Identifying Deficiencies of Standard Accretion Disk Theory: Lessons from a Mean-Field Approach

Turbulent viscosity is frequently used in accretion disk theory to replace the microphysical viscosity in order to accomodate the observational need for in- stabilities in disks that lead to enhanced transport. However, simply replacing the microphysical transport coefficient by a single turbulent transport coeffi- cient hides the fact that the procedure should formally arise as part of a closure in which the hydrodynamic or magnetohydrodynamic equations are averaged, and correlations of turbulent fluctuations are replaced by transport coefficients. Here we show how a mean field approach leads quite naturally two transport coefficients, not one, that govern mass and angular momentum transport. In particular, we highlight that the conventional approach suffers from a seemingly inconsistent neglect of turbulent diffusion in the surface density equation. We constrain these new transport coefficients for specific cases of inward, outward, and zero net mass transport. In addition, we find that one of the new transport terms can lead to oscillations in the mean surface density which then requires a constant or small inverse Rossby number for disks to maintain a monotonic power-law surface density.Comment: 11 page

Sustained Turbulence in Differentially Rotating Magnetized Fluids at Low Magnetic Prandtl Number

We show for the first time that sustained turbulence is possible at low magnetic Prandtl number for Keplerian flows with no mean magnetic flux. Our results indicate that increasing the vertical domain size is equivalent to increasing the dynamical range between the energy injection scale and the dissipative scale. This has important implications for a large variety of differentially rotating systems with low magnetic Prandtl number such as protostellar disks and laboratory experiments.Comment: 5 pages, 6 figures, submitted to ApJ, changes made in response to referee repor

The Stability of Weakly Collisional Plasmas with Thermal and Composition Gradients

Over the last decade, substantial efforts have been devoted to understanding the stability properties, transport phenomena, and long-term evolution of weakly-collisional, magnetized plasmas which are stratified in temperature. These studies have improved our understanding of the physics governing the intra-cluster medium (ICM), but assumed that ICM is a homogeneous. This, however, might not be a good approximation if heavy elements sediment in the inner region of the galaxy cluster. In this paper, we analyze the stability of a weakly-collisional, magnetized plane-parallel atmosphere which is stratified in both temperature and composition. This allows us to discuss for the first time the dynamics of weakly-collisional environments where heat conduction, momentum transport, and ion-diffusion are anisotropic with respect to the direction of the magnetic field. We show that, depending on the relative signs and magnitudes of the gradients in the temperature and the mean molecular weight, the plasma can be subject to a wide variety of unstable modes which include modifications to the magnetothermal instability (MTI), the heat-flux-driven buoyancy instability (HBI), and overstable gravity modes previously studied in homogeneous media. We discuss the astrophysical implications of our findings for a representative galaxy cluster where helium has sedimented. Our findings suggest that the core insulation that results from the magnetic field configurations that arise as a natural consequence of the HBI, which would be MTI stable in a homogeneous medium, could be alleviated if the mean molecular weight gradient is steep enough, i.e., $(\nabla \mu)/\mu > (\nabla T)/T$. This study constitutes a first step toward understanding the interaction between magnetic turbulence and the diffusion of heavy elements in the ICM. (abridged)Comment: 16 pages, 7 figures. This article supersedes arXiv:1111.3372 (5 pages, 3 figures). The present version of this article is published in Ap

Spectral Analysis of Non-Ideal MRI Modes: The effect of Hall diffusion

The effect of magnetic field diffusion on the stability of accretion disks is a problem that has attracted considerable interest of late. In particular, the Hall effect has the potential to bring about remarkable changes in the dynamical behavior of disks that are without parallel. In this paper, we conduct a systematic examination of the linear eigenmodes in a weakly magnetized differentially rotating gas with special focus on Hall diffusion. We first develop a geometrical representation of the eigenmodes and provide a detailed quantitative description of the polarization properties of the oscillatory modes under the combined influence of the Coriolis and Hall effects. We also analyze the effects of magnetic diffusion on the structure of the unstable modes and derive analytical expressions for the kinetic and magnetic stresses and energy densities associated with the non-ideal MRI. Our analysis explicitly demonstrates that, if the dissipative effects are relatively weak, the kinetic stresses and energies make up the dominant contribution to the total stress and energy density when the equilibrium angular momentum and magnetic field vectors are anti-parallel. This is in sharp contrast to what is observed in the case of the ideal or dissipative MRI. We conduct shearing box simulations and find very good agreement with the results derived from linear analysis. As the modes in consideration are also exact solutions of the non-linear equations, the unconventional nature of the kinetic and magnetic stresses may have significant implications for the non-linear evolution in some regions of protoplanetary disks.Comment: 16 pages, 11 figures, accepted by Ap

