Dynamical feedback of self-generated magnetic fields in cosmic ray modified shocks

We present a semi-analytical kinetic calculation of the process of non-linear diffusive shock acceleration (NLDSA) which includes the magnetic field amplification due to cosmic ray induced streaming instability, the dynamical reaction of the amplified magnetic field and the possible effects of turbulent heating. The approach is specialized to parallel shock waves, and the parameters we chose are the ones appropriate to forward shocks in supernova remnants. Our calculation allows us to show that the net effect of the amplified magnetic field is to enhance the maximum momentum of accelerated particles while reducing the concavity of the spectra, with respect to the standard predictions of NLDSA. This is mainly due to the dynamical reaction of the amplified field on the shock, which notably reduces the modification of the shock precursor. The total compression factors which are obtained for parameters typical of supernova remnants are R tot 7-10, in good agreement with the values inferred from observations. The strength of the magnetic field produced through excitation of streaming instability is found in good agreement with the values inferred for several remnants if the thickness of the X-ray rims is interpreted as due to severe synchrotron losses of high-energy electrons. We also discuss the relative role of turbulent heating and magnetic dynamical reaction in driving the reduction of the precursor modification.