Cosmic ray (CR) currents through magnetised plasma drive strong instabilities producing amplification of the magnetic field. This amplification helps explain the CR energy spectrum as well as observations of supernova remnants and radio galaxy hot spots. Using magnetohydrodynamic (MHD) simulations, we study the behaviour of the non-resonant hybrid (NRH) instability (also known as the Bell instability) in the case of CR currents perpendicular and parallel to the initial magnetic field. We demonstrate that extending simulations of the perpendicular case to 3D reveals a different character to the turbulence from that observed in 2D. Despite these differences, in 3D the perpendicular NRH instability still grows exponentially far into the non-linear regime with a similar growth rate to both the 2D perpendicular and 3D parallel situations. We introduce some simple analytical models to elucidate the physical behaviour, using them to demonstrate that the transition to the non-linear regime is governed by the growth of thermal pressure inside dense filaments at the edges of the expanding loops. We discuss our results in the context of supernova remnants and jets in radio galaxies. Our work shows that the NRH instability can amplify magnetic fields to many times their initial value in parallel and perpendicular shocks.
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