  We introduce a new functional inequality, which is a modification of
log-Harnack inequality established in [20] and [29], and prove that it implies
the asymptotically strong Feller property (ASF). This inequality seems to
generalize the criterion for ASF in [Proposition 3.12,14]. As a example, we
show by an asymptotic coupling that 2D stochastic Navier-Stokes equation driven
by highly degenerate but \emph{essentially elliptic} noises satisfies our
modified log-Harnack inequality.
