Joint Instability of Latitudinal Differential Rotation and Concentrated Toroidal Fields below the Solar Convection Zone

Abstract

Motivated by observations of sunspot and active-region latitudes that suggest that the subsurface toroidal Ðeld in the Sun occurs in narrow latitude belts, we analyze the joint instability of solar latitudi-nal di †erential rotation and the concentrated toroidal Ðeld below the base of the convection zone, extending the work of Gilman & Fox (hereafter GF). We represent the proÐle of the toroidal Ðeld by Gaussian functions whose width is a variable parameter and solve the two-dimensional perturbation equations of GF by relaxation methods. We reproduce the results of GF for broad proÐles, and we Ðnd instability for a wide range of amplitudes of di †erential rotation and toroidal Ðelds (103È106 G Ðelds at the base of the solar convection zone), as well as a wide range of toroidal-Ðeld bandwidths. We show that the combination of concentrated toroidal Ðelds and solar-type latitudinal di †erential rotation is again unstable, not only to longitudinal wavenumber m \ 1 as in GF, but also to m [ 1 for sufficiently narrow toroidal-Ðeld proÐles. For a Ðxed peak Ðeld strength, the growth rate Ðrst increases as the toroidal-Ðeld band is narrowed, reaching a peak for bandwidths between 10¡ and 20¡ in latitude, depending on the peak Ðeld strength, and then decreases to a cut-o † in the instability for toroidal Ðeld bands of 3¡È4¡. Irrespective of bandwidth, the di †erential rotation is the primary energy source for the instability for weak Ðelds, and the toroidal Ðeld is the primary source for strong Ðelds. The weaker (stronger) the peak toroidal Ðeld is, the narrower (broader) is the bandwidth for which the toroidal Ðeld becomes the primary energy source. The Reynolds, Maxwell, and mixed stresses required to extract energy from the di †erential rotation and toroidal Ðeld are most active in the neighborhood of the singular or turning points of the perturbation equations. This Ðrst study focuses on toroidal Ðelds that peak near 45¡ latitude , as in GF ; later papers will place the toroidal-Ðeld peak at a wide variety of latitudes, as we might expect to occur at di †erent phases of a sunspot cycle.