We investigate magnetic field amplification driven by the nonresonant hybrid (NRH, or Bell) instability and its impact on cosmic-ray (CR) acceleration at the reverse shocks of ultrafast outflows (UFOs) from active galactic nuclei. Previous kinetic studies by particle-in-cell simulations have demonstrated that when the maximum CR energy is near the injection scale, the NRH instability efficiently amplifies the magnetic field up to the saturation level. However, the efficiency of the NRH instability decreases as the maximum energy increases, since the CR current is carried by escaping CRs near the maximum energy. We employ a one-dimensional MHD-CR framework solving telegraph-type diffusion–convection equations to trace the coupled evolution of CRs, magnetic fields, and shock dynamics under realistic parameters. We find a distinct transition with magnetic field strength. For weak background fields (B ₀ ≲ 10 ^−4 G), the NRH instability efficiently amplifies upstream turbulence, driving a self-regulated state where E becomes independent of the initial strength of the magnetic turbulence. In contrast, for stronger background fields (B ₀ ≳ 10 ^−3 G), the escaping CR current is too weak to drive the NRH instability, and magnetic turbulence further decays through parametric instabilities, potentially reducing the acceleration efficiency. We give a physical interpretation for the transition and discuss conditions for PeV to EeV acceleration at UFO reverse shocks.
Nishiura et al. (Thu,) studied this question.