Cascade structures and scaling exponents in a dynamical model of turbulence: Measurements and comparison

A detailed examination of the cascade statistics and scaling exponents is carried out for a dynamical-system model of fully developed turbulence called the GOY shell model. The convergence in time of the probability density functions and moments of the velocity fluctuations and their scaling exponents is studied with particular care. With a large sample size (5×${10}^9$), we demonstrate that there exists a finite cutoff for the velocity fluctuations at each inertial-range wave-number shell and the properties of the cutoff determine the scaling exponents of all moments. This cutoff represents the most intermittent structures in the cascade dynamics and exhibits a power-law dependence on wave number. The accurately determined scaling exponents permit a detailed comparison with various phenomenological models describing the statistics of the energy cascade. The consideration of the first and second derivatives of the scaling exponents with respect to the order of the moments p provides the evidence that the hierarchical-structure model [She and Leveque, Phys. Rev. Lett. 72, 336 (1994)] predicts the best functional dependence on p of the scaling exponents in the GOY shell model.