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 analyzed with high temporal resolution. There is overall consistency between the codes. The density profiles agree within about 5% for the Eulerian simulations while 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%. The spread is larger in the tidal torques acting on the planet which agree within a factor 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.Comment: 32 pages, accepted for publication in MNRA

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.

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.

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.

We calculate the ionization fraction of protostellar α discs, taking into account vertical temperature structure and the possible presence of trace metal atoms. Both thermal and X‐ray ionization are considered. Previous investigations of layered discs used radial power‐law models with isothermal vertical structure. But α models are used to model accretion, and the present work is a step towards a self‐consistent treatment. The extent of the magnetically uncoupled (‘dead’) zone depends sensitively on α, on the assumed accretion rate and on the critical magnetic Reynolds number, below which magnetohydrodynamic (MHD) turbulence cannot be self‐sustained. Its extent is extremely model‐dependent. It is also shown that a tiny fraction of the cosmic abundance of metal atoms can dramatically affect the ionization balance. Gravitational instabilities are an unpromising source of transport, except in the early stages of disc formation.

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

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 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. 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.

Aims. We study the properties of MHD turbulence driven by the magnetorotational instability (MRI) in accretion disks. To do this we perform a series of numerical simulations for which the resolution is gradually increased. Methods. 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. Results. Performing a set of numerical simulations in a fixed computational domain with increasing resolution, we demonstrate that turbulent activity decreases as resolution increases. The highest resolution considered is 256 grid cells per scale height. We quantify the turbulent activity by measuring the rate of angular momentum transport through evaluating the standard α parameter. We find α = 0.004 when (N x , N y , N z ) = (64, 100, 64), α = 0.002 when (N x , N y , N z ) = (128, 200, 128) and α = 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. Conclusions. 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.