One of the fundamental features of fully developed 3D turbulence is the non-Gaussian statistics at small scales. The energy transfer from the largest scales toward the smallest scales is related to the non-zero skewness and the large kurtosis of the PDFs. Kolmogorov's theory fails to describe the intermittency behavior because it neglects the presence of fluctuations in the energy transfer. Multifractal model (MF) is a theoretical approach that gives indeed a nice agreement with the experimental / numerical results and describes faithfully the large exponential tails observed in the PDFs for small increments. This model is characterized by a fractal dimension D, which is a function of the singularity h (scaling exponent). My question is:
What is the best practical approach to follow in order to derive an explicit formulation of this fractal dimension as a function of h and how to set the singularity limits h_min and h_max?
I would suggest that rather than using a structure function formalism directly, you follow (as a starting point), Muzy et al. (1991) PRL and use the continuous wavelet transform, the scaling of the modulus maxima and a Legendre transform to get you there. However, there are other forms of estimator based on oscillating singularity measures that give you point-by-point estimators rather than D(h). You can then form D(h) empirically from these if you wish. We do this in Keylock et al. 2012 J. Geophys. Res.
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