The role of Lagrangian chaos in the creation of multifractal measures
In this paper we review and discuss the role of Lagrangian chaos in creating multifractal measures associated with spatial intermittency. In particular, we consider (1) passive scalars, (2) the kinematic magnetic dynamo problem, and (3) the stability of high Reynolds number fluid flows. In all cases we take the underlying flow to be smooth in the sense that it has no fractal or power law properties of its own (its wavenumber power spectrum is peaked at some low value and decays with increasing wavenumber much faster than a power law, e.g., exponentially). Nevertheless, chaos in the Lagrangian dynamics of such flows can lead to fractals and power law wavenumber spectra for relevant physical quantities, such as the distribution of the passive scalar (problem (1)), the magnetic field (problem (2)), and the vorticity field (problem (3)). A key concept used in the quantitative treatment of these problems is that of finite time Lyapunov exponents. This paper will illustrate how consideration of finite time Lyapunov exponents can be used to quantitatively analyze these situations. Thus we provide a connection between quantities of physical interest and ergodic dynamical characterization of the underlying chaos of the flow. In the case of problem (3), we also comment on implications for high Reynolds number fluid turbulence.
KeywordsVorticity Field Passive Scalar Fluid Element Wavenumber Spectrum Multifractal Measure
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- E. Ott, Chaos in Dynamical Systems (Cambridge University Press, 1994), pp. 78–85.Google Scholar
- Ref. 9, Sec. 9.4.Google Scholar
- Ref. 9, Chapter 5, and references therein; see also E. Ott and T. Tél,Chaos 3, 417 (1993).Google Scholar
- V. I. Arnold, Ya. B. Zeldovich, A. A. Ruzmaikin, and D. D. Sokolov, Sov. Phys. JETP 54, 1083 (1981).Google Scholar
- A. D. Gilbert, N. F. Otani, and S. Childress, in Theory of Solar and Planetary Dynamics, edited by M. R. E. Proctor, P. C. Matthews, and A. M. Rucklidge (Cambridge University Press, New York, 1993), pp. 129–136.Google Scholar
- E.g., see Ref. 9, p. 143.Google Scholar