  The present work is concerned about two-dimensional stochastic convective
Brinkman-Forchheimer (2D SCBF) equations perturbed by a white noise (non
degenerate) in smooth bounded domains in $\R^{2}$. We establish two important
properties of the Markov semigroup associated with the solutions of 2D SCBF
equations (for the absorption exponent $r=1,2,3$), that is, irreducibility and
strong Feller property. These two properties implies the uniqueness of
invariant measures and ergodicity also. Then, we discuss about the ergodic
behavior of 2D SCBF equations by providing a Large Deviation Principle (LDP)
for the occupation measure for large time (Donsker-Varadhan), which describes
the exact rate of exponential convergence.
