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Register a new userAn introductory guide to fluid models with anisotropic temperatures Part 2 -- Kinetic theory, Pade approximants and Landau fluid closures
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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 Pade approximants of $R(zeta)$ up to the 8th-pole order, with many given in an analytic form.
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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 modeling 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 (non Landau fluid) closures. Part 2 is dedicated to Landau fluid closures. Here in Part 1, we discuss the CGL fluid model in great detail, together with fluid models that contain dispersive effects introduced by the Hall term and by the finite Larmor radius (FLR) corrections to the pressure tensor. We consider dispersive effects introduced by the non-gyrotropic heat flux vectors. We investigate the parallel and oblique firehose instability, and show that the non-gyrotropic 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 4th-moment level by prescribing a specific form for the distribution function. For the bi-Maxwellian distribution, such a closure is known as the normal closure. We also discuss a fluid closure for the bi-kappa distribution. Finally, by considering one-dimensional Maxwellian fluid closures at higher-order 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 4th-order moment, Landau fluid closures are required.
Magnetic reconnection (MR) plays a fundamental role in plasma dynamics under many different conditions, from space and astrophysical environments to laboratory devices. High-resolution in-situ measurements from space missions allow to study naturally occurring MR processes in great detail. Alongside direct measurements, numerical simulations play a key role in investigating the fundamental physics underlying MR. The choice of an adequate plasma model to be employed in numerical simulations, while also compromising with their computational cost, is crucial to efficiently address the problem. We consider a new plasma model that includes a refined electron response within the hybrid-kinetic framework (kinetic ions, fluid electrons). The extent to which this new model can reproduce a full-kinetic description of 2D MR, with particular focus on its robustness during the non-linear stage, is evaluated. We perform 2D simulations of MR with moderate guide field by means of three different plasma models: a hybrid-Vlasov-Maxwell model with isotropic, isothermal electrons, a hybrid-Vlasov-Landau-fluid (HVLF) model where an anisotropic electron fluid is equipped with a Landau-fluid closure, and a full-kinetic one. When compared to the full-kinetic case, the HVLF model effectively reproduces the main features of MR, as well as several aspects of the associated electron micro-physics and its feedback onto proton dynamics. This includes the global evolution of MR and the local physics occurring within the so-called electron-diffusion region, as well as the evolution of species pressure anisotropy. In particular, anisotropy driven instabilities (such as firehose, mirror, and cyclotron instabilities) play a relevant role in regulating electrons anisotropy during the non-linear stage of MR. As expected, the HVLF model captures all these features, except for the electron-cyclotron instability.
Incorporation of kinetic effects such as Landau damping into a fluid framework was pioneered by Hammett and Perkins PRL 1990, by obtaining closures of the fluid hierarchy, where the gyrotropic heat flux fluctuations or the deviation of the 4th-order gyrotropic fluid moment, are expressed through lower-order fluid moments. To obtain a closure of a fluid model expanded around a bi-Maxwellian distribution function, the usual plasma dispersion function $Z(zeta)$ that appears in kinetic theory or the associated plasma response function $R(zeta)=1 + zeta Z(zeta)$, have to be approximated with a suitable Pade approximant in such a way, that the closure is valid for all $zeta$ values. Such closures are rare, and the original closures of Hammett and Perkins are often employed. Here we present a complete mapping of all plausible Landau fluid closures that can be constructed at the level of 4th-order moments in the gyrotropic limit and we identify the most precise closures. Furthermore, by considering 1D closures at higher-order moments, we show that it is possible to reproduce linear Landau damping in the fluid framework to any desired precision, thus showing convergence of the fluid and collisionless kinetic descriptions.
We present the results of deriving the Israel-Stewart equations of relativistic dissipative fluid dynamics from kinetic theory via Grads 14-moment expansion. Working consistently to second order in the Knudsen number, these equations contain several new terms which are absent in previous treatments.
The kinetic study of plasma sheaths is critical, among other things, to understand the deposition of heat on walls, the effect of sputtering, and contamination of the plasma with detrimental impurities. The plasma sheath also provides a boundary condition and can often have a significant global impact on the bulk plasma. In this paper, kinetic studies of classical sheaths are performed with the continuum code, Gkeyll, that directly solves the Vlasov-Poisson/Maxwell equations. The code uses a novel version of the finite-element discontinuous Galerkin (DG) scheme that conserves energy in the continuous-time limit. The electrostatic field is computed using the Poisson equation. Ionization and scattering collisions are included, however, surface effects are neglected. The aim of this work is to introduce the continuum-kinetic method and compare its results to those obtained from an already established finite-volume multi-fluid model also implemented in Gkeyll. Novel boundary conditions on the fluids allow the sheath to form without specifying wall fluxes, so the fluids and fields adjust self-consistently at the wall. The work presented here demonstrates that the kinetic and fluid results are in agreement for the momentum flux, showing that in certain regimes, a multi-fluid model can be a useful approximation for simulating the plasma boundary. There are differences in the electrostatic potential between the fluid and kinetic results. Further, the direct solutions of the distribution function presented here highlight the non-Maxwellian distribution of electrons in the sheath, emphasizing the need for a kinetic model.
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