Summary: The Batchelor-Prandtl theory of high-Reynolds number steady two-dimensional flows predicts uniform vorticity in regions of closed streamlines. The same argument applies to the gradients of a passive tracer and to the magnetic field in a conducting fluid which are also expelled from the eddies. In this case the problem is linear and allows more detailed analysis of the homogenization process which turns out to be much faster than diffusion.
Here we consider the expulsion of the magnetic flux. A vertical line vortex is ‘switched on’ in a horizontal magnetic field. When boundaries are present, we find a simple analytical solution for the steady state. In an infinite domain we find a family of similarity solutions with the field lines forming a spiral structure propagating away from the vortex. The similarity solution approximates the evolution well in the bounded domain and explains the scaling laws of various physical quantities at intermediate times when the expulsion process is in progress. This model of the flux expulsion is relevant when the magnetic Prandtl number is large, so that the line vortex spreads much more slowly than the ‘hole’ in the field created around it.