The Tayler instability in liquid metal batteries and dynamo theory

The Tayler instability (TI), a kink-type current-driven instability, is discussed as a crucial ingredient of an alternative α-Ω stellar dynamo model. In the framework of the Tayler-Spruit dynamo, a finite current helicity generated by the TI may produce an α-effect, transforming an azimuthal into a poloidal magnetic field. Going from astrophysics to engineering, the TI may be relevant for Liquid Metal Batteries, too. In the context of renewable energies, such batteries have recently received considerable interest for large scale energy storage. Their main advantages (cheap raw materials, long life-time, low cost) result from their simple construction. Built as a stable density stratification of two liquid metals separated by a molten salt layer, such batteries can be easily scaled up. In that case, currents in the order of a few kA will appear and possibly trigger the TI. If the resulting fluid flow becomes too strong, it may disrupt the electrolyte layer, leading to a battery failure.
A quasi-static numerical model, using an integro-differential approach for the coupling of velocity and magnetic field, is presented and used to simulate the TI in the medium to low magnetic Prandtl number range (1e−3 to 1e−6 ). The properties of the instability (growth rate, velocities in saturation), the influence of the geometric aspect ratio and scaling laws are explored. The relevance of the TI for Liquid Metal Batteries as well as several possible ountermeasures are discussed. Further, the influence of different axial boundary conditions and the interaction of TI and electro-vortex flow is analysed.
The saturation mechanism of the TI at high magnetic Prandtl numbers is commonly explained by the β-effect. We propose a second, hydrodynamic saturation mechanism for the TI at low magnetic Prandtl numbers. This allows us to explain the moderate fluid velocities, observed by Seilmayer et al. in a liquid metal TI experiment. Finally, we analyse the occurence of kinetic and current helicity at magnetic Prandtl numbers between 1e−3 and 1e−6 and show the limit of the quasi-static approximation. We finish with a description of helicity waves of the saturated TI. It’s occurence, frequency and amplitude are characterised in the astrophysical context.