Approximate Riemann Solvers and Robust High-Order Finite Volume Schemes for Multi-Dimensional Ideal MHD Equations

We design stable and high-order accurate finite volume schemes for the ideal MHD equations in multi-dimensions. We obtain excellent numerical stability due to some new elements in the algorithm. The schemes are based on three- and five-wave approximate Riemann solvers of the HLL-type, with the novelty that we allow a varying normal magnetic field. This is achieved by considering the semi-conservative Godunov-Powell form of the MHD equations. We show that it is important to discretize the Godunov-Powell source term in the right way, and that the HLL-type solvers naturally provide a stable upwind discretization. Second-order versions of the ENO- and WENO-type reconstructions are proposed, together with precise modifications necessary to preserve positive pressure and density. Extending the discrete source term to second order while maintaining stability requires non-standard techniques, which we present. The first- and second-order schemes are tested on a suite of numerical experiments demonstrating impressive numerical resolution as well as stability, even on very fine meshe

Statistical analysis of the mass-to-flux ratio in turbulent cores: effects of magnetic field reversals and dynamo amplification

We study the mass-to-flux ratio (M/\Phi) of clumps and cores in simulations of supersonic, magnetohydrodynamical turbulence for different initial magnetic field strengths. We investigate whether the (M/\Phi)-ratio of core and envelope, R = (M/\Phi)_{core}/(M/\Phi)_{envelope} can be used to distinguish between theories of ambipolar diffusion and turbulence-regulated star formation. We analyse R for different Lines-of-Sight (LoS) in various sub-cubes of our simulation box. We find that, 1) the average and median values of |R| for different times and initial magnetic field strengths are typically greater, but close to unity, 2) the average and median values of |R| saturate at average values of |R| ~ 1 for smaller magnetic fields, 3) values of |R| < 1 for small magnetic fields in the envelope are caused by field reversals when turbulence twists the field lines such that field components in different directions average out. Finally, we propose two mechanisms for generating values |R| ~< 1 for the weak and strong magnetic field limit in the context of a turbulent model. First, in the weak field limit, the small-scale turbulent dynamo leads to a significantly increased flux in the core and we find |R| ~< 1. Second, in the strong field limit, field reversals in the envelope also lead to values |R| ~< 1. These reversals are less likely to occur in the core region where the velocity field is more coherent and the internal velocity dispersion is typically subsonic.Comment: 12 pages, 8 figures, accepted for publication in MNRA

Magnetic fields during the early stages of massive star formation - I. Accretion and disk evolution

We present simulations of collapsing 100 M_\sun mass cores in the context of massive star formation. The effect of variable initial rotational and magnetic energies on the formation of massive stars is studied in detail. We focus on accretion rates and on the question under which conditions massive Keplerian disks can form in the very early evolutionary stage of massive protostars. For this purpose, we perform 12 simulations with different initial conditions extending over a wide range in parameter space. The equations of magnetohydrodynamics (MHD) are solved under the assumption of ideal MHD. We find that the formation of Keplerian disks in the very early stages is suppressed for a mass-to-flux ratio normalised to the critical value \mu below 10, in agreement with a series of low-mass star formation simulations. This is caused by very efficient magnetic braking resulting in a nearly instantaneous removal of angular momentum from the disk. For weak magnetic fields, corresponding to \mu > 10, large-scale, centrifugally supported disks build up with radii exceeding 100 AU. A stability analysis reveals that the disks are supported against gravitationally induced perturbations by the magnetic field and tend to form single stars rather than multiple objects. We find protostellar accretion rates of the order of a few 10^-4 M_\sun yr^-1 which, considering the large range covered by the initial conditions, vary only by a factor of ~ 3 between the different simulations. We attribute this fact to two competing effects of magnetic fields. On the one hand, magnetic braking enhances accretion by removing angular momentum from the disk thus lowering the centrifugal support against gravity. On the other hand, the combined effect of magnetic pressure and magnetic tension counteracts gravity by exerting an outward directed force on the gas in the disk thus reducing the accretion onto the protostars.Comment: 22 pages, 17 figures, accepted for publication in MNRAS, updated to final versio

