THE RENORMALIZATION-GROUP METHOD IN STATISTICAL HYDRODYNAMICS

This paper gives a first principles formulation of a renormalization g roup (RG) method appropriate to study of turbulence in incompressible fluids governed by Navier-Stokes equations. The present method is a mo mentum-shell RG of Kadanoff-Wilson type based upon the Martin-Siggia-R ose (MSR) field-theory formulation of stochastic dynamics. A simple se t of diagrammatic rules are developed which are exact within perturbat ion theory (unlike the well-known Ma-Mazenko prescriptions). It is als o shown that the claim of Yakhot and Orszag (1986) is false that highe r-order terms are irrelevant in the epsilon expansion RG for randomly forced Navier-Stokes (RFNS) with power-law force spectrum ($) over cap F(k)=D(0)k(-d+(4-epsilon)). In fact, as a consequence of Galilei cova riance, there are an infinite number of higher-order nonlinear terms m arginal by power counting in the RG analysis of the power-law RFNS, ev en when epsilon much less than 4. The difficulty does not occur in the Forster-Nelson-Stephen (FNS) RG analysis of thermal fluctuations in a n equilibrium NS fluid, which justifies a linear regression law for d> 2. On the other hand, the problem occurs also at the nontrivial fixed point in the FNS Model A, or its Burgers analog, when d<2. The margina l terms can still be present at the strong-coupling fixed point in tru e NS turbulence. If so, infinitely many fixed points may exist in turb ulence and be associated to a somewhat surprising phenomenon: nonunive rsality of the inertial-range scaling laws depending upon the dissipat ion-range dynamics.