Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Wave dynamics governing vortex breakdown in smooth Euler flows

Tsuyoshi Yoneda

TL;DR

This work addresses vortex breakdown in smooth 3D incompressible Euler flows by examining a vortex carried by a prescribed Lagrangian flow. It constructs a Lagrangian moving frame along the vortex axis (via Frenet–Serret geometry) and analyzes transverse particle motion through coordinates alpha1 and alpha2, avoiding singular pressure representations. The authors prove that the transverse coordinates satisfy second-order wave equations, namely $alpha1''(t)$ and $alpha2''(t)$, revealing intrinsic wave dynamics governing breakdown and defining a geometric breakdown criterion based on the relative orientation of transported and base vortex axes. By deriving these wave dynamics directly from the full Euler equations in a smooth setting, the paper links geometric vortex motion to dispersive-wave behavior and contrasts with filament-based reductions such as LIA.

Abstract

We consider the three-dimensional incompressible Euler equations in a setting where a vortex is transported by a prescribed Lagrangian flow. We show that vortex breakdown is governed by wave dynamics generated by the underlying transport flow. The key idea is to avoid any singular integral representation of the pressure term and instead construct an effective Lagrangian coordinate system in which the associated Lie bracket vanishes identically.

Wave dynamics governing vortex breakdown in smooth Euler flows

TL;DR

This work addresses vortex breakdown in smooth 3D incompressible Euler flows by examining a vortex carried by a prescribed Lagrangian flow. It constructs a Lagrangian moving frame along the vortex axis (via Frenet–Serret geometry) and analyzes transverse particle motion through coordinates alpha1 and alpha2, avoiding singular pressure representations. The authors prove that the transverse coordinates satisfy second-order wave equations, namely and , revealing intrinsic wave dynamics governing breakdown and defining a geometric breakdown criterion based on the relative orientation of transported and base vortex axes. By deriving these wave dynamics directly from the full Euler equations in a smooth setting, the paper links geometric vortex motion to dispersive-wave behavior and contrasts with filament-based reductions such as LIA.

Abstract

We consider the three-dimensional incompressible Euler equations in a setting where a vortex is transported by a prescribed Lagrangian flow. We show that vortex breakdown is governed by wave dynamics generated by the underlying transport flow. The key idea is to avoid any singular integral representation of the pressure term and instead construct an effective Lagrangian coordinate system in which the associated Lie bracket vanishes identically.
Paper Structure (2 sections, 1 theorem, 43 equations)

This paper contains 2 sections, 1 theorem, 43 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['criterion']}

Key Result

Theorem 1

The quantities $\alpha_1$ and $\alpha_2$, which govern the occurrence of vortex breakdown, satisfy the following second--order wave equations:

Theorems & Definitions (5)

  • Remark 1
  • Remark 2
  • Theorem 1
  • Remark 3
  • Remark 4