2008ApJ...684.1174N

We present results of two- and three-dimensional particle-in-cell simulations of magnetic turbulence production by isotropic cosmic-ray ions drifting upstream of supernova remnant shocks. The studies aim at testing recent predictions of a strong amplification of short-wavelength magnetic field and at studying the subsequent evolution of the magnetic turbulence and its back-reaction on cosmic-ray trajectories. For our parameters an oblique filamentary mode grows more rapidly than nonresonant parallel modes analytically found in the limit ω≪Ω_i, and the growth rate is slower than is estimated for the parallel plane wave mode. The evolved oblique filamentary mode was also observed in MHD simulations to dominate in the nonlinear phase, when the structures are already isotropic. We thus confirm the generation of the turbulent magnetic field due to the drift of cosmic-ray ions in the upstream plasma, but as our main result find that the amplitude of the turbulence saturates at about δB/B∼1. The back-reaction of the magnetic turbulence on the particles leads to an alignment of the bulk flow velocities of the cosmic rays and the background medium, which accounts for the saturation of the instability at moderate amplitudes of the magnetic field. Previously published MHD simulations have assumed a constant cosmic-ray current and no energy or momentum flux in the cosmic rays, which excludes a back-reaction of the generated magnetic field on cosmic rays, and thus the saturation of the field amplitude is artificially suppressed. This may explain the continued growth of the magnetic field in the MHD simulations. A strong magnetic field amplification to amplitudes δB≫B₀ has not been demonstrated yet.