Wavelet analysis of turbulence reveals the multifractal nature of the Richardson cascade

Abstract

THE central problem of fully developed turbulence is the energy-cascading process. It has resisted all attempts at a full physical understanding or mathematical formulation. The main reasons for this failure are related to the large hierarchy of scales involved, the highly nonlinear character inherent in the Navier–Stokes equations, and the spatial intermittency of the dynamically active regions. Richardson1 has described the interplay between large and small scales with the words 'Big whirls have little whirls which feed on their velocity ... and so on to viscosity", and the phenomenon so described is known as the Richardson cascade. This local interplay also forms the basis of a theory by Kolmogorov2. It was later realized that the cascade ought to be intermittent, and Mandelbrot3 has given a fractal description of the intermittency of the fine structure. A particular case that emphasizes the dynamical aspect of the fractal models is the β-model4, in which the flux of energy is transferred to only a fixed fraction β of the eddies of smaller scales. More recently, a multifractal model of the fine-scale intermittency has been introduced by Parisi and Frisch5; this accounts for the more complex cascading process suggested by the experimental data on inertial-range structure functions obtained by Anselmet et al.6. These statistical models are insufficient, however, because they cannot give any topological information about the intermittent structures in real space. Here we use the wavelet transform7 to analyse the velocity field of wind-tunnel turbulence at very high Reynolds numbers8. This 'space-scale' analysis is shown to provide the first visual evidence of the celebrated Richardson cascade1, and reveals in particular its fractal character3. The results also indicate that the energy-cascading process has remarkable similarities with the deterministic construction rules of non-homogeneous Cantor sets9,10.