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We prove the global existence of classical solutions to a class of forceddrift-diffusion equations with $L^2$ initial data and divergence free driftvelocity $\{u^\nu\}_{\nu_\ge0}\subset L^\infty_t BMO^{-1}_x$, and we obtainstrong convergence of solutions as the viscosity $\nu$ vanishes. We then applyour results to a family of active scalar equations which includes the threedimensional magneto-geostrophic $\{$MG$^\nu\}_{\nu\ge0}$ equation that has beenproposed by Moffatt in the context of magnetostrophic turbulence in the Earth'sfluid core. We prove the existence of a compact global attractor$\{\mathcal{A}^\nu\}_{\nu\ge0}$ in $L^2(\mathbb{T}^3)$ for the MG$^\nu$equations including the critical equation where $\nu=0$. Furthermore, we obtainthe upper semicontinuity of the global attractor as $\nu$ vanishes.

Solutions to a class of forced drift-diffusion equations with applications to the magneto-geostrophic equations