Functional renormalization-group approach to decaying turbulence

TL;DR

This paper extends the functional renormalization-group approach to various decaying turbulence models, providing analytical and numerical insights into energy cascades, dissipation anomalies, and correlation functions across different dimensions.

Contribution

It develops a unified FRG framework for decaying turbulence, including Burgers, Navier-Stokes, and Surface-Quasi-Geostrophic models, with explicit calculations and new analytical results.

Findings

Reproduction of FRG equations for Burgers turbulence, extending previous work.

Identification of energy cascade failure linked to shock formation and dimensional reduction.

Numerical solutions in 2D show energy conservation and inverse cascade consistent with Batchelor's scaling.

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, zeta_2=1, in the inertial range, instead of Kolmogorov's zeta_2=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.