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 well-posed (cf. [15]). 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. Nonlinearity. Volume 24, Number 11 (2011), 3019–3042.