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Confinement results near point vortices on the rotating sphere

Martin Donati, Emeric Roulley

TL;DR

We address the confinement of absolute vorticity near a finite set of point vortices for the Euler equations on the rotating sphere, extending planar confinement results to curved geometry. The analysis hinges on the Biot–Savart law on $\,oldsymbol{S}^2$, a decomposition of the vorticity into sharply concentrated blobs, and rigorous control of vorticity moments to propagate localization. We prove almost-sure collision impossibility, establish a logarithmic-in-time confinement bound for general configurations, and derive conditional, configuration-dependent power-law confinement under stability hypotheses; the latter are shown to be realizable by explicit point-vortex configurations. The work unifies and improves proofs from planar settings within the spherical geometry, clarifying how curvature and rotation influence long-time localization and offering concrete constructions (e.g., polar counter-rotating pairs and four-vortex stations) that realize the theoretical bounds. Collectively, the results illuminate the realistic persistence of concentrated vortices on rotating planets and provide a toolkit for further exploring vortex crystals on curved surfaces.

Abstract

We study the Euler equation on the rotating sphere in the case where the absolute vorticity is initially sharply concentrated around several points. We follow the literature already concerning vorticity confinement for the planar Euler equations, and obtain similar results on the rotating sphere, with new challenges due to the geometry. More precisely, we show the improbability of collisions for point-vortices, logarithmic in time absolute vorticity confinement for general configurations, the optimality of this last result in general, and the existence of configurations with power-law long confinement. We take this opportunity to write a unified, self-contained, and improved version of all the proofs, previously scattered across multiple papers on the planar case, with detailed exposition for pedagogical clarity.

Confinement results near point vortices on the rotating sphere

TL;DR

We address the confinement of absolute vorticity near a finite set of point vortices for the Euler equations on the rotating sphere, extending planar confinement results to curved geometry. The analysis hinges on the Biot–Savart law on , a decomposition of the vorticity into sharply concentrated blobs, and rigorous control of vorticity moments to propagate localization. We prove almost-sure collision impossibility, establish a logarithmic-in-time confinement bound for general configurations, and derive conditional, configuration-dependent power-law confinement under stability hypotheses; the latter are shown to be realizable by explicit point-vortex configurations. The work unifies and improves proofs from planar settings within the spherical geometry, clarifying how curvature and rotation influence long-time localization and offering concrete constructions (e.g., polar counter-rotating pairs and four-vortex stations) that realize the theoretical bounds. Collectively, the results illuminate the realistic persistence of concentrated vortices on rotating planets and provide a toolkit for further exploring vortex crystals on curved surfaces.

Abstract

We study the Euler equation on the rotating sphere in the case where the absolute vorticity is initially sharply concentrated around several points. We follow the literature already concerning vorticity confinement for the planar Euler equations, and obtain similar results on the rotating sphere, with new challenges due to the geometry. More precisely, we show the improbability of collisions for point-vortices, logarithmic in time absolute vorticity confinement for general configurations, the optimality of this last result in general, and the existence of configurations with power-law long confinement. We take this opportunity to write a unified, self-contained, and improved version of all the proofs, previously scattered across multiple papers on the planar case, with detailed exposition for pedagogical clarity.
Paper Structure (25 sections, 29 theorems, 411 equations, 3 figures)

This paper contains 25 sections, 29 theorems, 411 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Point-vortices and vorticity confinement on the plane
  3. Barotropic model
  4. Point-vortices on the rotating sphere
  5. Main results
  6. Improbability of point vortex collisions: proof of Theorem \ref{['theo:improbabilite']}
  7. Estimates on vortex evolutions
  8. Estimates on the vorticity moments
  9. Growth of the support
  10. Logarithmic confinement results
  11. Proof of Theorem \ref{['theo:general_conf']}
  12. Optimality of the bound
  13. Power-law confinement result
  14. Super-stability hypotheses
  15. The conditional theorem
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $(z_i^0)_{1\leqslant i\leqslant N}$ be $N$ pairwise distinct points of $\mathbb{R}^2$ and $(\Gamma_i)_{1\leqslant i \leqslant N}$ be some non-vanishing intensities such that the solution of the point-vortex dynamics PVS with initial datum $(z_i^0)_{1\leqslant i\leqslant N}$ is global in time and Let $\omega_0^\varepsilon$ satisfying Hypothesis hyp:omega0. Then for every $\beta<1/2$ there exist

Figures (3)

  • Figure 1: Numerical simulation of the Navier-Stokes equations with Reynold's number $10^5$, and an initial data concentrated near the vortex crystal \ref{['eq:VCN']} with $N=8$ and $\kappa =1$. Credit: Matthieu Brachet, Université de Poitiers, CNRS, LMA, Poitiers, France.
  • Figure 2: Maximum real part of the eigenvalues of $\mathrm{D} \mathcal{F}(\mathbf{X}^*)$ for the $N=4$ stationary configuration in the case $a=0.1$ (left), $a=0.3$ (right), depending on $\gamma$.
  • Figure 3: Maximum real part of the eigenvalues of $\mathrm{D} \mathcal{F}(\mathbf{X}^*)$ for the $N=4$ stationary configuration for values of $a$ close to 1, depending on $\gamma$.

Theorems & Definitions (54)

  • Theorem 1.1: BM18
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 2.1
  • Theorem 2.1
  • proof
  • Lemma 2.1
  • proof : Proof of Lemma \ref{['lem:bound_improb']}
  • ...and 44 more