Dynamics of vortex cap solutions on the rotating unit sphere

TL;DR

This work provides the first rigorous analytical construction of uniformly rotating vortex-cap solutions on the rotating unit sphere, near zonal flat-cap states. By recasting the dynamics as contour-dynamics problems on interfaces and employing Crandall–Rabinowitz bifurcation theory, the authors establish the existence of $\mathbf{m}$-fold rotating patches for both one-interface (M=2) and two-interface (M=3) configurations; the former yields branches with bifurcation points $c_{\mathbf{m}}(\tilde{\gamma})=\tilde{\gamma}-(\omega_N-\omega_S)\frac{\mathbf{m}-1}{2\mathbf{m}}$ corresponding to shifted Burbea frequencies, while the latter requires large $\mathbf{m}$ and non-degeneracy to avoid spectral collisions, producing (at least) two branches of vortex strips. The analysis hinges on detailed spectral properties of the linearized nonlocal operators, the Fredholm structure, and the compactness of singular integral components (e.g., Hilbert transform in Hölder spaces). These results extend the vortex-patch program to curved geometries and rotating backgrounds, shedding light on geophysical-like vortex structures on planetary scales. Practically, the work provides rigorous conditions under which rotating multi-layer patches exist on a sphere, offering a mathematical basis for observed spherical vortex configurations in geophysical and planetary contexts.

Abstract

In this work, we analytically study the existence of periodic vortex cap solutions for the homogeneous and incompressible Euler equations on the rotating unit 2-sphere, which was numerically conjectured by Dritschel-Polvani and Kim-Sakajo-Sohn. Such solutions are piecewise constant vorticity distributions, subject to the Gauss constraint and rotating uniformly around the vertical axis. The proof is based on the bifurcation from zonal solutions given by spherical caps. For the one--interface case, the bifurcation eigenvalues correspond to Burbea's frequencies obtained in the planar case but shifted by the rotation speed of the sphere. The two--interfaces case (also called band type or strip type solutions) is more delicate. Though, for any fixed large enough symmetry, and under some non-degeneracy conditions to avoid spectral collisions, we achieve the existence of at most two branches of bifurcation.

Figures (4)

• Figure 1: Convention colatitude/longitude for spherical coordinates.
• Figure 2: Representation of one interface (in red) vortex cap solutions with $6$-fold symmetry.
• Figure 3: Representation of two interfaces (in red and blue) vortex cap solutions with $6$-fold symmetry.
• Figure 4: The shaded regions represent the admissible couples $(\theta_1,\theta_2)\in(0,\pi)^2$ with $\theta_1<\theta_2.$

Theorems & Definitions (27)

• Lemma 1.1
• Lemma 1.2
• Theorem 1.1
• Remark 1.1
• Theorem 1.2
• Remark 1.2
• Lemma 2.1
• proof
• Proposition 2.1
• proof