We establish the existence, uniqueness and attraction properties of an
ergodic invariant measure for the Boussinesq Equations in the presence of a
degenerate stochastic forcing acting only in the temperature equation and only
at the largest spatial scales. The central challenge is to establish time
asymptotic smoothing properties of the Markovian dynamics corresponding to this
system. Towards this aim we encounter a Lie bracket structure in the associated
vector fields with a complicated dependence on solutions. This leads us to
develop a novel H\"{o}rmander-type condition for infinite-dimensional systems.
Demonstrating the sufficiency of this condition requires new techniques for the
spectral analysis of the Malliavin covariance matrix.Comment: 57 pages, 4 figure
