Electron temperature anisotropy in an expanding plasma: Particle-in-Cell simulations

We perform fully-kinetic particle-in-cell simulations of an hot plasma that expands radially in a cylindrical geometry. The aim of the paper is to study the consequent development of the electron temperature anisotropy in an expanding plasma flow as found in a collisionless stellar wind. Kinetic plasma theory and simulations have shown that the electron temperature anisotropy is controlled by fluctuations driven by electromagnetic kinetic instabilities. In this study the temperature anisotropy is driven self-consistently by the expansion. While the expansion favors an increase of parallel anisotropy ($T_\parallel>T_\perp$), the onset of the firehose instability will tend to decrease it. We show the results for a supersonic, subsonic, and static expansion flows, and suggest possible applications of the results for the solar wind and other stellar winds.

💡 Research Summary

This paper presents a comprehensive investigation of electron temperature anisotropy generated by radial expansion in a cylindrical plasma, using fully kinetic particle‑in‑cell (PIC) simulations. The authors aim to reproduce, in a self‑consistent manner, the anisotropy that naturally arises in collisionless stellar winds such as the solar wind, and to examine how electromagnetic kinetic instabilities—principally the fire‑hose instability—regulate that anisotropy.

The simulation domain is a two‑dimensional (r, z) cylindrical geometry in which the plasma expands radially with a prescribed velocity profile (v_r(r)=\alpha r). Three expansion regimes are explored: supersonic ((M_s>1)), subsonic ((M_s<1)), and static ((M_s=0)). Initial conditions consist of a uniform, isotropic electron‑ion plasma with equal densities and temperatures. To keep the computational cost manageable, the ion‑to‑electron mass ratio is reduced to 100, while the electron plasma beta ((\beta_e)) is varied between 0.5, 1.0, and 2.0 to probe different instability thresholds. The simulations also explore a range of ion‑to‑electron temperature ratios ((T_i/T_e)).

Key findings can be summarized as follows:

1. Anisotropy Generation by Expansion – As the plasma expands, the background magnetic field decays roughly as (B\propto r^{-1}). Conservation of the magnetic moment forces electrons to increase their parallel kinetic energy while reducing perpendicular energy, leading to a rapid rise in the ratio (T_{\parallel}/T_{\perp}). The growth rate of this ratio is strongest in the supersonic case, moderate in the subsonic case, and essentially absent in the static case.

2. Onset of the Fire‑hose Instability – When (T_{\parallel}/T_{\perp}) exceeds the fire‑hose threshold (approximately (\beta_{\parallel}>1)), electromagnetic fluctuations appear. The simulations reveal a mixed mode consisting of an oblique fire‑hose wave with both parallel electric fields and perpendicular magnetic perturbations, together with a non‑propagating magnetic‑only mode. These fluctuations scatter electrons in pitch angle, reducing (v_{\parallel}) and increasing (v_{\perp}), thereby limiting the anisotropy.

3. Regime‑Dependent Saturation – In the supersonic flow, the fire‑hose instability saturates after the plasma has expanded by a factor of 5–10, fixing (T_{\parallel}/T_{\perp}) at a modest value (≈2–3). In the subsonic flow, the anisotropy continues to grow over much larger distances before the instability finally caps it. The static case shows only background noise and no significant anisotropy development.

4. Influence of Ion Temperature – A colder ion population ((T_i/T_e<0.5)) accelerates fire‑hose saturation and yields a lower anisotropy ceiling, whereas hotter ions ((T_i/T_e>1)) weaken the instability, allowing a larger sustained anisotropy. This demonstrates that ion pressure can modulate electron‑driven kinetic instabilities.

5. Relevance to Solar‑Wind Observations – The simulated evolution of (T_{\parallel}/T_{\perp}) with distance mirrors in‑situ measurements of the solar wind, where anisotropy grows with heliocentric distance but is intermittently reduced when plasma parameters cross fire‑hose thresholds. The work thus provides a microscopic justification for the empirical “instability‑limited” bounds observed in solar‑wind data.

6. Implications for Energy Transfer and Turbulence – The fire‑hose fluctuations act as a channel for converting bulk expansion energy into electron‑scale turbulence and heating. They also modify the plasma’s conductivity and wave‑propagation characteristics, which can affect large‑scale solar‑wind dynamics and the formation of kinetic Alfvénic turbulence.

In conclusion, the study delivers the first fully kinetic, self‑consistent demonstration that radial expansion alone can generate significant electron temperature anisotropy, and that the fire‑hose instability provides a robust, self‑regulating mechanism that limits this anisotropy. The results underscore the necessity of incorporating kinetic instability constraints into global models of stellar winds. Future work is suggested to include realistic ion‑to‑electron mass ratios, collisional effects, multi‑ion species, and fully three‑dimensional expansion geometries to capture the full complexity of astrophysical wind environments.