Attraction-based computation of hyperbolic Lagrangian coherent structures

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January 2015, 2(1): 83-93. doi: 10.3934/jcd.2015.2.83

Attraction-based computation of hyperbolic Lagrangian coherent structures

Daniel Karrasch ^1, , Mohammad Farazmand ^2, and George Haller ^1,

1.	ETH Zürich, Institute of Mechanical Systems, Leonhardstrasse 21, 8092 Zürich, Switzerland, Switzerland
2.	ETH Zürich, Institute of Mechanical Systems, Rämistrasse 101, 8092 Zürich, Switzerland

Received May 2014 Revised October 2014 Published August 2015

Abstract
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Recent advances enable the simultaneous computation of both attracting and repelling families of Lagrangian Coherent Structures (LCS) at the same initial or final time of interest. Obtaining LCS positions at intermediate times, however, has been problematic, because either the repelling or the attracting family is unstable with respect to numerical advection in a given time direction. Here we develop a new approach to compute arbitrary positions of hyperbolic LCS in a numerically robust fashion. Our approach only involves the advection of attracting material surfaces, thereby providing accurate LCS tracking at low computational cost. We illustrate the advantages of this approach on a simple model and on a turbulent velocity data set.

Keywords: mixing, Lagrangian Coherent Structures, Transport, tracking, stability., nonautonomous dynamical systems.

Mathematics Subject Classification: Primary: 37C60; Secondary: 37N1.

Citation: Daniel Karrasch, Mohammad Farazmand, George Haller. Attraction-based computation of hyperbolic Lagrangian coherent structures. Journal of Computational Dynamics, 2015, 2 (1) : 83-93. doi: 10.3934/jcd.2015.2.83

References:

[1]	M. Farazmand, D. Blazevski and G. Haller, Shearless transport barriers in unsteady two-dimensional flows and maps, Physica D, 278-279 (2014), 44-57. doi: 10.1016/j.physd.2014.03.008.
[2]	M. Farazmand and G. Haller, Computing Lagrangian coherent structures from their variational theory, Chaos, 22 (2012), 013128. doi: 10.1063/1.3690153.
[3]	M. Farazmand and G. Haller, Attracting and repelling Lagrangian coherent structures from a single computation, Chaos, 23 (2013), 023101. doi: 10.1063/1.4800210.
[4]	M. Farazmand and G. Haller, How coherent are the vortices of two-dimensional turbulence?, submitted preprint, arXiv:1402.4835.
[5]	G. Haller, Lagrangian Coherent Structures, Annual Review of Fluid Mechanics, 47 (2015), 137-161. doi: 10.1146/annurev-fluid-010313-141322.
[6]	G. Haller and F. J. Beron-Vera, Geodesic theory of transport barriers in two-dimensional flows, Physica D, 241 (2012), 1680-1702. doi: 10.1016/j.physd.2012.06.012.
[7]	G. Haller and T. Sapsis, Lagrangian coherent structures and the smallest finite-time Lyapunov exponent, Chaos, 21 (2011), 023115, 7pp. doi: 10.1063/1.3579597.
[8]	G. Haller and G. Yuan, Lagrangian coherent structures and mixing in two-dimensional turbulence, Physica D, 147 (2000), 352-370. doi: 10.1016/S0167-2789(00)00142-1.
[9]	D. Karrasch, Attracting Lagrangian coherent structures on Riemannian manifolds, Chaos, 25 (2015), 087411.
[10]	A. M. Mancho, D. Small, S. Wiggins and K. Ide, Computation of stable and unstable manifolds of hyperbolic trajectories in two-dimensional, aperiodically time-dependent vector fields, Physica D, 182 (2003), 188-222. doi: 10.1016/S0167-2789(03)00152-0.
[11]	K. Onu, F. Huhn and G. Haller, LCS Tool: A computational platform for Lagrangian coherent structures, Journal of Computational Science, 7 (2015), 26-36. doi: 10.1016/j.jocs.2014.12.002.
[12]	R. Peikert and F. Sadlo, Height Ridge Computation and Filtering for Visualization, in Visualization Symposium, 2008. PacificVIS '08. IEEE Pacific, 2008, 119-126. doi: 10.1109/PACIFICVIS.2008.4475467.
[13]	B. Schindler, R. Peikert, R. Fuchs and H. Theisel, Ridge Concepts for the Visualization of Lagrangian Coherent Structures, in Topological Methods in Data Analysis and Visualization II (eds. R. Peikert, H. Hauser, H. Carr and R. Fuchs), Mathematics and Visualization, Springer, 2012, 221-235. doi: 10.1007/978-3-642-23175-9_15.
[14]	K.-F. Tchon, J. Dompierre, M.-G. Vallet, F. Guibault and R. Camarero, Two-dimensional metric tensor visualization using pseudo-meshes, Engineering with Computers, 22 (2006), 121-131. doi: 10.1007/s00366-006-0012-3.

