  Observations of non-thermal emission from several supernova remnants suggest
that magnetic fields close to the blastwave are much stronger than would be
naively expected from simple shock compression of the field permeating the
interstellar medium (ISM).
  We present a simple model which is capable of achieving sufficient magnetic
field amplification to explain the observations. We propose that the cosmic-ray
pressure gradient acting on the inhomogeneous ISM upstream of the supernova
blastwave induces strong turbulence upstream of the supernova blastwave. The
turbulence is generated through the differential acceleration of the upstream
ISM which occurs as a result of density inhomogeneities in the ISM. This
turbulence then amplifies the pre-existing magnetic field.
  Numerical simulations are presented which demonstrate that amplification
factors of 20 or more are easily achievable by this mechanism when reasonable
parameters for the ISM and supernova blastwave are assumed. The length scale
over which this amplification occurs is that of the diffusion length of the
highest energy non-thermal particles.
