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.