Joint Instability of Latitudinal Differential Rotation and Toroidal Magnetic Fields below the Solar Convection Zone
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Abstract
Below the convection zone, where the stratification is radiatively controlled, large-scale motions should be mainly horizontal, i.e., in spherical shells due to the stabilizing effect of negative buoyancy on radial displacements. Watson showed that the observed surface solar differential rotation is at the threshold for instability to horizontal disturbances. Therefore, since helioseismology tells us the latitudinal differential rotation below the convection zone is less than the surface value, the profile should be stable too. We show that in the presence of a broad, nonuniform toroidal field the solar differential rotation is unstable. This is true for a wide range of kinetic and magnetic energies of the unperturbed state, from well below equipartition to well above it. We find instability for essentially all values of differential rotation and toroidal fields for which we are able to find converged solutions. The instability appears to occur only for longitudinal wave number 1. Disturbance symmetries about the equator and profiles in latitude depend on the amplitude of the toridal field. Peak e-folding times are a few months. The primary energy source for the instability is differential rotation for low field strengths and the toroidal field for high field strengths. The mechanism of energy release from the differential rotation is the poleward transport of angular momentum, by the Maxwell stress rather than the Reynolds stress. For the profiles studied, the Reynolds stress is almost always trying to rebuild differential rotation, the exact opposite of the nonmagnetic case. Second-order perturbation theory predicts that the unstable modes produce zonal jets and fine structure in the toroidal field, the latitude of which migrates toward the equator with increasing magnetic field strength. The instability we have found may play a role in the solar dynamo, although being two-dimensional, it cannot produce a dynamo by itself. Mixing of angular momentum caused by the instability could allow achievement of equilibrium of the solar tachocline hypothesized by Spiegel & Zahn.