  We present results of 2D and 3D PIC simulations of magnetic turbulence
production by isotropic cosmic-ray ions drifting upstream of SNR shocks. The
studies aim at testing recent predictions of a strong amplification of short
wavelength magnetic field and at studying the evolution of the magnetic
turbulence and its backreaction on cosmic rays. We observe that an oblique
filamentary mode grows more rapidly than the non-resonant parallel modes found
in analytical theory, and the growth rate of the field perturbations is much
slower than is estimated for the parallel plane-wave mode, possibly because in
our simulations we cannot maintain omega << Omega_i, the ion gyrofrequency, to
the degree required for the plane-wave mode to emerge. The evolved oblique
filamentary mode was also observed in MHD simulations to dominate in the
nonlinear phase. 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 dB/B~1. The
backreaction of the turbulence on the particles leads to an alignment of the
bulk-flow velocities of the cosmic rays and the background medium, which is an
essential characteristic of cosmic-ray modified shocks. It accounts for the
saturation of the instability at moderate field amplitudes. Previously
published MHD simulations have assumed a constant cosmic-ray current and no
energy or momentum flux in the cosmic rays, which excludes a backreaction 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 dB >> B0 has not been demonstrated yet.
