Manfred Schultz

Experimental evidence for magnetorotational instability in a Taylor-Couette flow under the influence of a helical magnetic field.

A recent paper [R. Hollerbach and G. Rudiger, Phys. Rev. Lett. 95, 124501 (2005)] has shown that the threshold for the onset of the magnetorotational instability (MRI) in a Taylor-Couette flow is dramatically reduced if both axial and azimuthal magnetic fields are imposed. In agreement with this prediction, we present results of a Taylor-Couette experiment with the liquid metal alloy GaInSn, showing evidence for the existence of the MRI at Reynolds numbers of order 1000 and Hartmann numbers of order 10.

Linear magnetohydrodynamic Taylor-Couette instability for liquid sodium.

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 nonaxisymmetric magnetorotational instability in a rotating liquid metal exposed to an azimuthal magnetic field.

The azimuthal version of the magnetorotational instability (MRI) is a nonaxisymmetric instability of a hydrodynamically stable differentially rotating flow under the influence of a purely or predominantly azimuthal magnetic field. It may be of considerable importance for destabilizing accretion disks, and plays a central role in the concept of the MRI dynamo. We report the results of a liquid metal Taylor-Couette experiment that shows the occurrence of an azimuthal MRI in the expected range of Hartmann numbers.

The stability of MHD Taylor-Couette flow with current-free spiral magnetic fields between conducting cylinders

We study the magnetorotational instability in cylindrical Taylor-Couette flow, with the (vertically unbounded) cylinders taken to be perfect conductors, and with externally imposed spiral magnetic fields. The azimuthal component of this field is generated by an axial current inside the inner cylinder, and may be slightly stronger than the axial field. We obtain an instability beyond the Rayleigh line, for Reynolds numbers of order 1000 and Hartmann numbers of order 10, and independent of the (small) magnetic Prandtl number. For experiments with Rout = 2Rin = 10 cm and Ωout = 0.27 Ωin, the instability appears for liquid sodium for axial fields of ∼20 Gauss and axial currents of ∼1200 A. For gallium the numbers are ∼50 Gauss and ∼3200 A. The vertical cell size is about twice the cell size known for nonmagnetic experiments. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Theory of current-driven instability experiments in magnetic Taylor-Couette flows.

We consider the linear stability of dissipative magnetic 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(phi)(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 Taylor-Couette flow is stabilized, until for moderately strong fields the m=1 azimuthal mode dramatically destabilizes the flow again so that a maximum value for the critical Reynolds number exists. For sufficiently strong fields (as measured by the Hartmann number) the toroidal field is always unstable, even for the nonrotating case with 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; hence such experiments would work with only modest rotation rates.

The Tayler instability of toroidal magnetic fields in a columnar gallium experiment

The nonaxisymmetric Tayler instability of toroidal magnetic fields due to axial electric currents is studied for conducting incompressible fluids between two coaxial cylinders without endplates. The inner cylinder is considered as so thin that the limit of Rin 0 can be computed. The magnetic Prandtl number is varied over many orders of magnitudes but the azimuthal mode number of the perturbations is fixed to m = 1. In the linear approximation the critical magnetic field amplitudes and the growth rates of the instability are determined for both resting and rotating cylinders. Without rotation the critical Hartmann numbers do not depend on the magnetic Prandtl number but this is not true for the corresponding growth rates. For given product of viscosity and magnetic diffusivity the growth rates for small and large magnetic Prandtl number are much smaller than those for Pm = 1. For gallium under the influence of a magnetic field at the outer cylinder of 1 kG the resulting growth time is 5 s. The minimum electric current through a container of 10 cm diameter to excite the instability is 3.20 kA. For a rotating container both the critical magnetic field and the related growth times are larger than for the resting column (© 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

