Computing the scaling exponents in fluid turbulence from first principles: the formal setup

We propose a scheme for the calculation from the Navier–Stokes equations of the scaling exponents ζn of the nth order correlation functions 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 scaling exponents ζn. As a consequence the scaling exponents are determined by solvability conditions and not from power counting. It is argued that in order to achieve such a formulation one must recognize that the many-point space-time correlation functions are not scale invariant in their time arguments. The assumption of full scale invariance leads unavoidably to Kolmogorov exponents. It is argued that the determination of all the scaling exponents ζn requires equations for infinitely many renormalized objects. One can however proceed in controlled successive approximations by successive 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 nth order correlation functions and Green’s function. In light of this important result we can safely use finite resummations to construct successive closures of the infinite hierarchy of equations. 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.