  The behaviour of a harmonically trapped dipolar Bose-Einstein condensate with
its dipole moments rotating at angular frequencies lower than the transverse
harmonic trapping frequency is explored in the co-rotating frame. We obtain
semi-analytical solutions for the stationary states in the Thomas-Fermi limit
of the corresponding dipolar Gross-Pitaevskii equation and utilise linear
stability analysis to elucidate a phase diagram for the dynamical stability of
these stationary solutions with respect to collective modes. These results are
verified via direct numerical simulations of the dipolar Gross-Pitaevskii
equation, which demonstrate that dynamical instabilities of the co-rotating
stationary solutions lead to the seeding of vortices that eventually relax into
a triangular lattice configuration. Our results illustrate that rotation of the
dipole polarization represents a new route to vortex formation in dipolar
Bose-Einstein condensates.
