  We consider the two-dimensional (2D) noisy Kuramoto model of synchronization
with short-range coupling and a Gaussian distribution of intrinsic frequencies,
and investigate its ordering dynamics following a quench. We consider both
underdamped (inertial) and over-damped dynamics, and show that the long-term
properties of this intrinsically out-of-equilibrium system do not depend on the
inertia of individual oscillators. The model does not exhibit any phase
transition as its correlation length remains finite, scaling as the inverse of
the standard deviation of the distribution of intrinsic frequencies. The quench
dynamics proceeds via domain growth, with a characteristic length that
initially follows the growth law of the 2D XY model, although is not given by
the mean separation between defects. Topological defects are generically free,
breaking the Berezinskii-Kosterlitz-Thouless scenario of the 2D XY model.
Vortices perform a random walk reminiscent of the self-avoiding random walk,
advected by the dynamic network of boundaries between synchronised domains;
featuring long-time super-diffusion, with the anomalous exponent $\alpha=3/2$.
