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.