# An introductory guide to fluid models with anisotropic temperatures Part 2 -- Kinetic theory, Pad\'e approximants and Landau fluid closures

## TL;DR

This paper explores Landau fluid closures in collisionless plasma models, demonstrating their ability to accurately reproduce Landau damping and bridging fluid and kinetic descriptions across various scales.

## Contribution

It systematically maps Landau fluid closures at the 4th-order moment level in 1D and 3D geometries, showing their validity and convergence with kinetic theory.

## Key findings

- Landau closures can reproduce Landau damping with arbitrary precision.
- Closure mappings are valid from astrophysical scales down to the Debye length.
- Padé approximants of the plasma response function are provided up to the 8th order.

## 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 $Z(\zeta)$, and the associated plasma response function $R(\zeta)=1+\zeta Z(\zeta)=-Z'(\zeta)/2$. 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 Pad\'e approximants of $R(\zeta)$ up to the 8th-pole order, with many given in an analytic form.

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Source: https://tomesphere.com/paper/1901.09360