Three-dimensional magneto-shear instabilities in the solar tachocline

Abstract

The solar tachocline straddles the base of the convection zone. In the radiative part, global scale latitudinal magneto-shear instabilities have been thought to be constrained to two dimensions by strongly stable stratification, and so two-dimensional (2D) instabilities have been examined in great detail recently. However, it is shown here that three-dimensional (3D) effects can be important. We generalize the linear 2D analysis to 3D in the Boussinesq thin layer approximation for models where the magnetic field is wrapped toroidally around the Sun, and the equilibrium field and flow are independent of depth. It is found that a very rapid 'polar kink instability' dominates the dynamics of broad magnetic field distributions if the polar Alfven angular speed α p exceeds the rotational angular velocity ω p , with maximum growth rate (α 2 p -ω 2 p ) 1 / 2 . This might typically produce an e-folding time as short as a few months. Interestingly, the instability only affects the m = 1 'tipping modes', twisting polar loops towards a vertical orientation. On the other hand, for α 2 p < ω 2 p , 3D instabilities are restricted to radial length scales in which perhaps just a few wavelengths could fit across the tachocline. These could supplement, or even dominate, the shallow-water modes examined recently by Gilman and Dikpati. An analysis of the role of a large Brunt-Vaisala frequency, as found in the radiative part of the tachocline, suggests that its main effect is to flatten the motions in the instabilities rather than to suppress them. Strong banded magnetic profiles are found to be susceptible to an instability similar to but distinct from the polar kink.