Journal of Fixed Point Theory and Applications

, Volume 7, Issue 2, pp 351–384 | Cite as

Horseshoes in hurricanes



The method of using Finite Time Liapunov Exponents (FTLE) to extract Lagrangian Coherent Structures (LCS) in aperiodic flows, as originally developed by Haller, is applied to geophysical flows. In this approach, the LCS are identified as surfaces of greatest separation that parse the flow into regions with different dynamical behavior. In this way, the LCS reveal the underlying skeleton of turbulence. The time-dependence of the LCS provides insight into the mechanisms by which fluid is transported from one region to another. Of especial interest in this study is the utility with which the FTLE-LCS method can be used to reveal homoclinic and horseshoe dynamics in aperiodic flows.

The FTLE-LCS method is applied to turbulent flow in hurricanes and reveals LCS that delineate sharp boundaries to a storm. Moreover, intersections of the LCS define lobes that mediate transport into and out of a storm through the action of homoclinic lobe dynamics. Using the FTLE-LCS method, the same homoclinic structures are seen to be a dominant transport mechanism in the Global Ocean, and provide insights into the role of mesoscale eddies in enhancing lateral mixing.

Mathematics Subject Classification (2010)

37C10 86-08 76-04 


Finite time Lyapunov exponents Lagrangian coherent structures homoclinic tangles hurricanes mesoscale ocean eddies transport mixing 