DIFFUSIVE MAGNETOHYDRODYNAMIC INSTABILITIES BEYOND THE CHANDRASEKHAR THEOREM

We consider the stability of axially unbounded cylindrical flows that contain a toroidal magnetic background field with the same radial profile as their azimuthal velocity. For ideal fluids, Chandrasekhar had shown the stability of this configuration if the Alfven velocity of the field equals the velocity of the background flow, i.e., if the magnetic Mach number . We demonstrate that magnetized Taylor–Couette flows with such profiles become unstable against non-axisymmetric perturbations if at least one of the diffusivities is finite. We also find that for small magnetic Prandtl numbers the lines of marginal instability scale with the Reynolds number and the Hartmann number. In the limit the lines of marginal instability completely lie below the line for and for they completely lie above this line. For any finite value of , however, the lines of marginal instability cross the line , which separates slow from fast rotation. The minimum values of the field strength and the rotation rate that are needed for the instability (slightly) grow if the rotation law becomes flat. In this case, the electric current of the background field becomes so strong that the current-driven Tayler instability (which also exists without rotation) appears in the bifurcation map at low Hartmann numbers.

CRITICAL FIELDS AND GROWTH RATES OF THE TAYLER INSTABILITY AS PROBED BY A COLUMNAR GALLIUM EXPERIMENT

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 on 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.

Dissipative Taylor-Couette flows under the influence of helical magnetic fields.

The linear stability of magnetohydrodynamic Taylor-Couette flows in axially unbounded cylinders is considered for magnetic Prandtl number unity. Magnetic background fields varying from purely axial to purely azimuthal are imposed, with a general helical field parametrized by β=B(ϕ)/B(z). We map out the transition from the standard magnetorotational instability (MRI) for β=0 to the nonaxisymmetric azimuthal magnetorotational instability for β→∞. For finite β, positive and negative wave numbers m , corresponding to right and left spirals, are no longer degenerate. For the nonaxisymmetric modes, the most unstable mode spirals in the opposite direction to the background field. The standard (β=0) MRI is axisymmetric for weak fields (including the instability with the lowest Reynolds number) but is nonaxisymmetric for stronger fields. If the azimuthal field is due in part to an axial current flowing through the fluid itself (and not just along the central axis), then it is also unstable to the nonaxisymmetric Tayler instability which is most effective without rotation. For purely toroidal fields the solutions for m=±1 are identical so that in this case no preferred helicity results. For large β the wave number m=-1 is preferred, whereas for β≲1 the mode with m=-2 is most unstable. The most unstable modes always spiral in the same direction as the background field. For background fields with positive and not too large β the kinetic helicity of the fluctuations proves to be negative for all the magnetic instabilities considered.

Eddy viscosity and turbulent Schmidt number by kink-type instabilities of toroidal magnetic fields

The potential of the nonaxisymmetric magnetic instability to transport angular momentum and to mix chemicals is probed considering the stability of a nea rly uniform toroidal field between conducting cylinders with different rotation rates. The flu id between the cylinders is assumed as incompressible and to be of uniform density. With a linear theory the neutral-stability maps for m = 1 are computed. Rigid rotation must be subAlfvenic to allow instability while for differential rotation with negative shear also an unstable domain with superAlfvenic rotation exists. The rotational quenching of the magnetic instabili ty is strongest for magnetic Prandtl numberPm = 1 and becomes much weaker forPm 6 1. The effective angular momentum transport by the instability is directed out- wards(inwards) for subrotation(superrotation). The resu lting magnetic-induced eddy viscosi- ties exceed the microscopic values by factors of 10-100. This is only true for superAlfvenic flows; in the strong-field limit the values remain much smalle r. The same instability also quenches concentration gradients of chemicals by its nonmag- netic fluctuations. The corresponding diffusion coefficien t remains always smaller than the magnetic-generated eddy viscosity. A Schmidt number of order 30 is found as the ratio of the effective viscosity and the diffusion coefficient. The m agnetic instability transports much more angular momentum than that it mixes chemicals.