Linear magnetohydrodynamic Taylor-Couette instability for liquid sodium

Abstract

The linear stability of MHD Taylor-Couette flow of infinite vertical extension is considered for liquid sodium with its small magnetic Prandtl number Pm of order 10(-5). The calculations are performed for a container with R(out)=2R(in), with an axial uniform magnetic field and with boundary conditions for both vacuum and perfect conductions. For resting outer cylinder subcritical excitation in comparison to the hydrodynamical case occurs for large Pm but it disappears for small Pm. For rotating outer cylinder the Rayleigh line plays an exceptional role. The hydromagnetic instability exists with Reynolds numbers exactly scaling with Pm(-1/2) so that the moderate values of order 10(4) (for Pm=10(-5)) result. For the smallest step beyond the Rayleigh line, however, the Reynolds numbers scale as 1/Pm leading to much higher values of order 10(6). Then it is the magnetic Reynolds number Rm that directs the excitation of the instability. It results as lower for insulating than for conducting walls. The magnetic Reynolds number has to exceed here values of order 10 leading to frequencies of about 20 Hz for the rotation of the inner cylinder if containers with (say) 10 cm radius are considered. With vacuum boundary conditions the excitation of nonaxisymmetric modes is always more difficult than the excitation of axisymmetric modes. For conducting walls, however, crossovers of the lines of marginal stability exist for both resting and rotating outer cylinders, and this might be essential for future dynamo experiments. In this case the instability also can onset as an overstability.

Experimental Evidence for Magnetorotational Instability in a Taylor-Couette Flow under the Influence of a Helical Magnetic Field

“…We now interpret these experimental data in the context of numerical predictions, obtained and cross-checked by various independent codes [5,12,13] for the solution of the linear eigenvalue problem in unbounded cylinders. First, Fig.…”

Stability and turbulent transport in Taylor–Couette flow from analysis of experimental data

“…The proposed explanation is that the magnetic field makes the flow adjoin the walls for longer distances, so that the viscous dissipation remains comparable to the Joule dissipation at all fields. A second observation is the importance of the magnetic Prandtl number P m = ν/κ m (κ m is the magnetic diffusivity) on the instability [36,37]. On general grounds, it seems that at small Prandtl numbers, the magnetic field stabilizes the flow in the supercritical case, while at large Prandtl numbers, the magnetic field destabilizes the flow.…”

End-effects in rapidly rotating cylindrical Taylor-Couette flow

“…According to the WKB analysis of Ji et al [3,4], and confirmed by the full solution of Rüdiger et al [7,8], the rotation rates must be so large that the Reynolds number Re (10 5 ) or gallium (10 6 ) that then lead to these large values of Re.…”