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Supernova blast wave shock is a very important site of cosmic-ray acceleration. However, the detailed physical process of acceleration, in particular, non-linear interplay between cosmic-ray streaming and magnetic field amplification has not been studied under a realistic environment. In this paper, using a unique and novel numerical method, we study cosmic-ray acceleration at supernova blast wave shock propagating in the interstellar medium with well-resolved magnetic field amplification by non-resonant hybrid instability (or Bell instability). We find that the magnetic field is mildly amplified under typical ISM conditions that leads to maximum cosmic-ray energy ~30 TeV for supernova remnants with age ~1000 years consistent with gamma-ray observations. The strength of the amplified magnetic field does not reach so-called saturation level, because cosmic-ray electric current towards the shock upstream has finite spatial extent, by which Bell instability cannot experience many e-folding times.
Inoue et al. (Fri,) studied this question.