Quasi-linear approach of bi-Kappa distributed electrons with dynamic <i>κ</i> parameter

Context. 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. Aims. 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. Methods. The case study is done on the electron-cyclotron (EMEC) instability generated by anisotropic bi-Kappa electrons with A = T ⊥ / T ∥ &gt; 1 (∥, ⊥ 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. Results. 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. Conclusions. Instability-driven relaxation only partially resolves temperature anisotropy in bi-Kappa electron VDs, as wave fluctuations generally act to further energize suprathermal electrons. The present results show a preliminary agreement with in situ observations in the solar wind, suggesting that the new QL model could provide a sufficiently explanatory theoretical basis for the kinetic instabilities in natural plasmas with Kappa-like distributions.