Quasilinear theory for inhomogeneous plasma

Abstract

This paper presents quasilinear theory (QLT) for a classical plasma interacting with inhomogeneous turbulence. The particle Hamiltonian is kept general; for example, relativistic, electromagnetic and gravitational effects are subsumed. A Fokker–Planck equation for the dressed ‘oscillation-centre’ distribution is derived from the Klimontovich equation and captures quasilinear diffusion, interaction with the background fields and ponderomotive effects simultaneously. The local diffusion coefficient is manifestly positive-semidefinite. Waves are allowed to be off-shell (i.e. not constrained by a dispersion relation), and a collision integral of the Balescu–Lenard type emerges in a form that is not restricted to any particular Hamiltonian. This operator conserves particles, momentum and energy, and it also satisfies the $\smash {H}$ -theorem, as usual. As a spin-off, a general expression for the spectrum of microscopic fluctuations is derived. For on-shell waves, which satisfy a quasilinear wave-kinetic equation, the theory conserves the momentum and energy of the wave–plasma system. The action of non-resonant waves is also conserved, unlike in the standard version of QLT. Dewar's oscillation-centre QLT of electrostatic turbulence (Phys. Fluids, vol. 16, 1973, p. 1102) is proven formally as a particular case and given a concise formulation. Also discussed as examples are relativistic electromagnetic and gravitational interactions, and QLT for gravitational waves is proposed.

Gauge-invariant gravitational waves in matter beyond linearized gravity

“…The energy-momentum of a linear wave produces nonlinear average forces on the underlying medium, and is itself quadratic in the wave amplitude. Defining it generally requires so-called quasilinear (QL) theory [15][16][17], specifically, QL theory of adiabatic (nonresonant) wave-medium interactions. Such interactions are commonly described using variational methods [18] that originate from Whitham's average-Lagrangian approach [19,20].…”

“…Here, we show how to do this using the tools that we have developed previously in [11,13,17,41,42]. Let us briefly describe this series of papers to put our work into perspective.…”

“…Let us briefly describe this series of papers to put our work into perspective. Our general formalism for studying dispersive GWs interacting with matter is outlined in [11] and, within a broader context, in [17]. The work [13] also specifies some elementary blocks of our general theory, such as the so-called ponderomotive potential and the gravitational susceptibility.…”

Quasilinear theory of Brillouin resonances in rotating magnetized plasmas

“…For an infinite homogeneous magnetized plasma at rest the quasilinear equation is well known (Rax 2011), and has proven particularly useful to evaluate energy absorption and current generation in thermonuclear plasmas (Fisch 1978(Fisch , 1987. The quasilinear equation for a cylindrical unmagnetized plasma at rest has also been derived (Kaufman 1971), whereas a generalized quasilinear theory for inhomogeneous plasma has recently been laid out (Dodin 2022). However, and while as mentioned above it appears to be key to important applications, the quasilinear theory for a rotating magnetized plasma has to our knowledge not been derived yet.…”

Hamiltonian formulations of quasilinear theory for magnetized plasmas

“…The reader is referred to a pedagogical review by Krommes (Krommes, 2002) on the theoretical foundations of plasma turbulence as well as a recent study on the validity of quasilinear theory (Crews and Shumlak, 2022). In addition, the mathematical foundations of quasilinear theory for inhomogeneous plasma can be found in the recent work by Dodin (Dodin, 2022).…”