On the spectrum of high-energy cosmic rays produced by supernova   remnants in the presence of strong cosmic-ray streaming instability and wave   dissipation
  The cosmic-ray streaming instability creates strong magnetohydrodynamic turbulence in the precursor of a SN shock. The level of turbulence determines the maximum energy of cosmic-ray particles accelerated by the diffusive shock acceleration mechanism. The present consideration continues our work Ptuskin & Zirakashvili (2003). It is assumed that the Kolmogorov type nonlinear wave interactions together with the ion-neutral collisions restrict the amplitude of random magnetic field. As a result, the maximum energy of accelerated particles strongly depends on the age of a SNR. The average spectrum of cosmic rays injected in the interstellar medium over the course of adiabatic SNR evolution (the Sedov stage) is approximately $Q(p)p^{2}\propto p^{-2}$ at energies larger than 10-30 Gev/nucleon and with the maximum particle energy that is close to the knee position in cosmic ray spectrum observed at $\sim4\times10^{15}$ eV. At earlier stage of SNR evolution - the ejecta dominated stage described by the Chevalier-Nadyozhin solution, the particles are accelerated to higher energies and have rather steep power-law distribution on momentum. These results suggest that the knee may mark the transition from the ejecta-dominated to the adiabatic evolution of SNR shocks which accelerate cosmic rays. 
V.S. Ptuskin, V.N. Zirakashvili