How it cramps the flow: three regimes for the saturation of parallel ion-beam instabilities

Motivated by recent advances in laboratory experiments on parallel ion-beam instabilities, we present a theoretical framework for and simulations of their evolution towards shock formation and Fermi acceleration. After reviewing the theory of beam instabilities with a focus on the so-called nonresonant or Bell instability, which we show to be due to the gyromotion of background ions, we contrast the saturation of three parameter regimes: (I) the left-handed "nonresonant" regime, (II) the right-handed beam-gyroresonant regime, (III) the balanced, mixed-turbulence regime.