Anomalous diffusion in quasi-geostrophic flow

  • J. S. Urbach
  • E. R. Weeks
  • H. L. Swinney
Flouids And Turbulence
Part of the Lecture Notes in Physics book series (LNP, volume 511)


We review a series of experimental investigations of anomalous transport in quasi-geostrophic flow. Tracer particles are tracked for long periods of time in two-dimensional flows comprised of chains of vortices generated in a rapidly rotating annular tank. The tracer particles typically follow chaotic trajectories, alternately sticking in vortices and flying long distances in the jets surrounding the vortices. Probability distribution functions (PDFs) are measured for the sticking and flight times. The flight PDFs are found to be power laws for most time-dependent flows with coherent vortices. In many cases the PDFs have a divergent second moment, indicating the presence of Lévy flights. The variance of an ensemble of particles is found to vary in time as σ 2t γ , with γ>1 (superdiffusion). The dependence of the variance exponent γ on the flight and sticking PDFs is studied and found to be consistent with calculations based on a continuous time random walk model.


Tracer Particle Anomalous Diffusion Chaotic Advection Chaotic Flow Vortex Chain 
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  1. [1]
    For reviews, see J. Klafter, M. F. Shlesinger, G. Zumofen, Beyond Brownian motion, Physics Today 49, 33 (Feb. 1996)CrossRefGoogle Scholar
  2. [1a]
    M. F. Shlesinger, G. M. Zaslavsky, J. Klafter, Strange kinetics, Nature 363, 31 (1993)CrossRefADSGoogle Scholar
  3. [1b]
    E. W. Montroll and M. F. Shlesinger, in Nonequilibrium Phenomena II: From Stochastics to Hydrodynamics, Studies in Statistical Mechanics, Vol. II, eds. J. L. Lebowitz and E. W. Montroll (North-Holland, Amsterdam, 1984), 1.Google Scholar
  4. [2]
    L. F. Richardson, Atmospheric diffusion shown on a distance-neighbour graph, Proc. Roy. Soc. (London) Ser. A 110 (1926) 709.ADSCrossRefGoogle Scholar
  5. [3]
    T. H. Solomon and J. P. Gollub, Passive transport in steady Rayleigh-Bénard convection, Phys. Fluids 31, 1372 (1988).CrossRefADSGoogle Scholar
  6. [4]
    H. Aref, Stirring by chaotic advection, J. Fluid Mech. 143, 1 (1984).zbMATHCrossRefADSMathSciNetGoogle Scholar
  7. [5]
    J. Pedlosky, Geophysical Fluid Dynamics, 2nd ed. (Springer-Verlag, New York, 1987).zbMATHGoogle Scholar
  8. [6]
    J. Sommeria, S. D. Meyers and H. L. Swinney, in Nonlinear Topics in Ocean Physics, ed. A. Osborne (North-Holland, Amsterdam, 1991), p. 227.Google Scholar
  9. [7]
    J. Sommeria, S. D. Meyers, and H. L. Swinney, Laboratory simulation of Jupiter's Great Red Spot, Nature 331 (1988) 689.CrossRefADSGoogle Scholar
  10. [8]
    J. Sommeria, S. D. Meyers, and H. L. Swinney, Laboratory model of a planetary eastward jet, Nature 337 (1989) 58.CrossRefADSGoogle Scholar
  11. [9]
    T. H. Solomon, E. R. Weeks, H. L. Swinney, Observation of anomalous diffusion and Lévy flights in a two-dimensional rotating flow, Phys. Rev. Lett. 71, 3975 (1993).CrossRefADSGoogle Scholar
  12. [10]
    T. H. Solomon, E. R. Weeks, H. L. Swinney, Chaotic advection in a two-dimensional flow: Lévy flights and anomalous diffusion, Physica D 76, 70 (1994).CrossRefADSzbMATHGoogle Scholar
  13. [11]
    E. R. Weeks, T. H. Solomon, J. S. Urbach, H. L. Swinney, Observation of Anomalous Diffusion and Lévy Flights, in: Lévy Flights and Related Topics in Physics, eds. M. F. Shlesinger, G. M. Zaslavsky and U. Frisch (Springer-Verlag, Heidelberg, 1995) pp. 51.CrossRefGoogle Scholar
  14. [12].
    R. Weeks, J. S. Urbach, H. L. Swinney, Anomalous diffusion in asymmetric random walks with a quasi-geostrophic flow example, Physica D 97, 219 (1996).CrossRefGoogle Scholar
  15. [13]
    M. S. Pervez and T. H. Solomon, Long-term tracking of neutrally buoyant tracer particles in two-dimensional fluid flows, Exp. Fluids 17, 135 (1994).CrossRefGoogle Scholar
  16. [14]
    T. H. Solomon, W. J. Holloway, H. L. Swinney, Shear flow instabilities and Rossby waves in barotropic flow in a rotating annulus, Phys. Fluids A 5, 1971 (1993).CrossRefADSGoogle Scholar
  17. [15]
    G. I. Taylor, Diffusion by continuous movements, Proc. Lon. Math. Soc. 2 20, 196 (1921).CrossRefGoogle Scholar
  18. [16]
    I. Mezic and S. Wiggins, On the dynamical origin of asymptotic t 2 dispersion of a nondiffusive tracer in incompressible laminar flows, Phys. Fluids 6, 2227 (1994).zbMATHCrossRefADSMathSciNetGoogle Scholar
  19. [17]
    D. del-Castillo-Negrete, Asymmetric transport and non-Gaussian statistics of passive scalars in vortices in shear, submitted to Phys. Fluids (1997).Google Scholar
  20. [18]
    M. F. Shlesinger, Asymptotic solutions of continuous-time random walks, J. Stat. Phys. 10, 421 (1974).CrossRefMathSciNetADSGoogle Scholar
  21. [19]
    J. Klafter and G. Zumofen, Lévy Statistics in a Hamiltonian System, Phys. Rev. E 49, 4873 (1994).CrossRefADSGoogle Scholar
  22. [20]
    W. Feller, An Introduction to Probability Theory and Its Applications, (John Wiley & Sons Inc., New York, 1966) Vol. 2, Chap XVI.8, p. 525.zbMATHGoogle Scholar
  23. [21]
    M. F. Shlesinger, Comment on “Stochastic process with ultraslow convergence to a Gaussian: The truncated Lévy flight”, Phys. Rev. Lett. 74, 4959 (1995).CrossRefADSGoogle Scholar
  24. [22]
    S. Venkataramani, T. M. Antonsen, E. Ott, Anomalous diffusion in bounded temporally irregular flows, Physica D, to appear.Google Scholar
  25. [23]
    S. Venkataramani, T. M. Antonsen, E. Ott, Lévy flights in fluid flows with no Kolmogorov-Arnold-Moser Surfaces, Phys. Rev. Lett. 78, 3864 (1997)zbMATHCrossRefADSMathSciNetGoogle Scholar
  26. [24]
    J. P. Bouchaud and A. Georges, Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications, Phys. Rep. 195, 127 (1990).CrossRefADSMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 1998

Authors and Affiliations

  • J. S. Urbach
    • 2
  • E. R. Weeks
    • 1
  • H. L. Swinney
    • 1
  1. 1.Center for Nonlinear Dynamics and Department of PhysicsUniversity of Texas at AustinAustinUSA
  2. 2.Department of PhysicsGeorgetown UniversityWashington, DCUSA

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