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.
Is the Taylor–Proudman theorem exact in unbounded domains? Case study of the three-dimensional stability of a vortex pair in a rapidly rotating fluid