Do Finite-Size Lyapunov Exponents Detect Coherent Structures?

Ridges of the Finite-Size Lyapunov Exponent (FSLE) field have been used as indicators of hyperbolic Lagrangian Coherent Structures (LCSs). A rigorous mathematical link between the FSLE and LCSs, however, has been missing. Here we prove that an FSLE ridge satisfying certain conditions does signal a nearby ridge of some Finite-Time Lyapunov Exponent (FTLE) field, which in turn indicates a hyperbolic LCS under further conditions. Other FSLE ridges violating our conditions, however, are seen to be false positives for LCSs. We also find further limitations of the FSLE in Lagrangian coherence detection, including ill-posedness, artificial jump-discontinuities, and sensitivity with respect to the computational time step.Comment: 22 pages, 7 figures, v3: corrects the z-axis labels of Fig. 2 (left) that appears in the version published in Chao

Fast and robust computation of coherent Lagrangian vortices on very large two-dimensional domains

We describe a new method for computing coherent Lagrangian vortices in two-dimensional flows according to any of the following approaches: black-hole vortices [Haller & Beron-Vera, 2013], objective Eulerian Coherent Structures (OECSs) [Serra & Haller, 2016], material barriers to diffusive transport [Haller et al., 2018, Haller et al., 2019], and constrained diffusion barriers [Haller et al., 2019]. The method builds on ideas developed previously in [Karrasch et al., 2015], but our implementation alleviates a number of shortcomings and allows for the fully automated detection of such vortices on unprecedentedly challenging real-world flow problems, for which specific human interference is absolutely infeasible. Challenges include very large domains and/or parameter spaces. We demonstrate the efficacy of our method in dealing with such challenges on two test cases: first, a parameter study of a turbulent flow, and second, computing material barriers to diffusive transport in the global ocean.Comment: 25 pages, 10 figures (partially of very low quality due to size constraint by arxiv.org), postprin

Hyperbolicity & Invariant Manifolds for Finite-Time Processes

The aim of this thesis is to introduce a general framework for what is informally referred to as finite-time dynamics. Within this framework, we study hyperbolicity of reference trajectories, existence of invariant manifolds as well as normal hyperbolicity of invariant manifolds called Lagrangian Coherent Structures. We focus on a simple derivation of analytical results. At the same time, our approach together with the analytical results has strong impact on the numerical implementation by providing calculable expressions for known functions and continuity results that ensure robust computation. The main results of the thesis are robustness of finite-time hyperbolicity in a very general setting, finite-time analogues to classical linearization theorems, an approach to the computation of so-called growth rates and the generalization of the variational approach to Lagrangian Coherent Structures

Linearization of Hyperbolic Finite-Time Processes

We adapt the notion of processes to introduce an abstract framework for dynamics in finite time, i.e.\ on compact time sets. For linear finite-time processes a notion of hyperbolicity namely exponential monotonicity dichotomy (EMD) is introduced, thereby generalizing and unifying several existing approaches. We present a spectral theory for linear processes in a coherent way, based only on a logarithmic difference quotient, prove robustness of EMD with respect to a suitable (semi-)metric and provide exact perturbation bounds. Furthermore, we give a complete description of the local geometry around hyperbolic trajectories, including a direct and intrinsic proof of finite-time analogues of the local (un)stable manifold theorem and theorem of linearized asymptotic stability. As an application, we discuss our results for ordinary differential equations on a compact time-interval.Comment: 32 page