On the ill/well-posedness and nonlinear instability of the magneto-geostrophic equations

We consider an active scalar equation that is motivated by a model for magneto-geostrophic dynamics and the geodynamo. We prove that the non-diffusive equation is ill-posed in the sense of Hadamard in Sobolev spaces. In contrast, the critically diffusive equation is globally well-posed (cf Friedlander and Vicol (2011 Ann. Inst. Henri Poincaré Anal. Non Linéaire 28 283-301)). In this case we give an example of a steady state that is nonlinearly unstable, and hence produces a dynamo effect in the sense of an exponentially growing magnetic field.