Theory of current instability experiments in magnetic Taylor-Couette flows

Abstract

We consider the linear stability of dissipative MHD Taylor-Couette flow with imposed toroidal magnetic fields. The inner and outer cylinders can be either insulating or conducting; the inner one rotates, the outer one is stationary. The magnetic Prandtl number can be as small as 10-5, approaching realistic liquid-metal values. The magnetic field destabilizes the flow, except for radial profiles of B${}_{\varphi }$(R) close to the current-free solution. The profile with B${}_{in}$=B${}_{out}$ (the most uniform field) is considered in detail. For weak fields the TC-flow is stabilized, until for moderately strong fields the m=1 azimuthal mode dramatically destabilizes the flow again. There is thus a maximum value for the critical Reynolds number. For sufficiently strong fields (as measured by the Hartmann number) the toroidal field is always unstable, even for Re=0. The electric currents needed to generate the required toroidal fields in laboratory experiments are a few kA if liquid sodium is used, somewhat more if gallium is used. Weaker currents are needed for wider gaps, so a wide-gap apparatus could succeed even with gallium. The critical Reynolds numbers are only somewhat larger than the nonmagnetic values, so such an experiment would require only modest rotation rates.

arXiv comment8 pages, 4 figures, submitted to Phys. Rev. E

References in corpus (11)

• Stable magnetic fields in stellar interiors
• Magnetorotational Instability in a Rotating Liquid Metal Annulus
• Experimental evidence for magnetorotational instability in a helical magnetic field
• New type of magneto-rotational instability in cylindrical Taylor-Couette flow
• The stability of toroidal fields in stars
• Experiments on the magnetorotational instability in helical magnetic fields
• MHD instability in differentially rotating cylindric flows
• The traveling wave MRI in cylindrical Taylor-Couette flow: comparing wavelengths and speeds in theory and experiment
• The first close-up of the "flip-flop" phenomenon in a single star
• The stability of axisymmetric Taylor-Couette flow in hydromagnetics
• Hydromagnetic Taylor--Couette flow: numerical formulation and comparison with experiment

Cited by in corpus (10)

• Thermal convection in a liquid metal battery
• How to circumvent the size limitation of liquid metal batteries due to the Tayler instability
• Stability and instability of hydromagnetic Taylor-Couette flows
• Shallow water modeling of rolling pad instability in liquid metal batteries
• Tayler instability of toroidal magnetic fields in MHD Taylor-Couette flows
• The Tayler instability of toroidal magnetic fields in a columnar gallium experiment
• Stratorotational instability in MHD Taylor-Couette flows
• The Tayler instability and the Hall effect in protoneutron stars
• The pinch-type instability of helical magnetic fields
• Nonaxisymmetric MHD instabilities of Chandrasekhar states in Taylor-Couette geometry