Joint instability of latitudinal differential rotation and concentrated toroidal fields below the solar convection zone. II. Instability of narrow bands at all latitudes

Assuming that concentrated toroidal field bands occur in the solar tachocline at different latitudes as the solar cycle progresses, we examine the joint instability of latitudinal differential rotation and coexisting narrow bands placed at a wide range of latitudes. Following the basic formalism developed by Gilman & Fox and employing the numerical technique of Dikpati & Gilman, we show that the instability exists for almost all phases of the solar cycle, i.e., for a wide range of latitudinal positions of the bands of field strengths between 500 G and 200 kG. Modes with longitudinal wavenumber m = 1 up to m = 7, depending on parameter values, are unstable for both kinds of symmetries. Mid-latitude bands are the most unstable; the instability disappears if the band is at very high or low latitudes. High-latitude bands are highly unstable, with e-folding growth times of a few months, even when the field strength is low (about a few hundred gauss); low-latitude bands are unstable with longer growth times (≥1 yr), but for high field strengths (10⁴-2 x 10⁵ G). We argue that, because of the instability, the high-latitude bands would undergo turbulent mixing in the latitudinal direction on a timescale that is short compared to their build-up time from shearing of poloidal field by the differential rotation, and thus they may not be buoyant enough to appear as active regions at the surface, but the low-latitude bands can build up simultaneously with this instability and can eventually manifest as active regions. Instability for modes with m ≥ 1 could help determine the longitude distribution of active regions; nonlinear changes in the toroidal field due to the instability may contribute to their decay.