  In this work, we discuss how the linear and non-linear advection terms modify
the scaling behavior of the continuous symmetry breaking and stabilize the
long-range order, even in $d=2$ far from equilibrium, by means of simple
scaling arguments. For an example of the liner advection, we consider the
$O(n)$ model in the steady shear. Our scaling analysis reveals that the model
can undergo the continuous symmetry breaking even in $d=2$ and, moreover,
predicts the upper critical dimension $d_{\rm up}=2$. These results are fully
consistent with a recent numerical simulation of the $O(2)$ model, where the
mean-field critical exponents are observed even in $d=2$. For an example of the
non-linear advection, we consider the Toner-Tu hydrodynamic theory, which was
introduced to explain polar-ordered flocks, such as the Vicsek model. Our
simple scaling argument reproduces the previous results by the dynamical
renormalization theory. Furthermore, we discuss the effects of the additional
non-linear terms discovered by the recent re-analysis of the hydrodynamic
equation. Our scaling argument predicts that the additional non-linear terms
modify the scaling exponents and, in particular, recover the isotropic scaling
reported in a previous numerical simulation of the Vicsek model. We discuss
that the critical exponents predicted by the naive scaling theory become exact
in $d=2$ by using a symmetry consideration and similar argument proposed by
Toner and Tu.
