Turbulent amplification of magnetic field and diffusive shock acceleration of cosmic rays

The diffusive shock acceleration of cosmic rays by supernova remnants depends upon the generation of magnetic fluctuations by cosmic rays upstream of the shock. Strongly driven, non-resonant, nearly purely growing modes grow more rapidly than the resonant Alfvén waves usually considered. Non-linear simulation shows that the magnetic field can be amplified from its seed value by orders of magnitude. The consequences for the maximum attainable cosmic ray energy in supernova remnants are explored.