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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.Analytical and numerical calculations coupling stochastic differential equation schemes with 1D spherical magnetohydrodynamics simulations are used to investigate, in the context of test particles, the issues regarding the turbulence evolution in both the forshock and post-shock regions. Stochastic second order Fermi acceleration induced by resonant modes, magnetic field relaxation and amplification, turbulence compression at the shock front, are considered to model the multi-wavelength filaments produced in SNRs. $\gamma$-ray emission is also considered through the Inverse Compton mechanism.