  We consider the three-dimensional magnetohydrodynamics (MHD) equations in the
presence of a spatially degenerate stochastic forcing as a model for
magnetostrophic turbulence in the Earth's fluid core. We examine the
multi-parameter singular limit of vanishing Rossby number $\epsilon$ and
magnetic Reynold's number $\delta$, and establish that: (i) the limiting
stochastically driven active scalar equation (with $\epsilon =\delta=0$)
possesses a unique ergodic invariant measure, and (ii) any suitable sequence of
statistically invariant states of the full MHD system converge weakly, as
$\epsilon,\delta \rightarrow 0$, to the unique invariant measure of the limit
equation. This latter convergence result does not require any conditions on the
relative rates at which $\varepsilon, \delta$ decay.
  Our analysis of the limit equation relies on a recently developed theory of
hypo-ellipticity for infinite-dimensional stochastic dynamical systems. We
carry out a detailed study of the interactions between the nonlinear and
stochastic terms to demonstrate that a H\"{o}rmander bracket condition is
satisfied, which yields a contraction property for the limit equation in a
suitable Wasserstein metric. This contraction property reduces the convergence
of invariant states in the multi-parameter limit to the convergence of
solutions at finite times. However, in view of the phase space mismatch between
the small parameter system and the limit equation, and due to the
multi-parameter nature of the problem, further analysis is required to
establish the singular limit. In particular, we develop methods to lift the
contraction for the limit equation to the extended phase space, including the
velocity and magnetic fields. Moreover, for the convergence of solutions at
finite times we make use of a probabilistic modification of the Gr\"onwall
inequality, relying on a delicate stopping time argument.
