  The magnetorotational instability (MRI) has been extensively studied in
circular magnetized disks, and its ability to drive accretion has been
demonstrated in a multitude of scenarios. There are reasons to expect eccentric
magnetized disks to also exist, but the behavior of the MRI in these disks
remains largely uncharted territory. Here we present the first simulations that
follow the nonlinear development of the MRI in eccentric disks. We find that
the MRI in eccentric disks resembles circular disks in two ways, in the overall
level of saturation and in the dependence of the detailed saturated state on
magnetic topology. However, in contrast with circular disks, the Maxwell stress
in eccentric disks can be negative in some disk sectors, even though the
integrated stress is always positive. The angular momentum flux raises the
eccentricity of the inner parts of the disk and diminishes the same of the
outer parts. Because material accreting onto a black hole from an eccentric
orbit possesses more energy than material tracing the innermost stable circular
orbit, the radiative efficiency of eccentric disks may be significantly lower
than circular disks. This may resolve the "inverse energy problem" seen in many
tidal disruption events.
