  We study the long-range order in two dimensions where an order parameter is
advected by laminar flows such as rotational, shear, and elongational flows.
Under these flows, we analyze an ordered state of the $O(N)$ scalar model in
the large-$N$ limit. We show that the stability of the ordered state depends on
the flow pattern; shear and elongational flows stabilize the long-range order
but rotational flow does not. We discuss the physical mechanism underlying our
results by connecting static correlations of fluctuations and their dynamics
based on the interaction representation used in quantum mechanics. We find that
advective transport induces superdiffusion under shear and elongational flows,
thereby stabilizing the long-range order.
