Dissipative and nonaxisymmetric standard-MRI in Kepler disks

Deviations from axial symmetry are necessary to maintain self-sustained MRI-turbulence. We define the parameters region where nonaxisymmetric MRI is excited and study dependence of the unstable modes structure and growth rates on the relevant parameters. We solve numerically the linear eigenvalue problem for global axisymmetric and nonaxisymmetric modes of standard-MRI in Keplerian disks with finite diffusion. For small magnetic Prandtl number the microscopic viscosity completely drops out from the analysis so that the stability maps and the growth rates expressed in terms of the magnetic Reynolds number Rm and the Lundquist number S do not depend on the magnetic Prandtl number Pm. The minimum magnetic field for onset of nonaxisymmetric MRI grows with Rm. For given S all nonaxisymmetric modes disappear for sufficiently high Rm. This behavior is a consequence of the radial fine-structure of the nonaxisymmetric modes resulting from the winding effect of differential rotation. It is this fine-structure which presents severe resolution problems for the numerical simulation of MRI at large Rm. For weak supercritical magnetic fields only axisymmetric modes are unstable. Nonaxisymmetric modes need stronger fields and not too fast rotation. If Pm is small its real value does not play any role in MRI.Comment: 4 pages, 6 figures, A&A Lette

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

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 $\alpha_{\phi\phi}$ 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 $\alpha\Omega$ 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.Comment: 6 pages, 7 figures, submitted to MNRA

The angular momentum transport by standard MRI in quasi-Kepler cylindric Taylor-Couette flows

The instability of a quasi-Kepler flow in dissipative Taylor-Couette systems under the presence of an homogeneous axial magnetic field is considered with focus to the excitation of nonaxisymmetric modes and the resulting angular momentum transport. The excitation of nonaxisymmetric modes requires higher rotation rates than the excitation of the axisymmetric mode and this the more the higher the azimuthal mode number m. We find that the weak-field branch in the instability map of the nonaxisymmetric modes has always a positive slope (in opposition to the axisymmetric modes) so that for given magnetic field the modes with m>0 always have an upper limit of the supercritical Reynolds number. In order to excite a nonaxisymmetric mode at 1 AU in a Kepler disk a minimum field strength of about 1 Gauss is necessary. For weaker magnetic field the nonaxisymmetric modes decay. The angular momentum transport of the nonaxisymmetric modes is always positive and depends linearly on the Lundquist number of the background field. The molecular viscosity and the basic rotation rate do not influence the related {\alpha}-parameter. We did not find any indication that the MRI decays for small magnetic Prandtl number as found by use of shearing-box codes. At 1 AU in a Kepler disk and a field strength of about 1 Gauss the {\alpha} proves to be (only) of order 0.005

Stratorotational instability in Taylor-Couette flow heated from above

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.Comment: 10 pages, 6 figures, revised version appeared in J. Fluid Mech. (2009), vol. 623, pp. 375--38

Stability of latitudinal differential rotation in stars

The question is addressed whether stellar differentially rotating radiative zones (like the solar tachocline) excite nonaxisymmetric r-modes which can be observed. To this end the hydrodynamical stability of latitudinal differential rotation is studied. The amount of rotational shear required for the instability is estimated in dependence of the character of radial stratification and the flow patterns excited by the instability are found. The eigenvalue equations for the nonaxisymmetric disturbances are formulated in 3D and then solved numerically. Radial displacements and entropy disturbances are included. The equations contain the 2D approximation of strictly horizontal displacements as a special limit. The critical magnitude of the latitudinal differential rotation for onset of the instability is considerably reduced in the 3D theory compared to the 2D approximation. The instability requires a subadiabatic stratification. It does not exist in the bulk of convection zone with almost adiabatic stratification but may switch on near its base in the region of penetrative convection. Growth rates and symmetry types of the modes are computed in dependence on the rotation law parameters. The S1 mode with its transequatorial toroidal vortices is predicted as the dominating instability mode. The vortices show longitudinal drift rates retrograde to the basic rotation which are close to that of the observed weak r-mode signatures at the solar surface.Comment: 5 pages, 6 figure