Takayoshi Sano

Magnetorotational Instability in Protoplanetary Disks. II. Ionization State and Unstable Regions

We investigate where magnetorotational instability operates in protoplanetary disks, which can cause angular momentum transport in the disks. We investigate the spatial distribution of various charged particles and the unstable regions for a variety of models for protoplanetary disks, taking into account the recombination of ions and electrons at grain surfaces, which is an important process in most parts of the disks. We find that for all the models there is an inner region that is magnetorotationally stable due to ohmic dissipation. This must make the accretion onto the central star nonsteady. For the model of the minimum-mass solar nebula, the critical radius, inside of which the disk is stable, is about 20 AU, and the mass accretion rate just outside the critical radius is 10-7-10-6 M☉ yr-1. The stable region is smaller in a disk of lower column density. Dust grains in protoplanetary disks may grow by mutual sticking and may sediment toward the midplane of the disks. We find that the stable region shrinks as the grain size increases or the sedimentation proceeds. Therefore, in the late evolutionary stages, protoplanetary disks can be magnetorotationally unstable even in the inner regions.

ANGULAR MOMENTUM TRANSPORT BY MAGNETOHYDRODYNAMIC TURBULENCE IN ACCRETION DISKS: GAS PRESSURE DEPENDENCE OF THE SATURATION LEVEL OF THE MAGNETOROTATIONAL INSTABILITY

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 α disk model. From our numerical experiments we find 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. Over the entire region of parameter space explored, turbulence driven by the MRI has many characteristic ratios such as that of the Maxwell stress to the magnetic pressure. We also find that the amplitudes of the spatial fluctuations in density and the time variability in the stress are characterized by the ratio of magnetic pressure to gas pressure in the nonlinear regime. 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 nonsteady magnetic reconnection.

The Effect of the Hall Term on the Nonlinear Evolution of the Magnetorotational Instability. II. Saturation Level and Critical Magnetic Reynolds Number

The nonlinear evolution of the magnetorotational instability (MRI) in weakly ionized accretion disks, including the effect of the Hall term and ohmic dissipation, is investigated using local three-dimensional MHD simulations and various initial magnetic field geometries. When the magnetic Reynolds number, ReM ≡ v/ηΩ (where vA is the Alfven speed, η is the magnetic diffusivity, and Ω is the angular frequency), is initially larger than a critical value ReM,crit, the MRI evolves into MHD turbulence in which angular momentum is transported efficiently by the Maxwell stress. If ReM < ReM,crit, however, ohmic dissipation suppresses the MRI, and the stress is reduced by several orders of magnitude. The critical value is in the range of 1-30 depending on the initial field configuration. The Hall effect does not modify the critical magnetic Reynolds number by much but enhances the saturation level of the Maxwell stress by a factor of a few. We show that the saturation level of the MRI is characterized by v/ηΩ, where vAz is the Alfven speed in the nonlinear regime along the vertical component of the field. The condition for turbulence and significant transport is given by v/ηΩ 1, and this critical value is independent of the strength and geometry of the magnetic field or the size of the Hall term. If the magnetic field strength in an accretion disk can be estimated observationally and the magnetic Reynolds number v/ηΩ is larger than about 30, this would imply that the MRI is operating in the disk.

The Effect of the Hall Term on the Nonlinear Evolution of the Magnetorotational Instability. I. Local Axisymmetric Simulations

The effect of the Hall term on the evolution of the magnetorotational instability (MRI) in weakly ionized accretion disks is investigated using local axisymmetric simulations. First, we show that the Hall term has important effects on the MRI when the temperature and density in the disk is below a few thousand K and between 1013 and 1018 cm-3, respectively. Such conditions can occur in the quiescent phase of dwarf nova disks, or in the inner part (inside 10-100 AU) of protoplanetary disks. When the Hall term is important, the properties of the MRI are dependent on the direction of the magnetic field with respect to the angular velocity vector Ω. If the disk is threaded by a uniform vertical field oriented in the same sense as Ω, the axisymmetric evolution of the MRI is an exponentially growing two-channel flow without saturation. When the field is oppositely directed to Ω, however, small scale fluctuations prevent the nonlinear growth of the channel flow and the MRI evolves into MHD turbulence. These results are anticipated from the characteristics of the linear dispersion relation. In axisymmetry on a field with zero-net flux, the evolution of the MRI is independent of the size of the Hall term relative to the inductive term. The evolution in this case is determined mostly by the effect of ohmic dissipation.

Turbulent Mixing and the Dead Zone in Protostellar Disks

We investigate the conditions for the presence of a magnetically inactive dead zone in protostellar disks using three-dimensional shearing-box MHD calculations, including vertical stratification, ohmic resistivity, and time-dependent ionization chemistry. Activity driven by the magneto-rotational instability fills the whole thickness of the disk at 5 AU, provided cosmic-ray ionization is present, small grains are absent, and the gas-phase metal abundance is sufficiently high. At 1 AU, the larger column density of 1700 g cm-2 means the midplane is shielded from ionizing particles and remains magneto-rotationally stable, even under the most favorable conditions considered. Nevertheless, the dead zone is effectively eliminated. Turbulence mixes free charges into the interior as they recombine, leading to a slight coupling of the midplane gas to the magnetic fields. Weak, large-scale radial fields diffuse to the midplane, where they are sheared out to produce stronger azimuthal fields. On average, the resulting midplane accretion stresses are just a few times less than in the surface layers.

