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

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Abstract
Motivated by observations of sunspot and active-region latitudes that suggest that the subsurface toroidal field in the Sun occurs in narrow latitude belts, we analyze the joint instability of solar latitudinal differential rotation and the concentrated toroidal field below the base of the convection zone, extending the work of Gilman & Fox (hereafter GF). We represent the profile of the toroidal field 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 profiles, and we find instability for a wide range of amplitudes of differential rotation and toroidal fields (103-106 G fields at the base of the solar convection zone), as well as a wide range of toroidal-field bandwidths. We show that the combination of concentrated toroidal fields and solar-type latitudinal differential rotation is again unstable, not only to longitudinal wavenumber m=1 as in GF, but also to m>1 for sufficiently narrow toroidal-field profiles. For a fixed peak field strength, the growth rate first increases as the toroidal-field band is narrowed, reaching a peak for bandwidths between 10° and 20° in latitude, depending on the peak field strength, and then decreases to a cut-off in the instability for toroidal field bands of 3°-4°. Irrespective of bandwidth, the differential rotation is the primary energy source for the instability for weak fields, and the toroidal field is the primary source for strong fields. The weaker (stronger) the peak toroidal field is, the narrower (broader) is the bandwidth for which the toroidal field becomes the primary energy source. The Reynolds, Maxwell, and mixed stresses required to extract energy from the differential rotation and toroidal field are most active in the neighborhood of the singular or turning points of the perturbation equations. This first study focuses on toroidal fields that peak near 45° latitude, as in GF; later papers will place the toroidal-field peak at a wide variety of latitudes, as we might expect to occur at different phases of a sunspot cycle.