Chaos, Transport and Diffusion

  • Guido Boffetta
  • Guglielmo Lacorata
  • Angelo Vulpiani
Part of the Understanding Complex Systems book series (UCS)


This chapter presents basic elements of chaotic dynamical system theory. The concept of Lyapunov exponent, predictability time and Lagrangian chaos are introduced together with examples. The second part is devoted to the discussion of Lagrangian chaos, in particular in two dimensions, and its relation with Eulerian properties of the flow. The last part of the chapter contains an introduction to diffusion and transport processes, with particular emphasis on the treatment of non-ideal cases.


Velocity Field Lyapunov Exponent Fluid Particle Point Vortex Coherent Lagrangian Structure 
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  1. 1.
    Aref, H.: Integrable, chaotic and turbulent vortex motion in two-dimensional flows. Ann. Rev. Fluid Mech. 15, 345–389 (1983)CrossRefADSMathSciNetGoogle Scholar
  2. 2.
    Aref, H.: Stirring by chaotic advection. J. Fluid Mech. 143, 1–21 (1984)CrossRefADSzbMATHMathSciNetGoogle Scholar
  3. 3.
    Aref, H., Balachandar, S.: Chaotic advection in a Stokes flow. Phys. Fluids 29, 3515–3521 (1986)CrossRefADSzbMATHMathSciNetGoogle Scholar
  4. 4.
    Arnold, V.I.: Sur une propriété topologique des applications globalement canoniques de la mécanique classique. C. R. Acad. Sci. Paris A 261, 3719–3722 (1965)Google Scholar
  5. 5.
    Artale, V., Boffetta, G., Celani, A., Cencini, M., Vulpiani, A.: Dispersion of passive tracers in closed basins: beyond the diffusion coefficient. Phys. Fluids A 9, 3162–3171 (1997)CrossRefADSzbMATHMathSciNetGoogle Scholar
  6. 6.
    Babiano, A., Boffetta, G., Provenzale, A., Vulpiani, A.: Chaotic advection in point vortex models and two-dimensional turbulence. Phys. Fluids A 6, 2465–2474 (1994)CrossRefADSzbMATHMathSciNetGoogle Scholar
  7. 7.
    Benettin, G., Giorgilli, A., Galgani, L., Strelcyn, J.M.: Lyapunov characteristic exponents for smooth dynamical systems and for Hamil- tonian systems; a method for computing all of them. Meccanica 15, 9–30 (1980)CrossRefADSzbMATHGoogle Scholar
  8. 8.
    Berti, S., Dos Santos, F.A., Lacorata, G., Vulpiani, A.: Lagrangian drifter dispersion in the southwestern Atlantic Ocean. J. Phys. Oceanogr. 41, 1659–1672 (2011)CrossRefADSGoogle Scholar
  9. 9.
    Biferale, L., Crisanti, A., Vergassola, M., Vulpiani, A.: Eddy diffusivities in scalar transport. Phys. Fluids 7, 2725–2734 (1995)CrossRefADSzbMATHMathSciNetGoogle Scholar
  10. 10.
    Boffetta, G., Celani, A.: Pair dispersion in turbulence. Physica A 280, 1–9 (2000)CrossRefADSGoogle Scholar
  11. 11.
    Boffetta, G., Cencini, M., Espa, S., Querzoli, G.: Chaotic advection and relative dispersion in an experimental convective flow. Phys. Fluids 12, 3160–3167 (2000)CrossRefADSGoogle Scholar
  12. 12.
    Boffetta, G., Celani, A., Cencini, M., Lacorata, G., Vulpiani, A.: Non-asymptotic properties of transport and mixing. Chaos 10(1), 50–60 (2000)CrossRefADSzbMATHMathSciNetGoogle Scholar
  13. 13.
    Boffetta, G., Lacorata, G., Redaelli, G., Vulpiani, A.: Barriers to transport: a review of different techniques. Physica D 159, 58–70 (2001)CrossRefADSzbMATHGoogle Scholar
  14. 14.
    Boffetta, G., Cencini, M., Falcioni, M., Vulpiani, A.: Predictability: a way to characterize complexity. Phys. Rep. 356(6), 367–474 (2002)CrossRefADSzbMATHMathSciNetGoogle Scholar
  15. 15.
    Boldrighini, C., Franceschini, V.: A five-mode truncation of the plane Navier-Stokes equations. Commun. Math. Phys. 64, 159–170 (1979)CrossRefADSzbMATHMathSciNetGoogle Scholar
  16. 16.
    Bouchaud, J.P., Georges, A.: Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications. Phys. Rep. 195, 127–293 (1990)CrossRefADSMathSciNetGoogle Scholar
  17. 17.
