On the background-gyroresonant character of the Bell instability

We show that the Bell instability, which is widely considered potentially important for cosmic-ray acceleration, is the low-frequency limit of a gyroresonant interaction between the protons of the interstellar medium and shear-Alfv\'en waves. At large cosmic-ray current densities, its growth rate is therefore limited by the proton gyrofrequency, and two modes emerge from the cold-beam dispersion relation. A third mode driven by electron gyroresonance is only weakly unstable at low current densities. We discuss implications for magnetic-field amplification and its saturation in the vicinity of supernova remnants.