Firehose instability in the space plasma with anisotropic Cairns-distribution electrons

We study the electron firehose mode propagating parallel to the ambient magnetic field in the space plasma with anisotropic Cairns-distribution electrons. The dispersion relation, the wave frequency and the growth rate of electron firehose mode are derived, and the condition for onset of the firehose instability is obtained. We show that the wave frequency and the growth rate both depend significantly on the parameters, such as the parallel electron beta , the nonthermal parameter Λ and the electron temperature anisotropy Ae , and the anisotropic Cairns-distribution electrons change the instability condition. The numerical analyses show that the wave frequency and the growth rate of the electron firehose mode increase with increase of the parameters. The results may be helpful for understanding the firehose instability in space plasma environments.

💡 Research Summary

The paper investigates the electron firehose instability (EFI) for waves propagating parallel to the ambient magnetic field in a plasma where the electrons follow an anisotropic Cairns distribution—a non‑thermal velocity distribution characterized by a “non‑thermal parameter” Λ that controls the population of suprathermal electrons. The authors begin by reviewing the importance of temperature anisotropies in space plasmas and the role of the electron firehose mode in reducing parallel‑to‑perpendicular temperature excesses, especially in solar‑flare and solar‑wind environments. While previous studies have largely employed bi‑Maxwellian or kappa distributions, this work introduces the Cairns distribution to capture high‑energy tails observed in many space‑plasma measurements.

The theoretical framework starts from the linearized Vlasov‑Maxwell system. Electrons are described by the anisotropic Cairns distribution (Eq. 1) with parallel and perpendicular thermal speeds v‖e and v⊥e, while ions are assumed to be isotropic Maxwellian (Eq. 8). The wave vector k is taken to be exactly parallel to the background magnetic field (θ = 0), so the left‑hand circularly polarized (L‑mode) is the relevant branch. By inserting the distribution functions into the general dispersion relation for parallel propagation (Eqs. 5‑6) and using the standard plasma dispersion function formalism, the authors separate the contributions of resonant ions (ξi < 1) and non‑resonant electrons (ξe ≫ 1). After a series of algebraic manipulations (detailed in Appendices A and B), they obtain a compact expression for the real part of the frequency (Eq. 15) and for the growth rate (Eq. 17). The instability threshold is derived as Eq. (18), which reduces to the familiar bi‑Maxwellian condition β‖Ae < 1 when Λ = 0, confirming consistency with earlier work.

Numerical solutions of the dispersion relation are performed using typical solar‑wind parameters: parallel electron beta β‖e ≈ 5, temperature anisotropy Ae = Te⊥/Te‖ ≈ 0.5, and a range of Λ values (0–0.2). Figures 1 and 2 illustrate how both the normalized wave frequency ωr/Ωe and the normalized growth rate γ/Ωe increase monotonically with Λ, with β‖e, and with Ae. The authors interpret these trends as a direct consequence of the extra free energy supplied by the suprathermal electron population: larger Λ means a higher fraction of energetic electrons, which enhances the drive of the firehose mode. Similarly, higher β‖e (i.e., higher plasma pressure relative to magnetic pressure) and larger temperature anisotropy provide more free energy, leading to higher growth rates.

The conclusions emphasize that the Cairns‑distributed electrons lower the firehose instability threshold and amplify both the wave frequency and growth rate compared with the Maxwellian case. The authors suggest that these results could be relevant for interpreting wave emission, particle acceleration, and turbulence generation in various space‑plasma environments.

Critical assessment: The paper makes a novel contribution by incorporating a non‑thermal electron distribution into the EFI analysis, which is indeed missing from the literature. However, several limitations reduce its impact. First, the study is confined to strictly parallel propagation; oblique firehose modes, which are often more unstable, are not examined, leaving an incomplete picture of the instability landscape. Second, the derivations contain numerous typographical errors and undefined symbols (e.g., ξe, ξi, Ae), which hinder reproducibility. Third, the numerical parameter space is narrow and lacks comparison with spacecraft observations that could validate the theoretical predictions. Fourth, the appendices compress lengthy integrals into a few lines, making it difficult for readers to follow the calculation steps. Finally, while the authors claim an “anisotropic Cairns distribution,” the anisotropy is introduced solely through the temperature ratio Ae, and the role of Λ is treated independently; a clearer discussion of how these two sources of free energy interact would improve physical insight.

Future work should extend the analysis to oblique angles, include multiple ion species, explore a broader range of β‖e and Λ values, and directly compare the theoretical growth rates with measured wave spectra from missions such as Parker Solar Probe or Solar Orbiter. Additionally, a more rigorous treatment of the dispersion integral, perhaps with numerical integration codes, would enhance the robustness of the results. Overall, the paper opens an interesting avenue for studying non‑thermal effects on firehose instability, but further development is needed to fully assess its relevance to real space‑plasma conditions.