Monte Carlo study of a generalized Coulomb Milne problem

Because of its relevance to the I-M coupling, we investigated the diffusion of a minor ion species through a non-uniform background major ion species. Fokker-Planck expression was used to represent the coulomb ion-ion collisions. 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 tilde {z}>0 and zero density in the regiontilde {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 1/16, which were relevant to H⁺ and O⁺ ions, the two most dominant ions in the terrestrial ionosphere. The minor ion velocity distribution (f_s) and the velocity moments (density ( n_s); drift velocity (tilde {u}_s ), parallel (tilde {T}_{s| |} ) and perpendicular (tilde {T}_sbot ) temperatures; and parallel (tilde {q}_s^{| |} ) and perpendicular (tilde {q}_s^bot ) heat fluxes) were computed. The following were the main conclusions of the study. (1) For the cases of (γ =16, 1), the distribution was close to Maxwellian at low altitudes due to 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 TRIAL RESTRICTION for these two cases. (2) The parallel temperature was reduced (TRIAL RESTRICTION quad < 1) due to the missing down-going ions effect. (3) Deep into the collision-dominated region, n_s was found to be linearly dependent on the normalized distance TRIAL RESTRICTION with a gradient (m = d n/dTRIAL RESTRICTION). As γ decreased from 16 to 1 to 1/16, m decreased from 2.0 to 1.7 to 0.75, respectively. (4) The case of (γ =1/16) exhibited some qualitative differences in contrast to the other two cases. For example, a double-humped f_s formed and a corresponding rapid decrease in TRIAL RESTRICTION and TRIAL RESTRICTION occurred near the transition region.