  We study a system of Skorokhod stochastic differential equations (SDEs)
modeling the pairwise dispersion (in spatial dimension $d=2$) of heavy
particles transported by a rough self-similar, turbulent flow with H\"{o}lder
exponent $h\in (0,1)$. Under the assumption that $h>0$ is sufficiently small,
we use Lyapunov methods and control theory to show that the Markovian system is
nonexplosive and has a unique, exponentially attractive invariant probability
measure. Furthermore, our Lyapunov construction is radially sharp and gives
partial confirmation on a predicted asymptotic behavior with respect to the
H\"{o}lder exponent $h$ of the invariant probability measure. A physical
interpretation of the asymptotics is that intermittent clustering is weakened
when the carrier flow is sufficiently rough.