Sustained Magnetorotational Turbulence in Local Simulations of Stratified Disks with Zero Net Magnetic Flux

We examine the effects of density stratification on magnetohydrodynamic turbulence driven by the magnetorotational instability in local simulations that adopt the shearing box approximation. Our primary result is that, even in the absence of explicit dissipation, the addition of vertical gravity leads to convergence in the turbulent energy densities and stresses as the resolution increases, contrary to results for zero net flux, unstratified boxes. The ratio of total stress to midplane pressure has a mean of ~0.01, although there can be significant fluctuations on long (>~50 orbit) timescales. We find that the time averaged stresses are largely insensitive to both the radial or vertical aspect ratio of our simulation domain. For simulations with explicit dissipation, we find that stratification extends the range of Reynolds and magnetic Prandtl numbers for which turbulence is sustained. Confirming the results of previous studies, we find oscillations in the large scale toroidal field with periods of ~10 orbits and describe the dynamo process that underlies these cycles.Comment: 13 pages, 18 figures, submitted to Ap

On Vertically Global, Horizontally Local Models for Astrophysical Disks

Disks with a barotropic equilibrium structure, for which the pressure is only a function of the density, rotate on cylinders in the presence of a gravitational potential, so that the angular frequency of such a disk is independent of height. Such disks with barotropic equilibria can be approximately modeled using the shearing box framework, representing a small disk volume with height-independent angular frequency. If the disk is in baroclinic equilibrium, the angular frequency does generally depend on height, and it is thus necessary to go beyond the standard shearing box approach. In this paper, we show that given a global disk model, it is possible to develop approximate models that are local in horizontal planes without an expansion in height with shearing-periodic boundary conditions. We refer to the resulting framework as the vertically global shearing box (VGSB). These models can be non-axisymmetric for globally barotropic equilibria but should be axisymmetric for globally baroclinic equilibria. We provide explicit equations for this VGSB which can be implemented in standard magnetohydrodynamic codes by generalizing the shearing-periodic boundary conditions to allow for a height-dependent angular frequency and shear rate. We also discuss the limitations that result from the radial approximations that are needed in order to impose height-dependent shearing periodic boundary conditions. We illustrate the potential of this framework by studying a vertical shear instability and examining the modes associated with the magnetorotational instability.Comment: 24 pages, 8 figures, updated to match published versio

On the Anisotropic Nature of MRI-Driven Turbulence in Astrophysical Disks

The magnetorotational instability is thought to play an important role in enabling accretion in sufficiently ionized astrophysical disks. The rate at which MRI-driven turbulence transports angular momentum is related to both the strength of the amplitudes of the fluctuations on various scales and the degree of anisotropy of the underlying turbulence. This has motivated several studies of the distribution of turbulent power in spectral space. In this paper, we investigate the anisotropic nature of MRI-driven turbulence using a pseudo-spectral code and introduce novel ways to robustly characterize the underlying turbulence. We show that the general flow properties vary in a quasi-periodic way on timescales comparable to 10 inverse angular frequencies motivating the temporal analysis of its anisotropy. We introduce a 3D tensor invariant analysis to quantify and classify the evolution of the anisotropic turbulent flow. This analysis shows a continuous high level of anisotropy, with brief sporadic transitions towards two- and three-component isotropic turbulent flow. This temporal-dependent anisotropy renders standard shell-average, especially when used simultaneously with long temporal averages, inadequate for characterizing MRI-driven turbulence. We propose an alternative way to extract spectral information from the turbulent magnetized flow, whose anisotropic character depends strongly on time. This consists of stacking 1D Fourier spectra along three orthogonal directions that exhibit maximum anisotropy in Fourier space. The resulting averaged spectra show that the power along each of the three independent directions differs by several orders of magnitude over most scales, except the largest ones. Our results suggest that a first-principles theory to describe fully developed MRI-driven turbulence will likely have to consider the anisotropic nature of the flow at a fundamental level.Comment: 13 pages, 13 figures, submitted to Ap