Local Three-dimensional Simulations of Magnetorotational Instability in Radiation-dominated Accretion Disks

We examine the small-scale dynamics of black hole accretion disks in which radiation pressure exceeds gas pressure. Local patches of disk are modeled by numerically integrating the equations of radiation MHD in the flux-limited diffusion approximation. The shearing-box approximation is used, and the vertical component of gravity is neglected. Magnetorotational instability (MRI) leads to turbulence in which accretion stresses are due primarily to magnetic torques. When radiation is locked to gas over the length and timescales of fluctuations in the turbulence, the accretion stress, density contrast, and dissipation differ little from those in the corresponding calculations, with radiation replaced by extra gas pressure. However, when radiation diffuses each orbit a distance that is comparable to the rms vertical wavelength of the MRI, radiation pressure is less effective in resisting squeezing. Large density fluctuations occur, and radiation damping of compressive motions converts P dV work into photon energy. The accretion stress in calculations having a net vertical magnetic field is found to be independent of opacity over the range explored and approximately proportional to the square of the net field. In calculations with zero net magnetic flux, the accretion stress depends on the portion of the total pressure that is effective in resisting compression. The stress is lower when radiation diffuses rapidly with respect to the gas. We show that radiation-supported Shakura-Sunyaev disks accreting via internal magnetic stresses are likely to have radiation marginally coupled to turbulent gas motions in their interiors.

Saturation and Thermalization of the Magnetorotational Instability: Recurrent Channel Flows and Reconnections

The nonlinear evolution and saturation mechanism of the magnetorotational instability (MRI) are investigated using three-dimensional resistive MHD simulations. A local shearing box is used for our numerical analysis, and the simulations begin with a purely vertical magnetic field. We find that the magnetic stress in the nonlinear stage of the MRI is strongly fluctuating. The time evolution shows the quasi-periodic recurrence of spike-shaped variations, typically for a few orbits, which correspond to the rapid amplification of the magnetic field by the nonlinear growth of a two-channel solution followed by decay through magnetic reconnections. The increase rate of the total energy in the shearing box system is analytically related to the volume-averaged torque in the system. We find that at the saturated state this energy gain of the system is balanced with the increase of thermal energy, mostly due to joule heating. The spike-shaped time evolution is a general feature of the nonlinear evolution of the MRI in disks threaded by vertical fields and can be seen if the effective magnetic Reynolds number is larger than about unity.

Self-sustained Ionization and Vanishing Dead Zones in Protoplanetary Disks

We analyze the ionization state of the magnetohydrodynamically turbulent protoplanetary disks and propose a new mechanism of sustaining ionization. First, we show that in the quasi-steady state of turbulence driven by magnetorotational instability in a typical protoplanetary disk with dust grains, the amount of energy dissipation should be sufficient for providing the ionization energy that is required for activating magnetorotational instability. Second, we show that in the disk with dust grains the energetic electrons that compose electric currents in weakly ionized gas can provide collisional ionization, depending on the actual saturation state of magnetorotational turbulence. On the other hand, we show that in the protoplanetary disks with the reduced effect of dust grains, the turbulent motion can homogenize the ionization degree, which can activate magnetorotational instability even in the absence of other ionization processes. The results in this Letter indicate that most of the regions in protoplanetary disks remain magnetically active, and we thus require a change in the theoretical modeling of planet formation.

Local Axisymmetric Simulations of Magnetorotational Instability in Radiation-dominated Accretion Disks

We perform numerical simulations of magnetorotational instability in a local patch of accretion disk in which radiation pressure exceeds gas pressure. Such conditions may occur in the central regions of disks surrounding compact objects in active galactic nuclei and Galactic X-ray sources. We assume axisymmetry and neglect vertical stratification. The growth rates of the instability on initially uniform magnetic fields are consistent with the linear analysis of Blaes & Socrates (2001). As is the case when radiation effects are neglected, the nonlinear development of the instability leads to transitory turbulence when the initial magnetic field has no net vertical flux. During the turbulent phase, angular momentum is transported outward. The Maxwell stress is a few times the Reynolds stress, and their sum is about 4 times the mean pressure in the vertical component of the magnetic field. For magnetic pressure exceeding gas pressure, turbulent fluctuations in the field produce density contrasts about equal to the ratio of magnetic to gas pressure. These are many times larger than in the corresponding gas pressure-dominated situation and may have profound implications for the steady state vertical structure of radiation-dominated disks. Diffusion of radiation from compressed regions damps turbulent motions, converting kinetic energy into photon energy.

Magnetic Field Amplification Associated with the Richtmyer-Meshkov Instability

The amplification of a magnetic field due to the Richtmyer-Meshkov instability (RMI) is investigated by two-dimensional MHD simulations. Single-mode analysis is adopted to reveal definite relation between the nonlinear evolution of RMI and the field enhancement. It is found that an ambient magnetic field is stretched by fluid motions associated with the RMI, and the strength is amplified significantly by more than two orders of magnitude. The saturation level of the field is determined by a balance between the amplified magnetic pressure and the thermal pressure after shock passage. This effective amplification can be achieved in a wide range of the conditions for the RMI such as the Mach number of an incident shock and the density ratio at a contact discontinuity. The results suggest that the RMI could be a robust mechanism of the amplification of interstellar magnetic fields and cause the origin of localized strong fields observed at the shock of supernova remnants.