Revisiting Bump-on-Tail Instability in Reduced Cairns and Reduced Vasyliunas-Cairns Distributed Plasmas

Abstract

Numerous spacecraft observations have revealed that particle velocity distributions in space plasmas often exhibit high-energy tails, which can be effectively represented by non-thermal models. In this study, the dispersion and damping characteristics of the bump-on-tail (BOT) instability are revisited using the Vlasov-Poisson formalism of plasma kinetic theory for an isotropic, unmagnetized, and collisionless plasma. The core population is modeled by a Maxwellian distribution, while the high-energy tails are described using one-dimensional reduced Cairns and hybrid Vasyliunas-Cairns (VC) distributions. The corresponding longitudinal dielectric response function is derived analytically and evaluated numerically using solar wind data. The effects of key physical parameters—such as temperature ratio, number density ratio, and the non-thermality indices $\alpha$ and $\kappa$—are systematically examined to assess their influence on the growth and damping rates of the BOT instability. The results demonstrate that the simultaneous presence of the two non-thermality parameters ($\alpha$ and $\kappa$) significantly alters the instability behavior. Moreover, temperature and density ratios are found to play a crucial role in modulating the instability characteristics. Overall, this investigation provides a comprehensive framework for understanding wave–particle interactions and energy transfer processes in non-thermal space plasma environments such as the solar wind and magnetosphere.