The transport of charged particles in a flowing medium

The propagating source method for solving the time-dependent Boltzmann equation describing particle propagation in a magnetically turbulent medium is extended to a more realistic case that includes focusing and adiabatic deceleration. The solutions correspond to beam propagation in the solar wind. Pitch-angle scattering away from 90degrees is described by standard quasi-linear theory (QLT), while scattering through 90degrees is approximated by a BGK operator representing a slow mirroring process. The detailed numerical technique for solving the Fokker-Planck equation for two particular spectra is presented. Comparisons are made between our modified QLT (MQLT) model and a BGK model, between highly anisotropic scattering and moderately anisotropic scattering, and between fast particles and slow particles. It is shown that: (1) for moderately anisotropic pitch-angle scattering, the initial ring-beam distribution finally evolves into a broad Gaussian distribution and the QLT isotropic and MQLT anisotropic models could be rather well approximated by the simple relaxation time operator. (2) For highly anisotropic pitch-angle scattering, a moving pulse with a spatially extended at tail is formed, and there exist some differences between the MQLT and BGK models. Specifically, at a particular pitch angle, the spatial distribution from MQLT model occupies a much wider region than that in the BGK model. (3) In the highly anisotropic scattering medium, more particles are cooled by adiabatic deceleration, some particles move a little faster, and the spatial distribution at a specific pitch angle is much more dispersed than that in the case of moderately anisotropic scattering. (4) Compared with the BGK model, the anisotropy persists for a little longer and some particles move a little slower; consequently, intensity profiles have a greater amplitude at later times in the MQLT model. (5) Finally, fast and slow particles have similar distribution characteristics, except that convection is much more important for slow particles.