Three-dimensional dynamo waves in a sphere

The term dynamo wave was introduced by Parker to describe oscillatory solutions of the mean field induction equation containing differential rotation and an alpha-effect from helicity. In this article, we examine dynamo waves generated by a class of steady, three-dimensional flows in a sphere without using the mean-field approximation. The flows are defined by two parameters: D the differential rotation, and M the meridional circulation. Contrary to expectations based on the mean-field equations, most 3-dimensional solutions are stationary. Dynamo waves occupy a very narrow band in the parameter space of flows. They behave like solutions to the mean-field equations in three important ways: the magnetic Reynolds number approaches the asymptotic value, meridional circulation produces steady solutions, and radial and azimuthal components of flux tend to occupy the same region of the sphere. The last observation explains why steady solutions, in which radial and azimuthal flux occupy different parts of the sphere, dominate in 3-dimensions: the third dimension facilitates the separation of the two components. This kinematic result may apply to dynamical solutions in which any change in flow that tends to concentrate radial and azimuthal field in the same place leads to oscillations or reversal. This is a possible mechanism by which the Earth's magnetic field reverses.