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 ℓn such that each structure transfers its energy to two substructures of scale ℓn+1=ℓn/λ. For this model, we construct exact inertial range solutions that display intermittency, i.e., 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.