Advertisement

Chaotic dynamics of passive particles in three-vortex system: Dynamical analysis

  • Leonid Kuznetsov
  • George M. Zaslavsky
Flouids And Turbulence
Part of the Lecture Notes in Physics book series (LNP, volume 511)

Abstract

Analytical study of a passive particle advection in three point-vortex system is described in details. We specify two extreme cases of strong and weak chaos depending on the geometry of 3-vortex system. Mappings are derived for both cases and domains of chaotic dynamics are calculated analytically. We discuss the origin of the coherent vortex cores — holes in the stochastic sea filled with KAM orbits, and obtain the expression for the core radius in case of strong chaos. A weak dependence of core radius on geometrical parameters is discovered. For the case of weak chaos the separatrix map is constructed and the stochastic layer width is estimated. Numerical simulations have been performed and it was found that there exists a fine structure of the coherent core boundary layer, which consists of islands and subislands. We also have found the stickiness of the advected particle to the boundaries of vortex cores.

Keywords

Vortex Core Point Vortex Passive Particle Chaotic Advection Poincare Section 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    H. Aref, Stirring by chaotic advection, J. Fluid Mech. 143, 1 (1984)zbMATHCrossRefADSMathSciNetGoogle Scholar
  2. [2]
    H. Aref, Chaotic advection of fluid particles, Phil. Trans. R. Soc. London A 333, 273 (1990)ADSGoogle Scholar
  3. [3]
    J. Ottino, The kinematics of Mixing: Stretching, Chaos, and Transport (Cambridge U. P., Cambrige, 1989)zbMATHGoogle Scholar
  4. [4]
    J. Ottino, Mixing, chaotic advection and turbulence, Ann. Rev. Fluid Mech. 22, 207 (1990)CrossRefADSMathSciNetGoogle Scholar
  5. [5]
    V. Rom-Kedar, A. Leonard and S. Wiggins, An analytical study of transport mixing and chaos in an unsteady vortical flow, J. Fluid Mech. 214, 347 (1990)zbMATHCrossRefADSMathSciNetGoogle Scholar
  6. [6]
    S. Wiggins, Chaotic Transport in Dynamical Systems (Springer-Verlag, New York, 1992)zbMATHGoogle Scholar
  7. [7]
    A. Crisanti, M. Falcioni, G. Paladin and A. Vulpiani, Lagrangian Chaos: Transport, Mixing and Diffusion in Fluids, La Rivista del Nuovo Cimento, 14, 1 (1991)MathSciNetCrossRefADSGoogle Scholar
  8. [8]
    A. Crisanti, M. Falcioni, A. Provenzale, P. Tanga and A. Vulpiani, Dynamics of passively advected impurities in simple two-dimensional flow models, Phys. Fluids A 4, 1805 (1992)CrossRefADSGoogle Scholar
  9. [9]
    G. M. Zaslavsky, R. Z. Sagdeev and A. A. Chernikov, Stochastic nature of streamlines in steady-state flows, Sov. Phys. JETP 67, 270 (1988)MathSciNetGoogle Scholar
  10. [10]
    G. M. Zaslavsky, R. Z. Sagdeev, D. A. Usikov and A. A. Chernikov, Weak chaos and quasiregular patterns (Cambridge University Press, Cambridge, 1991)CrossRefGoogle Scholar
  11. [11]
    H. Aref and N. Pomphrey, Integrable and chaotic motion of four vortices, Phys. Lett. A 78, 297 (1980)CrossRefADSMathSciNetGoogle Scholar
  12. [12]
    S. L. Ziglin, Nonintegrability of a problem on the motion of four point vortices, Sov. Math. Dokl. 21, 296 (1980)zbMATHGoogle Scholar
  13. [13]
    Z. Neufeld and T. Tél, The vortex dynamics analogue of the restricted threebody problem: advection in the field of three identical point vortices, J. Phys. A: Math. Gen. 30, 2263 (1997)zbMATHCrossRefADSGoogle Scholar
  14. [14]
    S. Boatto and R. T. Pierrehumbert, Dynamics of a passive tracer in a velocity field of four identical vortices, unpublishedGoogle Scholar
  15. [15]
    T. H. Solomon and J. P. Gollub, Chaotic particle transport in time-dependent Rayleigh-Bénard convection, Phys. Rev. A 38, 6280 (1988)CrossRefADSGoogle Scholar
  16. [16]
    T. H. Solomon and J. P. Gollub, Passive transport in steady Rayleigh-Bénard convection, Phys. Fluids 31, 1372 (1988)CrossRefADSGoogle Scholar
  17. [17]
    V.V. Melezhko, M.Yu. Konstantinov, A.A. Gurzhi and T.P. Konovaljuk, Advection of a vortex pair atmosphere in a velocity field of point vortices, Phys. Fluids A 4, 2779 (1992)CrossRefADSMathSciNetGoogle Scholar
  18. [18]
    L. Zanetti and P. Franzese, Advection by a point vortex in a closed domain, Eur. J. Mech., B/Fluids 12, 43 (1993)Google Scholar
  19. [19]
    G. Boffetta, A. Celani and P. Franzese, Trapping of passive tracers in a point vortex system, J. Phys. A: Math. Gen. 29, 3749 (1996)zbMATHCrossRefADSMathSciNetGoogle Scholar
  20. [20]
