| Popis: |
The study of magnetohydrodynamic (MHD) instabilities occurring in liquid metals, with imposed differential rotation and magnetic field, is of fundamental importance in the astrophysical context. MHD instabilities are especially relevant in planets or stars, where electrically conducting flows are confined within their interiors. Such environments could be modeled by solving the Navier-Stokes and induction equations with appropriate conditions in a spherical shell composed of two concentric spheres. In particular, we consider the case where the liquid metal (GaInSn in our case), bounded by a stationary outer sphere and a uniformly rotating inner sphere, is subjected to an axial magnetic field. When the aspect ratio of the radii of the two spheres is fixed, only two parameters, namely, the Reynolds number (associated with the differential rotation) and the Hartmann number (associated with the applied magnetic field strength), govern the dynamics of the system (see [1,2] for full details). For the magnetized spherical Couette system, three different types of instabilities have so far been identified and characterized by means of numerical simulations (e.g. [1,3]), and also in experiments (e.g. [2,4]). These instabilities can each be described as a hydrodynamic radial jet instability, a return flow instability, and a Kelvin-Helmholtz-like Shercliff layer instability. We provide an overview of these instabilities with a focus on the description and analysis of the different spatio-temporal symmetries of the MHD flow. In particular, numerical and experimental bifurcation diagrams of nonlinear waves in the quasi-laminar regime (with moderate differential rotation) are presented and some numerical tools, related to nonlinear dynamics and chaos theory [5], are outlined. These tools include the numerical continuation of periodic solutions and their stability assessment, time series analysis such as the computation of the fundamental frequencies in one or several spatial dimensions, time dependent frequency spectra, and Poincaré sections. Our results show how periodic and quasiperiodic MHD flows with two, three and even four incommensurable frequencies, as well as MHD chaotic flows, are developed following a sequence of bifurcations from the base state. The knowledge of the different routes to chaos is of fundamental importance in turbulence theory. In addition, by taking into account the symmetries of the solutions several regions of multistability (and also hysteretic behavior) are identified in the parameter space with a good agreement between simulations and experiments, both in their temporal and spatial structures. Although unstable MHD flows are not experimentally realized, their numerical computation as in [1,6] provides a more complete picture of the dynamics and aids the understanding of transient and hysteretic behaviors in experiments. This work is funded by the European Research Council (ERC), Horizon 2020 research and innovation programme (grant agreement No. 787544). The authors wish to thank Kevin Bauch for technical support. 1. Garcia, F. and Stefani, F., Continuation and stability of rotating waves in the magnetized spherical Couette system: Secondary transitions and multistability. – Proc. R. Soc. A (474), 2018. – p. 20180281. 2. Ogbonna, J., Garcia, F., Gundrum, T., Seilmayer, M. and Stefani, F., Experimental investigation of the return flow instability in magnetized spherical Couette flows. – Phys. Fluids (32), 2020. – p. 124119. 3. Travnikov, V., Eckert, K. and Odenbach, S., Influence of an axial magnetic field on the stability of spherical Couette flows with different gap widths. – Acta Mech. (219), 2011. – p. 255. 4. Kasprzyk, C., Kaplan, E., Seilmayer, M. and Stefani, F., Transitions in a magnetized quasi-laminar spherical Couette flow. – Magnetohydrodynamics (53), 2017. – p. 393. 5. Kuznetsov, Y. A., Elements of Applied Bifurcation Theory, 2nd Edition – Springer, New York, 1998. 6. Garcia, F., Seilmayer, M., Giesecke, A. and Stefani, F., Four-frequency solution in a magnetohydrodynamic Couette flow as a consequence of azimuthal symmetry breaking. – Phys. Rev. Lett. (125), 2020. – p. 264501. |
