EIGENFUNCTION SOLUTION OF BOLTZMANNS EQUATION FOR THE CASE OF A FOCUSING MAGNETIC-FIELD WITH FINITE HELICITY

A seventh-order eigenfunction expansion is used to solve Boltzmann's equation for the case of a focusing magnetic field, with constant focusing length, and with finite magnetic helicity. A time constraint that determines when the solution is valid is derived, and when this constraint is satisfied, comparison with numerical models shows excellent agreement. Results from this model are also compared with solutions which follow from a first- and a third-order expansion of Boltzmann's equation. A time constraint for the third-order expansion is also derived, and it is shown that results for both pulse speeds and densities agree extremely well when this time constraint is satisfied. The modified telegrapher's equation resulting from the first-order expansion for the case of no focusing predicts pulse speeds that are typically larger than the actual pulse speeds, and densities that bear little resemblance to the actual densities for the choice of parameters considered in this paper. Furthermore it is shown that the pulse speeds predicted by both the seventh- and the third-order expansion are time dependent, and that the speeds predicted by the first-order expansion coincide with these speeds only if the dispersion in the pulse is small.