  Dilute ultracold quantum gases form an ideal and highly tunable system in
which superuidity can be studied. Recently quantum turbulence in Bose-Einstein
condensates was reported [PRL 103, 045310 (2009)], opening up a new
experimental system that can be used to study quantum turbulence. A novel
feature of this system is that vortex cores now have a finite size. This means
that the vortices are no longer one dimensional features in the condensate, but
that the radial behaviour and excitations might also play an important role in
the study of quantum turbulence in Bose-Einstein condensates. In this paper we
investigate these radial modes using a simplified variational model for the
vortex core. This study results in the frequencies of the radial modes, which
can be compared with the frequencies of the thoroughly studied Kelvin modes.
From this comparison we find that the lowest (l=0) radial mode has a frequency
in the same order of magnitude as the Kelvin modes. However the radial modes
still have a larger energy than the Kelvin modes, meaning that the Kelvin modes
will still constitute the preferred channel for energy decay in quantum
turbulence.
