Axisymmetric interaction between a vortex ring and a free surface

The initial motion of a horizontal vortex ring with upwelling flow in its center is studied analytically by a small-time expansion. The toroidal vortex ring is put impulsively into an inviscid fluid near a free surface. The vortex coordinates and surface elevation are calculated up to third order in time, including the leading gravitational effects. The first-order problem is solved exactly, whereas the higher-order problems are covered only in the limits of small and large radius-to-depth ratio. The leading-order effects of the nonlinear interaction with the free surface are analogous to the case of a two-dimensional vortex pair. To the leading order, vortex stretching tends to move the vortex ring inwards and upwards.