Description
In astrophysical shocks, instabilities driven by the streaming of energetic particles into the upstream medium amplify magnetic fields and generate turbulence, enabling efficient cosmic-ray (CR) acceleration (Malkov et al., 2010). However, a complete and self-consistent picture of CR acceleration, spanning the initial generation of magnetic fields through their subsequent amplification and the resulting particle acceleration, remains incomplete. These magnetic fields are expected to arise from the interplay, competition, and nonlinear evolution of several instabilities, notably the Bell (Bell, 2004) and firehose instabilities. These interactions happen across a wide range of scales, making both analytical and numerical investigations challenging. Our work aims at providing a tentative picture of such interplay through a simplified analytical and numerical framework, offering insight into mechanisms responsible for large-scale magnetic-field amplification.
We present kinetic simulations performed with the SMILEI particle-in-cell (PIC) code (Derouillat et al., 2018) to study the nonlinear evolution of the Bell and firehose instabilities in counter-streaming monoenergetic proton beams, used here as a minimal model for cosmic-ray–driven turbulence. By varying the beam density ratio, we probe regimes where either the Bell or the firehose instability dominates.
Our results constitute, to our knowledge, the first observation in fully kinetic PIC simulations of nonlinear self-coupling of the dominant linear mode, leading to growth of large-scale magnetic fields at twice the linear growth rate, in agreement with earlier magnetohydrodynamic simulations (Bykov et al., 2013). We develop an analytical interpretation of this behavior based on the growth of density perturbations, which drive nonlinear electrostatic modes that subsequently feed back into electromagnetic modes. This mechanism provides a potential pathway for the generation of large-scale magnetic fluctuations required to approach Bohm diffusion regimes for CRs, and thus for efficient particle acceleration in astrophysical shocks.