We consider the cumulant generating function of the logarithm of the distance between two infinitesimally close trajectories of a chaotic system. Its long-time behavior is given by the generalized Lyapunov exponent γ(k) providing the logarithmic growth rate of the k−th moment of the distance. The Legendre transform of γ(k) is a large deviations function that gives the probability of rare fluctuations where the logarithmic rate of change of the distance is much larger or much smaller than the mean rate defining the first Lyapunov exponent. The only non-trivial zero of γ(k) is at minus the correlation dimension of the attractor which for incompressible flows reduces to the space dimension. We describe here general properties constraining the form of γ(k) and the Gallavotti-Cohen type relations that hold when there is symmetry under time-reversal. This demands studying joint growth rates of infinitesimal distances and volumes. We demonstrate that quartic polynomial approximation for γ(k) does not violate the Marcinkiewicz theorem on invalidity of polynomial form for the generating function. We propose that this quartic approximation will fit many experimental situations, not having the effective time-reversibility and the short correlation time properties of the quadratic Grassberger-Procaccia estimates. We take the existing γ(k) for turbulent channel flow and demonstrate that the quartic fit is nearly perfect. The violation of time-reversibility for the Lagrangian trajectories of the incompressible Navier-Stokes turbulence below the viscous scale is considered. We demonstrate how the fit can be used for finding the correlation dimensions of strange attractors via easily measurable quantities. We provide a simple formula via the Lyapunov exponents, holding in quadratic approximation, and describe the construction of the quartic approximation. A different approximation scheme for finding the correlation dimension from expansion in the flow compressibility is also provided.

Most theoretical results about turbulent mixing have been obtained for ideal flows that are delta-correlated in time. As is often believed, those ideal flows are, with regard to mixing, very similar to real flows with a finite correlation time. However, recent results show that these two cases may differ considerably. In this article we study the effects of finite correlation time in a chaotic smooth statistically isotropic two-dimensional velocity field. As mixing is predominantly determined by the statistics of the stretching of material elements (e.g. lines "painted" onto a liquid), in this article we focus on the characteristics describing such stretching: finite-time Lyapunov exponents and the Lyapunov dimension. For these quantities, we derive analytical expressions as functions of the correlation time and the compressibility of the velocity field, and we investigate these expressions numerically. The results agree well with numerical results of other authors, and are useful for understanding several physical phenomena, e.g. patchiness of pollution spreading on an ocean and kinematic magnetic dynamos.

Majority of theoretical results regarding turbulent mixing are based on the model of ideal flows with zero correlation time. We discuss the reasons why such results may fail for real flows and develop a scheme which makes it possible to match real flows to ideal flows. In particular we introduce the concept of mixing dimension of flows which can take fractional values. For real incompressible flows, the mixing dimension exceeds the topological dimension; this leads to a local inhomogeneity of mixing -a phenomenon which is not observed for ideal flows and has profound implications, for instance impacting the rate of bimolecular reactions in turbulent flows. Finally, we build a model of compressible flows which reproduces the anomalous Lyapunov exponent values observed for time-correlated flows by Boffetta et al (2004), and provide a qualitative explanation of this phenomenon.

No abstract