  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 noticeably reduces the modification of the shock precursor.
The total compression factors which are obtained for parameters typical of
supernova remnants are $R_{tot}\sim 7-10$, 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 are
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.
