Evidence of intermittent cascades from discrete hierarchical dissipation in turbulence

We present the results of a search of log-periodic corrections to scaling in the moments of the square of velocity increments taken as proxies of the energy dissipation rate in experiments at high Reynolds number (≈2500) of three-dimensional fully developed turbulence. A simple dynamical representation of the Richardson–Kolmogorov cartoon of a cascade shows that standard averaging techniques erase by their very construction the possible existence of log-periodic corrections to scaling associated with a discrete hierarchy. To remedy this drawback, we introduce a novel “canonical” averaging that we test extensively on synthetic examples constructed to mimic the interplay between a weak log-periodic component and rather strong multiplicative and phase noises. Tests are also presented on 1/f5/3 random-phase noise, on the multifractal random walk model, and on lattice and non-lattice multifractal models. Our extensive tests confirm the remarkable observation of statistically significant log-periodic corrections to scaling, with a preferred scaling ratio for length scales compatible with the value γ=2. A strong confirmation of this result is provided by the identification of up to five harmonics of the fundamental log-periodic undulations, associated with up to five levels of the underlying hierarchical dynamical structure. A natural interpretation of our results is that the Richardson–Kolmogorov mental picture of a cascade becomes a realistic description if one allows for intermittent births and deaths of discrete cascades at varying scales.