  We studied a rotating Bose-Einstein condensate confined in ring trap
configurations that can be produced starting with a bubble trap confinement,
approximated by a Mexican hat and shifted harmonic oscillator potentials. Using
a variational technique and perturbation theory, we determined the vortex
configurations in this system by varying the interparticle interaction and the
angular velocity of the atomic cloud. We found that the phase diagram of the
system has macrovortex structures for small positive values of the interaction
parameter, and the charge of the central vortex increases with rotation.
Strengthening the atomic interaction makes the macrovortex unstable, and it
decays into multiple singly-charged vortices that arrange themselves in a
lattice configuration. We also look for experimentally realizable methods to
determine the vortex configuration without relying upon absorption imaging
since the structures are not always visible in the latter. More specifically,
we study how the vortex distribution affects the collective modes of the
condensate by solving the Gross-Pitaevskii equation numerically and by
analytical predictions using the sum-rule approach for the frequencies of the
modes. These results reveal important signatures to characterize the
macrovortices and vortex lattice transitions in the experiments.
