# Discontinuous transition from direct to inverse cascade in three-dimensional turbulence

## TL;DR

This study investigates a modified 3D Navier-Stokes system that exhibits a sudden, discontinuous shift from forward to inverse energy cascade as a parameter is varied, revealing complex turbulence behavior and multiple states.

## Contribution

It demonstrates a discontinuous transition in energy cascade direction in a Navier-Stokes variant with identical invariants, highlighting the role of triadic interactions.

## Key findings

- Discontinuous transition from forward to inverse cascade at a critical parameter value.
- Diverging fluctuations near the critical point.
- Purely isotropic 3D flows can support inverse energy transfer.

## Abstract

Inviscid invariants of flow equations are crucial in determining the direction of the turbulent energy cascade. In this work we investigate a variant of the three dimensional Navier-Stokes equations that shares exactly the same ideal invariants (energy and helicity) and the same symmetries (under rotations, reflexions and scale transforms) as the original equations. It is demonstrated that the examined system displays a change in the direction of the energy cascade when varying the value of a free parameter which controls the relative weights of the triadic interactions between different helical Fourier modes. The transition from a forward to inverse cascade is shown to occur at a critical point in a discontinuous manner with diverging fluctuations close to criticality. Our work thus supports the observation that purely isotropic and three dimensional flow configurations can support inverse energy transfer when interactions are altered and that inside all turbulent flows there is a competition among forward and backward transfer mechanism which might lead to multiple energy-containing turbulent states.