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Do Finite-Size Lyapunov Exponents detect coherent structures?
Daniel Karrasch1, George Haller1
1Institute of Mechanical Systems, ETH Zurich, Tannenstrasse 3, 8092 Zurich, Switzerland.
Chaos (Woodbury, N.Y.)
|January 7, 2014
Summary
Finite-Size Lyapunov Exponent (FSLE) ridges can indicate hyperbolic Lagrangian Coherent Structures (LCSs), but only under specific mathematical conditions. Otherwise, FSLE ridges may be false positives, revealing limitations in detecting Lagrangian coherence.
Area of Science:
- Fluid dynamics
- Dynamical systems theory
- Chaos theory
Background:
- Lagrangian Coherent Structures (LCSs) are fundamental for understanding fluid transport.
- Finite-Size Lyapunov Exponent (FSLE) ridges are commonly used to identify hyperbolic LCSs.
- A rigorous mathematical connection between FSLE and LCSs has been lacking.
Purpose of the Study:
- To establish a rigorous mathematical link between FSLE ridges and hyperbolic LCSs.
- To identify conditions under which FSLE ridges accurately indicate LCSs.
- To explore limitations of FSLE in detecting Lagrangian coherence.
Main Methods:
- Theoretical analysis of FSLE and Finite-Time Lyapunov Exponent (FTLE) fields.
- Mathematical proof establishing conditions for FSLE ridge to LCS correspondence.
- Investigation of FSLE properties such as ill-posedness and discontinuity.
Main Results:
- FSLE ridges meeting specific criteria signal nearby FTLE ridges, indicating hyperbolic LCSs.
- FSLE ridges not meeting these criteria can be false positives for LCSs.
- FSLE exhibits limitations including ill-posedness, artificial discontinuities, and time step sensitivity.
Conclusions:
- The study provides a rigorous mathematical foundation for using FSLE to detect LCSs.
- Identifies conditions for accurate LCS detection using FSLE and highlights potential pitfalls.
- Suggests caution when employing FSLE due to identified limitations in Lagrangian coherence analysis.
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