On the 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 non-linear theory leads to a real Ginzburg-Landau equation for the disturbance amplitude, like in our previous idealized analysis. For small magnetic Prandtl number (P \ll 1), the saturation amplitude scales as P^{2/3} while the magnitude of angular momentum transport scales as P^{4/3}. The difference with the previous scalings (~P^{1/2} and P respectively) is attributed to the emergence of radial boundary layers. Away from those, steady-state non-linear saturation is achieved through a modest reduction in the destabilizing shear. These results will be useful to understand MRI laboratory experiments and associated numerical simulations.