Anomalous Scaling from Controlled Closure in a Shell Model of Turbulence

Details Anomalous Scaling from Controlled Closure in a Shell Model of Turbulence Anomalous Scaling from Controlled Closure in a Shell Model of Turbulence

1998-10-15

arxiv.org/abs/chao-dyn/9810021v1

Nonlinear Sciences

Chaotic Dynamics

Phys. Fluids, special issue in the honour of R.H. Kraichnan, submitted, REVTex, 22 pages, 15 eps figures

Scientific paper

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 $\epsilon$. It is demonstrated, using numerical simulations of up to 25 copies and 28 shells that in the $N\to \infty$ limit but for $0<\epsilon\le 1$ 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 $\epsilon$ 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 $\epsilon^4$ and neglects terms of order $\epsilon^6$. Preliminary analysis 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 agreement with the numerical simulations.

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