Anomalous Scaling and Fusion Rules in Hydrodynamic Turbulence

Abstract

It is shown that statistical properties of developed hydrodynamic turbulence are characterized by an infinite set of independent anomalous exponents which describes the scaling behavior of hydrodynamic fields constructed from the second and larger powers of the velocity derivatives. A physical mechanism responsible for anomalous scaling, ``telescopic multi-step eddy interaction", is discovered and investigated. The essence of this mechanism is the existence of a very large number $(R/η)^{Δ_j}\gg 1$ of channels of interaction of large eddies of scale $R$ in the inertial interval with eddies of viscous scale $η$ via a set of eddies of all intermediate scales between $R$ and $η$. The description of this mechanism based on the NS equation in the quasi Lagrangian representation is presented. In the diagrammatic expansion of the correlation function of the energy dissipation field $K_ {\varepsilon \varepsilon}(R)$, we have found an infinite series of logarithmically diverging diagrams. Their summation leads to a renormalization of the normal Kolmogorov-41 dimensions. For a description of the scaling of various hydrodynamic fields an infinite set of primary fields $O_n$ with independent scaling exponents $Δ_n$ was introduced. We have proposed a symmetry classification of the fields $O_n$ enabling one to predict relations between scaling the behavior of different correlation functions.