Alexei G. Kritsuk

The Statistics of Supersonic Isothermal Turbulence

We present results of large-scale three-dimensional simulations of supersonic Euler turbulence with the piecewise parabolic method and multiple grid resolutions up to 2048 3 points. Our numerical experiments describe non-magnetized driven turbulent o ws with an isothermal equation of state and an rms Mach number of 6. We discuss numerical resolution issues and demonstrate convergence, in a statistical sense, of the inertial range dynamics in simulations on grids larger than 512 3 points. The simulations allowed us to measure the absolute velocity scaling exponents for the rst time. The inertial range velocity scaling in this strongly compressible regime deviates substantially from the incompressible Kolmogorov laws. The slope of the velocity power spectrum, for instance, is -1:95 compared to -5=3 in the incompressible case. The exponent of the third-order velocity structure function is 1:28, while in incompressible turbulence it is known to be unity. We propose a natural extension of Kolmogorov’s phenomenology that takes into account compressibility by mixing the velocity and density statistics and preserves the Kolmogorov scaling of the power spectrum and structure functions of the density-weighted velocity v 1=3 u. The low-order statistics of v appear to be invariant with respect to changes in the Mach number. For instance, at Mach 6 the slope of the power spectrum of v v is -1:69, and the exponent of the third-order structure function of v v is unity. We also directly measure the mass dimension of the ifractali density distribution in the inertial subrange, Dm 2:4, which is similar to the observed fractal dimension of molecular clouds and agrees well with the cascade phenomenology. Subject headings: hydrodynamics o instabilities o ISM: structure o methods: numerical o turbulence

ENZO: AN ADAPTIVE MESH REFINEMENT CODE FOR ASTROPHYSICS

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 codes parallel performance, and discuss the Enzo collaborations code development methodology.

Introducing Enzo, an AMR Cosmology Application

In this paper we introduce Enzo, a 3D MPI-parallel Eulerian block-structured adaptive mesh refinement cosmology code. Enzo is designed to simulate cosmological structure formation, but can also be used to simulate a wide range of astrophysical situations. Enzo solves dark matter N-body dynamics using the particle-mesh technique. The Poisson equation is solved using a combination of fast fourier transform (on a periodic root grid) and multigrid techniques (on non-periodic subgrids). Euler’s equations of hydrodynamics are solved using a modified version of the piecewise parabolic method. Several additional physics packages are implemented in the code, including several varieties of radiative cooling, a metagalactic ultraviolet background, and prescriptions for star formation and feedback. We also show results illustrating properties of the adaptive mesh portion of the code. Information on profiling and optimizing the performance of the code can be found in the contribution by James Bordner in this volume.

On the Density Distribution in Star-forming Interstellar Clouds

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.

Two regimes of turbulent fragmentation and the stellar initial mass function from primordial to present-day star formation

The Padoan and Nordlund model of the stellar initial mass function (IMF) is derived from low-order statistics of supersonic turbulence, neglecting gravity (e.g., gravitational fragmentation, accretion, and merging). In this work, the predictions of that model are tested using the largest numerical experiments of supersonic hydrodynamic (HD) and magnetohydrodynamic (MHD) turbulence to date (~10003 computational zones) and three different codes (Enzo, Zeus, and the Stagger code). The model predicts a power-law distribution for large masses, related to the turbulence-energy power-spectrum slope and the shock-jump conditions. This power-law mass distribution is confirmed by the numerical experiments. The model also predicts a sharp difference between the HD and MHD regimes, which is recovered in the experiments as well, implying that the magnetic field, even below energy equipartition on the large scale, is a crucial component of the process of turbulent fragmentation. These results suggest that the stellar IMF of primordial stars may differ from that in later epochs of star formation, due to differences in both gas temperature and magnetic field strength. In particular, we find that the IMF of primordial stars born in turbulent clouds may be narrowly peaked around a mass of order 10 M☉, as long as the column density of such clouds is not much in excess of 1022 cm-2.

COMPARING NUMERICAL METHODS FOR ISOTHERMAL MAGNETIZED SUPERSONIC TURBULENCE

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.

THE TWO STATES OF STAR-FORMING CLOUDS

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-Alfv?nic and one trans-Alfv?nic. 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(?, 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 Alfv?n 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-Alfv?nic nature of star-forming clouds is discussed.

The Power Spectrum of Turbulence in NGC 1333: Outflows or Large-Scale Driving?

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 & 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 ??, in the range of scales 0.06 pc ?? ? 1.5 pc, with slope ? = 1.85 ? 0.04. The estimated energy injection scale of stellar outflows in NGC 1333 is ?inj 0.3 pc, well resolved by the observations. There is no evidence of the flattening of the energy spectrum above the scale ?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 <?< 7.9 pc, with no feature above ?inj. We conclude that the observed turbulence in NGC 1333 does not appear to be driven primarily by stellar outflows.

A supersonic turbulence origin of Larson's laws

We revisit the origin of Larsons scaling laws describing the structure and kinematics of molecular clouds. Our analysis is based on recent observational measurements and data from a suite of six simulations of the interstellar medium, including effects of self-gravity, turbulence, magnetic field, and multiphase thermodynamics. Simulations of isothermal supersonic turbulence reproduce observed slopes in linewidth-size and mass-size relations. Whether or not self-gravity is included, the linewidth-size relation remains the same. The mass-size relation, instead, substantially flattens below the sonic scale, as prestellar cores start to form. Our multiphase models with magnetic field and domain size 200 pc reproduce both scaling and normalization of the first Larson law. The simulations support a turbulent interpretation of Larsons relations. This interpretation implies that: (i) the slopes of linewidth-size and mass-size correlations are determined by the inertial cascade; (ii) none of the three Larson laws is fundamental; (iii) instead, if one is known, the other two follow from scale invariance of the kinetic energy transfer rate. It does not imply that gravity is dynamically unimportant. The self-similarity of structure established by the turbulence breaks in star-forming clouds due to the development of gravitational instability in the vicinity of the sonic scale. The instability leads to the formation of prestellar cores with the characteristic mass set by the sonic scale. The high-end slope of the core mass function predicted by the scaling relations is consistent with the Salpeter power-law index.

The power spectrum of supersonic turbulence in perseus

We test a method of estimating the power spectrum of turbulence in molecular clouds based on the comparison of power spectra of integrated intensity maps and single-velocity-channel maps, suggested by A. Lazarian and D. Pogosyan. We use synthetic 13CO data from non-LTE radiative transfer calculations based on density and velocity fields of a simulation of supersonic hydrodynamic turbulence. We find that the method yields the correct power spectrum with good accuracy. We then apply the method to the Five College Radio Astronomy Observatory 13CO map of the Perseus region, from the COMPLETE Web site. We find a power-law power spectrum with slope β = 1.81 ± 0.10. The values of β as a function of velocity resolution are also confirmed using the lower resolution map of the same region obtained with the AT&T Bell Laboratories antenna. Because of its small uncertainty, this result provides a useful constraint for numerical codes used to simulate molecular cloud turbulence.