Impact of Cosmic Ray Distribution on the Growth and Saturation of Bell Instability

Cosmic rays (CRs) streaming in weakly magnetized plasmas can drive large-amplitude magnetic fluctuations via nonresonant streaming instability (NRSI), or Bell instability. Using one-dimensional kinetic simulations, we investigate how mono-energetic and power-law CR momentum distributions influence the growth and saturation of NRSI. The linear growth is governed solely by the CR current and is largely insensitive to the CR distribution. However, the saturation depends strongly on the CR distribution and is achieved through CR isotropization, which quenches the driving current. Mono-energetic CRs effectively amplify the magnetic field and isotropize. For power-law distributions, the lowest-energy CRs dominate current relaxation and magnetic growth, while the highest-energy CRs remain weakly scattered, limiting their contribution to saturation. In the absence of low-energy CRs, high-energy particles amplify magnetic fields effectively and isotropize. Accounting for these effects, we provide a modified saturation prescription valid for both relativistic and nonrelativistic CRs. We propose a layered CR-confinement scenario upstream of astrophysical shocks, relevant to particle acceleration to high energies.