Geodesic theory of transport barriers in two-dimensional flows

We introduce a new approach to locating key material transport barriers in two-dimensional, non-autonomous dynamical systems, such as unsteady planar fluid flows. Seeking transport barriers as minimally stretching material lines, we obtain that such barriers must be shadowed by minimal geodesics under the Riemannian metric induced by the Cauchy-Green strain tensor. As a result, snapshots of transport barriers can be explicitly computed as trajectories of ordinary differential equations. Using this approach, we locate hyperbolic barriers (generalized stable and unstable manifolds), elliptic barriers (generalized KAM curves) and parabolic barriers (generalized shear jets) in temporally aperiodic flows defined over a finite time interval. Our approach also yields a metric (geodesic deviation) that determines the minimal computational time scale needed for a robust numerical identification of generalized Lagrangian Coherent Structures (LCSs). As we show, an extension of our transport barrier theory to non-Euclidean flow domains, such as a sphere, follows directly. We illustrate our main results by computing key transport barriers in a chaotic advection map, and in a geophysical model flow with chaotic time dependence.