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  1. Aurell 1997.
    Aurell E., Boffetta G., Crisanti A., Paladin G., Vulpiani A.: Predictability in the large: An extension of the concept of Lyapunov exponent. J. Phys. A 30, 1–26 (1997)MATHCrossRefMathSciNetGoogle Scholar
  2. Beigie 1991.
    Beigie D., Leonard A., Wiggins S.: Chaotic transport in the homoclinic and heteroclinic tangle regions of quasiperiodically forced two-dimensional dynamical systems. Nonlinearity 4, 775–819 (1991)MATHCrossRefMathSciNetGoogle Scholar
  3. Bryan 2008.
    F. O. Bryan, Introduction: Ocean modeling—eddy or not. In: Ocean Modeling in an Eddying Regime, M. Hecht and H. Hasumi (eds.), American Geophysical Union, 2008, 1–3.Google Scholar
  4. Campbell 2005.
    Campbell D.K., Rosenau P., Zaslavsky G.M.: Introduction: The Fermi–Pasta–Ulam problem—The first fifty years. Chaos 15, 015101 (2005)CrossRefGoogle Scholar
  5. Chelton 2007.
    Chelton D.B., Schlax M.G., Samelson R.M., de Szoeke R.A.: Global observations of large oceanic eddies. Geophys. Res. Lett. 34, L15606 (2007)CrossRefGoogle Scholar
  6. Chen 2007.
    Chen S.S., Zhao W., Donelan M.A., Price J.F., Walsh E.J.: The CBLAST-Hurricane Program and the next-generation fully coupled atmosphere-wave-ocean models for hurricane research and prediction. Bull. Amer. Meteor. Soc 88, 311–317 (2007)CrossRefGoogle Scholar
  7. d’Ovidio et al. 2004
    d’Ovidio F., Fernández V., Hernández-Garca E., López C.: Mixing structures in the Mediterranean Sea from finite-size Lyapunov exponents. Geophys. Res. Lett. 31, L17203 (2004)CrossRefGoogle Scholar
  8. Dellnitz et al. 2005
    Dellnitz M., Junge O., Koon W.S., Lekien F., Lo M.W., Marsden J.E., Padberg K., Preis R., Ross S., Thiere B.: Transport in dynamical astronomy and multibody problems. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 15, 699–727 (2005)MATHCrossRefMathSciNetGoogle Scholar
  9. Drevillon 2008.
    Drvillon M.: The GODAE/Mercator-Ocean global ocean forecasting system: Results, applications and prospects. J. Operational Oceanogr. 1, 51–57 (2008)Google Scholar
  10. Emanuel 2001.
    Emanuel K.: Contribution of tropical cyclones to meridional heat transport by the oceans. J. Geophys. Res. 106, 14771–14781 (2001)CrossRefGoogle Scholar
  11. Emanuel 2007.
    Emanuel K.: Environmental factors affecting tropical cyclone power dissipation. J. Climate 20, 5497–5509 (2007)CrossRefGoogle Scholar
  12. Fox-Kemper 2008.
    B. Fox-Kemper and D. Menemenlis, Can large eddy simluation techniques improve mesoscale rich ocean models? In: Ocean Modeling in an Eddying Regime, M. Hecht and H. Hasumi (eds.), American Geophysical Union, 2008, 319–337.Google Scholar
  13. Franco 2007.
    Franco E., Pekarek D., Peng J., Dabiri J.: Geometry of unsteady fluid transport during fluid-structure interactions. J. Fluid Mech. 589, 125–145 (2007)MATHCrossRefMathSciNetGoogle Scholar
  14. Froyland 2007.
    Froyland G., Padberg K., England M.H., Treguier A.M.: Detection of coherent oceanic structures via transfer operators. Phys. Rev. Lett. 98, 224503 (2007)CrossRefGoogle Scholar
  15. Froyland and Padberg 2009.
    Froyland G., Padberg K.: Almost invariant sets and invariant manifold—connecting probabilistic and geometric descriptions of coherent structures in flows. Phys. D 238, 1507–1523 (2009)MATHCrossRefMathSciNetGoogle Scholar
  16. Gawlik et al. 2009
    Gawlik E., Marsden J., Du Toit P., Campagnola S.: Lagrangian coherent structures in the planar elliptic restricted three-body problem. Celestial Mech. Dynam. Astronom. 103, 227–249 (2009)MATHCrossRefMathSciNetGoogle Scholar
  17. Gent 1990.
    Gent P.R., Mcwilliams J.C.: Isopycnal mixing in ocean circulation models. J. Phys. Oceanogr. 20, 150–155 (1990)CrossRefGoogle Scholar
  18. Haller and Yuan 2000.
    Haller G., Yuan G.: Lagrangian coherent structures and mixing in two-dimensional turbulence. Phys. D 147, 352–370 (2000)MATHCrossRefMathSciNetGoogle Scholar
  19. Haller 2001.
    Haller G.: Distinguished material surfaces and coherent structures in three-dimensional fluid flows. Phys. D 149, 248–277 (2001)MATHCrossRefMathSciNetGoogle Scholar
  20. Haller 2002.
    Haller G.: Lagrangian coherent structures from approximate velocity data. Phys. Fluids 14, 1851–1861 (2002)CrossRefMathSciNetGoogle Scholar
  21. Hecht et al. 2008a
    Hecht M.W., Holm D.D., Petersen M.R., Wingate B.A.: The LANS-α and Leray turbulence parameterizations in primitive equation ocean modeling. J. Phys. A 41, 344009 (2008)CrossRefMathSciNetGoogle Scholar
  22. Hecht et al. 2008b
    Hecht M.W., Holm D.D., Petersen M.R., Wingate B.A.: Implementation of the LANS-α turbulence model in a primitive equation ocean model. J. Comput. Phys. 227, 5691–5716 (2008)MATHCrossRefMathSciNetGoogle Scholar
  23. Houze et al. 2007
    Houze J., Robert A., Chen S.S., Smull B.F., Lee W., Bell M.M.: Hurricane intensity and eyewall replacement. Science 315, 1235–1239 (2007)CrossRefGoogle Scholar
  24. Ide et al. 2002
    Ide K., Small D., Wiggins S.: Distinguished hyperbolic trajectories in time-dependent fluid flows: Analytical and computational approach for velocity fields defined as data sets. Nonlinear Process. Geophys. 9, 237–263 (2002)CrossRefGoogle Scholar
  25. Junge et al. 2004
    O. Junge, J. E. Marsden, and I. Mezic, Uncertainty in the dynamics of conservative maps. In: Proceedings of the 43rd IEEE Connference on Decision and Control, 2004, 2225–2230.Google Scholar
  26. Kuo et al. 2003
    B. Kuo, et al., Hurricane Isabel Data Produced by the Weather Research and Forecast (WRF) Model, Courtesy of NCAR, and the U.S. National Science Foundation.Google Scholar
  27. Lekien 2003.
