Bell Instability and Cosmic-Ray Acceleration in Active Galactic Nuclei Ultrafast Outflow Shocks

We investigate magnetic-field amplification driven by the nonresonant hybrid (NRH or Bell) instability and its impact on cosmic-ray (CR) acceleration at reverse shocks of ultrafast outflows (UFOs) from active galactic nuclei (AGN). Previous kinetic studies by particle-in-cell simulations have demonstrated that when maximum CR energy is near the injection scale, NRH instability efficiently amplifies magnetic field up to the saturation level. However, the efficiency of NRH instability goes down as maximum energy increase since 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 ($B_{0}\!\lesssim\!10^{-4}\,\mathrm{G}$), NRH instability efficiently amplifies upstream turbulence, driving a self-regulated state where $E_{\max}$ becomes independent of initial strength of magnetic turbulence. In contrast, for stronger background fields ($B_{0}\!\gtrsim\!10^{-3}\,\mathrm{G}$), the escaping CR current is too weak to drive NRH instability, and magnetic turbulence further decays through parametric instabilities, potentially reducing the acceleration efficiency. We give the physical interpretation for the transition and discuss conditions for PeV--EeV acceleration at UFO reverse shocks.