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 ( { \cal P } _ { m } \ll 1 ) , the saturation amplitude scales as { \cal P } _ { m } ^ { 2 / 3 } while the magnitude of angular momentum transport scales as { \cal P } _ { m } ^ { 4 / 3 } .
The difference with the previous scalings ( \propto { \cal P } _ { m } ^ { 1 / 2 } and { \cal P } _ { m } 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 .