Vortex atmospheres of traveling vortices: rigorous definition, existence, and topological classification
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| Format: | Preprint |
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2025
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| _version_ | 1866909956683333632 |
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| author | Choi, Kyudong Jeong, In-Jee Sim, Young-Jin |
| author_facet | Choi, Kyudong Jeong, In-Jee Sim, Young-Jin |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_10412 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Vortex atmospheres of traveling vortices: rigorous definition, existence, and topological classification Choi, Kyudong Jeong, In-Jee Sim, Young-Jin Analysis of PDEs 35Q31, 35Q35 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. |
| title | Vortex atmospheres of traveling vortices: rigorous definition, existence, and topological classification |
| topic | Analysis of PDEs 35Q31, 35Q35 |
| url | https://arxiv.org/abs/2512.10412 |