Hybrid simulations of the proton beam instabilities in the young solar wind. The formation of hammerhead-like distributions
Parker Solar Probe (PSP) observations in the young solar wind reveal new properties of both plasma particle velocity distributions (VDs) and associated electromagnetic (EM) wave fluctuations. The quasilinear (QL) kinetic theory of plasma wave instabilities has recently shown that new hammerhead (HH) proton distributions can be generated by the relaxation of proton beams through the instabilities of right-handed (RH) polarized waves. Such RH waves have indeed been reported in association with HH distributions. In this paper, new results from hybrid simulations of proton-beam-plasma systems with properties typical of those observed to excite EM-RH wave instabilities are presented. From the long-term evolution of these systems, it is found that beam relaxation is driven by instabilities and growing wave fluctuations, leading to HH-type features in the velocity distributions. The production of these features, as well as their prominence, depends on the magnetic power of the waves generated by the instabilities and, therefore, implicitly on the available free energy, quantified by the plasma beta parameter and the relative beam drift. The simulation results capture the self-consistent evolution of the instabilities and their nonlinear development. Linear theory, together with simulations, helps identify the nature of the unstable modes and the plasma conditions under which they arise. The good agreement with quasi-linear (QL) theory further indicates that it can serve as a computationally efficient complementary framework for interpreting the associated wave-particle interactions.