Dust settling in local simulations of turbulent protoplanetary disks

In this paper, we study the effect of MHD turbulence on the dynamics of dust particles in protoplanetary disks. We vary the size of the particles and relate the dust evolution to the turbulent velocity fluctuations. We performed numerical simulations using two Eulerian MHD codes, both based on finite difference techniques: ZEUS--3D and NIRVANA. These were local shearing box simulations incorporating vertical stratification. Both ideal and non ideal MHD simulations with midplane dead zones were carried out. The codes were extended to incorporate different models for the dust as an additional fluid component. Good agreement between results obtained using the different approaches was obtained. The simulations show that a thin layer of very small dust particles is diffusively spread over the full vertical extent of the disk. We show that a simple description obtained using the diffusion equation with a diffusion coefficient simply expressed in terms of the velocity correlations accurately matches the results. Dust settling starts to become apparent for particle sizes of the order of 1 to 10 centimeters for which the gas begins to decouple in a standard solar nebula model at 5.2 AU. However, for particles which are 10 centimeters in size, complete settling toward a very thin midplane layer is prevented by turbulent motions within the disk, even in the presence of a midplane dead zone of significant size. These results indicate that, when present, MHD turbulence affects dust dynamics in protoplanetary disks. We find that the evolution and settling of the dust can be accurately modelled using an advection diffusion equation that incorporates vertical settling. The value of the diffusion coefficient can be calculated from the turbulent velocity field when that is known for a time of several local orbits.Comment: 15 pages, 16 figures, accepted in Astronomy & Astrophysic

Radiation Magnetohydrodynamics In Global Simulations Of Protoplanetary Disks

Our aim is to study the thermal and dynamical evolution of protoplanetary disks in global simulations, including the physics of radiation transfer and magneto-hydrodynamic (MHD) turbulence caused by the magneto-rotational instability. We develop a radiative transfer method based on the flux-limited diffusion approximation that includes frequency dependent irradiation by the central star. This hybrid scheme is implemented in the PLUTO code. The focus of our implementation is on the performance of the radiative transfer method. Using an optimized Jacobi preconditioned BiCGSTAB solver, the radiative module is three times faster than the MHD step for the disk setup we consider. We obtain weak scaling efficiencies of 70% up to 1024 cores. We present the first global 3D radiation MHD simulations of a stratified protoplanetary disk. The disk model parameters are chosen to approximate those of the system AS 209 in the star-forming region Ophiuchus. Starting the simulation from a disk in radiative and hydrostatic equilibrium, the magnetorotational instability quickly causes MHD turbulence and heating in the disk. For the disk parameters we use, turbulent dissipation heats the disk midplane and raises the temperature by about 15% compared to passive disk models. A roughly flat vertical temperature profile establishes in the disk optically thick region close to the midplane. We reproduce the vertical temperature profile with a viscous disk models for which the stress tensor vertical profile is flat in the bulk of the disk and vanishes in the disk corona. The present paper demonstrates for the first time that global radiation MHD simulations of turbulent protoplanetary disks are feasible with current computational facilities. This opens up the windows to a wide range of studies of the dynamics of protoplanetary disks inner parts, for which there are significant observational constraints.Comment: Accepted to A&

MHD simulations of the magnetorotational instability in a shearing box with zero net flux: the case Pm=4

This letter investigates the transport properties of MHD turbulence induced by the magnetorotational instability at large Reynolds numbers Re when the magnetic Prandtl number Pm is larger than unity. Three MHD simulations of the magnetorotational instability (MRI) in the unstratified shearing box with zero net flux are presented. These simulations are performed with the code Zeus and consider the evolution of the rate of angular momentum transport as Re is gradually increased from 3125 to 12500 while simultaneously keeping Pm=4. To ensure that the small scale features of the flow are well resolved, the resolution varies from 128 cells per disk scaleheight to 512 cells per scaleheight. The latter constitutes the highest resolution of an MRI turbulence simulation to date. The rate of angular momentum transport, measured using the alpha parameter, depends only very weakly on the Reynolds number: alpha is found to be about 0.007 with variations around this mean value bounded by 15% in all simulations. There is no systematic evolution with Re. For the best resolved model, the kinetic energy power spectrum tentatively displays a power-law range with an exponent -3/2, while the magnetic energy is found to shift to smaller and smaller scales as the magnetic Reynolds number increases. A couple of different diagnostics both suggest a well-defined injection length of a fraction of a scaleheight. The results presented in this letter are consistent with the MRI being able to transport angular momentum efficiently at large Reynolds numbers when Pm=4 in unstratified zero net flux shearing boxes.Comment: 5 pages, 4 figures, accepted in Astronomy and Astrophysic

