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Abstract
A new method to describe hyperbolic patterns in two-dimensional flows is proposed. The method is based on the Covariant Lyapunov Vectors (CLVs), which have the properties of being covariant with the dynamics, and thus, being mapped by the tangent linear operator into another CLVs basis, they are norm independent, invariant under time reversal and cannot be orthonormal. CLVs can thus give more detailed information about the expansion and contraction directions of the flow than the Lyapunov vector bases, which are instead always orthogonal. We suggest a definition of Hyperbolic Covariant Coherent Structures (HCCSs), which can be defined on the scalar field representing the angle between the CLVs. HCCSs can be defined for every time instant and could be useful to understand the long-term behavior of particle tracers. We consider three examples: a simple autonomous Hamiltonian system, as well as the non-autonomous “double gyre” and Bickley jet, to see how well the angle is able to describe particular patterns and barriers. We compare the results from the HCCSs with other coherent patterns defined on finite time by the Finite Time Lyapunov Exponents (FTLEs), to see how the behaviors of these structures change asymptotically.
Highlights
In the paradigm of chaotic advection, the trajectories of passive tracers can be complex even when the velocity field of the flow is simple
The Hyperbolic LCS (HLCS) found by using the Finite Time Lyapunov Exponents (FTLEs) are, by definition, strongly related to the time interval chosen for the study, while the Hyperbolic Covariant Coherent Structures (HCCSs), which correspond to instantaneous structures of maximum hyperbolicity, are independent of the time interval considered for their computation, as shown in these two experiments
We have here proposed a new definition and a new computational framework to determine hyperbolic structures in a two-dimensional flow based on covariant Lyapunov vectors
Summary
In the paradigm of chaotic advection, the trajectories of passive tracers can be complex even when the velocity field of the flow is simple. Even flows with a complicated time-dependent structure allow for the formation of coherent patterns that influence the evolution of tracers These structures are common in nature, appearing both at short and long time scales, as well as small and large spatial scales. Geodesic theories are able to detect two other kinds of LCSs called parabolic and elliptic, which are of no interest for the present work All these methods such as FTLEs, FSLEs or the variational and geodesic theories aim to find particular structures on the flow, among these the most repelling, or attracting, structures in the flow on a finite time interval, the HLCS.
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