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 rig-orous 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 co-herence detection, including ill-posedness, artificial jump-discontinuities, and sensitivity with respect to the computational time step. Originally developed as a diagnostic for multi-scale mixing, the Finite-Size Lyapunov Exponent (FSLE) has also been broadly used to detect coherent structures in dynamical systems. This use of the FSLE is motivated by a heuristic analogy with the Finite-Time Lyapunov Exponent (FTLE), a classic measure of particle separation. Here we derive conditions under which this analogy is mathematically justified. We also show by examples, however, that the FSLE field has several shortcomings when applied to coherent structure detection. 1