Scaling and dissipation in the GOY shell model

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Abstract
This is a paper about multifractal scaling and dissipation in a shell model of turbulence, called the Gledzer–Ohkitani–Yamada (GOY) model. This set of equations describes a one-dimensional cascade of energy towards higher wave vectors. When the model is chaotic, the high-wave-vector velocity is a product of roughly independent multipliers, one for each logarithmic momentum shell. The appropriate tool for studying the multifractal properties of this model is shown to be the energy flux on each shell rather than the velocity on each shell. Using this quantity, one can obtain better measurements of the deviations from Kolmogorov scaling (in the GOY dynamics) than were available up to now. These deviations are seen to depend upon the details of inertial-range structure of the model and hence are not universal. However, once the conserved quantities of the model are fixed to have the same scaling structure as energy and helicity, these deviations seem to depend only weakly upon the scale parameter of the model. The connection between multifractality in the velocity distribution and multifractality in the dissipation is analyzed. Arguments suggest that the connection is universal for models of this character, but the model has a different behavior from that of real turbulence. Also, the scaling behavior of time correlations of shell velocities, of the dissipation, and of Lyapunov indices are predicted. These scaling arguments can be carried over, with little change, to multifractal models of real turbulence.
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