Moment-transport equations for wave-particle interactions in the magnetosphere

The particle transport equation which describes radial diffusion combined with quasi-linear diffusion by waves is analytically intractable, and is usually dealt with by fixing a priori either the radial transport term or the wave diffusion term. However, by taking moments of the transport equation it is possible to estimate the wave intensity and wave diffusion coefficient in a simple way in terms of the particle transport and loss processes. Substituting these estimates in the moment equations, one derives a series of radial transport equations (in general, non-linear) for moments of the distribution function: the number density, the average energy, and the average perpendicular energy.