  Cosmic-ray streaming instabilities at supernova shocks are discussed in the
quasilinear diffusion formalism which takes into account the feedback effect of
wave growth on the cosmic ray streaming motion. In particular, the nonresonant
instability that leads to magnetic field amplification in the short wavelength
regime is considered. The linear growth rate is calculated using kinetic theory
for a streaming distribution. We show that the nonresonant instability is
actually driven by a compensating current in the background plasma. The
nonresonant instability can develop into a nonlinear regime generating
turbulence. The saturation of the amplified magnetic fields due to particle
diffusion in the turbulence is derived analytically. It is shown that the
evolution of parallel and perpendicular cosmic-ray pressures is predominantly
determined by nonresonant diffusion. However, the saturation is determined by
resonant diffusion which tends to reduce the streaming motion through pitch
angle scattering. The saturated level can exceed the mean background magnetic
field.