MHD simulations of the magnetorotational instability in a shearing box with zero net flux. I. The issue of convergence

We study the properties of MHD turbulence driven by the magnetorotational instability (MRI) in accretion disks. We adopt the local shearing box model and focus on the special case for which the initial magnetic flux threading the disk vanishes. We employ the finite difference code ZEUS to evolve the ideal MHD equations. Performing a set of numerical simulations in a fixed computational domain with increasing resolution, we demonstrate that turbulent activity decreases as resolution increases. We quantify the turbulent activity by measuring the rate of angular momentum transport through evaluating the standard alpha parameter. We find alpha=0.004 when (N_x,N_y,N_z)=(64,100,64), alpha=0.002 when (N_x,N_y,N_z)=(128,200,128) and alpha=0.001 when (N_x,N_y,N_z)=(256,400,256). This steady decline is an indication that numerical dissipation, occurring at the grid scale is an important determinant of the saturated form of the MHD turbulence. Analysing the results in Fourier space, we demonstrate that this is due to the MRI forcing significant flow energy all the way down to the grid dissipation scale. We also use our results to study the properties of the numerical dissipation in ZEUS. Its amplitude is characterised by the magnitude of an effective magnetic Reynolds number Re_M which increases from 10^4 to 10^5 as the number of grid points is increased from 64 to 256 per scale height. The simulations we have carried out do not produce results that are independent of the numerical dissipation scale, even at the highest resolution studied. Thus it is important to use physical dissipation, both viscous and resistive, and to quantify contributions from numerical effects, when performing numerical simulations of MHD turbulence with zero net flux in accretion disks at the resolutions normally considered.Comment: 10 pages, 15 figures, accepted in A&A. Numerical results improved, various numerical issues addressed (boundary conditions, box size, run durations

Diffusive Migration of Low-Mass Proto-planets in Turbulent Disks

Torque fluctuations due to magnetorotational turbulence in proto-planetary disks may greatly influence the migration patterns and survival probabilities of nascent planets. Provided that the turbulence is a stationary stochastic process with finite amplitude and correlation time, the resulting diffusive migration can be described with a Fokker-Planck equation, which we reduce to an advection-diffusion equation. We calibrate the coefficients with existing turbulent-disk simulations and mean-migration estimates, and solve the equation both analytically and numerically. Diffusion tends to dominate over advection for planets of low-mass and those in the outer regions of proto-planetary disks, whether they are described by the Minimum Mass Solar Nebula (MMSN) or by T-Tauri alpha disks. Diffusion systematically reduces the lifetime of most planets, yet it allows a declining fraction of them to survive for extended periods of time at large radii. Mean planet lifetimes can even be formally infinite (e.g. in an infinite steady MMSN), though median lifetimes are always finite. Surviving planets may linger near specific radii where the combined effects of advection and diffusion are minimized, or at large radii, depending on model specifics. The stochastic nature of migration in turbulent disks challenges deterministic planet formation scenarios and suggests instead that a wide variety of planetary outcomes are possible from similar initial conditions. This would contribute to the diversity of (extrasolar) planetary systems.Comment: 31 pages, 7 figures, accepted for publication in Ap

On the Minimum Energy Configuration of a Rotating Barotropic Fluid: A Response to Narayan & Pringle astro-ph\/0208161

The authors respond to the commentary listed in the title

MHD simulations of the magnetorotational instability in a shearing box with zero net flux. II. The effect of transport coefficients

We study the influence of the choice of transport coefficients (viscosity and resistivity) on MHD turbulence driven by the magnetorotational instability (MRI) in accretion disks. We follow the methodology described in paper I: we adopt an unstratified shearing box model and focus on the case where the net vertical magnetic flux threading the box vanishes. For the most part we use the finite difference code ZEUS, including explicit transport coefficients in the calculations. However, we also compare our results with those obtained using other algorithms (NIRVANA, the PENCIL code and a spectral code) to demonstrate both the convergence of our results and their independence of the numerical scheme. We find that small scale dissipation affects the saturated state of MHD turbulence. In agreement with recent similar numerical simulations done in the presence of a net vertical magnetic flux, we find that turbulent activity (measured by the rate of angular momentum transport) is an increasing function of the magnetic Prandtl number Pm for all values of the Reynolds number Re that we investigated. We also found that turbulence disappears when the Prandtl number falls below a critical value Pm_c that is apparently a decreasing function of Re. For the limited region of parameter space that can be probed with current computational resources, we always obtained Pm_c>1. We conclude that the magnitudes of the transport coefficients are important in determining the properties of MHD turbulence in numerical simulations in the shearing box with zero net flux, at least for Reynolds numbers and magnetic Prandtl numbers that are such that transport is not dominated by numerical effects and thus can be probed using current computational resources.Comment: 10 pages, 13 figures, accepted in A&A. Numerical results improved, minor changes in the tex

