Towards a Nonperturbative Theory of Hydrodynamic Turbulence:Fusion Rules, Exact Bridge Relations and Anomalous Viscous Scaling Functions

Abstract

In this paper we derive here, on the basis of the NS eqs. a set of fusion rules for correlations of velocity differences when all the separation are in the inertial interval. Using this we consider the standard hierarchy of equations relating the $n$-th order correlations (originating from the viscous term in the NS eq.) to $n+1$'th order (originating from the nonlinear term) and demonstrate that for fully unfused correlations the viscous term is negligible. Consequently the hierarchic chain is decoupled in the sense that the correlations of $n+1$'th order satisfy a homogeneous equation that may exhibit anomalous scaling solutions. Using the same hierarchy of eqs. when some separations go to zero we derive a second set of fusion rules for correlations with differences in the viscous range. The latter includes gradient fields. We demonstrate that every n'th order correlation function of velocity differences ${\cal F}_n(\B.R_1,\B.R_2,\dots)$ exhibits its own cross-over length $η_{n}$ to dissipative behavior as a function of, say, $R_1$. This length depends on $n$ {and on the remaining separations} $R_2,R_3,\dots$. When all these separations are of the same order $R$ this length scales like $η_n(R)\sim η(R/L)^{x_n}$ with $x_n=(ζ_n-ζ_{n+1}+ζ_3-ζ_2)/(2-ζ_2)$, with $ζ_n$ being the scaling exponent of the $n$'th order structure function. We derive a class of exact scaling relations bridging the exponents of correlations of gradient fields to the exponents $ζ_n$ of the $n$'th order structure functions. One of these relations is the well known ``bridge relation" for the scaling exponent of dissipation fluctuations $μ=2-ζ_6$.