Helical magnetic background fields with 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 as the helicity of the large-scale field. These helicities and also the corresponding $alpha$-effect scale with the current helicity of the background field. The $alpha$-tensor is highly anisotropic as the components $alpha_{phiphi}$ and $alpha_{zz}$ have opposite signs. The amplitudes of the azimuthal $alpha$-effect computed with the cylindrical 3D MHD code are so small that the operation of an $alphaOmega$ 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 $alpha$-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 mega-years.
We investigate the instability and nonlinear saturation of temperature-stratified Taylor-Couette flows in a finite height cylindrical gap and calculate angular-momentum transport in the nonlinear regime. The model is based on an incompressible fluid
in Boussinesq approximation with a positive axial temperature gradient applied. While both ingredients itself, the differential rotation as well as the stratification due to the temperature gradient, are stable, together the system becomes subject of the stratorotational instability and nonaxisymmetric flow pattern evolve. This flow configuration transports angular momentum outwards and will therefor be relevant for astrophysical applications. The belonging viscosity $alpha$ coefficient is of the order of unity if the results are adapted to the size of an accretion disc. The strength of the stratification, the fluids Prandtl number and the boundary conditions applied in the simulations are well-suited too for a laboratory experiment using water and a small temperature gradient below five Kelvin. With such a rather easy realizable set-up the SRI and its angular momentum transport could be measured in an experiment.
The stability problem of MHD Taylor-Couette flows with toroidal magnetic fields is considered in dependence on the magnetic Prandtl number. Only the most uniform (but not current-free) field with B_in = B_out has been considered. For high enough Hart
mann numbers the toroidal field is always unstable. Rigid rotation, however, stabilizes the magnetic (kink-)instability. The axial current which drives the instability is reduced by the electromotive force induced by the instability itself. Numerical simulations are presented to probe this effect as a possibility to measure the turbulent conductivity in a laboratory. It is shown numerically that in a sodium experiment (without rotation) an eddy diffusivity 4 times the molecular diffusivity appears resulting in a potential difference of ~34 mV/m. If the cylinders are rotating then also the eddy viscosity can be measured. Nonlinear simulations of the instability lead to a turbulent magnetic Prandtl number of 2.1 for a molecular magnetic Prandtl number of 0.01. The trend goes to higher values for smaller Pm.
Toroidal magnetic fields subject to the Tayler instability can transport angular momentum. We show that the Maxwell and Reynolds stress of the nonaxisymmetric field pattern depend linearly on the shear in the cylindrical gap geometry. Resulting angul
ar momentum transport also scales linear with shear. It is directed outwards for astrophysical relevant flows and directed inwards for superrotating flows with dOmega/dR>0. We define an eddy viscosity based on the linear relation between shear and angular momentum transport and show that its maximum for given Prandtl and Hartmann number depends linear on the magnetic Reynolds number Rm. For Rm=1000 the eddy viscosity is of the size of 30 in units of the microscopic value.
We investigate in isothermal MHD simulations the instability of toroidal magnetic fields resulting by the action of z-dependent differential rotation on a given axial field B^0 in a cylindrical enclosure where in particular the helicity of the result
ing nonaxisymmetric flow is of interest. The idea is probed that helicity H is related to the external field and the differential rotation as H ~ B^0_i B^0_j Omega_i,j. The observed instability leads to a nonaxisymmetric solution with dominating mode m=1. With the onset of instability both kinematic and current helicity are produced which fulfill the suggested relation. Obviously, differential rotation dOmega/dz only needs an external axial field B^0_z to produce remarkable amounts of the helicities. Any regular time-dependency of the helicity could not be found. The resulting axial alpha-effect is mainly due to the current helicity, the characteristic time scale between both the values is of order of the rotation time. If the axial field is switched off then the helicity and the alpha-effect disappear.
Azimuthal magnetorotational instability is a mechanism that generates nonaxisymmetric field pattern. Nonlinear simulations in an infinite Taylor-Couette system with current-free external field show, that not only the linearly unstable mode m=1 appear
s, but also an inverse cascade transporting energy into the axisymmetric field is possible. By varying the Reynolds number of the flow and the Hartmann number for the magnetic field, we find that the ratio between axisymmetric (m=0) and dominating nonaxisymmetric mode (m=1) can be nearly free chosen. On the surface of the outer cylinder this mode distribution appears similarly, but with weaker axisymmetric fields. We do not find significant differences in the case that a constant current within the flow is added.
