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Understanding the thermal structure of protoplanetary disks is crucial for modeling planet formation and interpreting disk observations. We present a new two-layer radiative transfer model for computing the thermal structure of axisymmetric irradiated disks. Unlike the standard two-layer model, our model accounts for the radial as well as vertical transfer of the starlight reprocessed at the disk surface. The model thus allows us to compute the temperature below “shadowed” surfaces receiving no direct starlight. Thanks to the assumed axisymmetry, the reprocessed starlight flux is given in a one-dimensional integral form that can be computed at a low cost. Furthermore, our model evolves the midplane temperature using a time-dependent energy equation and can therefore treat thermal instabilities. We apply our global two-layer model to disks with a planetary induced gap and confirm that the model reproduces the disks’ temperature profiles obtained from more computationally expensive Monte Carlo radiative transfer calculations to an accuracy of less than 20%. We also apply the model to study the long-term behavior of the thermal wave instability in irradiated disks. Being simple and computationally efficient, the global two-layer model will be suitable for studying the interplay between disks’ thermal evolution and dust evolution.

We present calculations of non-LTE, relativistic accretion disk models applicable to the high/soft state of black hole X-ray binaries. We include the effects of thermal Comptonization and bound-free and free-free opacities of all abundant ion species. Taking into account the relativistic propagation of photons from the local disk surface to an observer at infinity, we present spectra calculated for a variety of accretion rates, black hole spin parameters, disk inclinations, and stress prescriptions. We also consider nonzero inner torques on the disk and explore different vertical dissipation profiles, including some that are motivated by recent radiation magnetohydrodynamic (MHD) simulations of magnetorotational turbulence. Bound-free metal opacity generally produces significantly less spectral hardening than previous models that only considered Compton scattering and free-free opacity. We find that the resulting effective photosphere usually lies at a small fraction of the total column depth, producing spectra that are remarkably independent of the stress prescription and vertical structure assumptions. We provide detailed comparisons between our models and the widely used multicolor disk model. Frequency-dependent discrepancies exist that may affect the parameters of other spectral components when this simpler disk model is used to fit modern X-ray data. For a given source, our models predict that the luminosity in the high/soft state should approximately scale with the fourth power of the empirically inferred maximum temperature, but with a slight hardening at high luminosities. This is in good agreement with observations.

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 a 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 standard equilibrium for radiation-dominated accretion disks has long been known to be viscously, thermally, and convectively unstable, but the nonlinear development of these instabilities--and hence the actual state of such disks--has not yet been identified. By performing local two-dimensional hydrodynamic simulations of disks, we demonstrate that convective motions can release heat sufficiently rapidly as to substantially alter the vertical structure of the disks. If the dissipation rate within a vertical column is proportional to its mass, the disk settles into a new configuration that is thinner than the standard radiation-supported equilibrium by a factor of 2. If, on the other hand, the vertically integrated dissipation rate is proportional to the vertically integrated total pressure, the disk is subject to the well-known thermal instability. Convection, however, biases the development of this instability toward collapse. The end result of such a collapse is a gas-pressure-dominated equilibrium at the original column density.

The vertical structure of a black hole accretion disk in which radiation dominates the total pressure is investigated using a three-dimensional radiation-MHD calculation, in the stratified shearing-box approximation. Magneto-rotational instability [Balbus, S. A. and Hawley, J. F., Astrophys. J. 376 (1991), 214] converts differential rotation to turbulent magnetic and kinetic energy. The gas is cooled by diffusion and advection of radiation through the vertical boundaries. Density is greatest near the midplane, while accretion stresses are greatest in optically-thick surface layers. The thermal time, and the ratio of radiation to gas pressure, are an order of magnitude smaller than in the corresponding Shakura-Sunyaev model [Shakura, N. I. and Sunyaev R. A., Astron. Astrophys. 24 (1973), 337] because dissipation occurs at low column density.

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 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. 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.