  Motivated by the recent experiment [Wright et al., Phys. Rev. A 88, 063633
(2013)], we investigate formation of vortices in an annular BEC stirred by a
narrow blue-detuned optical beam. In the framework of a two-dimensional mean
field model, we study the dissipative dynamics of the condensate with
parameters matched to the experimental conditions. Vortex-antivortex pairs
appear near the center of the stirrer in the bulk of the condensate for slow
motion of the stirring beam. When the barrier angular velocity is above some
critical value, an outer edge surface mode develops and breaks into the
vortices entering the condensate annulus. We determine the conditions for
creation of the vortex excitations in the stirred toroidal condensate and
compare our results with the experimental observations.
