Abstract:
The Tayler instability is a kink-type flow instability which occurs when the electrical current through a conducting fluid exceeds a certain critical value. Originally studied in the astrophysical context, the instability was recently shown to be also a limiting factor for the upward scalability of liquid metal batteries. In this paper, we continue our efforts to simulate this instability for liquid metals within the framework of an integro-differential equation approach. The original solver is enhanced by multi-domain support with Dirichlet-Neumann partitioning for the static boundaries. Particular focus is laid on the detailed influence of the axial electrical boundary conditions on the characteristic features of the Tayler instability, and, secondly, on the occurrence of electro-vortex flows and their relevance for liquid metal batteries.

Abstract:
Recently, a new type of battery has been proposed that relies on the principle of self-assembling of a liquid metalloid positive electrode, a liquid electrolyte, and a liquid metal negative electrode. While this configuration has been claimed to allow arbitrary up-scaling, there is a size limitation of such a system due to a current-driven kink-type instability that is known as the Tayler instability. We characterize this instability in large-scale self-assembled liquid metal batteries and discuss various technical means how it can be avoided.

Abstract:
In the current-driven, kink-type Tayler instability (TI) a sufficiently strong azimuthal magnetic field becomes unstable against non-axisymmetric perturbations. The TI has been discussed as a possible ingredient of the solar dynamo mechanism and a source of the helical structures in cosmic jets. It is also considered as a size limiting factor for liquid metal batteries. We report on a liquid metal TI experiment using a cylindrical column of the eutectic alloy GaInSn to which electrical currents of up to 8 kA are applied. We present results of external magnetic field measurements that indicate the occurrence of the TI in good agreement with numerical predictions. The interference of TI with the competing large scale convection, resulting from Joule heating, is also discussed.

Abstract:
This paper presents numerical analysis a pinch-type instability in a semi-infinite planar layer of inviscid conducting liquid bounded by solid walls and carrying a uniform electric current. The instability resembles the Tayler instability in astrophysics and can presumably disrupt the operation of the recently developed liquid metal batteries (Wang et al. 2014 Nature 514, 348). We show that the instability in liquid metals, which are relatively poor conductors, significantly differs from that in a well conducting fluid. In the latter, instability is dominated by the current perturbation resulting from the advection of the magnetic field. In the former, the instability is dominated by the magnetic field perturbation resulting from the diffusion of the electric current perturbation. As a result, in liquid metals, instability develops on the magnetic response time scale, which depends on the conductivity, and is much longer than the Alfv\'en time scale, on which the instability develops in a well conducting fluid. The instability threshold in viscous fluid resulting from our model is comparable with the numerical as well as experimental results for liquid metals in cylindrical geometries.

Abstract:
The chiral symmetry breaking properties of the Tayler instability are discussed. Effective amplitude equations are determined in one case. This model has three free parameters that are determined numerically. Comparison with chiral symmetry breaking in biochemistry is made.

Abstract:
The Tayler instability is a kink-type, current driven instability that plays an important role in plasma physics but might also be relevant in liquid metal applications with high electrical currents. In the framework of the Tayler-Spruit dynamo model of stellar magnetic field generation, the question of spontaneous helical (chiral) symmetry breaking during the saturation of the Tayler instability has received considerable interest. Focusing on fluids with low magnetic Prandtl numbers, for which the quasistatic approximation can be applied, we utilize an integro-differential equation approach in order to investigate the saturation mechanism of the Tayler instability. Both the exponential growth phase and the saturated phase are analyzed in terms of the action of the alpha and beta effects of mean-field magnetohydrodynamics. In the exponential growth phase we always find a spontaneous chiral symmetry breaking which, however, disappears in the saturated phase. For higher degrees of supercriticality, we observe helicity oscillations in the saturated regime. For Lundquist numbers in the order of one we also obtain chiral symmetry breaking of the saturated magnetic field.

Abstract:
The nonaxisymmetric 'kink-type' Tayler instability (TI) of toroidal magnetic fields is studied for conducting incompressible fluids of uniform density between two infinitely long cylinders rotating around the same axis. It is shown that for resting cylinders the critical Hartmann number for the unstable modes does not depend on Pm. By rigid rotation the instability is suppressed where the critical ratio of the rotation velocity and the Alfven velocity of the field (only) slightly depends on the magnetic Prandtl number Pm. For Pm=1 the rotational quenching of TI takes its maximum. Rotation laws with negative shear (i.e. d\Omega/dR<0) strongly destabilize the toroidal field if the rotation is not too fast. For sufficiently high Reynolds numbers of rotation the suppression of the nonaxisymmetric magnetic instability always dominates. The angular momentum transport of the instability is anticorrelated with the shear so that an eddy viscosity can be defined which proves to be positive. For negative shear the Maxwell stress of the perturbations remarkably contributes to the angular momentum transport. We have also shown the possibility of laboratory TI experiments with a wide-gap container filled with fluid metals like sodium or gallium. Even the effect of the rotational stabilization can be reproduced in the laboratory with electric currents of only a few kAmp.

Abstract:
We present axisymmetric numerical simulations of the solar interior, including the convection zone and an extended radiative interior. We find that differential rotation in the convection zone induces a toroidal field from an initially purely poloidal field. This toroidal field becomes unstable to the axisymmetric Tayler instability and undergoes equatorward propagating toroidal field reversals. These reversals occur in the absence of a dynamo and without accompanying poloidal field reversals. The nature and presence of such reversals depends sensitively on the initial poloidal field strength imposed, with north-south symmetric reversals only seen at a particular initial field strength. Coupled with a dynamo mechanism which regenerates the poloidal field this could be one ingredient in the sunspot cycle.

Abstract:
We study spontaneous breakdown of chiral symmetry during the nonlinear evolution of the Tayler instability. We start with an initial steady state of zero helicity. Within linearized perturbation calculations, helical perturbations of this initial state have the same growth rate for either sign of helicity. Direct numerical simulations (DNS) of the fully nonlinear equations, however, show that an infinitesimal excess of one sign of helicity in the initial perturbation gives rise to a saturated helical state. We further show that this symmetry breaking can be described by weakly nonlinear finite--amplitude equations with undetermined coefficients which can be deduced solely from symmetry consideration. By fitting solutions of the amplitude equations to data from DNS we further determine the coefficients of the amplitude equations.

Abstract:
Many astrophysical phenomena (such as the slow rotation of neutron stars or the rigid rotation of the solar core) can be explained by the action of the Tayler instability of toroidal magnetic fields in the radiative zones of stars. In order to place the theory of this instability on a safe fundament it has been realized in a laboratory experiment measuring the critical field strength, the growth rates as well as the shape of the supercritical modes. A strong electrical current flows through a liquid-metal confined in a resting columnar container with an insulating outer cylinder. As the very small magnetic Prandtl number of the gallium-indium-tin alloy does not influence the critical Hartmann number of the field amplitudes, the electric currents for marginal instability can also be computed with direct numerical simulations. The results of this theoretical concept are confirmed by the experiment. Also the predicted growth rates of the order of minutes for the nonaxisymmetric perturbations are certified by the measurements. That they do not directly depend on the size of the experiment is shown as a consequence of the weakness of the applied fields and the absence of rotation.