Variational principles for Hill’s spherical vortex and nearly spherical vortices

Wan, Yieh Hei

doi:10.1090/S0002-9947-1988-0946444-X

Variational principles for Hill’s spherical vortex and nearly spherical vortices
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Trans. Amer. Math. Soc. 308 (1988), 299-312 Request permission

Abstract:

In this paper, vortex rings are regarded as axisymmetric motions without swirl of an incompressible inviscid fluid in space, with vorticity confined to their finite cores. The main results of this paper are (H) Hill’s spherical vortex is a "nondegenerate" local maximum of the energy function subject to a fixed impulse, among vortex rings. (N) Norbury’s nearly spherical vortex is a "nondegenerate" local maximum of the energy function subject to a fixed impulse, and a fixed circulation. Estimates are established to overcome the discontinuity of vorticity distributions, and the singular behavior of Stoke’s stream functions near the axis of symmetry. The spectral analysis involves the use of Legendre’s functions.

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