Joint Instability of Latitudinal Differential Rotation and Toroidal Magnetic Fields below the Solar Convection Zone. II. Instability for Toroidal Fields that Have a Node between the Equator and Pole

We generalize results of Gilman and Fox to unperturbed toroidal fields that have a node somewhere between the equator and the pole as we speculate the Sun's field to have for most phases of its magnetic cycle. We use the same solution method as in Gilman and Fox, namely Legendre polynomial expansion and matrix inversion to solve for the eigenvalues and eigenfunctions. The solutions are structured around certain singular or critical points of the suitably transformed and combined vorticity and induction equations. There are singular points at the poles, and singularities where ω0-cr=±α0, in which ω0 is the local rotation rate, cr is the longitudinal phase speed of an unstable wave, and α0 is an angular measure of the toroidal field. We survey the instability as a function of toroidal field profile and amplitude as well as differential rotation amplitude, thereby examining reference states that could be characteristic of most phases of the solar cycle, and most depths within the rotational shear layer just below the base of the solar convection zone. As found in Gilman and Fox, instability occurs for a wide range of both toroidal fields and differential rotations. Differential rotation is again the primary energy source for growing modes when the toroidal field is weak, and the toroidal field is the primary source when it is strong. Unlike in Gilman and Fox, here modes of both symmetries about the equator are unstable for low and high toroidal fields, and for high fields a second antisymmetric mode appears. Which mode symmetry is favored for low fields depends in detail on the relative amplitudes of differential rotation and toroidal field. For low toroidal fields (unstable) modes of both symmetries are energetically active (extracting energy from the unperturbed state) only poleward of the node and an adjacent singularity, but are coupled to energetically neutral velocity perturbations equatorward of that singular point. In transition to higher field strengths, those velocity patterns are damped out when two additional singular points appear in the system, but the energetically active high-latitude disturbances remain. By contrast the second antisymmetric mode is energetically active equatorward of the toroidal field node and closely adjacent singular points, but is coupled to an energetically neutral pattern of both velocities and magnetic fields on the poleward side. As in Gilman and Fox, we find narrow-latitude bands of sharp changes in both differential rotation and toroidal magnetic field that migrate toward the equator with increasing field strength, but are bounded in their migration by the latitude of the toroidal field node. These sharp changes are always at the locations of the singular points of the system and represent narrow domains where both kinetic and magnetic energy are being extracted from the reference state to drive the instability. We interpret the instability as a form of resonant overreflection between singular points, analogous to what happens in stratified shear flow, as described for example by Lindzen. The instability may contribute to determining the latitudinal and longitudinal distribution of active regions and other large-scale, magnetic features on the Sun, as well as enable a degree of synchronization of the evolution of the solar cycle between low latitudes and high, and between north and south hemispheres.