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 theor y 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 o ne other. We first derive here, on the basis of two fundamental assump tions, 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 relati ng the nth-order correlations (originating from the viscous term in th e Navier-Stokes equations) to (n+1)th order (originating from the nonl inear term) and demonstrate that for fully unfused correlations the vi scous term is negligible. Consequently the hierarchic chain of equatio ns is decoupled in the sense that the correlations of (n+1)th order sa tisfy a homogeneous equation that may exhibit anomalous scaling soluti ons. 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 nth-order correlation function of velocity differences F-n(R(1),R(2),. ..) 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 remai ning separations R(2),R(3),.... When all these separations are of the same order R this length scales as eta(n)(R)similar to eta(R/L)(x)n wi th )=(zeta(n)-zeta(n+1)+zeta(3)-zeta(2))/(2-zeta(2)), with zeta(n) bei ng the scaling exponent of the nth-order structure function. We derive a class of exact scaling relations bridging the exponents of correlat ions of gradient fields to the exponents zeta(n) of the nth-order stru cture functions. One of these relations is the well known ''bridge rel ation'' for the scaling exponent of dissipation fluctuations mu=2-zeta 6.