  A non-resonant instability for the amplification of the interstellar magnetic
field in young Supernova Remnant (SNR) shocks was predicted by Bell (2004), and
is thought to be relevant for the acceleration of cosmic ray (CR) particles.
For this instability, the CRs streaming ahead of SNR shock fronts drive
electromagnetic waves with wavelengths much shorter than the typical CR Larmor
radius, by inducing a current parallel to the background magnetic field. We
explore the nonlinear regime of the non-resonant mode using Particle-in-Cell
(PIC) hybrid simulations, with kinetic ions and fluid electrons, and analyze
the saturation mechanism for realistic CR and background plasma parameters. In
the linear regime, the observed growth rates and wavelengths match the
theoretical predictions; the nonlinear stage of the instability shows a strong
reaction of both the background plasma and the CR particles, with the
saturation level of the magnetic field varying with the CR parameters. The
simulations with CR-to-background density ratios of n_CR/n_b=10^(-5) reveal the
highest magnetic field saturation levels, with energy also being transferred to
the background plasma and to the perpendicular velocity components of the CR
particles. The results show that amplification factors >10 for the magnetic
field can be achieved, and suggest that this instability is important for the
generation of magnetic field turbulence, and for the acceleration of CR
particles.
