Computing the Scaling Exponents in Fluid Turbulence from First Principles: the Formal Setup

Abstract

We propose a scheme for the calculation from the NS equations of the scaling exponents $ζ_n$ of the $n$th order correlators in fully developed hydrodynamic turbulence. The scheme is nonperturbative and constructed to respect the fundamental rescaling symmetry of the Euler equation. It constitutes an infinite hierarchy of coupled equations that are obeyed identically with respect to scaling for any set of $ζ_n$. As a consequence $ζ_n$ are determined by solvability conditions and not from power counting. It is arguedthat in order to achieve such a formulation one must recognize that the many-point space-time correlators are not scale invariant in time arguments. The assumption of full scale invariance leads unavoidably to K41 exponents. It is argued the determination of of $ζ_n$ requires equations for infinitely many renormalized objects. One ca n however proceed in controlled successive approximations by truncations of the infinite hierarchy of equations. Clues as to how to truncate without reintroducing power counting can be obtained from renormalized perturbation theory. To this aim we show that the fully resummed perturbation theory is equivalent in its contents to the exact hierarchy of equations obeyed by the $n$th order correlators and Green's function. In light of this important result we can safely use finite resummations to construct successive closures. This paper presents the conceptual and technical details of the scheme. The analysis of the high-order closure procedures which do not destroy the rescaling symmetry and the actual calculations for truncated models will be presented in a forthcoming paper in collaboration with V. Belinicher.