  We consider the celebrated Kuramoto model with nearest neighbour
interactions, arranged on a two-dimensional square lattice in presence of two
kinds of noise - annealed and quenched. We focus on both the steady state and
relaxation dynamics of the model. The bare model with annealed noise on finite
$2D$ lattice, in the stationary state, exhibits a crossover from
synchronization to desynchronization as noise strength varies. Finite-size
scaling (FSS) analysis reveals that, in the thermodynamic limit, this crossover
becomes a true phase transition, which is Kosterlitz-Thouless ($KT$)-type
analogous to that of $2D$ $XY$ model. On the other hand, when the noise is
quenched, it does not show any kind of synchronization-desynchronization phase
transition in the thermodynamic limit. But we do observe a crossover from low
noise-strength synchronization to high noise-strength desynchronization in
finite lattices. We analyze the crossover phenomena through the linear
stability of the stationary state solutions and obtain the crossover
noise-strength from the onset of local instability of the unsynchronized one.
The relaxation dynamics also differs for these two types of noise. In case of
annealed noise, the system, in the critically ordered phase, exhibits algebraic
relaxation which is described by the phenomenological Edwards-Wilkinson (EW)
model of growing surface, yielding the same dynamic exponent $z =2$. In
disordered phase, the system shows an exponential decay. On the contrary, the
system with quenched noise, as opposed to the annealed one, always relaxes to
the stationary state exponentially. Both the system-size and noise-strength
dependency of the average relaxation time in the synchronized regime are also
investigated.
