American Institute of Mathematical Sciences

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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

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.  Google Scholar

[2]

M. Farazmand and G. Haller, Computing Lagrangian coherent structures from their variational theory, Chaos, 22 (2012), 013128. doi: 10.1063/1.3690153.  Google Scholar

[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.  Google Scholar

[4]

M. Farazmand and G. Haller, How coherent are the vortices of two-dimensional turbulence?,, submitted preprint, ().   Google Scholar

[5]

G. Haller, Lagrangian Coherent Structures, Annual Review of Fluid Mechanics, 47 (2015), 137-161. doi: 10.1146/annurev-fluid-010313-141322.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

D. Karrasch, Attracting Lagrangian coherent structures on Riemannian manifolds, Chaos, 25 (2015), 087411. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[2]

M. Farazmand and G. Haller, Computing Lagrangian coherent structures from their variational theory, Chaos, 22 (2012), 013128. doi: 10.1063/1.3690153.  Google Scholar

[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.  Google Scholar

[4]

M. Farazmand and G. Haller, How coherent are the vortices of two-dimensional turbulence?,, submitted preprint, ().   Google Scholar

[5]

G. Haller, Lagrangian Coherent Structures, Annual Review of Fluid Mechanics, 47 (2015), 137-161. doi: 10.1146/annurev-fluid-010313-141322.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

D. Karrasch, Attracting Lagrangian coherent structures on Riemannian manifolds, Chaos, 25 (2015), 087411. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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