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In this work, we consider the two and three dimensional stochastic convectiveBrinkman-Forchheimer (SCBF) equations and examine some asymptotic behaviors ofits strong solution. We establish the asymptotic log-Harnack inequality for thetransition semigroup associated with the SCBF equations driven by additive aswell as multiplicative degenerate noise via the asymptotic coupling method. Asapplications of the asymptotic log-Harnack inequality, we derive the gradientestimate, asymptotic irreducibility, asymptotic strong Feller property,asymptotic heat kernel estimate and ergodicity. Whenever the absorptionexponent $r\in(3,\infty)$, the asymptotic log-Harnack inequality is obtainedwithout any restriction on the Brinkman coefficient (effective viscosity)$\mu>0$, the Darcy coefficient $\alpha>0$ and the Forchheimer coefficient$\beta>0$.

Asymptotic log-Harnack inequality for the stochastic convective Brinkman-Forchheimer equations with degenerate noise