A Lagrangian analysis of advection-diffusion equation for a three dimensional chaotic flow

Physics of Fluids

Volumn 11, Issue 6, 1999, Pages 1418-1434

  Tang, X Z ^a   Boozer, A H ^b,c  

^a PRINCETON PLASMA PHYSICS LABORATORY   (United States)

^b Department of Earth and Environmental Sciences   (United States)

^c MAX PLANCK INSTITUT FÜR PLASMAPHYSIK   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0001166724     PISSN: 10706631     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.870006     Document Type: Article

References (39)

1. L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Pergamon, Oxford, 1959), Chap. VI.
2. X. Z. Tang and A. H. Boozer, "Finite time Lyapunov exponent and advection-diffusion equation," Physica D 95, 283 (1996).
3. J.-P. Eckmann and D. Ruelle, "Ergodic theory of chaos and strange attractors," Rev. Mod. Phys. 57, 617 (1985).
4. V. I. Oseledec, "A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems," Trans. Moscow Math. Soc. 19, 197 (1968).
5. H. Aref, "Stirring by chaotic advection," J. Fluid Mech. 143, 1 (1984).
6. R. R. Prasad, C. Meneveau, and K. R. Sreenivasan, "Multifractal nature of the dissipation field of passive scalars in fully turbulent flows," Phys. Rev. Lett. 61, 74 (1988); J. C. Sommerer and E. Ott, "Particles floating on a moving fluid - a dynamically comprehensible physical fractal," Science 259, 335 (1993).
7. R. R. Prasad, C. Meneveau, and K. R. Sreenivasan, "Multifractal nature of the dissipation field of passive scalars in fully turbulent flows," Phys. Rev. Lett. 61, 74 (1988); J. C. Sommerer and E. Ott, "Particles floating on a moving fluid - a dynamically comprehensible physical fractal," Science 259, 335 (1993).
8. T. H. Solomon and J. P. Gollub, "Passive transport in steady Rayleigh-Benard convection," Phys. Rev. A 38, 6280 (1988); R. P. Behringer, S. D. Meyers, and H. L. Swinney, "Chaos and mixing in a geostrophic flow," Phys. Fluids A 3, 1243 (1991); T. H. Solomon, E. R. Weeks, and H. L. Swinney, "Observation of anomalous diffusion and levy flights in a 2-dimensional rotating flow," Phys. Rev. Lett. 71, 3975 (1993).
9. T. H. Solomon and J. P. Gollub, "Passive transport in steady Rayleigh-Benard convection," Phys. Rev. A 38, 6280 (1988); R. P. Behringer, S. D. Meyers, and H. L. Swinney, "Chaos and mixing in a geostrophic flow," Phys. Fluids A 3, 1243 (1991); T. H. Solomon, E. R. Weeks, and H. L. Swinney, "Observation of anomalous diffusion and levy flights in a 2-dimensional rotating flow," Phys. Rev. Lett. 71, 3975 (1993).
10. T. H. Solomon and J. P. Gollub, "Passive transport in steady Rayleigh-Benard convection," Phys. Rev. A 38, 6280 (1988); R. P. Behringer, S. D. Meyers, and H. L. Swinney, "Chaos and mixing in a geostrophic flow," Phys. Fluids A 3, 1243 (1991); T. H. Solomon, E. R. Weeks, and H. L. Swinney, "Observation of anomalous diffusion and levy flights in a 2-dimensional rotating flow," Phys. Rev. Lett. 71, 3975 (1993).
11. G. O. Foutain, D. V. Khakhar, and J. M. Ottino, "Visualization of three dimensional chaos," Science 281, 683 (1998).
12. J. D. Meiss, "Symplectic maps, variational principles, and transport," Rev. Mod. Phys. 64, 795 (1992).
13. J. M. Ottino, "Mixing, chaotic advection, and mixing," Annu. Rev. Fluid Mech. 22, 207 (1990); J. M. Ottino, The Kinematics of Mixing: Stretching, Chaos and Transport (Cambridge U.P., Cambridge, 1989); S. Wiggins, Chaotic Transport in Dynamical System (Springer-Verlag, New York, 1992).
14. J. M. Ottino, "Mixing, chaotic advection, and mixing," Annu. Rev. Fluid Mech. 22, 207 (1990); J. M. Ottino, The Kinematics of Mixing: Stretching, Chaos and Transport (Cambridge U.P., Cambridge, 1989); S. Wiggins, Chaotic Transport in Dynamical System (Springer-Verlag, New York, 1992).
15. J. M. Ottino, "Mixing, chaotic advection, and mixing," Annu. Rev. Fluid Mech. 22, 207 (1990); J. M. Ottino, The Kinematics of Mixing: Stretching, Chaos and Transport (Cambridge U.P., Cambridge, 1989); S. Wiggins, Chaotic Transport in Dynamical System (Springer-Verlag, New York, 1992).
