Modeling the Saturation of the Bell Instability Using Hybrid Simulations

The nonresonant streaming instability (Bell instability) plays a pivotal rolein the acceleration and confinement of cosmic rays (CRs); yet, the exactmechanism responsible for its saturation and the magnitude of the finalamplified magnetic field have not been assessed from first-principles. Using asurvey of hybrid simulations (with kinetic ions and fluid electrons), we studythe evolution of the Bell instability as a function of the parameters of the CRpopulation. We find that, at saturation, the magnetic pressure in the amplifiedfield is comparable with the initial CR anisotropic pressure, rather than withthe CR energy flux as previously argued. These results provide a predictiveprescription for the total magnetic field amplification expected in the manyastrophysical environments where the Bell instability is important.