The coupling of quasi-linear pitch angle and energy diffusion

Wave–particle interactions are described in the quasi-linear formulation by bounce-averaged diffusion coefficients for equatorial pitch angle α₀ and for momentum p, along with a mixed or cross diffusion coefficient D_α0p. Because the cross terms complicate the associated time-dependent diffusion equation, and because D_α0p has a somewhat unfamiliar character (e.g., it may be negative), it has frequently been omitted in numerical simulations of multidimensional diffusion. Generally, D_α0p becomes increasingly important for small values of f_pe/f_ce, as does D_pp, and for small widths of the wave frequency and wavenormal angle distributions. Here we use very simple models of diffusion to investigate the effect of the cross terms, and to demonstrate numerical problems associated with them.