  Recent observations of greatly amplified magnetic fields ($\delta B/B\sim
100$) around supernova shocks are consistent with the predictions of the
non-linear theory of particle acceleration (NLT), if the field is generated
upstream of the shock by cosmic ray induced streaming instability. The high
acceleration efficiencies and large shock modifications predicted by NLT need
however to be mitigated to confront observations, and this is usually assumed
to be accomplished by some form of turbulent heating. We show here that
magnetic fields with the strength inferred from observations have an important
dynamical role on the shock, and imply a shock modification substantially
reduced with respect to the naive unmagnetized case. The effect appears as soon
as the pressure in the turbulent magnetic field becomes comparable with the
pressure of the thermal gas. The relative importance of this unavoidable effect
and of the poorly known turbulent heating is assessed. More specifically we
conclude that even in the cases in which turbulent heating may be of some
importance, the dynamical reaction of the field cannot be neglected, as instead
is usually done in most current calculations.
