Direct and inverse cascades scaling in real shell models of turbulence

Shell models provide a simplified mathematical framework that captures essential features of incompressible fluid turbulence, such as the energy cascade and scaling of the fluid observables. We perform a precision analysis of the direct and inverse cascades in shell models of turbulence, where the velocity field is a real-valued function. We calculate the leading hundred anomalous scaling exponents, the marginal probability distribution functions of the velocity field at different shells, as well as the correlations between different shells. We find that the structure functions in both cascades exhibit a linear Kolomogorov scaling in the inertial range. We argue that the underlying reason for having no intermittency, is the strong correlations between the velocity fields at different shells. We analyze the tails of velocity distribution functions, which offer new insights to the structure of fluid turbulence.