The propagation of charged particles in a focussing magnetic field with random components

Boltzmann's equation which describes the evolution of the particle distribution function in a focussing magnetic field with finite helicity, is solved by expanding the distribution function in terms of orthogonal focussing eigenfunctions. The present work advances upon previous work by carrying the expansion of the particle distribution function to a higher order (Ν=7 compared to Ν=3), the adding of magnetic helicity, and the injection of a delta function instead of a Gaussian as initial distribution. Results from this model compare very well with those from other known numerical models, provided that a time constraint, which is a direct consequence of the truncation of the eigenfunction expansion, is satisfied. This model, which gives the solution of Boltzmann's equation to a very high degree of accuracy, is used to evaluate the densities predicted by three lower order (and hence easier to implement) models. Two of these models, where the anisotropic component of the distribution function is approximated to first and second order respectively, follow from the Born approximation technique, while the third follows from a truncated eigenfunction expansion of the particle distribution function. It is shown that the latter two models, which include the effect of the dispersion of the so-called coherent pulses, give a better description of the isotropic density than the model which ignores the effect. The main use of this dispersionless model is that it provides a zeroth order approximation to the speed of the coherent pulses in the presence of helicity and focussing. When the dispersion in the pulse is small, its speed is shown to be predicted quite well by this simple model. (author). 67 refs