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last change 2002 December 19, R. Arlt
MHD instability

The missing chapter in Chandrasekhar's book...

...would have been about the magneto-rotational instability. The transport of specific angular momentum is a delicate problem in various astronomical objects. The formation of stars and planets as well as the rotation of stellar interiors, also the luminosity of galactic nuclei imply transporting angular momentum inside out. Instabilities may provide such transport.

Cosmic rotation curves are not easily accessible to terrestrial experiments. A close analogue is the Taylor-Couette flow - a fluid between rotating cylinders. If the rotation speed of the cylinders differs a lot, the flow is unstable for higher than a critical Reynolds number. Chandrasekhar studied such a flow with magnetic fields and could not find an earlier onset of instability. He actually made an approximation valid for very low ratios of viscosity to diffusivity (magn. Prandtl number, upper curves in the graph, the Hartmann number is a measure of the magnetic field strength).

Instability in Taylor-Couette flow

Now we studied MHD Taylor-Couette flows for large Prandtl numbers, and there is indeed a subcritical excitation of Taylor vortices due to the presence of magnetic fields. The graphs for Prandtl number = 1 and = 10 have a minimum. The magnetic Prandtl numbers in astrophysical objects are indeed orders of magnitudes larger than those of terrestrial conducting fluids (Hg 10^-7, Ga 10^-6, Na 10^-5).

Other than in this experimental Taylor-Couette flow, astrophysical rotation profiles are hydrodynamically very stable. The magnetic fields will be essential for angular-momentum transport. The Taylor-Couette flow may, however, be a chance to see the magneto-rotational instability in experiments. First projects are being started to get cosmic objects into the laboratory!

(credits: G. Rüdiger, R. Arlt, AIP)

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