Nonautonomous flows as open dynamical systems: characterising escape rates and time-varying boundaries

Abstract

A Lagrangian coherent structure (LCS) in a nonautonomous flow can be viewed as an open dynamical system, from which there is time-dependent escape or entry. A difficulty with this viewpoint is formulating a definition for the time-dependent boundary of the LCS, since it does not correspond to an entity across which there is zero transport. Complementary to this is the question of how to determine the escape rate—the time-dependent fluid flux—across this purported boundary. These questions are addressed within the context of nonautonomously perturbed two-dimensional compressible flow. The LCS boundaries are thought of in terms of time-varying stable and unstable manifolds, whose primary locations are quantified. A definition for the time-varying flux across these is offered, and computationally tractable formulæ with a strong relation to Melnikov functions are provided. Simplifications of these formulæ for frequently considered situations (incompressibility, time-periodic perturbations) are demonstrated to be easily computable using Fourier transforms. Explicit connections to lobe areas and the average flux are also provided.