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The cosmic-ray streaming instability creates strong magnetohydrodynamicturbulence in the precursor of a SN shock. The level of turbulence determinesthe maximum energy of cosmic-ray particles accelerated by the diffusive shockacceleration mechanism. The present consideration continues our work Ptuskin &Zirakashvili (2003). It is assumed that the Kolmogorov type nonlinear waveinteractions together with the ion-neutral collisions restrict the amplitude ofrandom magnetic field. As a result, the maximum energy of accelerated particlesstrongly depends on the age of a SNR. The average spectrum of cosmic raysinjected 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 largerthan 10-30 Gev/nucleon and with the maximum particle energy that is close tothe knee position in cosmic ray spectrum observed at $\sim4\times10^{15}$ eV.At earlier stage of SNR evolution - the ejecta dominated stage described by theChevalier-Nadyozhin solution, the particles are accelerated to higher energiesand have rather steep power-law distribution on momentum. These results suggestthat the knee may mark the transition from the ejecta-dominated to theadiabatic evolution of SNR shocks which accelerate cosmic rays.Comment: 12 pages, 2 figures, LATE

astronomy and astrophysics

On the spectrum of high-energy cosmic rays produced by supernova remnants in the presence of strong cosmic-ray streaming instability and wave dissipation