  We study the three dimensional active scalar equation called the
magneto-geostropic equation which was proposed by Moffatt and Loper as a model
for the geodynamo processes in the Earth's fluid core. When the viscosity of
the fluid is positive, the constitutive law that relates the drift velocity
$u(x,t)$ and the scalar temperature $\theta(x,t)$ produces two orders of
smoothing. We study the implications of this property. For example, we prove
that in the case of the non-diffusive ($\varepsilon_\kappa=0$) active scalar
equation, initial data $\theta_0\in L^3$ implies the existence of unique,
global weak solutions. If $\theta_0\in W^{s,3}$ with $s>0$, then the solution
$\theta(x,t)\in W^{s,3}$ for all time. In the case of positive diffusivity
($\varepsilon_\kappa>0$), even for singular initial data $\theta_0\in L^3$, the
global solution is instantaneously $C^\infty$-smoothed and satisfies the
drift-diffusion equation classically for all $t>0$. We demonstrate, via a
particular example, that the viscous magneto-geostrophic equation permits
exponentially growing "dynamo type" instabilities.
