The strange eigenmode in Lagrangian coordinates

Chaos

Volumn 14, Issue 3, 2004, Pages 531-538

  Thiffeault, Jean Luc ^a  

^a IMPERIAL COLLEGE LONDON   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 7444222348     PISSN: 10541500     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.1759431     Document Type: Article

References (27)

1. T. M. Antonsen, Jr., Z. Fan, E. Ott, and E. Garcia-Lopez, "The role of chaotic orbits in the determination of power spectra," Phys. Fluids 8, 3094 (1996).
2. D. T. Son, "Turbulent decay of a passive scalar in the Batchelor limit: Exact results from a quantum-mechanical approach," Phys. Rev. E 59, R3811 (1999).
3. E. Balkovsky and A. Fouxon, "Universal long-time properties of Lagrangian statistics in the Batchelor regime and their application to the passive scalar problem," Phys. Rev. E 60, 4164 (1999).
4. D. R. Fereday, P. H. Haynes, A. Wonhas, and J. C. Vassilicos, "Scalar variance decay in chaotic advection and Batchelor-regime turbulence," Phys. Rev. E 65, 035301(R) (2002).
5. A. Wonhas and J. C. Vassilicos, "Mixing in fully chaotic flows," Phys. Rev. E 66, 051205 (2002).
6. A. Pikovsky and O. Popovych, "Persistent patterns in deterministic mixing flows," Europhys. Lett. 61, 625 (2003).
7. R. T. Pierrehumbert, "Tracer microstructure in the large-eddy dominated regime," Chaos, Solitons Fractals 4, 1091 (1994).
8. J.-L. Thiffeault and S. Childress, "Chaotic mixing in a torus map," Chaos 13, 502 (2003).
9. W. Liu and G. Haller, "Strange eigenmodes and decay of variance in the mixing of diffusive tracers," Physica D 188, 1 (2004).
10. J. Sukhatme and R. T. Pierrehumbert, "Decay of passive scalars under the action of single scale smooth velocity fields in bounded two-dimensional domains: From non-self-similar probability distribution functions to self-similar eigenmodes," Phys. Rev. E 66, 056032 (2002).
11. S. Childress and A. D. Gilbert, Stretch, Twist, Fold: The Fast Dynamo (Springer-Verlag, Berlin, 1995).
12. M. Pollicott, "On the rate of mixing of Axiom A flows," Invent. Math. 81, 413 (1981).
13. M. Pollicott, "Meromorphic extensions of generalised zeta functions," Invent. Math. 85, 147 (1986).
14. D. Ruelle, "Resonances of chaotic dynamical systems," Phys. Rev. Lett. 56, 405 (1986).
15. D. Rothstein, E. Henry, and J. P. Gollub, "Persistent patterns in transient chaotic fluid mixing," Nature (London) 401, 770 (1999).
16. G. A. Voth, T. C. Saint, G. Dobler, and J. P. Gollub, "Mixing rates and symmetry breaking in two-dimensional chaotic flow," Phys. Fluids 15, 2560 (2003).
17. Y. B. Zeldovich, A. A. Ruzmaikin, S. A. Molchanov, and D. D. Sokoloff, "Kinematic dynamo problem in a linear velocity field," J. Fluid Mech. 144, 1 (1984).
18. V. I. Arnold, Mathematical Methods of Classical Mechanics, 2nd ed. (Springer-Verlag, New York, 1989).
19. J.-L. Thiffeault, "Advection-diffusion in Lagrangian coordinates," Phys. Lett. A 309, 415 (2003).
20. T. Kato, Perturbation Theory for Linear Operators (Springer-Verlag, Berlin, 1980).
21. E. Merzbacher, Quantum Mechanics (Wiley, New York, 1970).
22. X. Z. Tang and A. H. Boozer, "Finite time Lyapunov exponent and advection-diffusion equation," Physica D 95, 283 (1996).
23. J.-L. Thiffeault and A. H. Boozer, "Geometrical constraints on finite-time Lyapunov exponents in two and three dimensions," Chaos 11, 16 (2001).
24. J.-L. Thiffeault, "Derivatives and constraints in chaotic flows: Asymptotic behaviour and a numerical method;" Physica D 172, 139 (2002).
25. E. Ott (private communication).
26. D. R. Fereday and P. H. Haynes, "Scalar decay in two-dimensional chaotic advection and Batchelor-regime turbulence" (preprint, 2003).
27. It is not possible to simply invoke an incompressible flow corresponding to a map: In general there is no incompressible flow of the same dimension whose trajectories agree with a given map. Formally, however, the replacement of the metric tensor by one corresponding to a map is well-defined mathematically.