  In this work, we consider the two and three dimensional stochastic convective
Brinkman-Forchheimer (SCBF) equations and examine some asymptotic behaviors of
its strong solution. We establish the asymptotic log-Harnack inequality for the
transition semigroup associated with the SCBF equations driven by additive as
well as multiplicative degenerate noise via the asymptotic coupling method. As
applications of the asymptotic log-Harnack inequality, we derive the gradient
estimate, asymptotic irreducibility, asymptotic strong Feller property,
asymptotic heat kernel estimate and ergodicity. Whenever the absorption
exponent $r\in(3,\infty)$, the asymptotic log-Harnack inequality is obtained
without any restriction on the Brinkman coefficient (effective viscosity)
$\mu>0$, the Darcy coefficient $\alpha>0$ and the Forchheimer coefficient
$\beta>0$.
