Scaling properties of a class of shell models

The scaling properties of a class of shell models are studied, via their velocity structure functions. The models all conserve energy and the fundamental symmetries of the Navier-Stokes equations. They also conserve a second quantity, which depends on the coefficients of the nonlinear terms, parametrized by ɛ. Models with ɛ varying from 0.2 to 10 are considered. All the models are found to display extended self-similarity, which allows a better estimate of the scaling exponents of the structure functions at any order. In most cases, deviations from Kolmogorov 1941 scaling are observed. As in fully developed turbulence, this intermittency is consistent with probability distribution functions resembling logarithmic Poisson distributions (exponential wings, negative skewness). Their signature is a hierarchical structure of the moments of the energy dissipation. The hierarchy is characterized by two main parameters, Δ and β, describing, respectively, the smallest dissipative scales and the degree of intermittency of the energy transfers. These parameters are measured in each shell model and the curves Δ and β as a function of ɛ are obtained. Two interesting transitions are obtained, for ɛ=0.39 and ɛ=1. In the first case, the system goes from nonintermittent to intermittent, with β and Δ going from β=1, Δ=0 to β<1, Δ≠0. This transition corresponds to the stable-unstable transition in some mapping characterizing the shell models. In the second case, discontinuities in both β and Δ are observed. This transition corresponds to the separation between models with ‘‘helicitylike’’ or ‘‘enstrophylike’’ conservation laws. Apart from these interesting transitions, no obvious correlation between the three parameters is found, indicating the complexity of the scaling character. Shell models, however, stand as a powerful tool to investigate the scale invariance, and the development of a complete theory which may help in understanding fully developed turbulence.