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Filamentation near monotone zonal vortex caps

Gian Marco Marin, Emeric Roulley

TL;DR

This paper addresses filamentation instabilities for vortex caps in the barotropic Euler equations on a rotating sphere. It combines $L^1$-stability of monotone zonal profiles with precise flow-confinement estimates to show that a small latitudinal perturbation near a monotone zonal cap induces linear-in-time longitudinal drift differences across nearby latitudes, causing the interface length to grow at least like $\kappa(T-T_0)$ for large times. The analysis leverages the Gauss constraint, explicit zonal cap stream-function structures, and stability bounds that control the velocity field under perturbations, ultimately establishing a rigorous mechanism for filamentation near spherical vortex caps. The results extend planar patch insights to spherical geometry and provide a quantitative instability mechanism relevant to geophysical flows on the rotating sphere, with potential implications for understanding large-scale atmospheric dynamics.

Abstract

We study the Euler equations on a rotating unit sphere, focusing on the dynamics of vortex caps. Leveraging the $L^1$-stability of monotone, longitude-independent profiles, we demonstrate that certain ill-prepared initial data within the vortex cap class exhibit an instability characterized by the growth of the interface perimeter. These configurations are nearly equivalent in area to a zonal vortex cap but are perturbed by a localized latitudinal bump. By comparing the longitudinal flows at points along the zonal interface and within the bump region, we track the induced stretching and capture the underlying instability mechanism.

Filamentation near monotone zonal vortex caps

TL;DR

This paper addresses filamentation instabilities for vortex caps in the barotropic Euler equations on a rotating sphere. It combines -stability of monotone zonal profiles with precise flow-confinement estimates to show that a small latitudinal perturbation near a monotone zonal cap induces linear-in-time longitudinal drift differences across nearby latitudes, causing the interface length to grow at least like for large times. The analysis leverages the Gauss constraint, explicit zonal cap stream-function structures, and stability bounds that control the velocity field under perturbations, ultimately establishing a rigorous mechanism for filamentation near spherical vortex caps. The results extend planar patch insights to spherical geometry and provide a quantitative instability mechanism relevant to geophysical flows on the rotating sphere, with potential implications for understanding large-scale atmospheric dynamics.

Abstract

We study the Euler equations on a rotating unit sphere, focusing on the dynamics of vortex caps. Leveraging the -stability of monotone, longitude-independent profiles, we demonstrate that certain ill-prepared initial data within the vortex cap class exhibit an instability characterized by the growth of the interface perimeter. These configurations are nearly equivalent in area to a zonal vortex cap but are perturbed by a localized latitudinal bump. By comparing the longitudinal flows at points along the zonal interface and within the bump region, we track the induced stretching and capture the underlying instability mechanism.
Paper Structure (11 sections, 7 theorems, 115 equations, 1 figure)

This paper contains 11 sections, 7 theorems, 115 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The model and associated literature
  3. Vortex cap solutions
  4. Main result and strategy of proof
  5. Preliminary properties
  6. Generic properties of the barotropic model and analysis of zonal caps
  7. Velocity estimates
  8. Proof of the filamentation
  9. Co-latitudinal and longitudinal evolutions
  10. General discussion
  11. Flow confinement for vortex cap solutions

Key Result

Theorem 1.1

$($Filamentation near monotone zonal vortex caps$)$ Let $N\in\mathbb{N}\setminus\{0,1\}$, $\mathtt{M}\geqslant1$ and Consider the monotone zonal cap with and There exists $\mu_0>0$, such that for all $\mu\in(-\mu_0,\mu_0),$ there exist $\kappa\triangleq\kappa(\mu)>0$ and $T_0\triangleq T_0(\mu)>0$ such that for all $T>T_0,$ there exists $\overline{\delta}\triangleq\overline{\delta}(\mu,\mathtt

Figures (1)

  • Figure 1: Illustration of the filamentation Theorem \ref{['thm filamentation']}.

Theorems & Definitions (18)

  • Definition 1.1: Vortex Cap
  • Remark 1.1
  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.2: $L^1$-stability of zonal profiles on the rotating sphere
  • Remark 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Remark 2.1
  • ...and 8 more