  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 investigate in some detail a simple model based
on turbulence generation by cosmic-ray pressure gradients. Previously this
model was investigated using 2D MHD simulations.
  Motivated by the well-known qualitative differences between 2D and 3D
turbulence, we further our investigations of this model using both 2D and 3D
simulations to study the influence of the dimensionality of the simulations on
the field amplification achieved. Further, since the model implies the
formation of shocks which can, in principle, be efficiently cooled by
collisional cooling we include such cooling in our simulations to ascertain
whether it could increase the field amplification achieved. Finally, we examine
the influence of different orientations of the magnetic field with respect to
the normal of the blastwave.
  We find that dimensionality has a slight influence on the overall
amplification achieved, but a significant impact on the morphology of the
amplified field. Collisional cooling has surprisingly little impact, primarily
due to the short time which any element of the ISM resides in the precursor
region for supernova blastwaves. Even allowing for a wide range of orientations
of the magnetic field, we find that the magnetic field can be expected to be
amplified by, on average, at least an order of magnitude in the precursors of
supernova blastwaves.
