  We study the magnetorotational instability (MRI) using nonmodal stability
techniques. Despite the spectral instability of many forms of the MRI, this
proves to be a natural method of analysis that is well-suited to deal with the
non-self-adjoint nature of the linear MRI equations. We find that the fastest
growing linear MRI structures on both local and global domains can look very
different to the eigenmodes, invariably resembling waves shearing with the
background flow (shear waves). In addition, such structures can grow many times
faster than the least stable eigenmode over long time periods, and be localized
in a completely different region of space. These ideas lead -- for both
axisymmetric and non-axisymmetric modes -- to a natural connection between the
global MRI and the local shearing box approximation. By illustrating that the
fastest growing global structure is well described by the ordinary differential
equations (ODEs) governing a single shear wave, we find that the shearing box
is a very sensible approximation for the linear MRI, contrary to many previous
claims. Since the shear wave ODEs are most naturally understood using nonmodal
analysis techniques, we conclude by analyzing local MRI growth over finite
time-scales using these methods. The strong growth over a wide range of
wave-numbers suggests that nonmodal linear physics could be of fundamental
importance in MRI turbulence.
