We consider the nonlinear outcome of gravitational instability in opticallythick disks with a realistic cooling function. We use a numerical model that is local, razor-thin, and unmagnetized. External illumination is ignored. Cooling is calculated from a one-zone model using analytic fits to low temperature Rosseland mean opacities. The model has two parameters: the initial surface density Σ o and the rotation frequency Ω. We survey the parameter space and find: (1) The disk fragments when τ c Ω ∼ 1, where τ c is an effective cooling time defined as the average internal energy of the model divided by the average cooling rate. This is consistent with earlier results that used a simplified cooling function.(2) The initial cooling time τ co for a uniform disk with Q = 1 can differ by orders of magnitude from τ c in the nonlinear outcome. The difference is caused by sharp variations in the opacity with temperature. The condition τ co Ω ∼ 1 therefore does not necessarily indicate where fragmentation will occur. (3) The largest difference between τ c and τ co is near the opacity gap, where dust is absent and hydrogen is largely molecular. (4) In the limit of strong illumination the disk is isothermal; we find that an isothermal version of our model fragments for Q 1.4. Finally, we discuss some physical processes not included in our model, and find that most are likely to make disks more susceptible to fragmentation. We conclude that disks with τ c Ω 1 do not exist.

We numerically evolve turbulence driven by the magnetorotational instability (MRI) in a 3D, unstratified shearing box and study its structure using two-point correlation functions. We confirm Fromang & Papaloizou's result that shearing box models with zero net magnetic flux are not converged; the dimensionless shear stress α is proportional to the grid scale. We find that the two-point correlation of B shows that it is composed of narrow filaments that are swept back by differential rotation into a trailing spiral. The correlation lengths along each of the correlation function principal axes decrease monotonically with the grid scale. For mean azimuthal field models, which we argue are more relevant to astrophysical disks than the zero net field models, we find that: α increases weakly with increasing resolution at fixed box size; α increases slightly as the box size is increased; α increases linearly with net field strength, confirming earlier results; the two-point correlation function of the magnetic field is resolved and converged, and is composed of narrow filaments swept back by the shear; the major axis of the two-point increases slightly as the box size is increased; these results are code independent, based on a comparison of ATHENA and ZEUS runs. The velocity, density, and magnetic fields decorrelate over scales larger than ∼ H, as do the dynamical terms in the magnetic energy evolution equations. We conclude that MHD turbulence in disks is localized, subject to the limitations imposed by the absence of vertical stratification, the use of an isothermal equation of state, finite box size, finite run time, and finite resolution.

We consider the formation and evolution of vortices in a hydrodynamic shearing-sheet model. The evolution is done numerically using a version of the ZEUS code. Consistent with earlier results, an injected vorticity field evolves into a set of long-lived vortices, each of which has a radial extent comparable to the local scale height. But we also find that the resulting velocity field has a positive shear stress, . This effect appears only at high resolution. The transport, which decays with time as t^-1/2, arises primarily because the vortices drive compressive motions. This result suggests a possible mechanism for angular momentum transport in low-ionization disks, with two important caveats: a mechanism must be found to inject vorticity into the disk, and the vortices must not decay rapidly due to three-dimensional instabilities.Comment: 8 pages, 10 figures (high resolution figures available in ApJ electronic edition

We consider the nonaxisymmetric linear theory of radially stratified disks. We work in a shearing-sheet-like approximation, in which the vertical structure of the disk is neglected, and develop equations for the evolution of a planewave perturbation comoving with the shear flow (a shearing wave, or ''shwave''). We calculate a complete solution set for compressive and incompressive short-wavelength perturbations in both the stratified and unstratified shearingsheet models. We develop expressions for the late-time asymptotic evolution of an individual shwave, as well as for the expectation value of the energy for an ensemble of shwaves that are initially distributed isotropically in k-space. We find that (1) incompressive, short-wavelength perturbations in the unstratified shearing sheet exhibit transient growth and asymptotic decay, but the energy of an ensemble of such shwaves is constant with time; (2) short-wavelength compressive shwaves grow asymptotically in the unstratified shearing sheet, as does the energy of an ensemble of such shwaves; (3) incompressive shwaves in the stratified shearing sheet have density and azimuthal velocity perturbations AE, v y $ t ÀRi (for jRijT1), where Ri N 2x /(q) 2 is the Richardson number, N 2 x is the square of the radial Brunt-Väisälä frequency, andq is the effective shear rate; and (4) the energy of an ensemble of incompressive shwaves in the stratified shearing sheet behaves asymptotically as Rit 1 4Ri for jRijT1. For Keplerian disks with modest radial gradients, jRij is expected to beT1, and there is therefore weak growth in a single shwave for Ri < 0 and near-linear growth in the energy of an ensemble of shwaves, independent of the sign of Ri.

We quantify the effects of electron thermal conduction on the properties of hot accretion flows, under the assumption of spherical symmetry. Electron heat conduction is important for low accretion rate systems where the electron cooling time is longer than the conduction time of the plasma, such as Sgr A Ã in the Galactic center. For accretion flows with density profiles similar to the Bondi solution [n(r) / r À3/2 ], we show that heat conduction leads to supervirial temperatures, implying that conduction significantly modifies the dynamics of the accretion flow. We then selfconsistently solve for the dynamics of spherical accretion in the presence of saturated conduction and electron heating. We find that the accretion rate onto the central object can be reduced by $1-3 orders of magnitude relative to the canonical Bondi rate. Electron conduction may thus be an important ingredient in explaining the low radiative efficiencies and low accretion rates inferred from observations of low-luminosity galactic nuclei. The solutions presented in this paper may also describe the nonlinear saturation of the magnetothermal instability in hot accretion flows.