A kinetic approach to cosmic-ray-induced streaming instability at supernova shocks

We show that a purely kinetic approach to the excitation of waves by cosmic rays in the vicinity of a shock front leads to predict the appearance of a non-Alfvenic fast-growing mode which has the same dispersion relation as that previously found by Bell in 2004 by treating the plasma in the magnetohydrodynamic approximation. The kinetic approach allows us to investigate the dependence of the dispersion relation of these waves on the microphysics of the current which compensates the cosmic ray flow. We also show that a resonant and a non-resonant mode may appear at the same time and one of the two may become dominant on the other depending on the conditions in the acceleration region. We discuss the role of the unstable modes for magnetic field amplification and particle acceleration in supernova remnants at different stages of the remnant evolution.