Turbulence statistics in a 2D vortex condensate

TL;DR

This paper investigates turbulence statistics in a 2D vortex condensate, revealing distinct mechanisms for momentum flux and turbulent energy, supported by long simulations and analytical modeling.

Contribution

It provides the first explicit formulas for turbulence statistics in a 2D vortex condensate, distinguishing the mechanisms governing momentum flux and turbulent energy.

Findings

Momentum flux is governed by a balance between forcing and mean-flow advection.

Turbulent energy is determined solely by mean-flow advection.

The paper offers the first direct evidence supporting the theoretical prediction for momentum flux.

Abstract

Disentangling the evolution of a coherent mean-flow and turbulent fluctuations, interacting through the non-linearity of the Navier-Stokes equations, is a central issue in fluid mechanics. It affects a wide range of flows, such as planetary atmospheres, plasmas or wall-bounded flows, and hampers turbulence models. We consider the special case of a two-dimensional flow in a periodic box, for which the mean-flow, a pair of box-size vortices called \emph{condensate}, emerges from turbulence through an inverse cascade process. As was recently shown, a perturbative closure describes correctly the condensate when turbulence is excited at small scales. In this context, we obtain explicit results for the statistics of turbulence, encoded in the Reynolds stress tensor. We demonstrate that the two components of the Reynolds stress, the momentum flux and the turbulent energy, are determined by different mechanisms. It was suggested previously that the momentum flux is fixed by a balance between forcing and mean-flow advection: using unprecedently long numerical simulations, we provide the first direct evidence supporting this prediction. By contrast, combining analytical computations with numerical simulations, we show that the turbulent energy is determined only by mean-flow advection, and obtain for the first time a formula describing its profile in the vortex.