The present article investigates magnetic amplification in the upstream medium of SNR blast wave through both resonant and non-resonant regimes of the streaming instability. It aims at a better understanding of the diffusive shock acceleration (DSA) efficiency considering various relaxation processes of the magnetic fluctuations in the downstream medium. Multi-wavelength radiative signatures coming from the SNR shock wave are used in order to put to the test the different downstream turbulence relaxation models. We confirm the result of Parizot et al (2006) that the maximum CR energies should not go well beyond PeV energies in young SNRs where X-ray filaments are observed. In order to match observational data, we derive an upper limit on the magnetic field amplitude insuring that stochastic particle reacceleration remain inefficient. Considering then, various magnetic relaxation processes, we present two necessary conditions to achieve efficient acceleration and X-ray filaments in SNRs: 1/the turbulence must fulfil the inequality $2-beta-delta_{rm d} ge 0$ where $beta$ is the turbulence spectral index while $delta_d$ is the relaxation length energy power-law index; 2/the typical relaxation length has to be of the order the X-ray rim size. We identify that Alvenic/fast magnetosonic mode damping does fulfil all conditions while non-linear Kolmogorov damping does not. Confronting previous relaxation processes to observational data, we deduct that among our SNR sample, the older ones (SN1006 & G347.3-0.5) fail to verify all conditions which means that their X-ray filaments are likely controlled by radiative losses. The younger SNRs, Cas A, Tycho and Kepler, do pass all tests and we infer that the downstream magnetic field amplitude is lying in the range of 200-300 $mu$ Gauss.
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