Revealing intermittency in experimental data with steep power spectra

The statistics of signal increments are commonly used in order to test for possible intermittent properties in experimental or synthetic data. However, for signals with steep power spectra [i.e., E(ω) ∼ ω−n with n ≥ 3], the increments are poorly informative and the classical phenomenological relationship between the scaling exponents of the second-order structure function and of the power spectrum does not hold. We show that in these conditions the relevant quantities to compute are the second or higher degree differences of the signal. Using this statistical framework to analyze a synthetic signal and experimental data of wave turbulence on a fluid surface, we accurately characterize intermittency of these data with steep power spectra. The general application of this methodology to study intermittency of experimental signals with steep power spectra is discussed. Introduction. – Since the prediction of Kolmogorov in 1941 [1], it is well known that the spatial power spectrum E(k) of a fluid particle velocity v in a turbulent flow is a power-law of the wave number k as k. The −5/3 exponent of the spatial power spectrum is related to the second-order moment of velocity increments S2(r) ≡ 〈[v(l + r) − v(l)] 〉 ∼ r, l and r being a position and a spatial separation [2]. The phenomenological relationship between both exponents comes from Fourier transform properties. It can be generalized to any stationary random processes: if the power spectrum of the process is E(k) ∼ k, then S2(r) ∼ r ζ2 with ζ2 = n − 1 [2]. This property allows to perform measurements in the real space to reach the power-law exponent of the power spectrum. The statistics of velocity increments are also crucial to characterize the intermittent nature of the velocity field using the scaling properties of structure functions: Sp(r) ≡ 〈[v(l + r) − v(l)] 〉 ∼ rp (p positive integer) [3]. A non-linear dependence of ζp versus p is the hallmark of intermittency. Steep power-law spectra (∼ ω or ∼ k with n close or larger than 3) of a process are observed in various situations: magnetohydrodynamics turbulence [4], atmospherics turbulence [5], gravity [6] or capillary [7] wave turbulence on a fluid surface, and direct cascade of twodimensional fluid turbulence [8, 9]. Whatever the corresponding signal measured in space or in time (e.g. fluid velocity or vorticity, surface-wave height, magnetic field, wind), such steep spectra mean that the measured signal is at least once continuously differentiable [2]. The signal differences or increments are thus poorly informative since they are dominated by the differentiable component of the signal. For instance, some numerical simulations of the power-law scaling of the energy spectrum in two-dimensional turbulence exhibited apparent contradictions with its reconstruction from spatial correlation measurement, i.e. ζ2 6= n − 1 (see Ref. [10]). Babiano et al. have systematically reconsidered the theoretical relations between second-order structure functions and energy spectra instead of phenomenological or dimensional arguments [10]. They showed that the apparent contradictions come from the fact that the relation ζ2 = n − 1 does not hold for steep power-law spectra. Indeed, due to the differentiable component, the exponent of the secondorder structure function is independent of the spectrum slope as soon as this one is steeper than −3; that is ζ2 = 2 p-1 ha l-0 05 04 91 6, v er si on 1 21 J ul 2 01 0 Author manuscript, published in "Europhysics Letters (EPL) 90 (2010) 50007" DOI : 10.1209/0295-5075/90/50007