Dispersion relations in hot magnetized plasmas

Abstract

In the framework of hot magnetized collisionless plasmas, dispersion relations have been extensively studied in the past [2, 11, 12, 22, 31, 32, 35]. This subject is still topical in plasma physics [17, 25, 30, 34, 39]. The aim of this article is to provide a rigorous derivation of the characteristic variety, based on some asymptotic analysis of the relativistic Vlasov-Maxwell system. Special emphasis is made on the modeling of Tokamaks, with spatial variations of the magnetic field and of the equilibrium distribution function. In order to take into account the inhomogeneities, the problem is formulated in terms of geometrical optics [27, 29]. This allows to unify, justify and extend the preceding results. New aspects are indeed included. For instance, the dielectric tensor is defined for real frequencies through singular integrals involving the Hilbert transform.

The relativistic Vlasov–Maxwell equations for strongly magnetized plasmas

“…The cold assumption is often used to model astrophysical plasmas or even plasmas located at the edge of fusion devices. As explained in [9,10], this implies that the distribution function f(·) is concentrated for velocities ξ such that |ξ| ∼ O(ε). This is reflected in Assumption 2.2 by the fact that the function M (·) is compactly supported (in ξ)…”

“…The impact of a strong external magnetic field. In view of a better understanding of what happens in magnetospheres [9] or fusion devices [10], it is important to consider the influence of a strong exterior inhomogeneous magnetic field, denoted by ε −1 B e (x). "Strong" because the parameter ε is small; in practice, the dimensionless number ε stands for the inverse of the electron cyclotron frequency; it is often of size 10 −4 .…”

“…It can also be viewed as a solution to the RVM system associated with the initial data (f s , E s , B s ) |t=0 = ε −2 M (ε −1 |ξ|) , 0 , 0 (2.16) Note that more general stationary solutions could be considered. In [10], a notion of shifted Maxwell-Boltzmann distribution is introduced. This allows to describe plasmas confined inside tokamaks.…”

No Cauchy datum needed for the unique solution in $\mathcal{S}^\prime(\mathbb{R}^{1+2d})$ of the linearized Vlasov-Maxwell system with the limiting absorption principle

“…By inspection of expression (13) we also notice that f ν ∈ S(R 3 ), but the sequence f ν is not uniformly bounded in S.…”

“…The same problem has been considered by Omnes for a bounded plasma [32]. More recently, Cheverry and Fontaine [14,13] have addressed the characteristic variety (or dispersion relation) for the linearized Maxwell-Vlasov system using asymptotic methods, but here we focus on the properties of the plasma constitutive relation as an operator.…”