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We consider an active scalar equation that is motivated by a model formagneto-geostrophic dynamics and the geodynamo. We prove that the non-diffusiveequation 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 anexample of a steady state that is nonlinearly unstable, and hence produces adynamo effect in the sense of an exponentially growing magnetic field.

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