The macroscopic dynamics of a kinetic equation involving a model wave-particle collision operator of plasma physics is investigated. The Chapman-Enskog asymptotics is first considered in the framework of a hydrodynamic scaling. The obtained macroscopic model still involves a kinetic variable, the particle energy in the rest frame of the fluid, but shares similarities with the compressible Navier-Stokes equation of gas dynamics. Then a diffusive scaling is examined under the hypothesis of small perturbations of a global equilibrium. In this case, the macroscopic model couples the usual incompressible Navier-Stokes with a diffusion equation for the energy distribution function of the particles, constrained by an extended version of the Boussinesq relation. In both cases, the effect of a Lorentz force term is developed, in the perspective of plasma physical modelling.

On the macroscopic dynamics induced by a model wave-particle collision operator

A global existence theorem is presented for a kinetic problem of the form ∂
 t
 
f+v·∇
 x
 
f=Q(f), f(t=0)=f
0, where Q(f) is a simplified model wave–particle collision operator extracted from quasilinear plasma physics. Evaluation of Q(f) requires the computation of the mean velocity of the distribution f. Therefore, the assumptions on the data are such that vacuum regions, where the mean velocity is not well defined, are excluded. Also the initial data are assumed to have bounded total energy. As additional results conservation laws for mass, momentum, and energy are derived, as well as an entropy dissipation law and the propagation of higher order moments.

Existence of Solutions of a Kinetic Equation Modeling Cometary Flows

This work is concerned with an extension of the classical compressible Euler model of fluid dynamics in which the fluid internal energy is a measure-valued quantity. This model can be derived from the hydrodynamic limit of a kinetic model involving a specific class of collision operators. In the present paper, we investigate diffusive corrections of this fluid dynamical model derived from a Chapman–Enskog expansion of the kinetic model, in the case where the collision time depends on the particle energy in the fluid frame. We show that the closure relations for the stress tensor and heat flux vector differ from their expression in the usual Navier–Stokes model. We argue why such a feature could be used as a tool towards an understanding of fluid turbulence from kinetic theory.

On the viscosity and thermal conduction of fluids with multivalued internal energy

Energetic charged particles or cosmic rays are found in space wherever the ambient matter density is small enough to allow them to exist. Their observed kinetic energies vary from just above the local thermal energies to in excess of 1020 eV. The energy spectrum is quite smooth, suggesting that a common mechanism is responsible for particles of all energies. Their arrival directions are distributed evenly in solid angle, with the observed anisotropies being quite small even at energies ∼ 1018 eV. Various scenarios for their acceleration are discussed, including 2nd-order Fermi acceleration, shocks and cosmic-ray viscosity. It is concluded that the most likely mechanism for the acceleration of most energetic particles to high energies is diffusive shock acceleration. This mechanism has the benefit of producing a power-law energy spectrum with a spectral index which is very insensitive to parameters and which is close in magnitude to that observed in a variety of contexts. It is also reasonably fast and efficient. Anomalous cosmic rays in the heliosphere are discussed as one example of the success of the shock-acceleration picture.

Acceleration and Transport of Energetic Charged Particles in Space

A kinetic equation with a relaxation time model for wave-particle collisions is considered. Similarly to the BGK-model of gas dynamics, it involves a projection onto the set of equilibrium distributions, nonlinearly dependent on moments of the distribution function. An earlier existence result is extended to bounded domains with reflecting boundaries and to initial conditions permitting vacuum regions. The long time behaviour is investigated. Convergence on compact time intervals (shifted to infinity) to the set of equilibrium solutions is proven. The set of smooth equilibrium solutions is computed.

Existence and Convergence to Equilibrium of a Kinetic Model for Cometary Flows

We present the fluid equations for typical astrophysical plasmas, for the case of no average magnetic field and nonrelativistic flow speeds, in which particular acceleration occurs. When combined with the particle transport equation presented earlier by Williams, et al., one obtains a fully self-consistent description of particle transport and smooth fluid flow (length scales significantly larger than the mean free path). A presumed scattering law is taken for particles of all energies, and there is a single distribution function as well. This model of the interaction of particle transport, including acceleration, and fluid dynamics is in terms of four unknowns: the fluid velocity vector and the isotropic part of the particle distribution function

J. R. Jokipii

Papers

A Single-Fluid, Self-consistent Formulation of Fluid Dynamics and Particle Transport