16. I. C. Percival, "Variational principles for invariant tori and cantori," in Nonlinear dynamics and beam-beam interaction, edited by M. Month and J. C. Herrera (AIP, New York, 1979); S. Aubry, "The new concept of transitions by breaking of analyticity in a crystallographic mode," in Soliton and condensed matter physics, Springer series in solid state physics, Vol. 8, edited by A. R. Bishop and T. Schneider (Springer-Verlag, New York, 1978).
17. I. C. Percival, "Variational principles for invariant tori and cantori," in Nonlinear dynamics and beam-beam interaction, edited by M. Month and J. C. Herrera (AIP, New York, 1979); S. Aubry, "The new concept of transitions by breaking of analyticity in a crystallographic mode," in Soliton and condensed matter physics, Springer series in solid state physics, Vol. 8, edited by A. R. Bishop and T. Schneider (Springer-Verlag, New York, 1978).
18. R. S. Mackay, J. D. Meiss, and I. C. Percival, "Transport in Hamiltonian systems," Physica D 13, 55 (1984).
19. J. D. Meiss and E. Ott, "Markov tree model of transport in area-preserving maps," Physica D 20, 387 (1986).
20. D. Beigie, A. Leonard, and S. Wiggins, "Invariant manifold templates for chaotic advection," Chaos Solitons Fractals 4, 749 (1994).
21. We note that previous analyses by others (Refs. 16 and 17) have employed a Lagrangian approach, but they are only locally valid, in contrast to the global (Lagrangian) coordinate transformation used in Ref. 2 and this paper.
22. J. M. Ottino, "Description of mixing with diffusion and reaction in terms of the concept of material interfaces," J. Fluid Mech. 114, 83 (1982).
23. D. Beigie, A. Leonard, and S. Wiggins, "A global study of enhanced stretching and diffusion in chaotic tangles," Phys. Fluids A 3, 1039 (1991).
24. E. Ott and T. M. Antonsen, Jr., "Chaotic fluid convection and the fractal nature of passive scalar gradients," Phys. Rev. Lett. 61, 2839 (1988); T. M. Antonsen, Jr. and E. Ott, "Multifractal power spectra of passive scalars convected by chaotic fluid flows," Phys. Rev. A 44, 851 (1991).
25. E. Ott and T. M. Antonsen, Jr., "Chaotic fluid convection and the fractal nature of passive scalar gradients," Phys. Rev. Lett. 61, 2839 (1988); T. M. Antonsen, Jr. and E. Ott, "Multifractal power spectra of passive scalars convected by chaotic fluid flows," Phys. Rev. A 44, 851 (1991).
26. A. J. Lichtenberg and M. A. Lieberman, Regular and Stochastic Motion (Spring-Verlag, New York, 1983).
27. G. K. Batchelor, "Small-scale variation of convected quantities like temperature in turbulent fluid: part 1. General discussion and the case of small conductivity," J. Fluid Mech. 5, 113 (1959).
28. X. Z. Tang and A. H. Boozer, "Hamiltonian structure of Hamiltonian chaos," Phys. Lett. A 236, 476 (1997).
29. D. P. Stern, "Euler Potentials," Am. J. Phys. 38, 494 (1970).
30. T. Dombre, U. Frisch, J. M. Greene, M. Hénon, A. Mehr, and A. M. Soward, "Chaotic streamlines in the ABC flows," J. Fluid Mech. 167, 353 (1986).
31. G. M. Zaslavsky, R. Z. Sagdeev, D. A. Usikov and A. A. Chernikov, Weak Chaos and Quasi-Regular Patterns (Cambridge U.P., Cambridge, 1991).
32. M. Feingold, L. P. Kadanoff, and O. Piro, "Passive scalars, three dimensional volume preserving maps, and chaos," J. Stat. Phys. 50, 529 (1988).
33. A. H. Boozer, "Plasma Confinement" in Encyclopedia of Physical Science and Technology (Academic, New York, 1992), Vol. 13, p. 1.
34. V. I. Arnold, Mathematical Methods of Classical Mechanics (Springer-Verlag, New York, 1989).
35. J. Pöschel, "Integrability of Hamiltonian systems on cantor set," Commun. Pure Appl. Math. XXXV, 653 (1982).
36. D. Ruelle, "Ergodic theory of differentiable dynamical systems," Publ. Math. IHES 50, 275 (1979).
37. D. Ruelle, Elements of Differentiable Dynamics and Bifurcation Theory (Academic, San Diego, 1989).
38. Leonhard Euler, "Sectio Secunda de Principiis Motus Fluidorum," Novi Comentarii Acad. Sci. Petropolitanae 14, 270 (1769); reprinted in Leonhardi Euleri Opera Omnia (Swiss Society for Natural Sciences, 1955), Ser. II, Vol. 13.
39. Leonhard Euler, "Sectio Secunda de Principiis Motus Fluidorum," Novi Comentarii Acad. Sci. Petropolitanae 14, 270 (1769); reprinted in Leonhardi Euleri Opera Omnia (Swiss Society for Natural Sciences, 1955), Ser. II, Vol. 13.