Finite-time Lagrangian transport analysis: Stable and unstable manifolds of hyperbolic trajectories and finite-time Lyapunov exponents

TL;DR

This paper explores finite-time Lagrangian transport in aperiodic flows, emphasizing the importance of finite-time hyperbolic trajectories and Lyapunov exponents for understanding transport barriers in geophysical flows.

Contribution

It introduces a comprehensive analysis of finite-time hyperbolic trajectories, their manifolds, and Lyapunov exponents, highlighting their role in characterizing transport in time-dependent flows.

Findings

Finite-time hyperbolic trajectories identify transport barriers.

Flow transitions occur when finite-time hyperbolicity is lost or gained.

Examples illustrate phenomena in finite-time dynamical systems.

Abstract

We consider issues associated with the Lagrangian characterisation of flow structures arising in aperiodically time-dependent vector fields that are only known on a finite time interval. A major motivation for the consideration of this problem arises from the desire to study transport and mixing problems in geophysical flows where the flow is obtained from a numerical solution, on a finite space-time grid, of an appropriate partial differential equation model for the velocity field. Of particular interest is the characterisation, location, and evolution of "transport barriers" in the flow, i.e. material curves and surfaces. We argue that a general theory of Lagrangian transport has to account for the effects of transient flow phenomena which are not captured by the infinite-time notions of hyperbolicity even for flows defined for all time. Notions of finite-time hyperbolic trajectories, their finite time stable and unstable manifolds, as well as finite-time Lyapunov exponent (FTLE) fields and associated Lagrangian coherent structures have been the main tools for characterizing transport barriers in the time-aperiodic situation. In this paper we consider a variety of examples, some with explicit solutions, that illustrate, in a concrete manner, the issues and phenomena that arise in the setting of finite-time dynamical systems. Of particular significance for geophysical applications is the notion of "flow transition" which occurs when finite-time hyperbolicity is lost, or gained. The phenomena discovered and analysed in our examples point the way to a variety of directions for rigorous mathematical research in this rapidly developing, and important, new area of dynamical systems theory.