    Á. Péntek, T. Tél and Z. Toroczkai, Chaotic advection in the velocity field of leapfrogging vortex pair, J. Phys. A: Math. Gen. 28, 2191 (1995)zbMATHCrossRefADSGoogle Scholar
  21. [21]
    V.V. Meleshko, Nonstirring of an inviscid fluid by a point vortex in a rectangle, Phys. Fluids 6, 6 (1994)zbMATHCrossRefADSMathSciNetGoogle Scholar
  22. [22]
    G. M. Zaslavsky, M. Edelman, B. A. Niyazov, Self-similarity, renormalization, and phase space nonuniformity of Hamiltonian chaotic dynamics, Chaos 7, 159 (1997)zbMATHCrossRefADSMathSciNetGoogle Scholar
  23. [23]
    H. Aref, Integrable, chaotic and turbulent vortex motion in two-dimensional flows, Ann. Rev. Fluid Mech. 15, 345 (1983)CrossRefADSMathSciNetGoogle Scholar
  24. [24]
    P.G. Saffman, Vortex Dynamics (Cambridge University Press, Cambridge, 1992)zbMATHGoogle Scholar
  25. [25]
    V.V. Melezhko, M.Yu. Konstantinov, Dinamika vikhrevykh struktur, (Naukova Dumka, Kiev, 1993) [in Russian]Google Scholar
  26. [26]
    A. Babiano, G. Boffetta, A. Provenzale and A. Vulpiani, Chaotic advection in point vortex models and two-dimensional turbulence, Phys. Fluids 6, 2465 (1994)zbMATHCrossRefADSMathSciNetGoogle Scholar
  27. [27]
    R. Benzi, G. Paladin, S. Patarnello, P. Santangelo and A. Vulpiani, Intermittency and coherent structures in two-dimensional turbulence, J. Phys A 19, 3771 (1986)CrossRefADSzbMATHGoogle Scholar
  28. [28]
    R. Benzi, S. Patarnello and P. Santangelo, Self-similar coherent structures in two-dimensional decaying turbulence, J. Phys A 21, 1221 (1988)CrossRefADSGoogle Scholar
  29. [29]
    J. B. Weiss, J.C. McWilliams, Temporal scaling behavior of decaying two-dimensional turbulence, Phys. Fluids A 5, 608 (1992)CrossRefADSMathSciNetGoogle Scholar
  30. [30]
    J.C. McWilliams, The emergence of isolated coherent vortices in turbulent flow, J. Fluid Mech. 146, 21 (1984)zbMATHCrossRefADSGoogle Scholar
  31. [31]
    J.C. McWilliams, The vortices of two-dimensional turbulence, J. Fluid Mech. 219, 361 (1990)CrossRefADSGoogle Scholar
  32. [32]
    D. Elhmaïdi, A. Provenzale and A. Babiano, Elementary topology of two-dimensional turbulence from a Lagrangian viewpoint and single particle dispersion, J. Fluid Mech. 257, 533 (1993)zbMATHCrossRefADSGoogle Scholar
  33. [33]
    G. F. Carnevale, J.C. McWilliams, Y. Pomeau, J. B. Weiss and W. R. Young, Evolution of Vortex Statistics in Two-Dimensional Turbulence, Phys. Rev. Lett. 66, 2735 (1991)CrossRefADSGoogle Scholar
  34. [34]
    E. A. Novikov, Yu. B. Sedov, Stochastic properties of a four-vortex system, Sov. Phys. JETP 48, 440 (1978)ADSGoogle Scholar
  35. [35]
    E. A. Novikov, Yu. B. Sedov, Stochastization of vortices, JETP Lett. 29, 667 (1979)ADSGoogle Scholar
  36. [36]
    H. Aref and N. Pomphrey, Integrable and chaotic motion of four vortices. I. The case of identical vortices, Proc. R. Soc. Lond. a 380, 359 (1982)ADSMathSciNetGoogle Scholar
  37. [37]
    V. K. Melnikov, On the stability of the center for time periodic perturbations, Trans. Moscow Math. Soc. 12, 1 (1963)Google Scholar
  38. [38]
    N. N. Filonenko, G. M. Zaslavsky, Stochastic instability of trapped particles and conditions of applicability of the quasi-linear approximation, Sov. Phys. JETP 25, 851 (1968)Google Scholar
  39. [39]
    A. J. Lichtenberg, M. A. Lieberman, Regular and chaotic dynamics (Springer-Verlag, New York, 1992)zbMATHGoogle Scholar
  40. [40]
    K.O. Friedrichs, Special Topics in Fluid Dynamics (Gordon and Breach, New York, 1966)zbMATHGoogle Scholar
  41. [41]
    E. A. Novikov, Dynamics and statistics of a system of vortices, Sov. Phys. JETP 41, 937 (1975)ADSGoogle Scholar
  42. [42]
    H. Aref, Motion of three vortices, Phys. Fluids 22, 393 (1979)zbMATHCrossRefADSGoogle Scholar
  43. [43]
    J.L. Synge, On the motion of three vortices, Can. J. Math. 1, 257 (1949)zbMATHMathSciNetCrossRefGoogle Scholar
  44. [44]
    J. Tavantzis and L. Ting, The dynamics of three vortices revisited, Phys. Fluids 31, 1392 (1988)zbMATHCrossRefADSMathSciNetGoogle Scholar
  45. [45]
    L. D. Landau, E. M. Lifshits, Mechanics (Pergamon Press, New York, 1976)Google Scholar
  46. [46]
    L. Kuznetsov and G.M. Zaslavsky, Hidden Renormalization Group for the Near-Separatrix Hamiltonian Dynamics, Phys. Reports 288, 457 (1997)CrossRefADSGoogle Scholar

Copyright information

© Springer-Verlag 1998

Authors and Affiliations

  • Leonid Kuznetsov
    • 2
  • George M. Zaslavsky
    • 1
    • 2
  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA
  2. 2.Department of PhysicsNew York UniversityNew YorkUSA

Personalised recommendations