The geometry and statistics of mixing in aperiodic flows

The relationship between statistical and geometric properties of particle motion in aperiodic, two-dimensional flows is examined. Finite-time-invariant manifolds associated with transient hyperbolic trajectories are shown to divide the flow into distinct regions with similar statistical behavior. In particular, numerical simulations of simple, eddy-resolving barotropic flows indicate that there exists a close correlation between such geometric structures and patchiness plots that describe the distribution of Lagrangian average velocity over initial conditions. For barotropic turbulence, we find that Eulerian velocity correlation time scales are significantly longer than their Lagrangian counterparts indicating the existence of well-defined Lagrangian structures. Identification of such structures shows a similar, close relationship between the invariant manifold geometry and patchiness calculations at intermediate time scales, where anomalous dispersion rates are found. (C) 1999 American Institute of Physics. [S1070-6631(99)02910-4].