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Stability of the Rankine Vortex and Perimeter Growth in Vortex Patches

John Brownfield

TL;DR

This work studies the stability and long-term geometry of Rankine vortex patches in 2D Euler flow on the plane. It leverages conserved quantities and the pseudo-energy $E(\omega)$ to derive quantitative $L^1$ stability bounds for Rankine patches, with strengthened results under $m$-fold symmetry and bounded angular momentum. The authors construct a simply connected perturbation with large angular momentum that remains close to the Rankine disk in $L^1$ but exhibits linear-in-time growth of the patch perimeter, using a universal-cover twisting argument and a bucket-decomposition to control boundary twisting. The results extend prior plane-based stability analyses and provide a mechanism for regularity loss via perimeter growth, connecting to observed filamentation phenomena in simulations.

Abstract

We prove that for $ω: \mathbb{R}^2 \to [0,1]$ sharing the same total vorticity and center of vorticity as the Rankine vortex, the $L^1$ deviation from the Rankine patch can be bounded by a function of the pseudo-energy deviation and the angular momentum of $ω$. In the case of $m-$fold symmetry, the dependence on the angular momentum can be dropped. Using this, we affirm the results of prior simulations by demonstrating linear in time perimeter growth for a simply connected perturbation of the Rankine vortex.

Stability of the Rankine Vortex and Perimeter Growth in Vortex Patches

TL;DR

This work studies the stability and long-term geometry of Rankine vortex patches in 2D Euler flow on the plane. It leverages conserved quantities and the pseudo-energy to derive quantitative stability bounds for Rankine patches, with strengthened results under -fold symmetry and bounded angular momentum. The authors construct a simply connected perturbation with large angular momentum that remains close to the Rankine disk in but exhibits linear-in-time growth of the patch perimeter, using a universal-cover twisting argument and a bucket-decomposition to control boundary twisting. The results extend prior plane-based stability analyses and provide a mechanism for regularity loss via perimeter growth, connecting to observed filamentation phenomena in simulations.

Abstract

We prove that for sharing the same total vorticity and center of vorticity as the Rankine vortex, the deviation from the Rankine patch can be bounded by a function of the pseudo-energy deviation and the angular momentum of . In the case of fold symmetry, the dependence on the angular momentum can be dropped. Using this, we affirm the results of prior simulations by demonstrating linear in time perimeter growth for a simply connected perturbation of the Rankine vortex.

Paper Structure

This paper contains 7 sections, 14 theorems, 84 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Stability of the Rankine Vortex
  4. Perimeter Growth of a Simply Connected Vortex Patch
  5. The Setup
  6. The Twisting
  7. Appendix

Key Result

Proposition 1.1

There exists $C_n>0$ such that for all $\omega : \mathbb{R}^n \to [0,1]$ with $\omega \in L^1(\mathbb{R}^n)$, where $E^*$ is the ball centered at the origin with $|E^*| = \int_{\mathbb{R}^n} \omega(x)dx$.

Figures (2)

  • Figure 1: A rough sketch of $\Omega_0$
  • Figure 2: Illustration of our setup over $\mathbb{R}\times\mathbb{R}^+$. Note that $N(t) = 1$.

Theorems & Definitions (21)

  • Proposition 1.1
  • Theorem 1.2: Stability under $m$-fold symmetry
  • Theorem 1.3: Stability under bounded angular momentum
  • Theorem 1.4: Linear in Time Perimeter Growth
  • Proposition 3.1
  • Proposition 3.2
  • Theorem : Restatement of \ref{['thm:mstability']}
  • proof
  • Theorem : Restatement of \ref{['thm:pgrowth']}
  • Lemma 4.1
  • ...and 11 more