Double Power-law Electron Spectra in Solar Flares Due to Temperature Anisotropy Instabilities

TL;DR

The paper investigates how temperature anisotropy instabilities in ALT regions of solar flares can drive stochastic electron acceleration, using 2D PIC simulations with shear-driven magnetic amplification to generate $T_e$ anisotropy. By varying the initial electron temperature $T_e^{init}$ and the ratio $f_e^{init} = omega_ce/omega_pe$, the authors show the resulting nonthermal electron spectra develop double power-law tails with breaks around 50–150 keV, with the spectral form controlled mainly by $f_e^{init}$ and the dominant unstable mode (OQES for low $f_e$ and PEMZ for high $f_e$). The fastest energization occurs during the exponential growth of the instabilities, and the final spectra show a weak dependence on the high-$ obreak omega_ce/init/s$ regime, arguing that ALT environments in real flares should exhibit similar acceleration efficiencies. The results connect well with observed flare spectra, offering a coherent framework for explaining the observed diversity and temporal evolution of nonthermal electrons in solar flares, while highlighting the need to incorporate transport effects and more varied ALT conditions in future work.

Abstract

Despite extensive observational and theoretical efforts, the physical processes responsible for shaping the diversity of accelerated electron spectra observed in solar flares remain poorly understood. We use 2D particle-in-cell (PIC) simulations of magnetized plasmas subject to continuous shear-driven magnetic amplification to investigate whether electron temperature anisotropy instabilities in above-the-loop-top (ALT) regions can account for this diversity. We explore how the resulting spectra depend on key plasma parameters: the initial electron temperature $T_e$ and the initial ratio of electron cyclotron to plasma frequencies, $f_e = ω_{ce}/ω_{pe}$. In our simulations, the adiabatic evolution of the plasma generates electron temperature anisotropy with the electron temperature perpendicular to the magnetic field being larger than the parallel temperature. This eventually drives electromagnetic instabilities capable of scattering and accelerating electrons. The simulations consistently produce nonthermal tails in the electron spectra whose hardness increases with the initial value of $f_e$, while depending only weakly on $T_e$. For runs in which $f_e \lesssim 1.2$, the spectra exhibit double power-law shapes with downward (knee-like) breaks, and the electron scattering is dominated by OQES modes. In runs with $f_e\gtrsim 1.5$, PEMZ modes dominate and produce harder double power-law spectra with upward (elbow-like) breaks. Cases that include the $f_e\sim 1.2-1.5$ transition yield nearly single power-laws that end with bump-like breaks. Our results support the role of temperature anisotropy instabilities in accelerating electrons in ALT regions, offering a promising framework to help explain the wide range of nonthermal electron spectra reported in solar flare observations.

Figures (7)

• Figure 1: Fields and electron temperatures for run Fe3Te2-1200 as functions of normalized time $t\cdot s$ (lower horizontal axes) and of the instantaneous $f_e$ (upper horizontal axes; using the average magnetic field at each time). Panel $\it{a}$ shows in solid-blue and solid-green lines the evolution of the energy in the $x$ and $y$ components of the mean magnetic field $\langle \textbf{B}\rangle$, as well as the energy in $\delta \textbf{B}$ in solid-red line. The solid-black and dashed-black lines show the contributions to the $\delta \textbf{B}$ energy given by the oblique ($\delta \textbf{B}_{ob}$) and quasi-parallel ($\delta \textbf{B}_{qp}$) modes. Panel $\it{b}$ shows in solid-black (solid-red) the evolution of the electron temperature perpendicular (parallel) to $\langle \textbf{B}\rangle$. The dashed-black (dashed-red) line shows the CGL prediction for the perpendicular (parallel) temperature. Finally, panel $\it{c}$ shows in solid- and dashed-blue lines the contributions to the energy in the fluctuating electric field $\delta \textbf{E}$ given by its electrostatic ($\delta \textbf{E}_{es}$) and electromagnetic ($\delta \textbf{E}_{em}$) components, respectively [all fields energies are in units of the initial magnetic energy]. This figure is adapted from Fig. 1 of Paper I.
• Figure 2: The 2D structure of $\delta \textit{B}_z$ at $t\cdot s = 1.6$ (panel $a$) and $t\cdot s = 3.1$ (panel $b$) for run Fe3Te2-1200. $\delta \textit{B}_z$ is normalized by $B_0$. The black arrows show the direction of the average magnetic field $\langle \textbf{B} \rangle$.
• Figure 3: The time evolution of the electron spectra for simulations with two values of initial electron temperature, $\Theta_e^{\textrm{init}} = 0.00438$ and $0.00875$ (upper and lower row), and four values of $f_e^{\textrm{init}} = 0.264$, $0.374$, $0.529$, and $0.748$. All runs use $\omega_{c,e}^{\textrm{init}}/s=1200$. Panels $a$-$h$ correspond to runs Fe1Te1-1200, Fe2Te1-1200, Fe3Te1-1200, Fe4Te1-1200, Fe1Te2-1200, Fe2Te2-1200, Fe3Te2-1200, and Fe4Te2-1200 from Table \ref{['tab:param']}.
• Figure 4: The evolution of the energy in magnetic fluctuations for the same runs shown in Fig. \ref{['fig:spectra']} separated by oblique and quasi-parallel modes. The magnetic energies in these two types of modes ($\delta B_{ob}^2$ and $\delta B_{qp}^2$) are shown in solid-black and dashed-black lines, respectively.
• Figure 5: The evolution of the energy in electric field fluctuations for the same runs shown in Figs. \ref{['fig:spectra']} and \ref{['fig:compdelb']} separated by electrostatic and electromagnetic components. The electric field energies in these two types of modes ($\delta E_{\textrm{es}}^2$ and $\delta E_{\textrm{em}}^2$) are shown in solid-blue and dashed-blue lines, respectively.