Quasi-linear approach of bi-Kappa distributed electrons with dynamic $κ$ parameter. EMEC instability

TL;DR

This work extends quasi-linear theory for EMEC (whistler) instability by allowing the Kappa index $κ$ to evolve in time, rather than remain fixed. A new QL framework ties $κ(t)$ to kurtosis $K(t)$, which itself depends on the evolving temperatures $T^{(κ)}_ot$ and $T^{(κ)}_\\parallel$, yielding a closed, energy-conserving set of equations that couple core and suprathermal electron populations to the self-generated waves. The results show that, in most regimes, $κ$ decreases (more suprathermalization) during saturation, although quasi-Maxwellian relaxation can occur at very low $β_\parallel$ and $κ_0>5$; instability-driven waves typically energize suprathermal electrons and only partially relax temperature anisotropy. These findings align with in situ solar wind observations and provide a more comprehensive kinetic framework for nonthermal plasmas with bi-Kappa distributions, with potential applicability to other EM modes and future extensions to asymmetries and heat flux.

Abstract

In recent years, significant progress has been made in the velocity-moment-based quasi-linear (QL) theory of waves and instabilities in plasmas with nonequilibrium velocity distributions (VDs) of the Kappa (or $κ$) type. However, the temporal variation of the parameter $κ$, which quantifies the presence of suprathermal particles, is not fully captured by such a QL analysis, and typically $κ$ remains constant during plasma dynamics. We propose a new QL modeling that goes beyond the limits of a previous approach, realistically assuming that the quasithermal core cannot evolve independently of energetic suprathermals. The case study is done on the electron-cyclotron (EMEC) instability generated by anisotropic bi-Kappa electrons with $A=T_\perp/T_\parallel > 1$ ($\parallel, \perp$ denoting directions with respect to the background magnetic field). The parameter $κ$ self-consistently varies through the QL equation of kurtosis (fourth-order moment) coupled with temporal variations of the temperature components, relaxing the constraint on the independence of the low-energy (core) electrons and suprathermal high-energy tails of VDs. The results refine and extend previous approaches. A clear distinction is made between regimes that lead to a decrease or an increase in the $κ$ parameter with saturation of the instability. What predominates is a decrease in $κ$, i.e., an excess of suprathermalization, which energizes suprathermal electrons due to self-generated wave fluctuations. Additionally, we found that VDs can evolve toward a quasi-Maxwellian shape (as $κ$ increases) primarily in regimes with low beta and initial kappa values greater than five. Instability-driven relaxation only partially resolves temperature anisotropy in bi-Kappa electron VDs, as wave fluctuations generally act to further energize suprathermal electrons.

Figures (7)

• Figure 1: QL runs with the same initial anisotropy, $A(0)=2$, three values of initial $\beta_\parallel (0)$ = 0.1 (left), 0.5 (middle), and 8.0 (right), and for eight different (initial) values of $\kappa (0)$ shown in various shades of brown. The first line (top) indicates the variation in $\kappa$, and the following ones indicate the variations in key parameters $\beta_\parallel (t)$ and $A (t)=T_\perp (t)/T_\parallel (t)$, and variations in wave properties $\delta B^2 (t)/B_0^2$ and $\gamma_{\rm max} (t)$, for both cases in which $\kappa$ varies in time and ones in which $\kappa$ is constant.
• Figure 2: QL runs with the same initial $\beta_\parallel (0)=0.1$, three values of initial $\kappa (0)$ = 2.6 (left), 4.0 (middle), and 8.0 (right), and for six different (initial) anisotropies, $T_\perp/T_\parallel (0)$, shown in various shades of brown. The first line (top) indicates the variation in $\kappa$, and the following ones indicate the variations in key parameters $\beta_\parallel (t)$ and $A (t)=T_\perp (t)/T_\parallel (t)$, and variations in wave properties $\delta B^2 (t)/B_0^2$ and $\gamma_{\rm max} (t)$, for both cases in which $\kappa$ varies in time and ones in which $\kappa$ is constant.
• Figure 3: QL runs with the same initial $\kappa (0)=8.0$, three values of initial anisotropy $A (0)$ = 2.0 (left), 3.9 (middle), and 6.0 (right), and for ten different (initial) values of $\beta_\parallel (0)$ shown in various shades of brown. The first line (top) indicates the variation in $\kappa$, and the following ones indicate the variations in key parameters $\beta_\parallel (t)$ and $A (t)=T_\perp /T_\parallel (t)$, and variations in wave properties $\delta B^2 (t)/B_0^2$ and $\gamma_{\rm max} (t)$, for both cases in which $\kappa$ varies in time and ones in which $\kappa$ is constant.
• Figure 4: QL dynamic paths in diagrams of $T_\perp/T_\parallel$ vs. $\beta_\parallel$, showing for each run time variations (color coded) in $\kappa(t)$ (top), the resonance factor (middle, see text), and the wave energy density (bottom). Each column groups runs with the same initial $\kappa (0)$ = 2.6 (left), 4.0 (middle), and 8.0 (right).
• Figure 5: Diagram of $T_\perp/T_\parallel$ vs. $\beta_\parallel$, showing final values of $\kappa(t_{end})$ (color-coded) obtained for the final states (filled circles), which reach the instability thresholds (light gray, $\gamma_{\rm max}/|\Omega_e|= 5 \times 10^{-3}$) predicted by linear theory for different initial values of $\kappa(0)$ (as is partially indicated).