| Summary: | 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mo>ℓ</mo><mi>n</mi></msub></semantics></math></inline-formula> such that each structure transfers its energy to two substructures of scale <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo>ℓ</mo><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mo>ℓ</mo><mi>n</mi></msub><mo>/</mo><mi>λ</mi></mrow></semantics></math></inline-formula>. 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.
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