Angular momentum transport and large eddy simulations in magnetorotational turbulence: the small Pm limit

Angular momentum transport in accretion discs is often believed to be due to magnetohydrodynamic turbulence mediated by the magnetorotational instability. Despite an abundant literature on the MRI, the parameters governing the saturation amplitude of the turbulence are poorly understood and the existence of an asymptotic behavior in the Ohmic diffusion regime is not clearly established. We investigate the properties of the turbulent state in the small magnetic Prandtl number limit. Since this is extremely computationally expensive, we also study the relevance and range of applicability of the most common subgrid scale models for this problem. Unstratified shearing boxes simulations are performed both in the compressible and incompressible limits, with a resolution up to 800 cells per disc scale height. The latter constitutes the largest resolution ever attained for a simulation of MRI turbulence. In the presence of a mean magnetic field threading the domain, angular momentum transport converges to a finite value in the small Pm limit. When the mean vertical field amplitude is such that {\beta}, the ratio between the thermal and magnetic pressure, equals 1000, we find {\alpha}~0.032 when Pm approaches zero. In the case of a mean toroidal field for which {\beta}=100, we find {\alpha}~0.018 in the same limit. Both implicit LES and Chollet-Lesieur closure model reproduces these results for the {\alpha} parameter and the power spectra. A reduction in computational cost of a factor at least 16 (and up to 256) is achieved when using such methods. MRI turbulence operates efficiently in the small Pm limit provided there is a mean magnetic field. Implicit LES offers a practical and efficient mean of investigation of this regime but should be used with care, particularly in the case of a vertical field. Chollet-Lesieur closure model is perfectly suited for simulations done with a spectral code.Comment: Accepted for publication in A&

The effect of MHD turbulence on massive protoplanetary disk fragmentation

Massive disk fragmentation has been suggested to be one of the mechanisms leading to the formation of giant planets. While it has been heavily studied in quiescent hydrodynamic disks, the effect of MHD turbulence arising from the magnetorotational instability (MRI) has never been investigated. This paper fills this gap and presents 3D numerical simulations of the evolution of locally isothermal, massive and magnetized disks. In the absence of magnetic fields, a laminar disk fragments and clumps are formed due to the effect of self--gravity. Although they disapear in less than a dynamical timescale in the simulations because of the limited numerical resolution, various diagnostics suggest that they should survive and form giant planets in real disks. When the disk is magnetized, it becomes turbulent at the same time as gravitational instabilities develop. At intermediate resolution, no fragmentation is observed in these turbulent models, while a large number of fragments appear in the equivalent hydrodynamical runs. This is because MHD turbulence reduces the strength of the gravitational instability. As the resolution is increased, the most unstable wavelengths of the MRI are better resolved and small scale angular momentum transport starts to play a role: fragments are found to form in massive and turbulent disks in that case. All of these results indicate that there is a complicated interaction between gravitational instabilities and MHD turbulence that influences disk fragmentation processes.Comment: 8 pages, 8 figures, accepted for publication in Astronomy & Astrophysic

Global MHD simulations of stratified and turbulent protoplanetary discs. I. Model properties

We present the results of global 3-D MHD simulations of stratified and turbulent protoplanetary disc models. The aim of this work is to develop thin disc models capable of sustaining turbulence for long run times, which can be used for on-going studies of planet formation in turbulent discs. The results are obtained using two codes written in spherical coordinates: GLOBAL and NIRVANA. Both are time--explicit and use finite differences along with the Constrained Transport algorithm to evolve the equations of MHD. In the presence of a weak toroidal magnetic field, a thin protoplanetary disc in hydrostatic equilibrium is destabilised by the magnetorotational instability (MRI). When the resolution is large enough (25 vertical grid cells per scale height), the entire disc settles into a turbulent quasi steady-state after about 300 orbits. Angular momentum is transported outward such that the standard alpha parameter is roughly 4-6*10^{-3}. We find that the initial toroidal flux is expelled from the disc midplane and that the disc behaves essentially as a quasi-zero net flux disc for the remainder of the simulation. As in previous studies, the disc develops a dual structure composed of an MRI--driven turbulent core around its midplane, and a magnetised corona stable to the MRI near its surface. By varying disc parameters and boundary conditions, we show that these basic properties of the models are robust. The high resolution disc models we present in this paper achieve a quasi--steady state and sustain turbulence for hundreds of orbits. As such, they are ideally suited to the study of outstanding problems in planet formation such as disc--planet interactions and dust dynamics.Comment: 19 pages, 29 figures, accepted in Astronomy & Astrophysic