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

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. 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.