EXACT RESUMMATIONS IN THE THEORY OF HYDRODYNAMIC TURBULENCE .1. THE BALL OF LOCALITY AND NORMAL SCALING

This paper is the first in a series of papers that aim at understandin g the scaling behavior of hydrodynamic turbulence. We present in this paper a perturbative theory for the structure functions and the respon se functions of the hydrodynamic velocity field in real space and time . Starting from the Navier-Stokes equations (at high Reynolds number R e) we show that the standard perturbative expansions that suffer from infrared divergences can be exactly resummed using the Belinicher-L'vo v transformation. After this exact (partial) resummation it is proven that the resulting perturbation theory is free of divergences, both in large and in small spatial separations. The hydrodynamic response and the correlations have contributions that arise from mediated interact ions which take place at some space-time coordinates. It is shown that the main contribution arises when these coordinates lie within a shel l of a ''ball of locality'' that is defined and discussed. We argue th at the real space-time formalism that is developed here offers a clear and intuitive understanding of every diagram in the theory, and of ev ery element in the diagrams. One major consequence of this theory is t hat none of the familiar perturbative mechanisms may ruin the classica l 1941 Kolmogorov (K41) scaling solution for the structure functions. Accordingly, corrections to the K41 solutions should be sought in nonp erturbative effects. These effects are the subjects of paper II (the f ollowing paper) and a future paper in this series that will propose a mechanism for anomalous scaling in turbulence, which in particular all ows a multiscaling of the structure functions.