Sebastien Fromang

A comparative study of disc–planet interaction

We perform numerical simulations of a disc-planet system using various grid-based and smoothed particle hydrodynamics (SPH) codes. The tests are run for a simple setup where Jupiter and Neptune mass planets on a circular orbit open a gap in a protoplanetary disc during a few hundred orbital periods. We compare the surface density contours, potential vorticity and smoothed radial profiles at several times. The disc mass and gravitational torque time evolution are analysed with high temporal resolution. There is overall consistency between the codes. The density profiles agree within about 5 per cent for the Eulerian simulations. The SPH results predict the correct shape of the gap although have less resolution in the low-density regions and weaker planetary wakes. The disc masses after 200 orbital periods agree within 10 per cent. The spread is larger in the tidal torques acting on the planet which agree within a factor of 2 at the end of the simulation. In the Neptune case, the dispersion in the torques is greater than for Jupiter, possibly owing to the contribution from the not completely cleared region close to the planet.

A high order Godunov scheme with constrained transport and adaptive mesh refinement for astrophysical magnetohydrodynamics

Aims. In this paper, we present a new method to perform numerical simulations of astrophysical MHD flows using the Adaptive Mesh Refinement framework and Constrained Transport. Methods. The algorithm is based on a previous work in which the MUSCL-Hancock scheme was used to evolve the induction equation. In this paper, we detail the extension of this scheme to the full MHD equations and discuss its properties. Results. Through a series of test problems, we illustrate the performances of this new code using two different MHD Riemann solvers (Lax-Friedrich and Roe) and the need of the Adaptive Mesh Refinement capabilities in some cases. Finally, we show its versatility by applying it to two completely different astrophysical situations well studied in the past years: the growth of the magnetorotational instability in the shearing box and the collapse of magnetized cloud cores. Conclusions. We have implemented a new Godunov scheme to solve the ideal MHD equations in the AMR code RAMSES. We have shown that it results in a powerful tool that can be applied to a great variety of astrophysical problems, ranging from galaxies formation in the early universe to high resolution studies of molecular cloud collapse in our galaxy.

Magnetic processes in a collapsing dense core I. Accretion and ejection

Context. To understand the star formation process, it is important to study the collapse of a prestellar dense core. Aims. We investigate the effect of the magnetic field during the first collapse up to the formation of the first core, focusing particularly on the magnetic braking and the launching of outflows. Methods. We perform 3D AMR high resolution numerical simulations of a magnetically supercritical collapsing dense core using the RAMSES MHD code and develop semi-analytical models that we compare with the numerical results. Results. We study in detail the various profiles within the envelope of the collapsing core for various magnetic field strengths. Even modest values of magnetic field strength modify the collapse significantly. This is largely due to the amplification of the radial and toroidal components of the magnetic field by the differential motions within the collapsing core. For a weak magnetic intensity corresponding to an initial mass-to-flux over critical mass-to-flux ratio, μ equals 20 a centrifugally supported disk forms. The strong differential rotation triggers the growth of a slowly expanding magnetic tower. For higher magnetic field strengths corresponding to μ = 2, the collapse occurs primarily along the field lines, therefore delivering weaker angular momentum into the inner part whereas at the same time, strong magnetic braking occurs. As a consequence no centrifugally supported disk forms. An outflow is launched from the central thermally supported core. Detailed comparisons with existing analytical predictions indicate that it is magneto-centrifugally driven. Conclusions. For cores having a mass-to-flux over critical mass-to-flux radio μ < 5, the magnetic field appears to have a significant impact. The collapsing envelope is denser and flatter than in the hydrodynamical case and no centrifugally supported disk forms. For values μ < 20, the magnetic field drastically modifies the disk evolution. In a companion paper, the influence of the magnetic field on the dense core fragmentation is studied.

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

Aims. 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. Methods. 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 operator split 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. Results. 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 Pmc 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 Pmc > 1. Conclusions. 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.

Thanatology in protoplanetary discs - The combined influence of Ohmic, Hall, and ambipolar diffusion on dead zones