    F. Lekien, Time-dependent dynamical systems and geophysical flows. PhD thesis, California Institute of Technology, 2003.Google Scholar
  28. Lekien et al. 2005
    Lekien F., Coulliette C., Mariano A.J., Ryan E.H., Shay L.K., Haller G., Marsden J.: Pollution release tied to invariant manifolds: A case study for the coast of Florida. Phys. D 210, 1–20 (2005)MATHCrossRefMathSciNetGoogle Scholar
  29. Lekien et al. 2007
    Lekien F., Shadden S.C., Marsden J.E.: Lagrangian coherent structures in n-dimensional systems. J. Math. Phys. 48, 065404 (2007)CrossRefMathSciNetGoogle Scholar
  30. Li et al. 2008
    Li Z., Chao Y., McWilliams J.C., Ide K.: A three-dimensional variational data assimilation scheme for the Regional Ocean Modeling System: Implementation and basic experiments. J. Geophys. Res. 113, C05002 (2008)CrossRefGoogle Scholar
  31. Lipinski and Mohseni 2010.
    Lipinski D., Mohseni K.: A ridge tracking algorithm and error estimate for efficient computation of Lagrangian coherent structures. Chaos 20, 017504 (2010)CrossRefGoogle Scholar
  32. Lorenz 1963.
    Lorenz E.N.: Deterministic non-periodic flow. J. Atmos. Sci. 20, 130–141 (1963)CrossRefGoogle Scholar
  33. MacKay et al. 1984
    MacKay R.S., Meiss J.D., Percival I.C.: Transport in Hamiltonian systems. Phys. D 13, 55–81 (1984)MATHCrossRefMathSciNetGoogle Scholar
  34. Mathur 2007.
    Mathur M., Haller G., Peacock T., Ruppert-Felsot J.E., Swinney H.L.: Uncovering the Lagrangian skeleton of turbulence. Phys. Rev. Lett. 98, 144502 (2007)CrossRefGoogle Scholar
  35. Odier 2007.
    P. Odier, Characterization of turbulent mixing in an Oceanic Overflow Facility. Abstract submitted to DFD07 Meeting of the American Physical Society, 2007.Google Scholar
  36. Peacock and Dabiri 2010.
    Peacock T., Dabiri J.: Introduction to Focus Issue: Lagrangian coherent structures. Chaos 20, 017501 (2010)CrossRefGoogle Scholar
  37. Peng and Dabiri 2009.
    Peng J., Dabiri J.O.: Transport of inertial particles by Lagrangian coherent structures: Application to predator-prey interaction in jellyfish feeding. J. Fluid Mech. 623, 75–84 (2009)MATHCrossRefGoogle Scholar
  38. Poincaré 1899.
    H. Poincaré, New Methods of Celestial Mechanics. Springer, 1899.Google Scholar
  39. Rom-Kedar et al. 1990
    Rom-Kedar V., Leonard A., Wiggins S.: An analytical study of transport, mixing and chaos in an unsteady vortical flow. J. Fluid Mech. 214, 347–394 (1990)MATHCrossRefMathSciNetGoogle Scholar
  40. Rom-Kedar et al. 1991
    Rom-Kedar V., Wiggins S.: Transport in two-dimensional maps: Concepts, examples, and a comparison of the theory of Rom-Kedar and Wiggins with the Markov model of MacKay, Meiss, Ott, and Percival. Phys. D 51, 248–266 (1991)MATHCrossRefMathSciNetGoogle Scholar
  41. Sadlo and Peikert 2007.
    Sadlo F., Peikert R.: Efficient visualization of Lagrangian coherent structures by filtered AMR ridge extraction. IEEE Trans. Vis. Comput. Graph. 13, 1456–1463 (2007)CrossRefGoogle Scholar
  42. Sapsis and Haller 2009.
    Sapsis T., Haller G.: Inertial particle dynamics in a hurricane. J. Atmosph. Sci. 66, 2481–2492 (2009)CrossRefGoogle Scholar
  43. Schiermeier 2007.
    Schiermeier Q.: Oceanography: Churn, churn, churn. Nature 447, 522–524 (2007)CrossRefGoogle Scholar
  44. Shadden 2005.
    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 212, 271–304 (2005)MATHCrossRefMathSciNetGoogle Scholar
  45. Shadden 2007.
    Shadden S.C., Katija K., Rosenfeld M., Marsden J.E., Dabiri J.O.: Transport and stirring induced by vortex formation. J. Fluid Mech. 593, 315–331 (2007)MATHCrossRefGoogle Scholar
  46. Shadden and Taylor 2008.
    Shadden S., Taylor C.: Characterization of coherent structures in the cardiovascular system. Ann. Biomed. Eng. 36, 1152–1162 (2008)CrossRefGoogle Scholar
  47. Smale 1967.
    Smale S.: Differentiable dynamical systems. Bull. Amer. Math. Soc. 73, 747–817 (1967)CrossRefMathSciNetGoogle Scholar
  48. Smale 1998.
    Smale S.: Finding a horseshoe on the beaches of Rio. Math. Intelligencer 20, 39–44 (1998)MATHCrossRefMathSciNetGoogle Scholar
  49. Solomon et al. 2007
    S. Solomon, D. Qin, M. Manning, Z. Chen, M. Marquis, K. B. Averyt, M. Tignor, and H. L. Miller (eds.), Climate Change 2007: The Physical Science Basis. Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change, Cambridge University Press, 2007.Google Scholar
  50. Sundermeyer et al. 2007
    Sundermeyer M.A., Terray E.A., Ledwell J.R., Cunningham A.G., LaRocque P.E., Banic J., Lillycrop W.J.: Threedimensional mapping of fluorescent dye using a scanning, depth-resolving airborne lidar. J. Atmos. Oceanic Technol. 24, 1050–1065 (2007)CrossRefGoogle Scholar
  51. Tallapragada 2008.
    Tallapragada P., Ross S.D.: Particle segregation by Stokes number for small neutrally buoyant spheres in a fluid. Phys. Rev. E 78, 036308 (2008)CrossRefGoogle Scholar
  52. Tanaka 2008.
    Tanaka M., Ross S.: Separatrices and basins of stability from time series data: An application to biodynamics. Nonlinear Dynam. 58, 1–21 (2009)MATHCrossRefMathSciNetGoogle Scholar
  53. Vellinga 2002.
    Vellinga M., Wood R.A.: Global climatic impacts of a collapse of the Atlantic thermohaline circulation. Climatic Change 54, 251–267 (2002)CrossRefGoogle Scholar
  54. Weinkauf 2007.
    Weinkauf T., Sahner J., Sahner J., Theisel H., Theisel H., Hege H.C.: Cores of swirling particle motion in unsteady flows. IEEE Trans. Vis. Comput. Graph. 13, 1759–1766 (2007)CrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  1. 1.Numerica CorporationLovelandUSA
  2. 2.California Institute of TechnologyControl and Dynamical SystemsPasadenaUSA

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