Compensation of instabilities in magnetic Taylor-Couette flow.

The axisymmetric linear stability of the Taylor-Couette flow with an azimuthal magnetic field is considered. It is shown that a flow with the combination of a linearly unstable rotation and a linearly unstable azimuthal magnetic field can be linearly stable. The flow stabilization takes place for both ideal and dissipative flows. For dissipative flow the stabilization exists only for a combination of counter-rotating cylinders and a counterdirecting azimuthal magnetic field on cylinders. The effect can be important for the problem of a plasma confinement by the magnetic field.

Rotational stabilization of pinch instabilities in Taylor-Couette flow.

The axisymmetric linear stability of dissipative Taylor-Couette flow with an azimuthal magnetic field is considered. The magnetic field can be unstable without a rotation. This is the well-known pinch type instability. The stable rotation stabilizes the unstable azimuthal magnetic field. The dissipative flow stability can be classified according to Michael's stability condition for an ideal flow. The dissipative effects stabilize the flow and an ideally unstable flow becomes really unstable only when both the angular velocity and the magnetic field exceed some critical values.

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.

Pinch instabilities in Taylor-Couette flow.

The linear stability of the dissipative Taylor-Couette flow with an azimuthal magnetic field is considered. Unlike ideal flows, the magnetic field is a fixed function of a radius with two parameters only: a ratio of inner to outer cylinder radii, eta, and a ratio of the magnetic field values on outer and inner cylinders, muB. The magnetic field with 0

Linear instability of magnetic Taylor-Couette flow with Hall effect.

The influence of the Hall effect on the linear marginal stability of a molecular hydrodynamic Taylor-Couette flow in the presence of an axial uniform magnetic field is considered. The Hall effect leads to the situation that the Taylor-Couette flow becomes unstable for any ratio of the angular velocities of the inner and outer cylinders. The instability, however, does not exist for both signs of the axial magnetic field B0. For positive shear dOmega/dR the Hall instability exists for negative Hartmann number and for negative shear dOmega/dR the Hall instability exists for positive Hartmann number. For negative shear, of course, the Hall instability combines with the magnetorotational instability, resulting in a rather complex bifurcation diagram. In this case the critical magnetic Reynolds numbers with Hall effect are much lower than without Hall effect. In order to verify the presented shear-Hall instability at the laboratory with experiments using liquid metals, one would need rather large magnetic fields ( approximately 10(7) G).

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.

Stability of axisymmetric Taylor-Couette flow in hydromagnetics.

The linear marginal instability of an axisymmetric magnetohydrodynamics Taylor-Couette flow of infinite vertical extension is considered. We are only interested in those vertical wave numbers for which the characteristic Reynolds number is minimum. For hydrodynamically unstable flows minimum Reynolds numbers exist even without a magnetic field, but there are also solutions with smaller characteristic Reynolds numbers for certain weak magnetic fields. The magnetic field, therefore, destabilizes the rotating flow by the so-called magnetorotational instability (MRI). The MRI, however, can only exist for hydrodynamically unstable flow if the magnetic Prandtl number, Pr, is not too small. For too small magnetic Prandtl numbers (and too strong magnetic fields) only the well-known magnetic suppression of the Taylor-Couette instability can be found. The MRI is even more pronounced for hydrodynamically stable flows. In this case we can always find a magnetic field amplitude where the characteristic Reynolds number is minimum. These critical values are computed for different magnetic Prandtl numbers and for three types of geometry (small, medium, and wide gaps between the rotating cylinders). In all cases the minimum Reynolds numbers are running with 1/Pr for small enough Pr so that the critical Reynolds numbers may easily exceed values of 10(6) for the magnetic Prandtl number of sodium (10(-5)) or gallium (10(-6)). The container walls are considered either electrically conducting or insulating. For insulating walls with small and medium-size gaps between the cylinders (i) the critical Reynolds number is smaller, (ii) the critical Hartmann number is higher, and (iii) the Taylor vortices are longer in the direction of the rotation axis. For wider gaps the differences in the results between both sets of boundary conditions become smaller and smaller.