Strange eigenmodes and decay of variance in the mixing of diffusive tracers

We prove the existence of asymptotic spatial patterns for diffusive tracers advected by unsteady velocity fields. The asymptotic patterns arise from convergence to a time-dependent inertial manifold in the underlying advection–diffusion equation. For time-periodic velocity fields, we find that the inertial manifold is spanned by a finite number of Floquet solutions, the strange eigenmodes, observed first numerically by Pierrehumbert. These strange eigenmodes only admit a regular asymptotic expansion in the diffusivity if the velocity field is completely integrable.