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.

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