Kelvin wave exchange between toroidal vortices

We study the interaction of a circular vortex ring with a helico-toroidal vortex

in an incompressible inviscid fluid. The vortices are thin tubes of equal

cross-section and circulation. The helico-toroidal vortex lies on the surface

of an immaterial torus (radius: r0, cross sectional radius: r1) and is

characterised by the number of turns, q, that it makes round the torus

centerline while making one turn round the torus symmetry axis to close

on itself. The vortex ring and the immaterial torus have the same

centerline. Under these initial conditions, the flow evolution depends only

on q, r1/r0 and a (the cross-sectional radius of the vortices). Numerical

simulations using the Rosenhead-Moore approximation to the Biot-Savart

law show that for small r1/r0 (<0.2) the vortices translate along and rotate

around the torus symmetry axis while changing shape almost periodically:

at regular intervals the helico-toroidal vortex becomes an almost circular

vortex ring and vice versa. The linear speed of the vortices decreases and

their angular speed increases with growing values of the parameters q and

r1/r0. The capacity of the vortices to carry fluid decreases while their

capacity to stir their surroundings increases as r1/r0 increases.