Anomalous Scaling Exponents in Nonlinear Models of Turbulence

Abstract

We propose a new approach to the old-standing problem of the anomaly of the scaling exponents of nonlinear models of turbulence. We achieve this by constructing, for any given nonlinear model, a linear model of passive advection of an auxiliary field whose anomalous scaling exponents are the same as the scaling exponents of the nonlinear problem. The statistics of the auxiliary linear model are dominated by 'Statistically Preserved Structures' which are associated with exact conservation laws. The latter can be used for example to determine the value of the anomalous scaling exponent of the second order structure function. The approach is equally applicable to shell models and to the Navier-Stokes equations.The calculation of the scaling exponents of structure functions of nonlinear turbulent velocity fields remains one of the major open problems of statistical physics [1]. Dimensional considerations appear to fail to provide the measured exponents, and present theory cannot even specify the mechanism for the so called "anomaly", i.e. the deviation of the scaling exponents from their dimensional estimates. Theoretical attempts to calculate the exponents were mainly based on perturbative expansions [2] or on closures of the infinite correlation function hierarchy [3]. The aim of this Letter is to propose a new idea to ascertain the anomaly of the scaling exponents in turbulence. In addition, we exhibit an alternative way to determine the anomalous scaling exponent of the second order structure function. The proposed approach is equally applicable to Navier-Stokes turbulence and to simplified models of turbulence, like nonlinear shell models. The only distinction is in the ease of numerical demonstration; for shell models we present adequate numerical confirmation of the proposed theory. For Navier-Stokes turbulence we present calculations at a resolution of 128 3 .The central idea is to construct a linear model whose scaling exponents are the same as those of the nonlinear problem. In this linear problem the exponents are universal to the forcing, and we understand the mechanism for the anomaly of the scaling exponents; we use this to show that also the nonlinear problem must have anomalous exponents. We exemplify the idea first in the context of the Navier-Stokes equations. Consider a model for two coupled vector fields u and w,Here ν is the kinematic viscosity, p andp are pressure fields imposing ∇ · u = ∇ · w = 0 and f andf are two uncorrelated Gaussian random forcing. Finally λ is a real number. Here and below we assume that the scaling exponents are universal to the forcing. We want to demonstrate their anomaly and to find their numerical values. For λ = 0 Eq. (1) reduces to the Navier-Stokes equations for u, whereas Eq. (2) becomes a linear equation for w, passively advected by u. This linear problem was referred to before as a "passive vector with pressure" [5,6,7]. It is known to exhibit anomalous scaling with exponents that are universal to the forcing. In addition, one understand the mechanism for the a...

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