Locality of vortex stretching for the 3D Euler equations

TL;DR

This paper investigates how large-scale stationary flows influence the stretching of small-scale vortex blobs in the 3D Euler equations, using Lagrangian coordinates and the Frobenius theorem to analyze the pressure term's locality.

Contribution

It introduces a novel Lagrangian coordinate framework to analyze vortex stretching and clarifies the conditions under which large-scale flows stretch small-scale vortices.

Findings

Identifies specific large-scale flows that effectively stretch small vortex blobs.

Develops a Lagrangian coordinate method to analyze pressure locality.

Provides insights into the alignment of vortex stretching with straining directions.

Abstract

We consider the 3D incompressible Euler equations under the following situation: small-scale vortex blob being stretched by a prescribed large-scale stationary flow. More precisely, we clarify what kind of large-scale stationary flows really stretch small-scale vortex blobs in alignment with the straining direction. The key idea is constructing a Lagrangian coordinate so that the Lie bracket is identically zero (c.f. the Frobenius theorem), and investigate the locality of the pressure term by using it.