Further results on multifractality in shell models

Abstract

Very long integrations, involving hundreds of millions of time steps, have been performed for the Gledzer–Ohkitana–Yamada ‘‘shell model’’ of fully developed turbulence, thereby allowing the computation of essentially noise-free structure functions at all inertial- and dissipation-range scales. Previously reported results by Jensen et al. [Phys. Rev. A 43, 798 (1991)] on the multifractal behavior of this model are confirmed. Oscillations in the structure functions are found to be genuine. An exact relation for certain cubic moments, equivalent to Kolmogorov’s four-fifth law, is derived and tested. The third-order structure function, here defined in terms of the third moment of shell amplitudes, is not directly determined by this relation and need not have its exponent equal to one. Significant discrepancies are actually found when the ratio between successive shell wave numbers is less than two.