THE SATURATION LIMIT OF THE MAGNETOROTATIONAL INSTABILITY
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Simulations of the magnetorotational instability (MRI) in a homogeneous
shearing box have shown that the asymptotic strength of the magnetic field
declines steeply with increasing resolution. Here I model the MRI driven dynamo
as a large scale dynamo driven by the vertical magnetic helicity flux. This
growth is balanced by large scale mixing driven by a secondary instability. The
saturated magnetic energy density depends almost linearly on the vertical
height of the typical eddies. The MRI can drive eddies with arbitrarily large
vertical wavenumber, so the eddy thickness is either set by diffusive effects,
by the magnetic tension of a large scale vertical field component, or by
magnetic buoyancy effects. In homogeneous, zero magnetic flux, simulations only
the first effect applies and the saturated limit of the dynamo is determined by
explicit or numerical diffusion. The exact result depends on the numerical
details, but is consistent with previous work, including the claim that the
saturated field energy scales as the gas pressure to the one quarter power
(which we interpret as an artifact of numerical dissipation). The magnetic
energy density in a homogeneous shearing box will tend to zero as the
resolution of the simulation increases, but this has no consequences for the
dynamo or for angular momentum transport in real accretion disks. The claim
that the saturated state depends on the magnetic Prandtl number may also be an
artifact of simulations in which microphysical transport coefficients set the
MRI eddy thickness. Finally, the efficiency of the MRI dynamo is a function of
the ratio of the Alfv\'en velocity to the product of the pressure scale height
and the local shear. As this approaches unity from below the dynamo reaches
maximum efficiency.