We present a detailed guide to advanced collisionless fluid models that incorporate kinetic effects into the fluid framework, and that are much closer to the collisionless kinetic description than traditional magnetohydrodynamics. Such fluid models are directly applicable to modelling the turbulent evolution of a vast array of astrophysical plasmas, such as the solar corona and the solar wind, the interstellar medium, as well as accretion disks and galaxy clusters. The text can be viewed as a detailed guide to Landau fluid models and it is divided into two parts. Part 1 is dedicated to fluid models that are obtained by closing the fluid hierarchy with simple (nonLandau fluid) closures. Part 2 is dedicated to Landau fluid closures. Here in Part 1, we discuss the fluid model of Chew–Goldberger–Low (CGL) in great detail, together with fluid models that contain dispersive effects introduced by the Hall term and by the finite Larmor radius corrections to the pressure tensor. We consider dispersive effects introduced by the nongyrotropic heat flux vectors. We investigate the parallel and oblique firehose instability, and show that the nongyrotropic heat flux strongly influences the maximum growth rate of these instabilities. Furthermore, we discuss fluid models that contain evolution equations for the gyrotropic heat flux fluctuations and that are closed at the fourthmoment level by prescribing a specific form for the distribution function. For the biMaxwellian distribution, such a closure is known as the ‘normal’ closure. We also discuss a fluid closure for the bikappa distribution. Finally, by considering onedimensional Maxwellian fluid closures at higherorder moments, we show that such fluid models are always unstable. The last possible non Landau fluid closure is therefore the ‘normal’ closure, and beyond the fourthorder moment, Landau fluid closures are required.
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An introductory guide to fluid models with anisotropic temperatures. Part 2. Kinetic theory, Padé approximants and Landau fluid closures
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 longwavelength lowfrequency limit. We start with a brief introduction to kinetic theory, where we discuss in detail the plasma dispersion function $Z(\unicode[STIX]{x1D701})$ , and the associated plasma response function $R(\unicode[STIX]{x1D701})=1+\unicode[STIX]{x1D701}Z(\unicode[STIX]{x1D701})=Z^{\prime }(\unicode[STIX]{x1D701})/2$ . We then consider a onedimensional (1D) (electrostatic) geometry and make a significant effort to map all possible Landau fluid closures that can be constructed at the fourthorder 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 ionacoustic mode, and the electron Landau damping of the Langmuir mode. We proceed by considering 1D closures at higherorder moments than the fourth 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 (longwavelength lowfrequency) limit and map all closures that are available at the fourthorder moment level. In appendix A, we provide comprehensive tables with Padé approximants of $R(\unicode[STIX]{x1D701})$ up to the eighthpole order, with many given in an analytic form.
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 NSFPAR ID:
 10143308
 Date Published:
 Journal Name:
 Journal of Plasma Physics
 Volume:
 85
 Issue:
 6
 ISSN:
 00223778
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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