The magnetorotational instability (MRI) is broadly accepted as the source of turbulence in astrophysical systems exhibiting strong shear. We present here a detailed analysis of the development of MRI in a model environment that has features similar to ongoing terrestrial experiments seeking to demonstrate the mechanism. We perform a linear and weakly nonlinear analysis of this instability with a background vertical magnetic field. We find that the amount of transport decays with decreasing magnetic Prandtl number. We offer some reflections upon these results and how they relate to the astrophysical problem.