# Sufficient conditions for dual cascade flux laws in the stochastic 2d Navier-Stokes equations

## TL;DR

This paper establishes rigorous mathematical conditions under which the dual cascade flux laws in 2D turbulence hold, aligning theoretical predictions with experimental observations of energy and enstrophy fluxes.

## Contribution

It provides the first rigorous sufficient conditions for the dual cascade flux laws in 2D Navier-Stokes equations, including cases with only one cascade.

## Key findings

- Conditions for energy flux to large scales established
- Conditions for enstrophy flux to small scales established
- Criteria showing these conditions are nearly necessary

## Abstract

We provide sufficient conditions for mathematically rigorous proofs of the third order universal laws capturing the energy flux to large scales and enstrophy flux to small scales for statistically stationary, forced-dissipated 2d Navier-Stokes equations in the large-box limit. These laws should be regarded as 2d turbulence analogues of the $4/5$ law in 3d turbulence, predicting a constant flux of energy and enstrophy (respectively) through the two inertial ranges in the dual cascade of 2d turbulence. Conditions implying only one of the two cascades are also obtained, as well as compactness criteria which show that the provided sufficient conditions are not far from being necessary. The specific goal of the work is to provide the weakest characterizations of the '0-th laws' of 2d turbulence in order to make mathematically rigorous predictions consistent with experimental evidence.