  We present a hydrodynamic model for a thin spherical shell of active nematic
liquid crystal with an arbitrary configuration of defects. The active flows
generated by defects in the director lead to the formation of stable vortices,
analogous to those seen in confined systems in flat geometries, which generate
an effective dynamics for four +1/2 defects that reproduces the tetrahedral to
planar oscillations observed in experiments. As the activity is increased and
the vortices become stronger, the defects are drawn more tightly into pairs,
rotating about antipodal points. We extend this situation to also describe the
dynamics of other configurations of defects. For example, two +1 defects are
found to attract or repel according to the local geometric character of the
director field around them, while additional pairs of opposite charge defects
can give rise to flow states containing more than two vortices. Finally, we
describe the generic relationship between defects in the orientation and
singular points of the flow, and suggest implications for the three-dimensional
nature of the flow and deformation in the shape of the shell.
