Functional renormalization-group approach to decaying turbulence
Andrei A. Fedorenko1, Pierre Le Doussal2, Kay Jörg Wiese2
1CNRS-Laboratoire de Physique de 'Ecole Normale Supérieure de Lyon, 46 allée d'Italie, 69007 Lyon, France.
2CNRS-Laboratoire de Physique Théorique de
l'Ecole Normale Supérieure,
24 rue Lhomond, 75005 Paris, France.
Abstract
We reconsider the functional renormalization-group (FRG) approach to decaying Burgers turbulence, and extend it to
decaying Navier-Stokes and Surface-Quasi-Geostrophic turbulence. The method is based on a renormalized small-time expansion, equivalent to a loop expansion, and naturally produces a dissipative anomaly and a cascade after a finite time. We explicitly calculate and analyze the one-loop FRG equations in the zero-viscosity limit as a function of
the dimension. For Burgers they reproduce the FRG equation obtained in the context of random manifolds, extending previous results of one of us. Breakdown of energy conservation due to shocks and the appearance of a direct energy cascade corresponds to failure of dimensional reduction in the context of disordered systems. For Navier-Stokes in three dimensions, the velocity-velocity correlation function acquires a linear dependence on the distance, ζ2 = 1, in the inertial range, instead of Kolmogorov's ζ2 = 2/3;
however the possibility remains for corrections at two- or higher-loop order.
In two dimensions, we obtain a numerical solution which conserves energy and exhibits an inverse cascade,
with explicit analytical results both for large and small distances, in agreement with the scaling proposed by Batchelor.
In large dimensions, the one-loop FRG equation for Navier-Stokes converges to that of Burgers.
arXiv:12012.2117 [pdf]
J. Stat. Mech. (2013) P04014 [pdf]
Copyright (C) by Kay Wiese. Last edited December 10, 2012.