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Abstract
The magnetorotational instability (MRI) is probably the main cause of turbulence and accretion in sufficiently ionized astrophysical disks. However, despite much theoretical and computational work, the nonlinear saturation of MRI is imperfectly understood. In Chap. 2 and Chap. 3 of this thesis we present non-ideal magnetohydrodynamic simulations of the Princeton MRI experiment. In vertically infinite or periodic cylinders, MRI saturates in a resistive current-sheet with a significant reduction of the mean shear, and with poloidal circulation scaling as the square root of resistivity. Angular momentum transport scales as the reciprocal square root of viscosity but only weakly depends on resistivity. For finite cylinders with insulating end caps, a method for implementing the fully insulating boundary condition is introduced. MRI grows with a clear linear phase from small amplitudes at rates in good agreement with linear analysis. In the final state one inflowing “jet” opposite to the usual Ekman “jet” is found near the inner cylinder. The MRI enhances the angular momentum transport at saturation. Under proper conditions, our experimental facility is a good platform to show that MRI could be suppressed by a strong magnetic field.
Recently, Hollerbach and Rüdiger have reported that MRI modes may grow at much reduced magnetic Reynolds number (Rem) and Lundquist number S in the presence of a helical background field, a current-free combination of axial and toroidal field. We have investigated these helical MRI modes in Chap. 4 and Chap. 5. In vertically infinite or periodic cylinders, resistive HMRI is a weakly destabilized hydrodynamic inertial oscillation propagating axially along the background Poynting flux. Growth rates are small, however, and require large axial currents. Furthermore, finite cylinders with insulating endcaps were shown to reduce the growth rate and to stabilize highly resistive, inviscid flows entirely, and the new mode is stable in Keplerian flow profiles regardless of end conditions. We also numerically investigate a traveling wave pattern observed in experimental magnetized Taylor-Couette flow at low magnetic Reynolds number. By accurately modeling viscous and magnetic boundaries in all directions, we reproduce the experimentally measured wave patterns and their amplitudes. Contrary to previous claims, the waves are shown to be transiently amplified disturbances launched by viscous boundary layers rather than globally unstable magnetorotational modes.
The experiment is complicated by the extremely large Reynolds number and by Ekman circulation and Stewartson layers, even though the experimental apparatus has been designed to minimize the circulation (e.g. by the use of independently controlled split endcaps). Understanding the role of the boundary layers is critical to this research. In Chap. 6 the magnetic field is found to inhibit the Ekman suction. While we quantitatively confirmed the conclusions of Gilman et al, the finite differential rotation cannot be neglected and modifies the linear Ekman layer. The width of the Ekman layer is reduced with increased magnetic field normal to the end plate. A uniformly-rotating region forms near the outer cylinder. The Stewartson layer penetrates deeper into the fluid with larger Reynolds number and stronger magnetic field. Furthermore a strong magnetic field leads to a steady Stewartson layer, at least in axisymmetric configuration.
