Shearless transport barriers in unsteady two-dimensional flows and maps

TL;DR

This paper introduces a variational principle to identify shearless transport barriers in unsteady 2D flows, revealing hyperbolic and parabolic Lagrangian Coherent Structures as key barrier types, with an algorithm for their detection demonstrated on various flow models.

Contribution

It extends the concept of shearless barriers to unsteady flows, classifies barriers into hyperbolic and parabolic types, and provides an automated detection algorithm.

Findings

Parabolic barriers are more observable and robust than hyperbolic barriers.

Both barrier types are null-geodesics of a Lorentzian metric derived from the Cauchy--Green tensor.

The detection method is successfully applied to non-twist maps and the Bickley jet.

Abstract

We develop a variational principle that extends the notion of a shearless transport barrier from steady to general unsteady two-dimensional flows and maps defined over a finite time interval. This principle reveals that hyperbolic Lagrangian Coherent Structures (LCSs) and parabolic LCSs (or jet cores) are the two main types of shearless barriers in unsteady flows. Based on the boundary conditions they satisfy, parabolic barriers are found to be more observable and robust than hyperbolic barriers, confirming widespread numerical observations. Both types of barriers are special null-geodesics of an appropriate Lorentzian metric derived from the Cauchy--Green strain tensor. Using this fact, we devise an algorithm for the automated computation of parabolic barriers. We illustrate our detection method on steady and unsteady non-twist maps and on the aperiodically forced Bickley jet.