Abstract: We show by using asymptotic techniques that the linear magneto-rotational instability, in a magnetized rotating shear flow in a thin annulus, nonlinearly saturates at an amplitude that scales like a power of the magnetic Prandtl number, A_s µ P_m^k with k = 1/2 – 2/3 depending on boundary conditions. This result has consequences for angular momentum transport and experimental results for these flows. In addition it calls for critical examination of numerica strategies to faithfull simulate such flows as well as accretion disks.