  We establish the convergence of statistically invariant states for the
stochastic Boussinesq Equations in the infinite Prandtl number limit and in
particular demonstrate the convergence of the Nusselt number (a measure of heat
transport in the fluid). This is a singular parameter limit significant in
mantle convection and for gasses under high pressure. The equations are subject
to a both temperature gradient on the boundary and internal heating in the bulk
driven by a stochastic, white in time, gaussian forcing. Here, the stochastic
source terms have a strong physical motivation for example as a model of
radiogenic heating. Our approach uses mixing properties of the formal limit
system to reduce the convergence of invariant states to an analysis of the
finite time asymptotics of solutions and parameter-uniform moment bounds. Here,
it is notable that there is a phase space mismatch between the finite Prandtl
system and the limit equation, and we implement methods to lift both finite and
infinite time convergence results to an extended phase space which includes
velocity fields. For the infinite Prandtl stochastic Boussinesq equations, we
show that the associated invariant measure is unique and that the dual
Markovian dynamics are contractive in an appropriate Kantorovich-Wasserstein
metric. We then address the convergence of solutions on finite time intervals,
which is still a singular perturbation. In the process we derive well-posed
equations which accurately approximate the dynamics up to the initial time when
the Prandtl number is large.
