In a uniform fluid, a quantized vortex line with circulation
h∕M can support long-wavelength helical traveling waves $\ensuremath{\propto}{e}^{i(kz\ensuremath{-}{\ensuremath{\omega}}_{k}t)}$ with the well-known Kelvin dispersion relation ${\ensuremath{\omega}}_{k}\ensuremath{\approx}(\ensuremath{\hbar}{k}^{2}∕2M)\mathrm{ln}(1∕\ensuremath{\mid}k\ensuremath{\mid}\ensuremath{\xi})$, where $\ensuremath{\xi}$ is the vortex-core radius. This result is extended to include the effect of a nonuniform harmonic trap potential, using a quantum generalization of the Biot-Savart law that determines the local velocity
V of each element of the vortex line. The normal-mode eigenfunctions form an orthogonal Sturm-Liouville set. Although the line's curvature dominates the dynamics, the transverse and axial trapping potential also affect the normal modes of a straight vortex on the symmetry axis of an axisymmetric Thomas-Fermi condensate. The leading effect of the nonuniform condensate density is to increase the amplitude along the axis away from the trap center. Near the ends, however, a boundary layer forms to satisfy the natural Sturm-Liouville boundary conditions. For a given applied frequency, the next-order correction renormalizes the local wave number
k(z) upward near the trap center, and
k(z) then increases still more toward the ends.
While quantized vortices and vortex lattices in Bose-Einstein condensates (BECs) have received much recent attention, the formation and dynamics of vortex dipoles have been nearly unexplored in experimental BECs. Vortex dipoles consist of a pair of vortices of opposite circulation moving under the influence of each other, and may be considered as metastable topological structures that carry linear momentum in a fluid. In this talk, a collaborative experimental and numerical study of the formation, dynamics, and lifetimes of vortex dipoles in BECs will be presented. In our work [1], single vortex dipoles were deterministically nucleated by causing highly oblate, harmonically trapped BECs to move past a repulsive obstacle above a supercritical flow velocity. Our measured critical velocity for vortex dipole shedding, and the subsequent dipole dynamics, are in good agreement with new numerical simulations. We find that vortex dipoles can survive for many seconds in the BEC without selfannihilation, and that multiply charged vortex dipoles can be formed with high enough flow velocities. References: 1. T.W. Neely, E.C. Samson, A.S. Bradley, M.J. Davis, and B.P. Anderson, preprint available at http://arxiv.org/abs/0912.3773.