A UNIFYING COMPARISON OF NEARLY SCATTER FREE TRANSPORT MODELS

Gombosi et al. (1993) recently derived a modified telegrapher's equati on for charged particle transport under the influence of isotropic sca ttering. This equation obeys causality and disallows upstream diffusio n for particles with random velocities smaller than the bulk flow velo city. The acausal diffusion equation was obtained to lowest order in t he expansion of smallness parameters. The paper by Gombosi et al. (199 3) prompted responses from Pauls et al. (1993) and Earl (1993). This p aper is written to explain the differences between the methods, assump tions, and results of Gombosi et al. (1993), Pauls et al. (1993), and Earl (1993) and presents a new method of obtaining approximate solutio ns. It is shown that the assumptions used by Gombosi et al. (1993) and Pauls et al. (1993) are physically equivalent. In these papers, a sec ond-order expansion is made by introducing smallness parameters, not b y truncating an eigenfunction series. Earl (1993) estimates the overal l behavior of a dispersion relation for the two lowest-frequency modes by truncating an eigenfunction series and by using empirical approxim ations motivated by a Monte Carlo simulation. Earl's (1993) approximat ions are mathematically not equivalent to the smallness parameter expa nsion introduced by Gombosi et al. (1993). We have developed a new sol ution 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 propagatio n and diffusion, it becomes clear that Earl's (1993) coherent pulse ve locity is smaller than the modified telegrapher's coherent velocity be cause diffusion limits the efficiency of coherent propagation. In our solution method, the solution of the modified telegrapher's equation i s obtained as the causal limit of solutions accurate to second order i n the smallness parameter expansion. In order to investigate the coher ent velocity, we have also developed ''wavenumber eigenfunctions'' whi ch account for all the pitch angle dependence in our Boltzmann equatio n. Using truncation, Earl (1993) obtains approximations for the wavenu mber dependence of the lowest two frequency modes, which correspond to two of the wavenumber eigenmodes. We find that a consequence of inclu ding only two wavenumber eigenmodes is that one obtains solutions whic h disobey causality at sufficiently short times. Furthermore, the cohe rent velocity of the two eigenmodes is strongly dependent on wavenumbe r and approaches the particle velocity in the limit of large wavenumbe r for both isotropic and anisotropic scattering processes. We conclude that Earl's (1993) solutions and solutions obtained using the new sol ution method implicitly assume weak acausality and reasonable behavior in the temporal regime, t < 4tau. The solutions are not strictly cons istent with the behavior of the lowest two frequency modes but have si milar behavior in the regime of low wavenumber.