  This paper is concerned about the stochastic convective Brinkman-Forchheimer
(SCBF) equations subjected to multiplicative pure jump noise in bounded or
periodic domains. Our first goal is to establish the existence of a pathwise
unique strong solution satisfying the energy equality (It\^o's formula) to the
SCBF equations. We resolve the issue of the global solvability of SCBF
equations, by using a monotonicity property of the linear and nonlinear
operators and a stochastic generalization of the Minty-Browder technique. The
major difficulty is that an It\^o's formula in infinite dimensions is not
available for such systems. This difficulty is overcame by approximating the
solution using approximate functions composing of 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. Due to technical
difficulties, we discuss about the global in time regularity results of such
strong solutions in periodic domains only. Once the system is well-posed, we
look for the asymptotic behavior of strong solutions. The exponential stability
results (in mean square and pathwise sense) for the stationary solutions is
established in this work for large effective viscosity. Moreover, a
stabilization result of the SCBF equations by using a multiplicative pure jump
noise is also obtained. Finally, we prove the existence of a unique ergodic and
strongly mixing invariant measure for the SCBF equations subject to
multiplicative pure jump noise, by using the exponential stability of strong
solutions.
