Abstract
We derive weak turbulence and quasilinear models for relativistic charged particle dynamics in pitch-angle and energy space, due to interactions with electromagnetic waves propagating (anti-)parallel to a uniform background magnetic field. We use a Markovian approach that starts from the consideration of single particle motion in a prescribed electromagnetic field. This Markovian approach has a number of benefits, including: 1) the evident self-consistent relationship between a more general weak turbulence theory and the standard resonant diffusion quasilinear theory (as is commonly used in e.g. radiation belt and solar wind modeling); 2) the general nature of the Fokker-Planck equation that can be derived without any prior assumptions regarding its form; 3) the clear dependence of the form of the Fokker-Planck equation and the transport coefficients on given specific timescales. The quasilinear diffusion coefficients that we derive are not new in and of themselves, but this concise derivation and discussion of the weak turbulence and quasilinear theories using the Markovian framework is physically very instructive. The results presented herein form fundamental groundwork for future studies that consider phenomena for which some of the assumptions made in this manuscript may be relaxed.
Highlights
The classic derivations of quasilinear theory (Drummond and Pines, 1962; Vedenov et al, 1962; Kennel and Engelmann, 1966; Lerche, 1968; Lyons, 1974; Summers, 2005) provide the form of the Fokker-Planck equation to describe the particle dynamics, and the diffusion coefficients that encode the effect of the resonant wave-particle interactions as a function of the background magnetic field strength, plasma refractive index, and electromagnetic wave spectral properties
These insights are one benefit of using this Markovian approach—and one can conclude that Eq 4 describes a particular subset of a more rich set of possible particle dynamics, that are described by Eq 1
In this paper we have presented new derivations of relativistic weak turbulence and quasilinear diffusion models
Summary
Quasilinear diffusion theory forms the basis of much of the modeling and interpretation of particle transport and energization due to interactions with electromagnetic waves; at terrestrial (Horne et al, 2005; Summers, 2005; Thorne, 2010) and planetary (Woodfield et al, 2014; Kollmann et al, 2018) radiation belts; in the solar atmosphere and solar wind (Steinacker and Miller, 1992; Vocks et al, 2005; Vocks, 2012; Verscharen and Chandran, 2013; Jeong et al, 2020); and for the dynamics of cosmic rays (Schlickeiser, 1989; Mertsch, 2020). Despite the fact that the electromagnetic perturbation considered is a stationary magnetic field only, the equations derived by Lemons (2012) do reproduce the standard form for pitch-angle diffusion by field-aligned propagating electromagnetic waves using the quasilinear theory - for the particular case of pitch-angle diffusion only. This corresponds to the subset of plasma environments in which the plasma frequency is significantly larger than the gyrofrequency (fpe ≫ fce, e.g., see Eq 8 in Summers and Thorne (2003)).
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