BIRTH OF ANOMALOUS SCALING IN A MODEL OF HYDRODYNAMIC TURBULENCE WITH A TUNABLE PARAMETER

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We introduce a model of hydrodynamic turbulence with a tunable parameter ε, which represents the ratio between deterministic and random components in the coupling between N identical copies of the turbulent field. To compute the anomalous scaling exponents ζn (of the nth order structure functions) for chosen values of ε, we consider a systematic closure procedure for the hierarchy of equations for the n-order correlation functions, in the limit N → ∞. The parameter ε regularizes the closure procedure, in the sense that discarded terms are of higher order in ε compared to those retained. It turns out that after the terms of O(1), the first nonzero terms are O(ε4). Within this ε-controlled procedure, we have a finite and closed set of scale-invariant equations for the 2nd and 3rd order statistical objects of the theory. This set of equations retains all terms of O(1) and O(ε4) and neglects terms of O(ε6). On this basis, we expect anomalous corrections δ ζn in the scaling exponents ζn to increase with εn. This expectation is confirmed by extensive numerical simulations using up to 25 copies and 28 shells for various values of εn. The simulations demonstrate that in the limit N → ∞, the scaling is normal for ε < ε cr with ε cr ≈ 0.6. We observed the birth of anomalous scaling at ε = ε cr with [Formula: see text] according to our expectation.