Towards a nonperturbative theory of hydrodynamic turbulence: Fusion rules, exact bridge relations, and anomalous viscous scaling functions

In this paper we address nonperturbative aspects of the analytic theory of hydrodynamic turbulence. Of paramount importance for this theory are the "fusion rules" that describe the asymptotic properties of $n$-point correlation functions when some of the coordinates tend toward one other. We first derive here, on the basis of two fundamental assumptions, a set of fusion rules for correlations of velocity differences when all the separations are in the inertial interval. Using this set of fusion rules we consider the standard hierarchy of equations relating the $n\mathrm{th}$-order correlations (originating from the viscous term in the Navier-Stokes equations) to $\mathrm{}(n+1\mathrm{})\mathrm{}\mathrm{th}$ order (originating from the nonlinear term) and demonstrate that for fully unfused correlations the viscous term is negligible. Consequently the hierarchic chain of equations is decoupled in the sense that the correlations of $\mathrm{}(n+1\mathrm{})\mathrm{}\mathrm{th}$ order satisfy a homogeneous equation that may exhibit anomalous scaling solutions. Using the same hierarchy of equations when some separations go to zero we derive, on the basis of the Navier-Stokes equations, a second set of fusion rules for correlations with differences in the viscous range. The latter includes gradient fields. We demonstrate that every $n\mathrm{th}$-order correlation function of velocity differences ${\mathcal{F}}_n(R_1, R_2, \dots )$ exhibits its own crossover length ${\eta }_n$ 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 as ${\eta }_n\mathrm{}\left(R\right)\mathrm{}\sim \eta {\left(\frac{R}{L}\right)}^{x_n}$ with $x_n=\frac{({\zeta }_n-{\zeta }_{n+1\mathrm{}}+{\zeta }_3-{\zeta }_2)}{(2-{\zeta }_2)}$, with ${\zeta }_n$ being the scaling exponent of the $n\mathrm{th}$-order structure function. We derive a class of exact scaling relations bridging the exponents of correlations of gradient fields to the exponents ${\zeta }_n$ of the $n\mathrm{th}$-order structure functions. One of these relations is the well known "bridge relation" for the scaling exponent of dissipation fluctuations $\mu =2-{\zeta }_6$.