[en] Many astrophysical applications involve magnetized turbulent flows with shock waves. Ab initio star formation simulations require a robust representation of supersonic turbulence in molecular clouds on a wide range of scales imposing stringent demands on the quality of numerical algorithms. We employ simulations of supersonic super-Alfvenic turbulence decay as a benchmark test problem to assess and compare the performance of nine popular astrophysical MHD methods actively used to model star formation. The set of nine codes includes: ENZO, FLASH, KT-MHD, LL-MHD, PLUTO, PPML, RAMSES, STAGGER, and ZEUS. These applications employ a variety of numerical approaches, including both split and unsplit, finite difference and finite volume, divergence preserving and divergence cleaning, a variety of Riemann solvers, and a range of spatial reconstruction and time integration techniques. We present a comprehensive set of statistical measures designed to quantify the effects of numerical dissipation in these MHD solvers. We compare power spectra for basic fields to determine the effective spectral bandwidth of the methods and rank them based on their relative effective Reynolds numbers. We also compare numerical dissipation for solenoidal and dilatational velocity components to check for possible impacts of the numerics on small-scale density statistics. Finally, we discuss the convergence of various characteristics for the turbulence decay test and the impact of various components of numerical schemes on the accuracy of solutions. The nine codes gave qualitatively the same results, implying that they are all performing reasonably well and are useful for scientific applications. We show that the best performing codes employ a consistently high order of accuracy for spatial reconstruction of the evolved fields, transverse gradient interpolation, conservation law update step, and Lorentz force computation. The best results are achieved with divergence-free evolution of the magnetic field using the constrained transport method and using little to no explicit artificial viscosity. Codes that fall short in one or more of these areas are still useful, but they must compensate for higher numerical dissipation with higher numerical resolution. This paper is the largest, most comprehensive MHD code comparison on an application-like test problem to date. We hope this work will help developers improve their numerical algorithms while helping users to make informed choices about choosing optimal applications for their specific astrophysical problems.

[en] We use deep adaptive mesh refinement simulations of isothermal self-gravitating supersonic turbulence to study the imprints of gravity on the mass density distribution in molecular clouds. The simulations show that the density distribution in self-gravitating clouds develops an extended power-law tail at high densities on top of the usual lognormal. We associate the origin of the tail with self-similar collapse solutions and predict the power index values in the range from -7/4 to -3/2 that agree with both simulations and observations of star-forming molecular clouds.

[en] Highlights: • Improved numerical stability and accuracy for turbulent flows with stochastic forcing. • Nonlinear filter methods are computationally efficient and highly parallelizable. • Desirable methods for exascale simulations involving subsonic turbulence. • Cross-scale energy flux is a robust indicator of spectral bandwidth of a method. Numerical simulations of forced turbulence in compressible fluids are challenging due to the multi-scale nature of the problem and conflicting requirements for numerical methods to accurately resolve the small scales and, at the same time, to handle shock waves and other discontinuities without generating spurious oscillations. Minimizing nonlinear instability and aliasing error while maintaining high accuracy before the simulation reaches the statistically stationary stage is computationally intensive. The goal of this work is to employ an efficient class of high order finite difference nonlinear filter methods for subsonic turbulence simulation with stochastic forcing. The 3D Euler equations for subsonic turbulence with temporally varying stochastic forcing at rms Mach numbers up to 0.6 are numerically solved using the Strang operator splitting of the homogeneous part of the Euler equations and the forcing terms. It was shown in Yee et al. (2013) that the Strang operator splitting is more stable than solving the full Euler equation with the source term included. The spatially seventh-order nonlinear filter methods with adaptive dissipation control developed by Yee & Sjögreen (2007, 2011) are used to solve the homogeneous system and an ODE solver is used to solve the forcing source terms. The nonlinear filter method includes a full time step of a spatially eighth-order central base method, using a third-order TVD Runge-Kutta time integration. The solution computed with the central base method is then nonlinearly filtered by an adaptive flow sensor and the dissipative portion of a seventh-order WENO with the Roe Riemann solver. In order to improve nonlinear stability of the base method without added numerical dissipation, the central base method discretizes the skew-symmetric split form of the inviscid flux derivatives. Both Ducros et al. and Kennedy-Gruber skew-symmetric split forms are tested. Numerical stability, computational efficiency, and effective spectral bandwidth of the nonlinear filter schemes are compared with those of second-order TVD and fifth- and seventh-order WENO methods. It is shown that the nonlinear filter method for this application is substantially more efficient, accurate and yields a superior spectral bandwidth compared to the standard TVD and WENO methods. The nonlinear filter method also demonstrates robust long-time integration for moderately compressible, statistically stationary turbulence with large-scale solenoidal forcing, including small-scale quantities such as enstrophy and mean-square dilatation.

[en] We present results of large-scale three-dimensional weakly magnetized supersonic turbulence simulations with an isothermal equation of state at grid resolutions up to 1024³ cells with the Piecewise Parabolic Method on a Local Stencil. The turbulence is driven by a large-scale isotropic solenoidal force in a periodic computational domain and fully develops in a few flow crossing times. We then evolve the flow for a number of flow crossing times and analyze various statistical properties of the saturated turbulent state. We show that the energy transfer rate in the inertial range of scales is surprisingly close to a constant, indicating that Kolmogorov's phenomenology for incompressible turbulence can be extended to magnetized supersonic flows. We also discuss numerical dissipation effects and convergence of different turbulence diagnostics as grid resolution refines from 256³ to 1024³ cells.

