The renormalization group method in statistical hydrodynamics

Abstract

This paper gives a first principles formulation of a renormalization group (RG) method appropriate to study of turbulence in incompressible fluids governed by Navier–Stokes equations. The present method is a momentum-shell RG of Kadanoff–Wilson type based upon the Martin–Siggia–Rose (MSR) field-theory formulation of stochastic dynamics. A simple set of diagrammatic rules are developed which are exact within perturbation theory (unlike the well-known Ma–Mazenko prescriptions). It is also shown that the claim of Yakhot and Orszag (1986) is false that higher-order terms are irrelevant in the ε expansion RG for randomly forced Navier–Stokes (RFNS) with power-law force spectrum F̂(k)=D0k−d+(4−ε). In fact, as a consequence of Galilei covariance, there are an infinite number of higher-order nonlinear terms marginal by power counting in the RG analysis of the power-law RFNS, even when ε≪4. The difficulty does not occur in the Forster–Nelson–Stephen (FNS) RG analysis of thermal fluctuations in an 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 marginal terms can still be present at the strong-coupling fixed point in true NS turbulence. If so, infinitely many fixed points may exist in turbulence and be associated to a somewhat surprising phenomenon: nonuniversality of the inertial-range scaling laws depending upon the dissipation-range dynamics.