    Bower, A.S.: A simple kinematic mechanism for mixing fluid parcels across a meandering jet. J. Phys. Oceanogr. 21, 173–180 (1991)CrossRefADSGoogle Scholar
  18. 18.
    Cardoso, O., Tabeling, P.: Anomalous diffusion in a linear array of vortices. Europhys. Lett. 7, 225–230 (1988)CrossRefADSGoogle Scholar
  19. 19.
    Chaiken, J., Chu, C.K., Tabor, M., Tan, Q.M.: Lagrangian turbulence in Stokes flow. Phys. Fluids 30, 687–694 (1987)CrossRefADSMathSciNetGoogle Scholar
  20. 20.
    Chandrasekhar, S.: Stochastic problems in physics and astronomy. Rev. Mod. Phys. 15, 1–89 (1943)CrossRefADSzbMATHMathSciNetGoogle Scholar
  21. 21.
    Chirikov, B.V.: A universal instability of many dimensional oscillator systems. Phys. Rep. 52, 263–379 (1979)CrossRefADSMathSciNetGoogle Scholar
  22. 22.
    Crisanti, A., Falcioni, M., Paladin, G., Vulpiani, A.: Lagrangian chaos: transport, mixing and diffusion in fluids. Riv. Nuovo Cim. 14, 1–80 (1991)CrossRefADSMathSciNetGoogle Scholar
  23. 23.
    d’Ovidio, F., Isern-Fontanet, J., Lopez, C., Hernandez-Garcia, E., Garcia-Ladona, E.: Comparison between Eulerian diagnostics and finite-size Lyapunov exponents computed from altimetry in the Algerian basin. Deep Sea Res. I 56, 15–31 (2009)CrossRefGoogle Scholar
  24. 24.
    Falcioni, M., Paladin, G., Vulpiani, A.: Regular and chaotic motion of fluid-particles in two-dimensional fluids. J. Phys. A: Math. Gen. 21, 3451–3462 (1988)CrossRefADSzbMATHMathSciNetGoogle Scholar
  25. 25.
    Garcia-Olivares, A., Isern-Fontanet, J. Garcia-Ladona, E.: Dispersion of passive tracers and finite-scale Lyapunov exponents in the Western Mediterranean Sea. Deep Sea Res. I Oceanogr. Res. Pap. 54(2), 253–268 (2007)CrossRefADSGoogle Scholar
  26. 26.
    Gardiner, C.W.: Handbook of Stochastic Methods. Springer, Berlin (1985)Google Scholar
  27. 27.
    Grassberger, P., Procaccia, I.: Estimation of the Kolmogorov entropy from a chaotic signal. Phys. Rev. A 28, 2591–2593 (1983)CrossRefADSGoogle Scholar
  28. 28.
    Havlin, S., Ben-Avraham, D.: Diffusion in disordered media. Adv. Phys. 36, 695–798 (1987)CrossRefADSGoogle Scholar
  29. 29.
    Hénon, M.: Sur la topologie des lignes de courant dans un cas particulier. C. R. Acad. Sci. Paris A 262, 312–314 (1966)Google Scholar
  30. 30.
    Hernandez-Carrasco, I., Lopez, C., Hernandez-Garcia, E., Turiel, A.: How reliable are finite-size Lyapunov exponents for the assessment of ocean dynamics? Ocean Model. 36, 208–218 (2011)CrossRefADSGoogle Scholar
  31. 31.
    Ichikawa, Y.H., Kamimura, T., Hatori, T.: Stochastic diffusion in the standard map. Physica D 29, 247–255 (1987)CrossRefADSGoogle Scholar
  32. 32.
    Isichenko, M.B.: Percolation, statistical topography, and transport in random media. Rev. Mod. Phys. 64, 961–1044 (1992)CrossRefADSMathSciNetGoogle Scholar
  33. 33.
    Joseph, B., Legras, B.: Relation between kinematic boundaries, stirring and barriers for the Antarctic Polar Vortex. J. Atmos. Sci. 59, 1198–1212 (2002)CrossRefADSGoogle Scholar
  34. 34.
    Koh, T.-Y., Legras, B.: Hyperbolic lines and the stratospheric Polar Vortex. Chaos 12(2), 382–394 (2002)CrossRefADSzbMATHMathSciNetGoogle Scholar
  35. 35.
    Lacasce, J.H.: Statistics from Lagrangian observations. Prog. Oceanogr. 77, 1–29 (2008)CrossRefADSGoogle Scholar
  36. 36.
    Lacasce, J.H., Ohlmann, C.: Relative dispersion at the surface of the Gulf of Mexico. J. Mar. Res. 61, 285–312 (2003)CrossRefGoogle Scholar
  37. 37.
