Joint instability of latitudinal differential rotation and concentrated toroidal fields below the solar convection zone

Motivated by observations of sunspot and active-region latitudes that sugge st that the subsurface toroidal field in the Sun occurs in narrow latitude belts, we analyze the joint instability of solar latitudinal differential r otation and the concentrated toroidal held below the base of the convection zone, extending the work of Gilman & Fox (hereafter GF). We represent the profile of the toroidal held by Gaussian functions whose width is a variabl e parameter and solve the two-dimensional perturbation equations of GF by r elaxation methods. We reproduce the results of GF for broad profiles, and w e find instability for a wide range of amplitudes of differential rotation and toroidal fields (10(3)-10(6) G fields at the base of the solar convecti on zone), as well as a wide range of toroidal-field bandwidths. We show tha t the combination of concentrated toroidal fields and solar-type latitudina l differential rotation is again unstable, not only to longitudinal wavenum ber m = 1 as in GF, but also to m >1 for sufficiently narrow toroidal-field profiles. For a fixed peak held strength, the growth rate first increases as the toroidal-field band is narrowed, reaching a peak for bandwidths betw een 10 degrees and 20 degrees in latitude, depending on the peak held stren gth, and then decreases to a cut-off in the instability for toroidal held b ands of 3 degrees-4 degrees. Irrespective of bandwidth, the differential ro tation is the primary energy source for the instability for weak fields, an d the toroidal field is the primary source for strong fields. The weaker (s tronger) the peak toroidal held is, the narrower (broader) is the bandwidth for which the toroidal field becomes the primary energy source. The Reynol ds, Maxwell, and mixed stresses required to extract energy from the differe ntial rotation and toroidal field are most active in the neighborhood of th e singular or turning points of the perturbation equations. This first stud y focuses on toroidal fields that peak near 45 degrees latitude, as in GF; later papers will place the toroidal-field peak at a wide variety of latitu des, as we might expect to occur at different phases of a sunspot cycle.