Exact Resummations in the Theory of Hydrodynamic Turbulence: III. Scenarios for Anomalous Scaling and Intermittency

Abstract

Elements of the analytic structure of anomalous scaling and intermittency in fully developed hydrodynamic turbulence are described. We focus here on the structure functions of velocity differences that satisfy inertial range scaling laws $S_n(R)\sim R^{ζ_n}$, and the correlation of energy dissipation $K_{εε}(R) \sim R^{-μ}$. The goal is to understand the exponents $ζ_n$ and $μ$ from first principles. In paper II of this series it was shown that the existence of an ultraviolet scale (the dissipation scale $η$) is associated with a spectrum of anomalous exponents that characterize the ultraviolet divergences of correlations of gradient fields. The leading scaling exponent in this family was denoted $Δ$. The exact resummation of ladder diagrams resulted in the calculation of $Δ$ which satisfies the scaling relation $Δ=2-ζ_2$. In this paper we continue our analysis and show that nonperturbative effects may introduce multiscaling (i.e. $ζ_n$ not being linear in $n$) with the renormalization scale being the infrared outer scale of turbulence $L$. It is shown that deviations from K41 scaling of $S_n(R)$ ($ζ_n\neq n/3$) must appear if the correlation of dissipation is mixing (i.e. $μ>0$). We derive an exact scaling relation $μ= 2ζ_2-ζ_4$. We present analytic expressions for $ζ_n$ for all $n$ and discuss their relation to experimental data. One surprising prediction is that the time decay constant $τ_n(R)\propto R^{z_n}$ of $S_n(R)$ scales independently of $n$: the dynamic scaling exponent $z_n$ is the same for all $n$-order quantities, $z_n=ζ_2$.