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References:

[1]	M. Farazmand, D. Blazevski and G. Haller, Shearless transport barriers in unsteady two-dimensional flows and maps, Physica D, 278-279 (2014), 44-57. doi: 10.1016/j.physd.2014.03.008.
[2]	M. Farazmand and G. Haller, Computing Lagrangian coherent structures from their variational theory, Chaos, 22 (2012), 013128. doi: 10.1063/1.3690153.
[3]	M. Farazmand and G. Haller, Attracting and repelling Lagrangian coherent structures from a single computation, Chaos, 23 (2013), 023101. doi: 10.1063/1.4800210.
[4]	M. Farazmand and G. Haller, How coherent are the vortices of two-dimensional turbulence?, submitted preprint, arXiv:1402.4835.
[5]	G. Haller, Lagrangian Coherent Structures, Annual Review of Fluid Mechanics, 47 (2015), 137-161. doi: 10.1146/annurev-fluid-010313-141322.
[6]	G. Haller and F. J. Beron-Vera, Geodesic theory of transport barriers in two-dimensional flows, Physica D, 241 (2012), 1680-1702. doi: 10.1016/j.physd.2012.06.012.
[7]	G. Haller and T. Sapsis, Lagrangian coherent structures and the smallest finite-time Lyapunov exponent, Chaos, 21 (2011), 023115, 7pp. doi: 10.1063/1.3579597.
[8]	G. Haller and G. Yuan, Lagrangian coherent structures and mixing in two-dimensional turbulence, Physica D, 147 (2000), 352-370. doi: 10.1016/S0167-2789(00)00142-1.
[9]	D. Karrasch, Attracting Lagrangian coherent structures on Riemannian manifolds, Chaos, 25 (2015), 087411.
[10]	A. M. Mancho, D. Small, S. Wiggins and K. Ide, Computation of stable and unstable manifolds of hyperbolic trajectories in two-dimensional, aperiodically time-dependent vector fields, Physica D, 182 (2003), 188-222. doi: 10.1016/S0167-2789(03)00152-0.
[11]	K. Onu, F. Huhn and G. Haller, LCS Tool: A computational platform for Lagrangian coherent structures, Journal of Computational Science, 7 (2015), 26-36. doi: 10.1016/j.jocs.2014.12.002.
[12]	R. Peikert and F. Sadlo, Height Ridge Computation and Filtering for Visualization, in Visualization Symposium, 2008. PacificVIS '08. IEEE Pacific, 2008, 119-126. doi: 10.1109/PACIFICVIS.2008.4475467.
[13]	B. Schindler, R. Peikert, R. Fuchs and H. Theisel, Ridge Concepts for the Visualization of Lagrangian Coherent Structures, in Topological Methods in Data Analysis and Visualization II (eds. R. Peikert, H. Hauser, H. Carr and R. Fuchs), Mathematics and Visualization, Springer, 2012, 221-235. doi: 10.1007/978-3-642-23175-9_15.
[14]	K.-F. Tchon, J. Dompierre, M.-G. Vallet, F. Guibault and R. Camarero, Two-dimensional metric tensor visualization using pseudo-meshes, Engineering with Computers, 22 (2006), 121-131. doi: 10.1007/s00366-006-0012-3.

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