Marcus Gellert

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.

Helicity and α-effect by current-driven instabilities of helical magnetic fields

Helical magnetic background fields with an adjustable pitch angle are imposed on a conducting fluid in a differentially rotating cylindrical container. The small-scale kinetic and current helicities are calculated for various field geometries, and shown to have the opposite sign to the helicity of the large-scale field. These helicities and also the corresponding α-effect scale with the current helicity of the background field. The α-tensor is highly anisotropic as the components αϕϕ and αzz have opposite signs. The amplitudes of the azimuthal α-effect computed with the cylindrical 3D magnetohydrodynamics code are so small that the operation of an αΩ-dynamo on the basis of the current-driven, kink-type instabilities of toroidal fields is highly questionable. In any case, the low value of the α-effect would lead to very long growth times of a dynamo in the radiation zone of the Sun and early-type stars of the order of megayears.

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)

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.

Bifurcations of rotating waves in rotating spherical shell convection.

The dynamics and bifurcations of convective waves in rotating and buoyancy-driven spherical Rayleigh-Bénard convection are investigated numerically. The solution branches that arise as rotating waves (RWs) are traced by means of path-following methods, by varying the Rayleigh number as a control parameter for different rotation rates. The dependence of the azimuthal drift frequency of the RWs on the Ekman and Rayleigh numbers is determined and discussed. The influence of the rotation rate on the generation and stability of secondary branches is demonstrated. Multistability is typical in the parameter range considered.

Stability and instability of hydromagnetic Taylor–Couette flows

Decades ago S. Lundquist, S. Chandrasekhar, P.H. Roberts and R. J.~Tayler first posed questions about the stability of Taylor-Couette flows of conducting material under the influence of large-scale magnetic fields. These and many new questions can now be answered numerically where the nonlinear simulations even provide the instability-induced values of several transport coefficients. The cylindrical containers are axially unbounded and penetrated by magnetic background fields with axial and/or azimuthal components. The influence of the magnetic Prandtl number

Nonaxisymmetric MHD Instabilities of Chandrasekhar States in Taylor-Couette Geometry

Pm

Azimuthal magnetorotational instability with super-rotation

on the onset of the instabilities is shown to be substantial. The potential flow subject to axial fields becomes unstable against axisymmetric perturbations for a certain supercritical value of the averaged Reynolds number