  The cosmic ray current-driven (CRCD) instability, predicted by Bell (2004),
consists of non-resonant, growing plasma waves driven by the electric current
of cosmic rays (CRs) that stream along the magnetic field ahead of both
relativistic and non-relativistic shocks. Combining an analytic, kinetic model
with one-, two-, and three-dimensional particle-in-cell simulations, we confirm
the existence of this instability in the kinetic regime and determine its
saturation mechanisms. In the linear regime, we show that, if the background
plasma is well magnetized, the CRCD waves grow exponentially at the rates and
wavelengths predicted by the analytic dispersion relation. The magnetization
condition implies that the growth rate of the instability is much smaller than
the ion cyclotron frequency. As the instability becomes non-linear, significant
turbulence forms in the plasma. This turbulence reduces the growth rate of the
field and damps the shortest wavelength modes, making the dominant wavelength,
\lambda_d, grow proportional to the square of the field. At constant CR
current, we find that plasma acceleration along the motion of CRs saturates the
instability at the magnetic field level such that v_A ~ v_{d,cr}, where v_A is
the Alfven velocity in the amplified field, and v_{d,cr} is the drift velocity
of CRs. The instability can also saturate earlier if CRs get strongly deflected
by the amplified field, which happens when their Larmor radii get close to
\lambda_d. We apply these results to the case of CRs in the upstream medium of
supernova remnants. Considering only the most energetic CRs that escape from
the shock, we obtain that the field amplification factor of ~10 can be reached.
This confirms the CRCD instability as a potentially important component of
magnetic amplification process in astrophysical shocks.
