Linear Analysis and Nonlinear Evolution of Two‐Dimensional Global Magnetohydrodynamic Instabilities in a Diffusive Tachocline

Abstract

We develop a more realistic two-dimensional model for global MHD instabilities in the solar tachocline, by including diffusion in the form of kinetic and magnetic drag (following Newton's cooling law formulation). This instability has previously been studied by us and others for an idealized tachocline with no kinematic viscosity and magnetic diffusivity. Since radial diffusion is more important than latitudinal diffusion in the thin solar tachocline, diffusive decay of flow and magnetic fields can be considered as proportional to those variables. We find that, for solar-like toroidal magnetic fields of ~100 kG, instability exists for a wide range of kinetic and magnetic drag parameters, providing a mechanism for enhanced angular momentum transport in latitudes, which could explain how thin the solar tachocline is. From a detailed parameter space survey, we set upper limits of 5 × 1011 and 3 × 1010 cm2 s-1 for kinematic viscosity and magnetic diffusivity, respectively, such that this instability occurs in the solar tachocline on a timescale shorter than a sunspot cycle. We find that magnetic drag has much more influence than kinetic drag in damping this instability. This happens because the sink due to magnetic drag dissipates perturbation magnetic energy faster than the vorticity sink from kinetic drag dissipates perturbation kinetic energy. Consequently, in the presence of a large enough magnetic drag, the nonsolar-like clamshell pattern, found by Cally to be an inevitable final state of a broad profile undergoing an ideal MHD tachocline instability, is suppressed, whereas a banded profile still tips with no reduction in tip angle. We discuss how tipping may affect various surface manifestations of magnetic features, such as the latitudes and orientations of bipolar active regions.