The renormalization group method in statistical hydrodynamics

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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.