Invariants, diffusion and topological change in incompressible Navier-Stokes equations

We discuss the effect that the presence of a small viscosity has on the evolution of fields that are transported unchanged in the absence of viscosity. We employ a diffusive Lagrangian formulation and show that the Cauchy invariant, the helicity density, the Jacobian determinant, and the virtual velocity obey parabolic equations that are well-behaved as long as the diffusive transformations are invertible. We call such quantities diffusive Lagrangian. We showby numerical calculations that the loss of invertibility of the diffusive transformation can occur, and that the time scale on which it does can be short even when the viscosity is small. We present quantitative evidence relating the loss of invertibility to the physicalphenomenon of vortex reconnection.