Abstract
Lagrangian coherent structures (LCS) can be extracted from time-dependent vector fields by means of ridges in the finite-time Lyapunov exponent (FTLE). While the LCS approach has proven successful in many areas and applications for the analysis of time-dependent topology, it is to some extent still an open problem how the finite time scope is appropriately chosen. One has to be aware, however, that the introduction of this finite time scope in the Lyapunov exponent, where the time scope was originally infinite, is largely responsible for the recent success of the FTLE in analysis of real-world data. Hence, there is no general upper bound for the time scope: it depends on the application and the goal of the analysis. There is, however, a clear need for a lower bound of the time scope because the FTLE converges to the eigenvalue of the rate of strain tensor as the time scope approaches zero. Although this does not represent a problem per se, it is the loss of important properties that causes ridges in such FTLE fields to lose the LCS property. LCS are time-dependent separatrices: they separate regions of different behavior over time. Thereby they behave like material constructs, advecting with the vector field and exhibiting negligible cross flow. We present a method for investigating and determining a lower bound for the FTLE time scope at isolated points of its ridges. Our approach applies the advection property to the points where attracting and repelling LCS intersect. These points are of particular interest because they are important in typical questions of Lagrangian topology. We demonstrate our approach with examples from dynamical systems theory and computational fluid dynamics.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Asimov, D.: Notes on the topology of vector fields and flows. Technical Report RNR-93-003, NASA Ames Research Center (1993)
Benettin, G., Galgani, L., Giorgilli, A., Strelcyn, J.M.: Lyapunov characteristic exponent for smooth dynamical systems and Hamiltonian systems; a method for computing all of them. Mechanica 15, 9–20 (1980)
Eberly, D.: Ridges in Image and Data Analysis. Computational Imaging and Vision. Kluwer Academic Publishers, Dordrecht (1996)
Garth, C., Gerhardt, F., Tricoche, X., Hagen, H.: Efficient computation and visualization of coherent structures in fluid flow applications. IEEE Trans. Visual. Comput. Graph. 13(6), 1464–1471 (2007)
Haller, G.: Finding finite-time invariant manifolds in two-dimensional velocity fields. Chaos, 10(1), 99–108 (2000)
Haller, G.: Distinguished material surfaces and coherent structures in three-dimensional fluid flows. Physica D 149, 248–277 (2001)
Helman, J., Hesselink, L.: Representation and display of vector field topology in fluid flow data sets. Computer 22(8), 27–36 (1989)
Kasten, J., Petz, C., Hotz, I., Noack, B., Hege, H.-C.: Localized finite-time Lyapunov exponent for unsteady flow analysis. In: Vision, Modeling, and Visualization, pp. 265–274 (2009)
Mathur, M., Haller, G., Peacock, T., Ruppert Felsot, J.E., Swinney, H.L.: Uncovering the Lagrangian skeleton of turbulence. Phys. Rev. Lett. 98(14) (2007), 144502
Sadlo, F., Peikert, R.: Efficient visualization of Lagrangian coherent structures by filtered AMR ridge extraction. IEEE Trans. Visual. Comput. Graph. 13(5), 1456–1463 (2007)
Sadlo, F., Weiskopf, D.: Time-dependent 2-D vector field topology: an approach inspired by Lagrangian coherent structures. Comput. Graph. Forum 29(1), 88–100 (2010)
Shadden, S.C., Lekien, F., Marsden, J.E.: Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows. Phys. D: Nonlinear Phenom. 212(3–4), 271–304 (2005)
Acknowledgements
This work has been supported by DFG within the Cluster of Excellence in Simulation Technology (EXC 310/1) and the Collaborative Research Centre SFB-TRR 75 at University Stuttgart.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Sadlo, F., Üffinger, M., Ertl, T., Weiskopf, D. (2012). On the Finite-Time Scope for Computing Lagrangian Coherent Structures from Lyapunov Exponents. In: Peikert, R., Hauser, H., Carr, H., Fuchs, R. (eds) Topological Methods in Data Analysis and Visualization II. Mathematics and Visualization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23175-9_18
Download citation
DOI: https://doi.org/10.1007/978-3-642-23175-9_18
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-23174-2
Online ISBN: 978-3-642-23175-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)