Dynamics of a vortex ring moving perpendicularly to the axis of a rotating fluid

Abstract

The dynamics of a vortex ring moving orthogonally to the rotation vector of a uniformly rotating fluid is analysed by laboratory experiments and numerical simulations. In the rotating system the vortex ring describes a curved trajectory, turning in the opposite sense to the system's anti-clockwise rotation. This behaviour has been explained by using the analogy with the motion of a sphere in a rotating fluid for which Proudman (1916) computed the forces acting on the body surface. Measurements have revealed that the angular velocity of the vortex ring in its curved trajectory is opposite to the background rotation rate, so that the vortex has a fixed orientation in an inertial frame of reference and that the curvature increases proportionally to the rotation rate.

The dynamics of the vorticity of the vortex ring is affected by the background rotation in such a way that the part of the vortex core in clockwise rotation shrinks while the anti-clockwise-rotating core part widens. By this opposite forcing on either side of the vortex core Kelvin waves are excited, travelling along the toroidal axis of the vortex ring, with a net mass flow which is responsible for the accumulation of passive scalars on the anti-clockwise-rotating core part. In addition, the curved motion of the vortex ring modifies its self-induced strain field, resulting in stripping of vorticity filaments at the front of the vortex ring from the anti-clockwise-rotating core part and at the rear from the core part in clockwise rotation. Vortex lines, being deflected by the main vortex ring due to induction of relative vorticity, are stretched by the local straining field and form a horizontally extending vortex pair behind the vortex ring. This vortex pair propagates by its self-induced motion towards the clockwise-rotating side of the vortex ring and thus contributes to the deformation of the ring core. The deflection of vortex lines from the main vortex ring persists during the whole motion and is responsible for the gradual erosion of the coherent toroidal structure of the initial vortex ring.