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 Dα0p. Because the cross terms complicate the associated time-dependent diffusion equation, and because Dα0p has a somewhat unfamiliar character (e.g., it may be negative), it has frequently been omitted in numerical simulations of multidimensional diffusion. Generally, Dα0p becomes increasingly important for small values of fpe/fce, as does Dpp, 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.