# From two-dimensional to three-dimensional turbulence through two-dimensional three-component flows

## TL;DR

This paper investigates the dynamics of 2D3C flows and their connection to 3D turbulence, combining analytical insights and numerical simulations to understand energy transfer and the transition from 2D to 3D turbulence.

## Contribution

It provides a detailed analytical framework for 2D3C flows, explores the role of helicity, and demonstrates how coupling multiple 2D3C flows can lead to 3D turbulence through numerical simulations.

## Key findings

- Homochiral 2D3C flows exhibit inverse energy cascade.
- Coupling orthogonal 2D3C flows results in stationary 3D turbulence.
- Adding 3D modes induces a transition from 2D to 3D turbulence.

## Abstract

The relevance of two-dimensional three-components (2D3C) flows goes well beyond their occurrence in nature, and a deeper understanding of their dynamics might be also helpful in order to shed further light on the dynamics of pure two-dimenional (2D) or three-dimensional (3D) flows and vice versa. The purpose of the present paper is to make a step in this direction through a combination of numerical and analytical work. The analytical part is mainly concerned with the behavior of 2D3C flows in isolation and the connection between the geometry of the nonlinear interactions and the resulting energy transfer directions. Special emphasis is given to the role of helicity. We show that a generic 2D3C flow can be described by two stream functions corresponding to the two helical sectors of the velocity field. The projection onto one helical sector (homochiral flow) leads to a full 3D constraint and to the inviscid conservation of the total (three dimensional) enstrophy and hence to an inverse cascade of the kinetic energy of the third component also. The coupling between several 2D3C flows is studied through a set of suitably designed direct numerical simulations (DNS), where we also explore the transition between 2D and fully 3D turbulence. In particular, we find that the coupling of three 2D3C flows on mutually orthogonal planes subject to small-scale forcing leads to stationary 3D out-of-equilibrium dynamics at the energy containing scales. The transition between 2D and 3D turbulence is then explored through adding a percentage of fully 3D Fourier modes in the volume.