The acceleration efficiency of Cosmic Ray (CR) particles in Supenova Remnant (SNR)
Shocks is highly dependent on the properties of the magnetic field present. Here, we explore
the magnetic field amplification mechanism originally proposed in , where the CRs
streaming along the background magnetic field drive a current, and induce the non-resonant
formation of small-scale wave modes.
Using hybrid simulations, with kinetic ions and fluid electrons, it is possible to explore the saturation phase of the instability, and to study the feedback mechanism of magnetic field turbulence on the CR trajectories. Our results show a significant magnetic field enhancement for most of the parameters scanned, with growth rates in agreement with linear theory. We find that the saturation of the instability differs when considering a kinetic driver, as opposed to an externally imposed current. Saturation is related to the enhancement of the magnetic field along the flow direction, and also to the reaction of the CR particles and the background plasma. We explore the non-linear growth phase and the saturation phase in detail, and discuss the relevance of the mechanism to the CR acceleration at SNR shocks.
 A. R. Bell, Mon. Not. R. Astron. Soc. 353, 550, 2004
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