Finding finite-time invariant manifolds in two-dimensional velocity fields

Abstract

For two-dimensional velocity fields defined on finite time intervals, we derive an analytic condition that can be used to determine numerically the location of uniformly hyperbolic trajectories. The conditions of our main theorem will be satisfied for typical velocity fields in fluid dynamics where the deformation rate of coherent structures is slower than individual particle speeds. We also propose and test a simple numerical algorithm that isolates uniformly finite-time hyperbolic sets in such velocity fields. Uniformly hyperbolic sets serve as the key building blocks of Lagrangian mixing geometry in applications.