A remark on the zeroth law and instantaneous vortex stretching on the incompressible 3D Euler equations

TL;DR

This paper investigates vortex tube behavior in 3D incompressible Euler equations, demonstrating that such behavior induces an energy cascade and a modified zeroth-law, with a derivation of Kolmogorov's law from a new perspective.

Contribution

It introduces a novel analysis of vortex-tube dynamics in Euler equations, linking vortex stretching to energy cascade without nonlinear scale interaction.

Findings

Vortex-tube behavior induces energy cascade in Euler flows.

A modified zeroth-law of turbulence is proved.

Kolmogorov's -5/3 law is derived from vortex dynamics perspective.

Abstract

By DNS of Navier-Stokes turbulence, Goto-Saito-Kawahara (2017) showed that turbulence consists of a self-similar hierarchy of anti-parallel pairs of vortex tubes, in particular, stretching in larger-scale strain fields creates smaller-scale vortices. Inspired by their numerical result, we examine the Goto-Saito-Kawahara type of vortex-tubes behavior using the 3D incompressible Euler equations, and show that such behavior induces energy cascade in the absence of nonlinear scale-interaction. From this energy cascade, we prove a modified version of the zeroth-law. In the Appendix, we derive Kolmogorov's $-5/3$-law from the GSK point of view.