| 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 talk, 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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