  We show theoretically that elongated vortex-antivortex dipoles can be created
controllably in trapped Bose-Einstein condensates, using known experimental
techniques. Vortex dipoles of sufficient length are unstable and cascade into
slow vortex rings which ultimately decay via sound emission. This instability
of antiparallel vortex line elements, which self-generates Kelvin waves on
vortex loops and in trapped atomic gases, may play a role in bridging the
Kelvin-wave and Kolmogorov-Richardson cascades of quantum turbulence.
