  We consider the advection of a passive scalar by a divergence free random
Gaussian field, white in time and H\"older regular in space (rough Kraichnan's
model), a well established synthetic model of passive scalar turbulence. By
studying the evolution of negative Sobolev norms, we show an anomalous
regularization effect induced by the dynamics: distributional initial
conditions immediately become functions of positive Sobolev regularity.
