A NEW APPROACH TO COMPUTING THE SCALING EXPONENTS IN FLUID TURBULENCEFROM FIRST PRINCIPLES

In this short paper we describe the essential ideas behind a new consi stent closure procedure for the calculation of the scaling exponents z eta(n) of the nth order correlation functions in fully developed hydro dynamic turbulence, starting from first principles. The closure proced ure is constructed to respect the fundamental rescaling symmetry of th e Euler equation, The starting point of the procedure is an infinite h ierarchy of coupled equations that are obeyed identically with respect to scaling for any set of scaling exponents zeta(n). This hierarchy w as discussed in detail in a recent publication [V.S. L'vov and I. Proc accia, Physica A (1998), in press, chao-dyn/9707015]. The scaling expo nents in this set of equations cannot be found from power counting. In this short paper we discuss in detail low order non-trivial closures of this infinite set of equations, and prove that these closures lead to the determination of the scaling exponents from solvability conditi ons. The equations under consideration after this closure are nonlinea r integro-differential equations, reflecting the nonlinearity of the o riginal Navier-Stokes equations. Nevertheless, they have a very specia l structure such that the determination of the scaling exponents requi res a procedure that is very similar to the solution of linens homogen eous equations, in which amplitudes are determined by fitting to the b oundary conditions in the space of scales. The renormalization scale t hat is necessary for any anomalous scaling appears at this point, The Holder inequalities on the scaling exponents select the renormalizatio n scale as the outer scale of turbulence L. (C) 1998 Elsevier Science B.V. All rights reserved.