Nonlinear saturation of the magnetorotational instability near threshold in a thin-gap Taylor-Couette setup.

We study the saturation near threshold of the axisymmetric magnetorotational instability (MRI) of a viscous, resistive, incompressible fluid in a thin-gap Taylor-Couette configuration. A vertical magnetic field, Keplerian shear, and no-slip conducting radial boundary conditions are adopted. The weakly nonlinear theory leads to a real Ginzburg-Landau equation for the disturbance amplitude, as in our previous idealized analysis. For small magnetic Prandtl number (Pm<<1) , the saturation amplitude scales as Pm2/3 while the magnitude of angular momentum transport scales as Pm4/3. The difference from the previous scalings (proportional to Pm1/2 and Pm respectively) is attributed to the emergence of radial boundary layers. Away from those, steady-state nonlinear saturation is achieved through a modest reduction in the destabilizing shear. These results will be useful in understanding MRI laboratory experiments and associated numerical simulations.

Weakly nonlinear analysis of the magnetorotational instability in a model channel flow.

We show by means of a perturbative weakly nonlinear analysis that the axisymmetric magnetorotational instability (MRI) of a viscous, resistive, incompressible rotating shear flow subject to a background vertical magnetic field in a thin channel gives rise to a Ginzburg-Landau equation for the disturbance amplitude. For small magnetic Prandtl number (P(m)), the saturation amplitude is proportional square root P(m) and the resulting momentum transport scales as R(-1), where R is the hydrodynamic Reynolds number. Simplifying assumptions, such as linear shear base flow, mathematically expedient boundary conditions, and continuous spectrum of the vertical linear modes, are used to facilitate this analysis. The asymptotic results are shown to comply with numerical calculations using a spectral code. They suggest that the transport due to the nonlinearly developed MRI may be very small in experimental setups with P(m)<<1.

Viscosity mechanisms in accretion disks

The self-sustained turbulence that develops in magnetized accretion disks is suppressed in the weakly ionized, quiescent disks of close binary stars. Because accretion still proceeds during quiescence, another viscosity mechanism operates in these systems. An anticorrelation of the recurrence times of SU UMa dwarf novae with their mass ratio supports spiral waves or shockwaves tidally induced by the companion star as the main process responsible for accretion in the quiescent disks. Other weakly ionized gaseous disks in systems lacking a massive companion must rely on yet another transport mechanism, or they could be essentially passive.