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arXiv · 2512.10412

Vortex atmospheres of traveling vortices: rigorous definition, existence, and topological classification

Kyudong Choi·In-Jee Jeong·Young-Jin Sim
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Abstract

In incompressible and inviscid fluids, the vortex atmosphere refers to the collection of fluid particles outside the support of a traveling vortex that are nevertheless carried along with it. This phenomenon has been recognized since the nineteenth century, e.g., in the classical works of O. Reynolds [Nature, 1876] and O. Lodge [Lond. Edinb. Dubl. Phil. Mag., 1885], yet rigorous mathematical definitions and proofs have remained largely undeveloped, with most subsequent studies relying on thin-core approximations or asymptotic analyses. In this paper, we give a rigorous definition of a vortex atmosphere and establish its existence and uniqueness. We further compare the planar atmosphere surrounding a 2D vortex dipole with the axisymmetric atmosphere surrounding a 3D vortex ring. In particular, we emphasize and prove the topological distinctions observed by W. Hicks [Lond. Edinb. Dubl. Phil. Mag., 1919]: under natural assumptions, every 2D dipole with its atmosphere forms an oval-shaped region, whereas for 3D rings, both spheroidal and toroidal configurations may occur. Our proof is based on showing that each atmosphere can be characterized precisely as a specific superlevel set of its corresponding stream function.

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Kyudong Choi, In-Jee Jeong, Young-Jin Sim. 2025-12-11. Vortex atmospheres of traveling vortices: rigorous definition, existence, and topological classification. https://arxiv.org/abs/2512.10412

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