Protoplanetary discs are poorly ionised due to their low temperatures and high column densities and are therefore subject to three “non-ideal” magnetohydrodynamic (MHD) effects: Ohmic dissipation, ambipolar diffusion, and the Hall effect. The existence of magnetically driven turbulence in these discs has been a central question since the discovery of the magnetorotational instability (MRI). Early models considered Ohmic diffusion only and led to a scenario of layered accretion, in which a magnetically “dead” zone in the disc midplane is embedded within magnetically “active” surface layers at distances of about 1–10 au from the central protostellar object. Recent work has suggested that a combination of Ohmic dissipation and ambipolar diffusion can render both the midplane and surface layers of the disc inactive and that torques due to magnetically driven outflows are required to explain the observed accretion rates. We reassess this picture by performing three-dimensional numerical simulations that include all three non-ideal MHD effects for the first time. We find that the Hall effect can generically “revive” dead zones by producing a dominant azimuthal magnetic field and a large-scale Maxwell stress throughout the midplane, provided that the angular velocity and magnetic field satisfy Ω·B > 0. The attendant large magnetic pressure modifies the vertical density profile and substantially increases the disc scale height beyond its hydrostatic value. Outflows are produced but are not necessary to explain accretion rates ≲ 10-7 M⊙ yr-1. The flow in the disc midplane is essentially laminar, suggesting that dust sedimentation may be efficient. These results demonstrate that if the MRI is relevant for driving mass accretion in protoplanetary discs, one must include the Hall effect to obtain even qualitatively correct results.

Dust settling in local simulations of turbulent protoplanetary disks

Aims. 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. Methods. 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. Results. 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. Conclusions. 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.

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

Aims. 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. Methods. 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 12 500 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. Results. The rate of angular momentum transport, measured using the α parameter, depends only very weakly on the Reynolds number: α is found to be about 7 × 10- 3 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. Conclusions. 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.

Local outflows from turbulent accretion disks

Aims. The aim of this paper is to investigate the properties of accretion disks threaded by a weak vertical magnetic field, with a particular focus on the interplay between magnetohydrodynamic (MHD) turbulence driven by the magnetorotational instability (MRI) and outflows that might be launched from the disk. Methods. For that purpose, we use a set of numerical simulations performed with the MHD code RAMSES in the framework of the shearing box model. We concentrate on the case of a rather weak vertical magnetic field such that the initial ratio β0 of the thermal and magnetic pressures in the disk midplane equals 10 4 . Results. As reported recently, we find that MHD turbulence drives an efficient outflow out of the computational box. We demonstrate a strong sensitivity of that result to the box size: enlargements in the radial and vertical directions lead to a reduction of up to an order of magnitude in the mass-loss rate. Such a dependence prevents any realistic estimates of disk mass-loss rates being derived using shearing-box simulations. We find however that the flow morphology is robust and independent of the numerical details of the simulations. Its properties display some features and approximate invariants that are reminiscent of the Blandford & Payne launching mechanism, but differences exist. For the magnetic field strength considered in this paper, we also find that angular momentum transport is most likely dominated by MHD turbulence, the saturation of which scales with the magnetic Prandtl number, the ratio of viscosity and resistivity, in a way that is in good agreement with expectations based on unstratified simulations. Conclusions. This paper thus demonstrates for the first time that accretion disks can simultaneously exhibit MRI-driven MHD turbulence along with magneto-centrifugally accelerated outflows. However, in contradiction with previously published results, such outflows probably have little impact on the disk dynamics.

On the accumulation of solid bodies in global turbulent protoplanetary disc models

We study the migration of solid bodies in turbulent protoplanetary accretion discs by means of global magnetohydrodynamic simulations. The bodies range in size from 5 cm up to 1 m, and so include objects whose migration is expected to be the most rapid due to gas drag interaction with the disc. As they drift inward through the disc, some of them are trapped in regions where gas pressure maxima are created by long-lived anticyclonic vortices. This accumulation is very efficient, locally increasing the dust-to-gas ratio by a factor >100 in some cases. We discuss the possible implications of this result for theories of planet formation.

Turbulent resistivity driven by the magnetorotational instability

Aims. We measure the turbulent resistivity in the nonlinear regime of the MRI, and evaluate the turbulent magnetic Prandtl number. Methods. We perform a set of numerical simulations with the Eulerian finite volume codes Athena and Ramses in the framework of the shearing box model. We consider models including explicit dissipation coefficients and magnetic field topologies such that the net magnetic flux threading the box in both the vertical and azimuthal directions vanishes. Results. We first demonstrate good agreement between the two codes by comparing the properties of the turbulent states in simulations having identical microscopic diffusion coefficients (viscosity and resistivity). We find the properties of the turbulence do not change when the box size is increased in the radial direction, provided it is elongated in the azimuthal direction. To measure the turbulent resistivity in the disk, we impose a fixed electromotive force on the flow and measure the amplitude of the saturated magnetic field that results. We obtain a turbulent resistivity that is in rough agreement with mean field theories like the Second Order Smoothing Approximation. The numerical value translates into a turbulent magnetic Prandtl number Pmt of order unity. Pmt appears to be an increasing function of the forcing we impose. It also becomes smaller as the box size is increased in the radial direction, in good agreement with previous results obtained in very large boxes. Conclusions. Our results are in general agreement with other recently published papers studying the same problem but using different methodology. Thus, our conclusion that Pmt is of order unity appears robust.