    Lacorata, G., Espa, S.: On the influence of a β-effect on Lagrangian diffusion. Geophys. Res. Lett. 39, L11605 (2012)CrossRefADSGoogle Scholar
  38. 38.
    Lacorata, G., Aurell, E., Vulpiani, A.: Drifter dispersion in the Adriatic Sea: Lagrangian data and chaotic model. Ann. Geophys. 19, 1–9 (2001)CrossRefGoogle Scholar
  39. 39.
    Lacorata, G., Aurell, E., Legras, B., Vulpiani, A.: Evidence for a \(k^{-5/3}\) spectrum from the EOLE Lagrangian balloon in the low stratosphere. J. Atmos. Sci. 61, 2936–2942 (2004)CrossRefADSGoogle Scholar
  40. 40.
    Lamb, H.: Hydrodynamics. Dover Publications, New York (1945)Google Scholar
  41. 41.
    Landau, L.D., Lifshitz, L.: Fluid Mechanics. Pergamon Press, New York (1987)zbMATHGoogle Scholar
  42. 42.
    Lee, J.: Triad-angle locking in low-order models of the 2D Navier-Stokes equations. Physica D 24, 54–70 (1987)CrossRefADSzbMATHMathSciNetGoogle Scholar
  43. 43.
    Lin, C.C.: On the motion of vortices in two dimensions. Proc. Natl. Acad. Sci. USA 27, 575–577 (1941)CrossRefADSGoogle Scholar
  44. 44.
    Lorenz, E.N.: Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130–141 (1963)CrossRefADSGoogle Scholar
  45. 45.
    Majda, A.J., Kramer, P.R.: Simplified models for turbulent diffusion: theory, numerical modeling and physical phenomena. Phys. Rep. 314, 237–574 (1999)CrossRefADSMathSciNetGoogle Scholar
  46. 46.
    Marsden, J.E., McCracken, M.: The Hopf Bifurcation and Its Applications. MIT Press, Cambridge, MA (1975)Google Scholar
  47. 47.
    Maxey, M.R., Riley, J.J.: Equation of motion for a small rigid sphere in a non uniform flow. Phys. Fluids 26, 883–889 (1983)CrossRefADSzbMATHGoogle Scholar
  48. 48.
    Moffat, H.K.: Transport effects associated with turbulence, with particular attention to the influence of helicity. Rep. Prog. Phys. 46, 621–664 (1983)CrossRefADSGoogle Scholar
  49. 49.
    Osborne, A.R., Kirwan, A.D., Provenzale, A., Bergamasco, L.: A search for chaotic behaviour in large and mesoscale motions in the Pacific Ocean. Physica D 23, 75–83 (1986)CrossRefADSGoogle Scholar
  50. 50.
    Ott, E.: Chaos in Dynamical Systems. Cambridge University Press, Cambridge (1993)zbMATHGoogle Scholar
  51. 51.
    Samelson, R.M.: Fluid exchange across a meandering jet. J. Phys. Oceanogr. 22, 431–440 (1992)CrossRefADSGoogle Scholar
  52. 52.
    Shraiman, B.I., Siggia, E.D.: Scalar Turbulence. Nature 405, 639–646 (2000)Google Scholar
  53. 53.
    Solomon, T.H., Gollub, J.P.: Passive transport in steady Rayleigh-Bénard convection. Phys. Fluids 31, 1372–1379 (1988)CrossRefADSGoogle Scholar
  54. 54.
    Taylor, G.I.: Diffusion by continuous movements. Proc. Lond. Math. Soc. 2, 196–211 (1921)Google Scholar
  55. 55.
    Tritton, D.J.: Physical Fluid Dynamics. Oxford Science Publications, Oxford (1988)Google Scholar
  56. 56.
    Vergassola, M.: In: Benest, D., Froeschlé, C. (eds.) Analysis and Modelling of Discrete Dynamical Systems, vol. 229. Gordon & Breach, Amsterdam (1998)Google Scholar
  57. 57.
    Wei, X., Ni, P., Zhan, H.: Monitoring cooling water discharge using Lagrangian coherent structures: a case study in Daya Bay, China. Mar. Pollut. Bull. 75, 105–113 (2013)CrossRefGoogle Scholar
  58. 58.
    Young, W., Pumir, A., Pomeau, Y.: Anomalous diffusion of tracer in convection rolls. Phys. Fluids 1, 462–469 (1989)CrossRefADSzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Guido Boffetta
    • 1
  • Guglielmo Lacorata
    • 2
  • Angelo Vulpiani
    • 3
  1. 1.Department of PhysicsUniversity of TorinoTorinoItaly
  2. 2.CNR ISACLecceItaly
  3. 3.Department of Physics and CNR ISCUniversity of Rome “La Sapienza”RomaItaly

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