Anomalous Scaling from Controlled Closure in a Shell Model of Turbulence

Abstract

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 $N\to \infty$ limit but for $0<ε\le 1$ 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 $N\to \infty$ 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 $ε^4$ and neglects terms of order $ε^6$. 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.