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arXiv · 1202.4588

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

Carlos Mejía-Monasterio·Paolo Muratore-Ginanneschi
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Abstract

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 Hölder exponent $4-2\,\varepsilon$ 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 $\varepsilon=2$ 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 $\varepsilon$-dependence of the scaling dimension of the eddy diffusivity at $\varepsilon=3/2$ when, according to perturbative renormalization, the velocity field becomes infra-red relevant.

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Carlos Mejía-Monasterio, Paolo Muratore-Ginanneschi. 2012-06-20. A non-perturbative renormalization group study of the stochastic Navier--Stokes equation. https://doi.org/10.1103/physreve.86.016315

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