When threaded by a weak magnetic field, a differentially rotating electrically conducting fluid may exhibit the Magnetorotational Instability (MRI), a likely mechanism for enhanced angular momentum transport in accretion disks. In this thesis, we investigate the MRI and its role in the transition to magnetohydrodynamic turbulence in laboratory liquid metal flows. In addition to presenting a basic WKB local linear analysis, we use two independently developed, global, nonlinear codes to study the problem of MRI in cylindrical geometry. We verify both codes by demonstrating their ability to simulate well-known nonlinear fluid phenomena, such as the development of Taylor vortices in unstable viscous Taylor-Couette flow. In the presence of magnetic fields, we demonstrate that both codes reproduce the correct MRI stability threshold.

Our numerical simulations predict the nonlinear saturation amplitude of excited MRI modes for a range of Prandtl numbers, and results indicate that in laboratory liquid metal investigations, these magnetic excitations saturate at a low level when compared to the background field strength.

We address the characteristics of saturated MRI excitations, and investigate their susceptibility to secondary instabilities, such as tearing modes. Finally, we predict the phenomenology of MRI near threshold in realistic cylindrical liquid metal experiments, including the effects of adding a toroidal field in the presence of endcaps. We comment on how the tools created during this research can be used to aid in the design of future experiments to investigate this transition region to magnetic turbulence.