Vortex atmospheres of traveling vortices: rigorous definition, existence, and topological classification
Kyudong Choi, In-Jee Jeong, Young-Jin Sim

TL;DR
This paper rigorously defines the vortex atmosphere in incompressible, inviscid fluids, proves its existence and uniqueness, and classifies topologically different configurations for 2D and 3D vortices.
Contribution
It provides the first rigorous mathematical definition and proof of the vortex atmosphere, including topological classification for different vortex types.
Findings
Vortex atmospheres are uniquely characterized as superlevel sets of stream functions.
2D vortex dipoles have oval-shaped atmospheres.
3D vortex rings can have spheroidal or toroidal atmospheres.
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.…
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Taxonomy
TopicsNavier-Stokes equation solutions · Quantum chaos and dynamical systems · Fluid dynamics and aerodynamics studies
