  We use De Giorgi techniques to prove H\"older continuity of weak solutions to
a class of drift-diffusion equations, with $L^2$ initial data and divergence
free drift velocity that lies in $L_{t}^{\infty}BMO_{x}^{-1}$. We apply this
result to prove global regularity for a family of active scalar equations which
includes the advection-diffusion equation that has been proposed by Moffatt in
the context of magnetostrophic turbulence in the Earth's fluid core.
