Arbitrary Order Hilbert Spectral Analysis: Definition And Application To Fully Developed Turbulence And Environmental Time Series

Empirical Mode Decomposition(EMD),or Hilbert-Huang Transform(HHT) is a novel general time-frequency analysis method for nonstationary and nonlinear time series, which was proposed by Huang et al.(1998,1999) more than ten years ago.During the last ten years,there have been more than 1000 papers applying this new method to various applications and research fields.In this thesis we apply this method to turbulence time series for the first time,and to environmental time series.It is found that the EMD acts a dyadic filter bank for fully developed turbulence.To characterize the intermittent properties of a scaling time series,we generalize the classical Hilbert spectral analysis to arbitrary order q,performing what we denoted "arbitrary order Hilbert spectral analysis".This provides a new frame to characterize scale invariance directly in an amplitude-frequency space,by taking a marginal integral of a joint pdf p(ω,A) of instantaneous frequencyωand amplitude A.We first validate the method by analyzing a simulated fractional Brownian motion time series,and by analyzing a synthesized multifractal nonstationary time series respectively for monoscaling and multifractal processes.Compared with the classical structure function approach,it is found numerically that the Hilbert-based methodology provides a more precise estimator for the intermittency parameter.Assuming statistical stationarity,we propose an analytical model for the autocorrelation function of velocity increments time seriesΔu_e(t),whereΔu_e(t)= u(t+e) - u(t),and e is the time increment.With this model,we prove analytically that,if a power law behaviour holds for the original variable,the location of the minimum values of the autocorrelation function is equal exactly to the time separation e when e belongs to scaling range.A power law behaviour for the minimum values is suggested by this model,and verified by a fractional Brownian motion simulation and a turbulent database.By defining a cumulative function for the autocorrelation function,the scale contribution is then characterized in the Fourier frequency space. It is found that the main contribution to the autocorrelation function comes from the large scale part.The same idea is applied to the second order structure function.It is found the second order structure function is strongly influenced by the large scale part,showing that it is not a good approach to extract the scaling exponent from a given scaling time series when the data possess energetic large scales.We then apply this Hilbert-based methodology to an experimental homogeneous and nearly isotropic turbulent database to characterize multifractal scaling properties of the velocity time series in fully developed turbulence.We obtain a scaling trend in the joint pdf p(ω,A) with a scaling exponent close to the Kolmogorov value.We recover the structure function scaling exponentsζ(q) in amplitude-frequency space for the first time.The isotropy hypothesis is then checked scale by scale in amplitudefrequency space.It is found that the generalized isotropy ratio decreases linearly with the order q.We also perform the analysis on a temperature(passive scalar) time series with strong ramp-cliff structures.For these data,the traditional structure function fails. However,the new method extracts a clear power law up to q=8.The scaling exponentξ(q) is quite close to the scaling exponentζ(q) of the longitudinal velocity in fully developed turbulence.We then consider the traditional Extended Self-Similarity(ESS)(Benzi et al., 1993b) and the hierarchy model(She & Lev(?)que,1994) under the Hilbert frame.For the case of ESS,we have here two special cases q=0 and q=3 to define the ESS in the Hilbert frame.Both of them work for the fully developed turbulence providing the same scaling exponents.Based on the turbulent database we have,it seems that the lognormal model with a proper chosen intermittency parameterμprovides a better prediction of the scaling exponents. We finally apply the new method to daily river flow discharge and surf zone marine turbulence to characterize the scale invariance under the Hilbert frame.