Quasi-linear approach of bi-Kappa distributed electrons with dynamic

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∥ > 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.