  We use high-quality VLA images of the Fanaroff & Riley Class I radio galaxy
3C 31 at six frequencies in the range 1365 to 8440MHz to explore the spatial
scale and origin of the rotation measure (RM) fluctuations on the line of sight
to the radio source. We analyse the distribution of the degree of polarization
to show that the large depolarization asymmetry between the North and South
sides of the source seen in earlier work largely disappears as the resolution
is increased. We show that the depolarization seen at low resolution results
primarily from unresolved gradients in a Faraday screen in front of the
synchrotron-emitting plasma. We establish that the residual degree of
polarization in the short-wavelength limit should follow a Burn law and we fit
such a law to our data to estimate the residual depolarization at high
resolution. We show that the observed RM variations over selected areas of 3C
31 are consistent with a power spectrum of magnetic fluctuations in front of 3C
31 whose power-law slope changes significantly on the scales sampled by our
data. The power spectrum can only have the form expected for Kolmogorov
turbulence on scales <5 kpc. On larger scales we find a flatter slope. We also
compare the global variations of RM across 3C 31 with the results of
three-dimensional simulations of the magnetic-field fluctuations in the
surrounding magnetoionic medium. We show that our data are consistent with a
field distribution that favours the plane perpendicular to the jet axis -
probably because the radio source has evacuated a large cavity in the
surrounding medium. We also apply our analysis techniques to the case of Hydra
A, where the shape and the size of the cavities produced by the source in the
surrounding medium are known from X-ray data. (Abridged)
