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Existence and stability of asymmetric Burgers vortices

Th. Gallay, C. E. Wayne

TL;DR

The paper proves the existence of non-axisymmetric Burgers vortices under a weak asymmetric strain by a perturbative construction around the axisymmetric Burgers vortex, using a fixed-point approach in a weighted function space. It develops a rigorous linearization framework with the operators ${\cal L},{\cal M},\Lambda$ and analyzes the large-$R$ (high Reynolds) limit to recover the known Gaussian core with $\lambda/R$-type corrections, in agreement with prior asymptotics. It also establishes stability of these asymmetric vortices against localized 2D perturbations via energy estimates, uniformly in the Reynolds number, and discusses expansions for small and large Reynolds numbers as well as potential extensions to larger asymmetry or 3D perturbations. Together, these results extend the understanding of Burgers-type vortices beyond axisymmetry and provide a solid mathematical foundation for their existence and robustness in weakly asymmetric turbulent-like flows.

Abstract

Burgers vortices are stationary solutions of the three-dimensional Navier-Stokes equations in the presence of a background straining flow. These solutions are given by explicit formulas only when the strain is axisymmetric. In this paper we consider a weakly asymmetric strain and prove in that case that non-axisymmetric vortices exist for all values of the Reynolds number. In the limit of large Reynolds numbers, we recover the asymptotic results of Moffatt, Kida and Ohkitani (1994). We also show that the asymmetric vortices are stable with respect to localized two-dimensional perturbations.

Existence and stability of asymmetric Burgers vortices

TL;DR

The paper proves the existence of non-axisymmetric Burgers vortices under a weak asymmetric strain by a perturbative construction around the axisymmetric Burgers vortex, using a fixed-point approach in a weighted function space. It develops a rigorous linearization framework with the operators and analyzes the large- (high Reynolds) limit to recover the known Gaussian core with -type corrections, in agreement with prior asymptotics. It also establishes stability of these asymmetric vortices against localized 2D perturbations via energy estimates, uniformly in the Reynolds number, and discusses expansions for small and large Reynolds numbers as well as potential extensions to larger asymmetry or 3D perturbations. Together, these results extend the understanding of Burgers-type vortices beyond axisymmetry and provide a solid mathematical foundation for their existence and robustness in weakly asymmetric turbulent-like flows.

Abstract

Burgers vortices are stationary solutions of the three-dimensional Navier-Stokes equations in the presence of a background straining flow. These solutions are given by explicit formulas only when the strain is axisymmetric. In this paper we consider a weakly asymmetric strain and prove in that case that non-axisymmetric vortices exist for all values of the Reynolds number. In the limit of large Reynolds numbers, we recover the asymptotic results of Moffatt, Kida and Ohkitani (1994). We also show that the asymmetric vortices are stable with respect to localized two-dimensional perturbations.

Paper Structure

This paper contains 5 sections, 11 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. Linearization in the symmetric case
  3. Large Reynolds number asymptotics
  4. Existence of asymmetric vortices
  5. Stability of asymmetric vortices

Key Result

Theorem 1.1

There exist \lambda_0 > 0 and K_0 > 0 such that, for all \lambda \in [0,\lambda_0] and all \alpha \in {\bf R}, equation (omeq2) has a unique stationary solution \omega^{\alpha,\lambda},{\bf u}^{\alpha,\lambda} such that \|\omega^{\alpha,\lambda} - \alpha G\|_Y \le K_0. Moreover, \|\omega^{\alpha,\la

Theorems & Definitions (12)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Proposition 3.1
  • Remark 3.2
  • Corollary 3.3
  • Proposition 3.4
  • Lemma 4.1
  • ...and 2 more