The evolution of macroscopic material closed filaments in a time-periodic chaotic 2D flow is simulated for cases with large, small, and very small islands of regular motion using an algorithm that preserves spatial continuity. The length of the stretched filament increases much faster than predicted by the Liapunov exponent. In chaotic regions, the filament asymptotically evolves into a self-similar structure with permanent spatial nonuniformities in density. Filament densities and local length scales corresponding to different times are described by families of frequency distributions with invariant shape that can be collapsed onto a single curve by means of a simple scaling.

This article addresses the scaling and spectral properties of the advection–diffusion equation in closed two-dimensional steady flows. We show that homogenization dynamics in simple model flows is equivalent to a Schrödinger eigenvalue problem in the presence of an imaginary potential. Several properties follow from this formulation: spectral invariance, eigenfunction localization, and a universal scaling of the dominant eigenvalue with respect to the Péclet number $\mbox{\it Pe}$. The latter property means that, in the high-$\mbox{\it Pe}$ range (in practice $\mbox{\it Pe},\geq 10^2 \hbox{--} 10^3$), the scaling exponent controlling the behaviour of the dominant eigenvalue with the Péclet number depends on the local behaviour of the potential near the critical points (local maxima/minima). A kinematic interpretation of this result is also addressed.