Joint Instability of Latitudinal Differential Rotation and Toroidal Magnetic Fields below the Solar Convection Zone. III. Unstable Disturbance Phenomenology and the Solar Cycle

Abstract

We analyze additional solutions for the two-dimensional instability of coexisting differential rotation and toroidal magnetic fields, organized in families with fixed ratios ER of magnetic to kinetic energy in the unperturbed state. Solutions are found for a wide range of differential rotation amplitudes found in the solar tachocline, for toroidal fields that have a node that ranges in latitude from the pole to the equator, as we expect to exist in the Sun through a sunspot cycle. Fixed ER is a proxy for nonlinear saturation of the solar dynamo due to the reaction of electromagnetic body forces. Since the saturation ratio is not known from either theory or observations, we find solutions in the range 0.1 ≤ ER ≤ 30, corresponding to peak toroidal fields in the solar tachocline of between about 8 × 103 to 1.4 × 105 G. We focus on properties of the unstable disturbances that could test the hypothesis that such disturbances in the solar tachocline provide a template for surface features. We show that the symmetry of magnetic pattern about the equator could switch at one or more phases of the magnetic cycle, and for high ER a switch could also occur between two antisymmetric patterns of different latitudinal profiles. In the former case, the pattern rotation rate would be unchanged, but there would be a sudden longitudinal phase shift in one or both hemispheres. In the latter case, there would be no phase shift but instead a substantial change in the rotation rate of the observed magnetic pattern. For a given mode symmetry and type, the rotation rate is the same at all latitudes, with the rate being close to that of the local rotation of the plasma at the latitude where the disturbance amplitude peaks. For ER ≲ 1, the disturbance magnetic patterns have significant tilts upstream away from the equator, reminiscent of similar patterns in synoptic magnetograms. Sharp changes with latitude in the differential rotation and toroidal field are associated with "critical points" in the system, where the Doppler-shifted disturbance rotation equals the local (angular) Alfvén speed. These migrate toward the equator with the toroidal field node but increasingly lag it. The higher the magnetic energy for a given differential rotation, the closer the equator is approached. If these sharp changes in differential rotation and toroidal field are related to the torsional oscillations and latitudes of sunspots, then these solutions favor large toroidal fields in the tachocline, of peak amplitude at least 6 × 104 G.