Estimating global and local scaling exponents in turbulent flows using discrete wavelet transformations

Abstract

High frequency longitudinal velocity (u) measurements were performed in the atmospheric surface layer to investigate the inertial subrange structure of turbulence. The u measurements, collected over a wide range of atmospheric stability conditions, were used to investigate local and global intermittency buildup in the inertial subrange. Global scaling exponents and other statistical properties were derived using nondecimated (NDWT) and critically sampled orthonormal (OWT) wavelet transformations. These statistical measures were contrasted to similar statistical measures derived by applying NDWT and OWT to an ensemble of fractional Brownian motion (fBm) time series with Hurst exponent of 1/3. Such comparisons permit direct assessment as to whether discrepancies in observed intermittency corrections are artifacts of wavelet transformations or limitations in sample size. This study demonstrated that both NDWT and OWT were able to resolve intermittency-based departures from global power laws observed in higher-order structure functions of turbulence time series. Particularly, global power laws inferred from OWT and NDWT were consistent with new intermittency correction results derived from the dynamics of the fourth order structure functions. This study is the first to report on the ensemble behavior of such a power law for a wide range of surface boundary conditions (e.g., variable surface heating and friction velocity). The wavelet computed global intermittency departures from the classical Kolmogorov theory (or K41) were marginally smaller than those computed by the traditional structure function approach. In terms of local exponents, we found that the application of NDWT to fBm time series resulted in a wide empirical frequency distribution of local scaling exponents (α). The latter finding clearly demonstrates that previous and present α determined by wavelet analysis cannot be used as evidence for multifractality in turbulence. We also demonstrated that the classical local regression estimation of α is theoretically impaired by heteroscedascity when the local scale is finite. While the spread in α does not reflect any multifractal signatures, the modes of the local α frequency distribution support findings from global exponent analysis. We found that the modes of the local α distribution are well reproduced by global intermittency models for u and by K41 for the fBm.