Anomalous Scaling from Controlled Closure in a Shell Model of Turbulence

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We present a model of hydrodynamic turbulence for which the program of computing the scaling exponents from first principles can be developed in a controlled fashion. The model consists of N suitably coupled copies of the "Sabra" shell model of turbulence. The couplings are chosen to include two components: random and deterministic, with a relative importance that is characterized by a parameter called ε. It is demonstrated, using numerical simulations of up to 25 copies and 28 shells that in the N→ ∞ limit but for 0<ε≤ 1 this model exhibits correlation functions whose scaling exponents are anomalous. The theoretical calculation of the scaling exponents follows verbatim the closure procedure suggested recently for the Navier-Stokes problem, with the additional advantage that in the N→ ∞ limit the parameter ε can be used to regularize the closure procedure. The main result of this paper is a finite and closed set of scale-invariant equations for the 2nd and 3rd order statistical objects of the theory. This set of equations takes into account terms up to order ε⁴ and neglects terms of order ε⁶. Preliminary analysis of this set of equations indicates a K41 normal scaling at ε= 0, with a birth of anomalous exponents at larger values of ε, in agreement with the numerical simulations.