Exact intermittent solutions in a turbulence multi branch shell model

Abstract

Reproducing complex phenomena with simple models marks our understanding of the phenomena themselves and this is what Jack Herring's work demonstrated multiple times. In that spirit, this work studies a turbulence shell model consisting of a hierarchy of structures of different scales $\ell_n$ such that each structure transfers its energy to two substructures of scale $\ell_{n+1} = \ell_n /λ$. For this model we construct exact inertial range solutions that display intermittency ie absence of self-similarity. Using a large ensemble of these solutions we investigate how the probability distributions of the velocity modes change with scale. It is demonstrated that while velocity amplitudes are not scale invariant their ratios are. Furthermore using large deviation theory we show how the probability distributions of the velocity modes can be re-scaled to collapse in a scale independent form. Finally, we discuss the implications the present results have for real turbulent flows.