Abstract
Finite-time and finite-size Lyapunov exponents are related concepts that have been used for the purpose of identifying transport structures in time-dependent flow. The preference for one or the other concept seems to be based more on a tradition within a scientific community than on proven advantages. In this study, we demonstrate that with the two concepts highly similar visualizations can be produced, by maximizing a simple similarity measure. Furthermore, we show that results depend crucially on the numerical implementation of the two concepts.
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References
E. Aurell, G. Boffetta, A. Crisanti, G. Paladin, A. Vulpiani, Growth of noninfinitesimal perturbations in turbulence. Phys. Rev. Lett. 77, 1262–1265 (1996)
G. Benettin, L. Galgani, A. Giorgilli, J.M. Strelcyn, Lyapunov characteristic exponent for smooth dynamical systems and hamiltonian systems: a method for computing all of them. Mechanica 15, 9–20 (1980)
J. Berntsen, User guide for a modesplit σ-coordinate ocean model. Version 4.1. Technical report, Department of Mathematics, University of Bergen, Norway, 2004
F.J. Beron-Vera, M.J. Olascoaga, M.G. Brown, H. Kocak, I.I. Rypina, Invariant-tori-like lagrangian coherent structures in geophysical flows. Chaos 20(1), 1–13 (2010)
G. Boffetta, G. Lacorata, G. Redaelli, A. Vulpiani, Detecting barriers to transport: a review of different techniques. Physica D 159, 58–70 (2001)
A. Bower, A simple kinematic mechanism for mixing fluid parcels across a meandering jet. J. Phys. Oceanogr. 21, 173–171 (1991)
S. Brunton, C. Rowley, Fast computation of finite-time Lyapunov exponent fields for unsteady flows. Chaos 20(017503), 017503-1–017503-12 (2010)
C. Coulliette, F. Lekien, J.D. Paduan, G. Haller, J.E. Marsden, Optimal pollution mitigation in monterey bay based on coastal radar data and nonlinear dynamics. Environ. Sci. Technol. 41(18), 6562–6572 (2007)
F. d’Ovidio, V. Fernández, E. Hernández-García, C. López, Mixing structures in the Mediterranean Sea from finite-size Lyapunov exponents. Geophys. Res. Lett. 31(17), L17203-1–L17203-4 (2004). doi:10.1029/2004GL020328
R. Fuchs, B. Schindler, R. Peikert, Scale-space approaches to FTLE ridges, in Topological Methods in Data Analysis and Visualization II, ed. by R. Peikert, H. Hauser, H. Carr, R. Fuchs (Springer, New York, 2012), pp. 283–296
I. Goldhirsch, P.L. Sulem, S.A. Orszag, Stability and Lyapunov stability of dynamical systems: a differential approach and a numerical method. Physica D 27(3), 311–337 (1987)
G. Haller, Finding finite-time invariant manifolds in two-dimensional velocity fields. Chaos 10(1), 99–108 (2000)
G. Haller, Distinguished material surfaces and coherent structures in three-dimensional fluid flows. Physica D 149, 248–277 (2001)
I. Hernández-Carracos, C. López, E. Hernández-García, A. Turiel, How reliable are finite-size Lyapunov exponents for the assessment of ocean dynamics? Ocean Model. 36(3–4), 208–218 (2011)
B. Joseph, B. Legras, Relation between kinematic boundaries, stirring, and barriers for the antarctic polar vortex. J. Atmos. Sci. 59, 1198–1212 (2002)
D. Karrasch, G. Haller, Do Finite-Size Lyapunov Exponents Detect Coherent Structures? (2013). http://arxiv.org/abs/1307.7888
J. Kasten, C. Petz, I. Hotz, B. Noack, H.C. Hege, Localized finite-time lyapunov exponent for unsteady flow analysis, in Vision Modeling and Visualization, vol. 1, ed. by M. Magnor, B. Rosenhahn, H. Theisel (Universität Magdeburg, Inst. f. Simulation u. Graph., 2009), pp. 265–274
T.Y. Koh, B. Legras, Hyperbolic lines and the stratospheric polar vortex. Chaos 12(2), 382–394 (2002)
A.J. Mariano, A. Griffa, T.M. Özgökmen, E. Zambianchi, Lagrangian analysis and predictability of coastal and ocean dynamics 2000. J. Atmos. Ocean. Technol. 19(7), 1114–1126 (2002)
A. Pobitzer, R. Peikert, R. Fuchs, B. Schindler, A. Kuhn, H. Theisel, K. Matković, H. Hauser, The state of the art in topology-based visualization of unsteady flow. Comput. Graph. Forum 30(6), 1789–1811 (2011)
F. Sadlo, R. Peikert, Efficient visualization of lagrangian coherent structures by filtered AMR ridge extraction. IEEE Trans. Vis. Comput. Graph. 13(5), 1456–1463 (2007)
R. Samelson, Fluid exchange across a meandering jet. J. Phys. Oceanogr. 22, 431–440 (1992)
B. Schindler, R. Peikert, R. Fuchs, H. Theisel, Ridge concepts for the visualization of lagrangian coherent structures, in Topological Methods in Data Analysis and Visualization II, ed. by R. Peikert, H. Hauser, H. Carr, R. Fuchs (Springer, New York, 2012), pp. 221–236
S.C. Shadden, F. Lekien, J.E. Marsden, Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows. Physica D: Nonlinear Phenom. 212(3–4), 271–304 (2005)
M. Üffinger, F. Sadlo, M. Kirby, C.D. Hansen, T. Ertl, FTLE computation beyond first order approximation, in Eurographics Short Papers, ed. by C. Andujar, E. Puppo, Eurographics Association, Cagliari, pp. 61–64, 2012
S. Wiggins, The dynamical systems approach to lagrangian transport in oceanic flows. Annu. Rev. Fluid Mech. 37, 295–328 (2005)
Acknowledgements
We wish to thank Tomas Torsvik, Uni Research, Uni Computing (Bergen, Norway), for the tidal flow data. This work was funded in part by the Seventh Framework Programme for Research of the European Commission, under FET-Open grant number 226042 (project SemSeg) and the Swiss National Science Foundation, under grant number 200020_140556.
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Peikert, R., Pobitzer, A., Sadlo, F., Schindler, B. (2014). A Comparison of Finite-Time and Finite-Size Lyapunov Exponents. In: Bremer, PT., Hotz, I., Pascucci, V., Peikert, R. (eds) Topological Methods in Data Analysis and Visualization III. Mathematics and Visualization. Springer, Cham. https://doi.org/10.1007/978-3-319-04099-8_12
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DOI: https://doi.org/10.1007/978-3-319-04099-8_12
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