζ s ) by R(ζ s ) = 1 + ζ s Z(ζ s ) with ζ s = ω/(2 1/2 |k z |v th s ). Different Padé approximants R ij (ζ e ) (for which we follow the notations of Hunana et al 2019) are used to estimate the electron response function.…”

“…Different Padé approximants (for which we follow the notations of Hunana et al. 2019) are used to estimate the electron response function.…”

“…This coefficient, of course, follows directly from taking the second order moment, in Hermite polynomials for the normalized parallel velocity, of the gyrokinetic equation (2.1), as discussed in appendix B. However, as pointed out by Keramidas Charidakos et al (2015), in the presence of background magnetic curvature such coefficient, in general, has to be adjusted in order to obtain a Hamiltonian structure. Finally, we consider the term on the right-hand side of (3.26), associated with the parallel heat flux.…”

Editorial: Adaptive Kinetic-Fluid Models for Plasma Simulations on Modern Computer Systems

“…Lautenbach and Grauer describe a hierarchical treatment of collisionless magnetized plasmas combining the Vlasov treatment for ions and electrons at the fine level with a 10-moment fluid model at the coarse level for the Geospace Environmental Modeling of magnetic reconnection. We wish to add that new closures for the kinetic-fluid simulations of collisionless plasma have been recently proposed in Hunana et al [8].…”

Model order reduction approach to the one-dimensional collisionless closure problem

“…Several methods have been suggested in the literature 10 . Asymptotic methods use a Taylor 2,11 or Padé 12,13 expansion of the kinetic response function kin for 1 and 1 to constrain the free parameters. As such, they explicitly choose an asymptotic frequency range, and are limited to it.…”