Existence, uniqueness, regularity and instability results for the viscous magneto-geostrophic equation

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 (${\epsilon }_{\kappa }=0$) 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 (${\epsilon }_{\kappa }>0$), 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. (paper)