By Oliver Allanson (Exeter University)
Quasilinear diffusion theory forms the basis of much of the modelling and interpretation of particle transport and energization due to interactions with electromagnetic waves; at terrestrial and planetary radiation belts; in the solar atmosphere and solar wind; and for the dynamics of cosmic rays.
We present a derivation of weak turbulence and quasilinear diffusion theories in energy and pitch-angle space that differs from the most standard methods of derivation (based upon the Vlasov equation ). We
- start from solutions to the single-particle Lorentz force equation
- expand the relevant equations of motion up to second order in a small parameter (magnitude of magnetic perturbations to background field)
- ensemble average the solutions to obtain the diffusion coefficients.
The approach used in this paper builds upon the work by , in which only pitch-angle dynamics were considered.
The main conclusions and results of this paper are as follows:
- A derivation and discussion of the general Fokker-Planck equation to describe stochastic charged particle dynamics. This equation includes all possible advective and diffusive dynamics, in principle. The form of the drift and diffusion coefficients are then to be determined on a system-by-system basis. We solve for the diffusive dynamics only, and leave investigations of the drift coefficients and drift-diffusion relations for future works
- The weak turbulence diffusion coefficients: i) display an interesting dependency on time (see Figure1); ii) and also explicitly incorporate the effects of non-resonant particles, as well as the standard effects of cyclotron-resonant particles
- We recover the standard form as used in the resonant-diffusion limit of relativistic quasilinear theory , when we consider elapsed timescales much greater than a gyroperiod
- Our new derivation has a number of benefits, including: 1) the relationship between a more general weak turbulence theory and the standard resonant diffusion quasilinear; 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.
See paper for full details: Allanson O, Elsden T, Watt C and Neukirch T (2022) Weak Turbulence and Quasilinear Diffusion for Relativistic Wave-Particle Interactions Via a Markov Approach. Front. Astron. Space Sci. 8:805699. doi: 10.3389/fspas.2021.805699
1: C. F. Kennel and F. Engelmann , "Velocity Space Diffusion from Weak Plasma Turbulence in a Magnetic Field", The Physics of Fluids 9, 2377-2388 (1966)
2: Don S. Lemons , "Pitch angle scattering of relativistic electrons from stationary magnetic waves: Continuous Markov process and quasilinear theory", Physics of Plasmas 19, 012306 (2012)
3: Glauert, S. A., and Horne, R. B. (2005), Calculation of pitch angle and energy diffusion coefficients with the PADIE code, J. Geophys. Res., 110, A04206