  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.
