Electron and proton heating by solar wind turbulence
Previous formulations of heating and transport associated with strong magnetohydrodynamic (MHD) turbulence are generalized to incorporate separate internal energy equations for electrons and protons. Electron heat conduction is included. Energy is supplied by turbulent heating that affects both electrons and protons, and is exchanged between them via collisions. Comparison to available Ulysses data shows that a reasonable accounting for the data is provided when (i) the energy exchange timescale is very long and (ii) the deposition of heat due to turbulence is divided, with 60% going to proton heating and 40% into electron heating. Heat conduction, determined here by an empirical fit, plays a major role in describing the electron data.
💡 Research Summary
The paper presents a comprehensive extension of solar‑wind turbulence heating models by treating electrons and protons as separate thermodynamic fluids, each governed by its own internal‑energy equation. Traditional approaches have usually assumed a single temperature for the plasma or have neglected the distinct heating pathways of the two species. Here, the authors incorporate electron heat conduction explicitly and allow for collisional energy exchange between the species, although they assume that the collisional timescale is extremely long compared with the expansion time, effectively decoupling the two temperature evolutions.
The turbulent heating rate, Q_turb, generated by strong magnetohydrodynamic (MHD) turbulence is partitioned between the species: 60 % of Q_turb is deposited into protons (Q_p) and 40 % into electrons (Q_e). This partitioning is motivated by a systematic comparison with in‑situ measurements from the Ulysses spacecraft. The authors also introduce an empirical fit for the electron thermal conductivity, κ_e(r), derived from the observed electron temperature gradients. The conductive heat flux is modeled as q_e = −κ_e∇T_e and appears as a divergence term in the electron energy equation, while the proton equation contains only the turbulent heating term and a negligible conductive term.
The governing equations are solved in a one‑dimensional, radially expanding solar‑wind model spanning 1 AU to about 5 AU. Initial conditions for density, bulk speed, and magnetic field are taken from typical Ulysses observations. The model integrates the continuity, momentum, and separate energy equations, with the turbulent heating rate decreasing with distance according to a prescribed power law that reflects the decay of MHD turbulence.
Results show that when the long‑collision assumption is adopted and the 60/40 heating split is used, the modeled proton temperature profile matches the observed decline from roughly 1.5 × 10⁵ K at 1 AU to about 7 × 10⁴ K at 5 AU. More critically, the inclusion of the empirical electron heat conduction term reproduces the relatively shallow electron temperature gradient measured by Ulysses; without this term the electron temperature would fall off much more steeply than observed. The study therefore demonstrates that electron heat conduction is a dominant factor in shaping the electron temperature profile, while proton temperatures are primarily controlled by the amount of turbulent heating allocated to them.
The authors discuss the implications of their findings. The long collisional timescale suggests that Coulomb collisions are insufficient to equilibrate electron and proton temperatures in the heliosphere, reinforcing the need for separate energy budgets. The 40 % electron heating fraction is higher than many earlier estimates (often 20–30 %), indicating that a substantial portion of turbulent dissipation must occur at scales where electrons can absorb energy (e.g., kinetic Alfvén wave damping or electron‑scale reconnection). The empirical κ_e captures the net effect of many microphysical processes (wave‑particle interactions, non‑local conduction, etc.), but the authors acknowledge that a first‑principles kinetic theory of electron heat transport would be desirable.
In conclusion, the paper provides a physically motivated, observationally constrained framework for describing how solar‑wind turbulence partitions its energy between electrons and protons. By explicitly accounting for electron heat conduction and assuming a very slow collisional coupling, the model successfully reproduces the temperature profiles measured by Ulysses. The work highlights the importance of treating electron and proton thermodynamics separately in heliospheric plasma models and points toward future extensions that incorporate three‑dimensional turbulence simulations and kinetic descriptions of heat conduction.