Structure functions, multifractality and intermittency

The results in the previous section were based on the assumption that the separation rate of the fluid elements is well represented by a single Lyapunov exponent. However, the finite-time Lyapunov exponents (Sect. 2.5.1) may deviate from the asymptotic value and this can affect average properties of the concentration fluctuations 

If the concentration field had exactly the same local Holder exponent (a) everywhere in space, the spatial averaging would be irrelevant and the scaling exponents would be given by (q = qa. This is a valid approximation for small q, but typically there are corrections that lead to an anomalous scaling when the fluctuations of the finitetime Lyapunov exponents is taken into account. As time increases 

Decay-type and Stable Reaction Dynamics in Flows 

In the limit l — 0 the dominant contribution comes from a saddle point and the scaling exponents are 

If q is relatively small then Xq is close to the average Lyapunov exponent and the distribution of the finite-time Lyapunov exponents around A°° can be approximated by a Gaussian form (2.76) which leads to 

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