Numerical solution of the time-dependent kinetic equation for anisotropic pitch-angle scattering

A new powerful numerical method is developed for solving the time-dependent kinetic equation describing the anisotropic pitch-angle scattering of charged particles. The model includes the effects of adiabatic focusing in a radial magnetic field, adiabatic deceleration, anisotropic pitch-angle scattering, and convection in a magnetized plasma and significantly generalizes a model introduced by in Kota 1994. The pitch-angle scattering is assumed to scatter slowly through 90 degrees. By applying Legendre polynomial expansions to the particle transport equation, an infinite series of first-order differential equations for the harmonics of the distribution function is obtained. By means of a characteristic method (together with operator splitting), the solution of distributions at certain harmonics is computed. Solutions exhibiting coherent pulses are obtained, and these are identical to the exact analytic results obtained by Kota. However, the approach presented here allows for arbitrarily anisotropic initial data to be prescribed, and it can also be used to study the dependence of particle distribution on pitch angle. It is shown that the presence of adiabatic focusing results in highly asymmetric particle propagation in opposite directions with typically more particles in the sunward hemisphere than in the antisunward hemisphere, although two oppositely propagating initial beams are introduced symmetrically. An abrupt transition can be found between hemispheres, and the distribution within each hemisphere is quasi-isotropic. The model and approach discussed here lend themselves to the study of the propagation and transport of charged particles and pickup ions.