The spiral wind-up and diffusive decay of a passive scalar in circular streamlines is considered. An accelerated diffusion mechanism operates to destroy scalar fluctuations on a time scale of order P1/3 times the turn-over time, where P is a Péclet number. The mechanism relies on differential rotation, that is, a non-zero gradient of angular velocity. However if the flow is smooth, the gradient of angular velocity necessarily vanishes at the centre of the streamlines, and the time scale becomes greater. The behaviour at the centre is analysed and it is found that scalar there is only destroyed on a time scale of order P1/2. Related results are obtained for magnetic field and for weak vorticity, a scalar coupled to the stream function of the flow. Some exact solutions are presented.