[en] Is the turbulence in cluster-forming regions internally driven by stellar outflows or the consequence of a large-scale turbulent cascade? We address this question by studying the turbulent energy spectrum in NGC 1333. Using synthetic ¹³CO maps computed with a snapshot of a supersonic turbulence simulation, we show that the velocity coordinate spectrum method of Lazarian and Pogosyan provides an accurate estimate of the turbulent energy spectrum. We then apply this method to the ¹³CO map of NGC 1333 from the COMPLETE database. We find that the turbulent energy spectrum is a power law, E(k) ∝ k ^-β, in the range of scales 0.06 pc ≤l ≤ 1.5 pc, with slope β = 1.85 ± 0.04. The estimated energy injection scale of stellar outflows in NGC 1333 is l_inj ∼ 0.3 pc, well resolved by the observations. There is no evidence of the flattening of the energy spectrum above the scale l_inj predicted by outflow-driven simulations and analytical models. The power spectrum of integrated intensity is also a nearly perfect power law in the range of scales 0.16 pc < l< 7.9 pc, with no feature above l_inj. We conclude that the observed turbulence in NGC 1333 does not appear to be driven primarily by stellar outflows.

[en] We examine the effects of self-gravity and magnetic fields on supersonic turbulence in isothermal molecular clouds with high-resolution simulations and adaptive mesh refinement. These simulations use large root grids (512³) to capture turbulence and four levels of refinement to follow the collapse to high densities, for an effective resolution of 8192³. Three Mach 9 simulations are performed, two super-Alfvénic and one trans-Alfvénic. We find that gravity splits the clouds into two populations, one low-density turbulent state and one high-density collapsing state. The low-density state exhibits properties similar to non-self-gravitating in this regime, and we examine the effects of varied magnetic field strength on statistical properties: the density probability distribution function is approximately lognormal, the velocity power spectral slopes decrease with decreasing mean field strength, the alignment between velocity and magnetic field increases with the field, and the magnetic field probability distribution can be fitted to a stretched exponential. The high-density state is well characterized by self-similar spheres: the density probability distribution is a power law, collapse rate decreases with increasing mean field, density power spectra have positive slopes, P(ρ, k)∝k, thermal-to-magnetic pressure ratios are roughly unity for all mean field strengths, dynamic-to-magnetic pressure ratios are larger than unity for all mean field strengths, the magnetic field distribution follows a power-law distribution. The high Alfvén Mach numbers in collapsing regions explain the recent observations of magnetic influence decreasing with density. We also find that the high-density state is typically found in filaments formed by converging flows, consistent with recent Herschel observations. Possible modifications to existing star formation theories are explored. The overall trans-Alfvénic nature of star-forming clouds is discussed.

[en] We study the clustering of inertial particles in turbulent flows and discuss its applications to dust particles in protoplanetary disks. Using numerical simulations, we compute the radial distribution function (RDF), which measures the probability of finding particle pairs at given distances, and the probability density function of the particle concentration. The clustering statistics depend on the Stokes number, St, defined as the ratio of the particle friction timescale, τ_p, to the Kolmogorov timescale in the flow. In agreement with previous studies, we find that, in the dissipation range, the clustering intensity strongly peaks at St ≅ 1, and the RDF for St ∼ 1 shows a fast power-law increase toward small scales, suggesting that turbulent clustering may considerably enhance the particle collision rate. Clustering at inertial-range scales is of particular interest to the problem of planetesimal formation. At these large scales, the strongest clustering is from particles with τ_p in the inertial range. Clustering of these particles occurs primarily around a scale where the eddy turnover time is ∼τ_p. We find that particles of different sizes tend to cluster at different locations, leading to flat RDFs between different particles at small scales. In the presence of multiple particle sizes, the overall clustering strength decreases as the particle size distribution broadens. We discuss particle clustering in two recent models for planetesimal formation. We argue that, in the model based on turbulent clustering of chondrule-size particles, the probability of finding strong clusters that can seed planetesimals may have been significantly overestimated. We discuss various clustering mechanisms in simulations of planetesimal formation by gravitational collapse of dense clumps of meter-size particles, in particular the contribution from turbulent clustering due to the limited numerical resolution.

[en] This paper describes the open-source code Enzo, which uses block-structured adaptive mesh refinement to provide high spatial and temporal resolution for modeling astrophysical fluid flows. The code is Cartesian, can be run in one, two, and three dimensions, and supports a wide variety of physics including hydrodynamics, ideal and non-ideal magnetohydrodynamics, N-body dynamics (and, more broadly, self-gravity of fluids and particles), primordial gas chemistry, optically thin radiative cooling of primordial and metal-enriched plasmas (as well as some optically-thick cooling models), radiation transport, cosmological expansion, and models for star formation and feedback in a cosmological context. In addition to explaining the algorithms implemented, we present solutions for a wide range of test problems, demonstrate the code's parallel performance, and discuss the Enzo collaboration's code development methodology