Instability of an Evolving Magnetic Field in the Solar Tachocline

The magnetic field observed on the solar photosphere, in the form of active regions, is believed to be generated in the tachocline, a thin region of strong velocity shear (both in radius and latitude) located between the convection and radiative zones. Although the details of how the generation process (the solar dynamo) occurs are still a matter of some debate, it is believed that the means by which the field escapes, eventually to appear at the surface, is through instability due to magnetic buoyancy.

Magnetic buoyancy instability, in both the linear and nonlinear regimes, has been studied for a number of years. However, as their starting point, such studies have almost always considered a static and unidirectional horizontal magnetic field, stratified with depth. In reality, the situation is more complicated: the toroidal field will be formed from the drawing out of the poloidal field by the shear flow (the "omega-effect" of mean field dynamo theory), and so will evolve with time. In this project we shall extend previous stability studies to investigate the instability of a time-evolving basic state, in which a weak vertical field is pulled out into a strong horizontal field, which can then be susceptible to magnetic buoyancy instability.

Mathematically, the instability of time-evolving basic states has received much less attention than that of static states, in which stability is determined via an eigenvalue problem. When the background state changes with time, there are important and subtle issues over how instability should be determined. We shall therefore start with the simpler problem of thermal convection with a time-dependent boundary temperature, in the process extending previous work on that problem. The knowledge thus gained will be invaluable in tackling the magnetic buoyancy problem.

The chief aim of the project will be to determine the nature of shear flows that can generate unstable magnetic fields, the growth time of any instability, and the spatial from of the unstable modes. This will be of importance in understanding the complex interaction between flows and magnetic fields in the solar tachocline, thereby also shedding light on the solar dynamo mechanism.