Resolution of a Few Problems in the Application of Quasilinear Theory to Calculating Diffusion Coefficients in Heliophysics

Abstract

A theory of quasilinear diffusion for obliquely propagating electromagnetic waves was developed in the 1960’s and applied in the 1970’s to model scattering of relativistic electrons by a prescribed distribution of waves. In the latter work, a transformation of variables, from wavevector space to the temporal frequency and tangent of the wave normal angle, was used so that simple Gaussian functions of frequency and tangent of the wave normal angle could be multiplied together to define the distribution of wave power, although arbitrary distributions of power in these two variables is also permitted. Finally, in 2005, previous work was consolidated and has been widely used in heliophysics studies that require computation of quasilinear diffusion coefficients. Here it is shown that this transformation is inppropriate when the precise wave vector distribution is known. The correct transformation is derived and used to produce diffusion coefficients that can differ by orders of magnitude from those computed using the inappropriate transformation. The differences are largest when the distribution of wave power extends to wave normal angles near the resonance cone. When the ratio of the plasma frequency to the gyrofrequency is large, only low energies (keV) are affected, but as the ratio decreases higher energies (MeV) also show differences. It is also shown that the derivation from the 1960’s uses a notation that results in the diffusion coefficients depending on the distribution of wave power with respect to the wave azimuthal angle whereas there should be no such dependence.

A New Model of Electron Pitch Angle Distributions and Loss Timescales in the Earth's Radiation Belts

“…While the Cunningham result merits further study, it has yet to be demonstrated that the differences are significant once the diffusion coefficients have been bounce-and driftaveraged and the different wave modes combined. For the reasons above, it is unlikely that the differences demonstrated in Cunningham (2023) would substantially alter the results presented here, but this does need to be tested thoroughly elsewhere.…”

“…These effects can be investigated using 2-and 3dimensional radiation belt models and this will be the subject of further work. Cunningham (2023) suggests that, in certain situations, some of the assumptions underlying the calculation of diffusion coefficients by the PADIE code and similar codes may be improved. At low values of the ratio of the plasma frequency to the gyro frequency, f pe /f ce = 1.5, and at wave-normal angles very close to the resonance cone this can lead to significant differences in the local diffusion coefficients at some energies and pitch angles.…”

“…While these result merit further study, it has yet to be demonstrated that the differences are significant once the diffusion coefficients have been bounce‐ and drift‐averaged and the different wave modes combined. Further, the f pe / f ce used to calculate the chorus diffusion coefficients employed here are typically ≥3, even in the most active conditions, and, unlike Cunningham (2023), these diffusion coefficient calculations reduce the wave power when the wave normal angle approaches the resonance cone to prevent the waves becoming electrostatic, see Horne et al. (2013).…”