A method for reconstructing the variance of a 3D physical field from 2D observations: Application to turbulence in the ISM

We introduce and test an expression for calculating the variance of a physical field in three dimensions using only information contained in the two-dimensional projection of the field. The method is general but assumes statistical isotropy. To test the method we apply it to numerical simulations of hydrodynamic and magnetohydrodynamic turbulence in molecular clouds, and demonstrate that it can recover the 3D normalised density variance with ~10% accuracy if the assumption of isotropy is valid. We show that the assumption of isotropy breaks down at low sonic Mach number if the turbulence is sub-Alfvenic. Theoretical predictions suggest that the 3D density variance should increase proportionally to the square of the Mach number of the turbulence. Application of our method will allow this prediction to be tested observationally and therefore constrain a large body of analytic models of star formation that rely on it.Comment: 8 pages, 9 figures, accepted for publication in MNRA

Magnetohydrodynamics on an unstructured moving grid

Magnetic fields play an important role in astrophysics on a wide variety of scales, ranging from the Sun and compact objects to galaxies and galaxy clusters. Here we discuss a novel implementation of ideal magnetohydrodynamics (MHD) in the moving mesh code AREPO which combines many of the advantages of Eulerian and Lagrangian methods in a single computational technique. The employed grid is defined as the Voronoi tessellation of a set of mesh-generating points which can move along with the flow, yielding an automatic adaptivity of the mesh and a substantial reduction of advection errors. Our scheme solves the MHD Riemann problem in the rest frame of the Voronoi interfaces using the HLLD Riemann solver. To satisfy the divergence constraint of the magnetic field in multiple dimensions, the Dedner divergence cleaning method is applied. In a set of standard test problems we show that the new code produces accurate results, and that the divergence of the magnetic field is kept sufficiently small to closely preserve the correct physical solution. We also apply the code to two first application problems, namely supersonic MHD turbulence and the spherical collapse of a magnetized cloud. We verify that the code is able to handle both problems well, demonstrating the applicability of this MHD version of AREPO to a wide range of problems in astrophysics.Comment: 11 pages, 9 figures, accepted by MNRA

A new Jeans resolution criterion for (M)HD simulations of self-gravitating gas: Application to magnetic field amplification by gravity-driven turbulence

Cosmic structure formation is characterized by the complex interplay between gravity, turbulence, and magnetic fields. The processes by which gravitational energy is converted into turbulent and magnetic energies, however, remain poorly understood. Here, we show with high-resolution, adaptive-mesh simulations that MHD turbulence is efficiently driven by extracting energy from the gravitational potential during the collapse of a dense gas cloud. Compressible motions generated during the contraction are converted into solenoidal, turbulent motions, leading to a natural energy ratio of E_sol/E_tot of approximately 2/3. We find that the energy injection scale of gravity-driven turbulence is close to the local Jeans scale. If small seeds of the magnetic field are present, they are amplified exponentially fast via the small-scale dynamo process. The magnetic field grows most efficiently on the smallest scales, for which the stretching, twisting, and folding of field lines, and the turbulent vortices are sufficiently resolved. We find that this scale corresponds to about 30 grid cells in the simulations. We thus suggest a new minimum resolution criterion of 30 cells per Jeans length in (magneto)hydrodynamical simulations of self-gravitating gas, in order to resolve turbulence on the Jeans scale, and to capture minimum dynamo amplification of the magnetic field. Due to numerical diffusion, however, any existing simulation today can at best provide lower limits on the physical growth rates. We conclude that a small, initial magnetic field can grow to dynamically important strength on time scales significantly shorter than the free-fall time of the cloud.Comment: 17 pages, 13 figures, ApJ accepted, more info at http://www.ita.uni-heidelberg.de/~chfeder/pubs/dynamo/dynamo.shtml?lang=e