  Magnetorotational instability (MRI) has a potential to generate the vigorous
turbulence in protoplanetary disks, although its turbulence strength and
accretion stress remains debatable because of the uncertainty of MRI with low
ionization fraction. We focus on the heating of electrons by strong electric
fields which amplifies nonideal magnetohydrodynamic effects. The heated
electrons frequently collide with and stick to dust grains, which in turn
decreases the ionization fraction and is expected to weaken the turbulent
motion driven by MRI. In order to quantitatively investigate the nonlinear
evolution of MRI including the electron heating, we perform
magnetohydrodynamical simulation with the unstratified shearing box. We
introduce a simple analytic resistivity model depending on the current density
by mimicking resistivity given by the calculation of ionization. Our simulation
confirms that the electron heating suppresses magnetic turbulence when the
electron heating occurs with low current density. We find a clear correlation
between magnetic stress and its current density, which means that the magnetic
stress is proportional to the squared current density. When the turbulent
motion is completely suppressed, laminar accretion flow is caused by ordered
magnetic field. We give an analytical description of the laminar state by using
a solution of linear perturbation equations with resistivity. We also propose a
formula that successfully predicts the accretion stress in the presence of the
electron heating.
