  The stochastic convective Brinkman-Forchheimer (SCBF) equations or the tamed
Navier-Stokes equations in bounded or periodic domains are considered in this
work. We show the existence of a pathwise unique strong solution (in the
probabilistic sense) satisfying the energy equality (It\^o formula) to the SCBF
equations perturbed by multiplicative Gaussian noise. We exploited a
monotonicity property of the linear and nonlinear operators as well as a
stochastic generalization of the Minty-Browder technique in the proofs. The
energy equality is obtained by approximating the solution using approximate
functions constituting the elements of eigenspaces of the Stokes operator in
such a way that the approximations are bounded and converge in both Sobolev and
Lebesgue spaces simultaneously. We further discuss about the global in time
regularity results of such strong solutions in periodic domains. The
exponential stability results (in mean square and pathwise sense) for the
stationary solutions is also established in this work for large effective
viscosity. Moreover, a stabilization result of the stochastic convective
Brinkman-Forchheimer equations by using a multiplicative noise is obtained.
Finally, we prove the existence of a unique ergodic and strongly mixing
invariant measure for the SCBF equations subject to multiplicative Gaussian
noise, by making use of the exponential stability of strong solutions.
