  We consider merger of two parallel toroidal atomic Bose-Einstein condensates
with different vorticities in a three-dimensional (3D) trap. In the
tunnel-coupling regime, Josephson vortices (rotational fluxons) emerge in the
barrier between the superflows. When the barrier is gradually eliminated, we
observe essentially three-dimensional evolution of quantum vortices, which may
include the development of the Kelvin-Helmholtz instability at the interface
between the rings, in the framework of a weakly dissipative Gross-Pitaevskii
equation. An initially more populated ring, carrying a persistent current, can
drag an initially non-rotating less populated one into the same vortex state.
The final state of the condensate crucially depends on an initial population
imbalance in the double-ring set, as well as on the shape of the 3D trapping
potential, oblate or prolate. In the prolate (axially elongated) configuration,
robust 3D hybrid structures may appear as a result of the merger of persistent
currents corresponding to different vorticities.
