Comparison of solutions to bi‐Maxwellian and Maxwellian transport equations for subsonic flows

Abstract

A systematic comparison has been conducted of the solutions to the bi‐Maxwellian‐based 16‐moment and the Maxwellian‐based 13‐moment transport equations for conditions corresponding to the steady state subsonic flow of a fully ionized electron‐proton plasma in the terrestrial ionosphere. The bi‐Maxwellian‐based equations can account for large temperature anisotropies and the flow of both parallel and perpendicular thermal energy, while the Maxwellian‐based equations account for small temperature anisotropies and only a total heat flow. The comparison was conducted for a range of lower boundary temperatures (2000–10,000 K) and temperature gradients (1‐4 K km−1) over the altitude range 1500–13,000 km. These boundary parameters are the most important in the subsonic flow regime since they produce different temperature anisotropies and heat flows, which are handled differently in the two formulations. The systematic comparison has led to the following conclusions: (1) For “low” boundary temperatures (T ∼ 2000 K) and low boundary temperature gradients (▽T ≲ 2 K km−1), the differences between the bi‐Maxwellian‐based 16‐moment and Maxwellian‐based 13‐moment solutions are negligibly small, (2) For intermediate boundary temperatures and temperature gradients, both the 16‐moment and the 13‐moment formulations predict electron and proton temperature anisotropies with perpendicular temperatures (T⊥) greater than parallel temperatures (T∥). However, there are significant differences with regard to the magnitude of the anisotropies predicted by the two formulations, (3) For high boundary temperatures (T≳ 8000 K) and high boundary temperature gradients (▽T∼4 K km−1), the two formulations predict different thermal structures for both the electrons and the protons. Specifically, the bi‐Maxwellian‐based equations predict temperature anisotropies with T∥ > T⊥ at high altitudes, whereas the Maxwellian‐based equations predict that T⊥ > T∥ at all altitudes, (4) The electron total heat flow profiles predicted by the two formulations are virtually identical for all cases considered, and the proton total heat flow profiles are similar, (5) In the 16‐moment formulation, the flow of parallel energy is almost always greater than the flow of perpendicular energy for both the electrons and the protons, (6) The 16‐moment solutions are very sensitive to small changes in the boundary temperature and heat flow values.