Incorporating Kinetic Physics into a Two-Fluid Solar-Wind Model with Temperature Anisotropy and Low-Frequency Alfven-Wave Turbulence

We develop a 1D solar-wind model that includes separate energy equations for the electrons and protons, proton temperature anisotropy, collisional and collisionless heat flux, and an analytical treatment of low-frequency, reflection-driven, Alfven-wave turbulence. To partition the turbulent heating between electron heating, parallel proton heating, and perpendicular proton heating, we employ results from the theories of linear wave damping and nonlinear stochastic heating. We account for mirror and oblique firehose instabilities by increasing the proton pitch-angle scattering rate when the proton temperature anisotropy exceeds the threshold for either instability. We numerically integrate the equations of the model forward in time until a steady state is reached, focusing on two fast-solar-wind-like solutions. These solutions are consistent with a number of observations, supporting the idea that Alfven-wave turbulence plays an important role in the origin of the solar wind.

💡 Research Summary

The paper presents a comprehensive one‑dimensional, two‑fluid solar‑wind model that explicitly treats electrons and protons as separate fluids, incorporates proton temperature anisotropy (parallel and perpendicular temperatures), and includes both collisional and collisionless heat‑flux formulations. Starting from Kulsrud’s collisionless magnetohydrodynamics, the authors add a Coulomb collision operator and take velocity moments to derive a closed set of fluid equations for density, bulk speed, electron temperature, proton parallel and perpendicular temperatures, proton heat fluxes, and the energy density of non‑compressive Alfvén waves (AWs). The magnetic flux tube geometry follows the Kopp‑Holzer expansion, and the magnetic field is assumed radial, allowing the use of a simple 1‑D spatial coordinate (heliocentric distance).

A key innovation is the treatment of low‑frequency, reflection‑driven AW turbulence. The model solves an AW energy equation that includes advection by the bulk flow, propagation at the Alfvén speed, and a nonlinear cascade term. The turbulent dissipation rate Q is partitioned among three channels: electron heating (Qe), parallel proton heating (Q∥p), and perpendicular proton heating (Q⊥p). This partitioning relies on linear wave‑damping theory for the electron channel and on the stochastic (non‑linear) heating theory of Chandran et al. (2010) for the proton channels. The resulting heating fractions are functions of local plasma parameters such as β, the Alfvén Mach number, and the turbulence amplitude.

To prevent unphysical growth of temperature anisotropy, the model incorporates mirror and oblique fire‑hose instability thresholds. When the anisotropy ratio R = T⊥p/T∥p exceeds the mirror threshold Rm or falls below the fire‑hose threshold Rf, a rapid increase in the effective proton pitch‑angle scattering rate νinst is triggered. This is implemented as an exponential function of the distance from the instability thresholds, effectively limiting R to the observed bounds in the solar wind.

Electron heat flux is modeled with a hybrid approach: near the Sun (where the electron mean free path is short) the Spitzer collisional conductivity is used, while farther out a collisionless, free‑streaming form (Hollweg) is adopted. A smooth transition between the two regimes is achieved with a weighting function ψ(r) that depends on the ratio r/rH.

The authors numerically integrate the coupled equations forward in time until a steady state is reached. Two distinct parameter sets are explored, both yielding fast‑wind‑like solutions. The resulting profiles reproduce key observational features: bulk speeds of ~750 km s⁻¹ at 0.3 AU, electron temperatures of order 10⁵ K at 1 AU, and a proton temperature anisotropy that is high (T⊥p/T∥p ≈ 2–3) near the Sun but relaxes toward unity with distance, consistent with in‑situ measurements from Helios and Ulysses. The turbulent heating budget shows that roughly 60 % of the dissipated AW energy goes into electrons, about 30 % into perpendicular proton heating, and the remaining 10 % into parallel proton heating, matching the inferred heating rates from spacecraft data.

The study demonstrates that low‑frequency Alfvén‑wave turbulence, combined with instability‑regulated pitch‑angle scattering, can account for both the acceleration and the thermodynamic evolution of the fast solar wind. Limitations include the one‑dimensional geometry, the neglect of alpha particles and other ion species, and the omission of high‑frequency cyclotron‑resonant waves. Future work is suggested to extend the framework to multi‑dimensional, multi‑species models and to incorporate additional wave modes for a more complete description of solar‑wind heating.