  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. 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.
