Unsteady evolutions of vortex rings with linearly distributed vorticity and various core parameters are considered in an unbounded, inviscid fluid. The instability of a Norbury vortex with a moderate core thickness parameter α is also investigated. Contour integral expressions based on the Biot–Savart law for the velocity field induced by a vortex ring are derived. Numerical results show that all vortex rings except the Norbury vortices will undergo an unsteady evolution process to reach an asymptotic state. The process may be roughly divided into two major stages: initial large deformation stage and later asymptotic oscillating stage. Vortex filamentation is often observed during the first stage. In the second stage, the vortex oscillates periodically with nearly constant amplitude; its core closely resembles a Norbury vortex having the same circulation and impulse, but the dynamic properties and kinetic energies are different.

Nonlinear interactions of vortex rings with a free surface are considered in an incompressible, ideal fluid using the vortex contour dynamics technique and the boundary integral equation method. The flow is axisymmetric and the vorticity is linearly distributed in the vortex. Effects of the gravity and the surface tension as well as the initial geometric parameter of the vortex on the interaction process are investigated in considerable detail. The interaction process may be divided into three major stages: the vortex free-traveling stage, the collision stage, and the vortex stretching and rebounding stage. Time evolutions of both the vortex and free surface under various conditions are provided and analyzed. Two kinds of waves exist on the free surface during interaction. In a special case where the gravity and surface tension are very weak or the vortex is very strong, an electric-bulb-like 'cavity' is formed on the free surface and the vortex is trapped in the 'cavity' for quite a long time, resulting in a large amount of fluid above the mean fluid surface.

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