Analysis of quasi-linear diffusion coefficients

This paper considers the quasi-linear treatment of either electrons or protons cyclotron-resonant with either whistler or ion cyclotron. waves. The dispersion relation used does not yield a simple, explicit expression for the resonant frequency omega. Detailed analysis is made of the conditions determining the resonant frequencies omega for a given wave normal angle theta; there may be up to three values of omega(theta). Criteria are derived which identify the frequencies and indicate when they are outside specified frequency cutoffs. From this analysis, much unnecessary computational effort to evaluate the diffusion coefficients may be avoided. For electron-whistler interactions in the radiation belts, the sensitivity of the local results to the parameters of the wave distribution is considered. It is shown how parameter-independent calculations may be used to analyze the dependence on the parameter values. This approach is extended to the bounce-averaged coefficients, which determine estimates of the pitch angle distribution and corresponding time to pitch angle scatter into the loss cone. Numerical results are obtained using recently reported parameters for whistler waves due to hiss, lightning, and VLF transmitters, as well as perturbations to these values.