A unifying comparison of nearly scatter free transport models

Abstract

Gombosi et al. (1993) recently derived a modified telegrapher's equation for charged particle transport under the influence of isotropic scattering. This equation obeys causality and disallows upstream diffusion for particles with random velocities smaller than the bulk flow velocity. The acausal diffusion equation was obtained to lowest order in the expansion of smallness parameters. The paper by Gombosi et al. (1993) prompted responses from Pauls et al. (1993) and Earl (1993). This paper is written to explain the differences between the methods, assumptions, and results of Gombosi et al. (1993), Pauls et al. (1993), and Earl (1993) and presents a new method of obtaining approximate solutions. It is shown that the assumptions used by Gombosi et al. (1993) and Pauls et al. (1993) are physically equivalent. In these papers, a second‐order expansion is made by introducing smallness parameters, not by truncating an eigenfunction series. Earl (1993) estimates the overall behavior of a dispersion relation for the two lowest‐frequency modes by truncating an eigenfunction series and by using empirical approximations motivated by a Monte Carlo simulation. Earl's (1993) approximations are mathematically not equivalent to the smallness parameter expansion introduced by Gombosi et al. (1993). We have developed a new solution method which is both functionally consistent with Earl's (1993) solutions and with the smallness parameter expansion. Because the new solution method involves both causal telegrapher's equation propagation and diffusion, it becomes clear that Earl's (1993) coherent pulse velocity is smaller than the modified telegrapher's coherent velocity because diffusion limits the efficiency of coherent propagation. In our solution method, the solution of the modified telegrapher's equation is obtained as the causal limit of solutions accurate to second order in the smallness parameter expansion. In order to investigate the coherent velocity, we have also developed “wavenumber eigenfunctions” which account for all the pitch angle dependence in our Boltzmann equation. Using truncation, Earl (1993) obtains approximations for the wavenumber dependence of the lowest two frequency modes, which correspond to two of the wavenumber eigenmodes. We find that a consequence of including only two wavenumber eigenmodes is that one obtains solutions which disobey causality at sufficiently short times. Furthermore, the coherent velocity of the two eigenmodes is strongly dependent on wavenumber and approaches the particle velocity in the limit of large wavenumber for both isotropic and anisotropic scattering processes. We conclude that Earl's (1993) solutions and solutions obtained using the new solution method implicitly assume weak acausality and reasonable behavior in the temporal regime, t < 4τ. The solutions are not strictly consistent with the behavior of the lowest two frequency modes but have similar behavior in the regime of low wavenumber.