A Monte Carlo study of the generalized Coulomb Milne problem

Because of its relevance to space plasma problems (such as the terrestrial polar wind), we investigated the diffusion of a minor ion species through a non-uniform background major ion species. A Fokker-Planck expression was used to represent the Coulomb collisions between the minor and the background ions. A change of variables was implemented in order to transform the problem into a simpler form where the background medium is uniform. This transformed problem described minor ions diffusing through a background of ions with constant density in the semi-infinite region z˜⩾0 and zero density in the region z˜<0. This problem was termed the generalized Coulomb Milne problem and was addressed by a Monte Carlo simulation. Three different minor-to-background mass ratios (γ) were considered, namely γ=16, 1, and 116, which were relevant to H+ and O+ ions, the two most dominant ions in the terrestrial ionosphere. The minor ion velocity distribution (fs) and the velocity moments (density (n˜s); drift velocity (u˜s), parallel (T˜s∥) and perpendicular (T˜s⊥) temperatures; and parallel (q˜s∥) and perpendicular (q˜s⊥) heat fluxes) were computed. For the cases when the minor species mass was comparable to, or larger than the background species mass (γ=16,1), the distribution  was close to Maxwellian at low altitudes due to Coulomb collisions, gradually formed a weak upward tail in the transition region, and eventually assumed a half-Maxwellian shape at the collisionless region. This was reflected in the enhancement of the flow and random energies, and the energy fluxes for these cases. Deep into the collision-dominated region, n˜s was found to be linearly dependent on the normalized distance z˜ with a gradient (m=dn˜/dz˜). As γ decreased from 16 to 1 to 116, m decreased from 2.0 to 1.7 to 0.75, respectively. For the case of a lighter minor ion species drifting through a heavier background ion species (e.g. γ=116), the ion outflow exhibited some interesting qualitatively different features. For example, a double-humped fs formed and a corresponding rapid change in the heat flux occurred near the transition region. The above characteristics can be explained if we remembered that the ions located in the tail of the distribution function were relatively collisionless, and their effects increased as γ decreased. It was also concluded that using a simple Maxwell molecule (instead of the more proper Coulomb) collision model in order to simulate the ion-ion collisions, could produce reasonable results only for the case of (γ⪢1).