Anomalous scaling from controlled closure in a shell model of turbulence

We present a model of hydrodynamic turbulence for which the program of comp uting the scaling exponents from first principles can be developed in a con trolled fashion. The model consists of N suitably coupled copies of the "Sa bra" shell model of turbulence. The couplings are chosen to include two com ponents: random and deterministic, with a relative importance that is chara cterized by a parameter called epsilon. It is demonstrated, using numerical simulations of up to 25 copies and 28 shells that in the N -->infinity lim it but for 0 <epsilon less than or equal to 1 this model exhibits correlati on functions whose scaling exponents are anomalous. The theoretical calcula tion of the scaling exponents follows verbatim the closure procedure sugges ted recently for the Navier-Stokes problem, with the additional advantage t hat in the N -->infinity limit the parameter epsilon can be used to regular ize the closure procedure. The main result of this paper is a finite and cl osed 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 t o order epsilon(4) and neglects terms of order epsilon(6). Preliminary anal ysis of this set of equations indicates a K41 normal scaling at epsilon=0, with a birth of anomalous exponents at larger values of epsilon, in agreeme nt with the numerical simulations. (C) 2000 American Institute of Physics. [S1070-6631(00)00204-X].