A non-perturbative renormalization group study of the stochastic Navier--Stokes equation

We study the renormalization group flow of the average action of the stochastic Navier–Stokes equation with power-law forcing. Using Galilean invariance we introduce a non-perturbative approximation adapted to the zero frequency sector of the theory in the parametric range of the H"older exponent $4-2\epsilon$ of the forcing where real-space local interactions are relevant. In any spatial dimension $d$, we observe the convergence of the resulting renormalization group flow to a unique fixed point which yields a kinetic energy spectrum scaling in agreement with canonical dimension analysis. Kolmogorov’s -5/3 law is, thus, recovered for $\epsilon =2$ as also predicted by perturbative renormalization. At variance with the perturbative prediction, the -5/3 law emerges in the presence of a \emph{saturation} in the $\epsilon$-dependence of the scaling dimension of the eddy diffusivity at $\epsilon =3\mathrm{/}2$ when, according to perturbative renormalization, the velocity field becomes infra-red relevant.