  The saturation level of the magnetorotational instability (MRI) is
investigated using three-dimensional MHD simulations. The shearing box
approximation is adopted and the vertical component of gravity is ignored, so
that the evolution of the MRI is followed in a small local part of the disk. We
focus on the dependence of the saturation level of the stress on the gas
pressure, which is a key assumption in the standard alpha disk model. From our
numerical experiments it is found that there is a weak power-law relation
between the saturation level of the Maxwell stress and the gas pressure in the
nonlinear regime; the higher the gas pressure, the larger the stress. Although
the power-law index depends slightly on the initial field geometry, the
relationship between stress and gas pressure is independent of the initial
field strength, and is unaffected by Ohmic dissipation if the magnetic Reynolds
number is at least 10. The relationship is the same in adiabatic calculations,
where pressure increases over time, and nearly-isothermal calculations, where
pressure varies little with time. Our numerical results are qualitatively
consistent with an idea that the saturation level of the MRI is determined by a
balance between the growth of the MRI and the dissipation of the field through
reconnection. The quantitative interpretation of the pressure-stress relation,
however, may require advances in the theoretical understanding of non-steady
magnetic reconnection.
