Stationary, isotropic and homogeneous two-dimensional turbulence: a first non-perturbative renormalization group approach

TL;DR

This paper applies a non-perturbative renormalization group approach to analyze the statistical properties of 2D turbulence, revealing new symmetries and deriving explicit forms of correlation functions in the stationary turbulent state.

Contribution

It uncovers two extended symmetries of the 2D Navier-Stokes action and derives their Ward identities, enabling exact solutions for correlation functions in the turbulence regime.

Findings

Derived explicit time dependence of the 2-point correlation function.

Identified two extended symmetries of the 2D Navier-Stokes action.

Provided insights into the constraints on multi-point correlation functions.

Abstract

We study the statistical properties of stationary, isotropic and homogeneous turbulence in two-dimensional (2D) flows, focusing on the direct cascade, that is on wave-numbers large compared to the integral scale, where both energy and enstrophy are provided to the fluid. Our starting point is the 2D Navier-Stokes equation in the presence of a stochastic forcing, or more precisely the associated field theory. We unveil two extended symmetries of the Navier-Stokes action which were not identified yet, one related to time-dependent (or time-gauged) shifts of the response fields and existing in both 2D and 3D, and the other to time-gauged rotations and specific to 2D flows. We derive the corresponding Ward identities, and exploit them within the non-perturbative renormalization group formalism, and the large wave-number expansion scheme developed in [Phys. Fluids {\bf 30}, 055102 (2018)]. We consider the flow equation for a generalized $n$-point correlation function, and calculate its leading order term in the large wave-number expansion. At this order, the resulting flow equation can be closed exactly. We solve the fixed point equation for the 2-point function, which yields its explicit time dependence, for both small and large time delays in the stationary turbulent state. On the other hand, at equal times, the leading order term vanishes, so we compute the next-to-leading order term. We find that the flow equations for simultaneous $n$-point correlation functions are not fully constrained by the set of extended symmetries, and discuss the consequences.