Vortex Rings with Swirl: Axisymmetric Solutions of the Euler Equations with Nonzero Helicity

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Abstract
This work introduces a new class of steady solutions of the axisymmetric Euler equations for an incompressible inviscid fluid. Each solution represents a three-dimensional vortex flow whose azimuthal components of vorticity and velocity are nonzero inside a toroidal region determined by the solution. The governing free-boundary problem is solved by variational techniques. The underlying variational principle is formulated from the natural invariants associated with the evolution equations for axisymmetric flows, and involves a family of invariants that generalizes the standard angular impulse and helicity integrals. A direct method is employed to prove the existence of steady solutions in a bounded domain and steadily translating solutions in space. Qualitative properties of these vortices are discussed and concentrated vortex rings with large swirl are shown to constitute a desingularization of the classical circular vortex filament.

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