The statistical properties of turbulence are considered to be universal at sufficiently small length scales, i. e., independent of boundary conditions and large-scale forces acting on the fluid. Analyzing data from numerical simulations of supersonic turbulent flow driven by external forcing, we demonstrate that this is not generally true for the two-point velocity statistics of compressible turbulence. However, a reformulation of the refined similarity hypothesis in terms of the mass-weighted velocity rho^(1/3)v yields scaling laws that are almost insensitive to the forcing. The results imply that the most intermittent dissipative structures are shocks closely following the scaling of Burgers turbulence.
We show that density-weighted moments of the dissipation rate, $ε_l$, averaged over a scale $l$, in supersonic turbulence can be successfully explained by the She and Lévêque model [Phys. Rev. Lett. {\bf 72}, 336 (1994)]. A general method is developed to measure the two parameters of the model, $γ$ and $d$, based directly on their physical interpretations as the scaling exponent of the dissipation rate in the most intermittent structures ($γ$) and the dimension of the structures ($d$). We find that the best-fit parameters ($γ=0.71$ and $d=1.90$) derived from the $ε_l$ scalings in a simulation …
We present the results of three-dimensional simulations of supersonic Euler turbulence with grid resolutions up to 1024^3 points. Our numerical experiments describe nonmagnetized driven turbulent flows with an isothermal equation of state and an rms Mach number of 6. We demonstrate that the inertial range scaling properties of turbulence in this strongly compressible regime deviate substantially from a Kolmogorov-like behavior previously recovered for mildly compressible transonic flows.
Jets and outflows from young stellar objects are proposed candidates to drive supersonic turbulence in molecular clouds. Here, we present the results from multi-dimensional jet simulations where we investigate in detail the energy and momentum deposition from jets into their surrounding environment and quantify the character of the excited turbulence with velocity probability density functions. Our study include jet--clump interaction, transient jets, and magnetised jets. We find that collimated supersonic jets do not excite supersonic motions far from the vicinity of the jet. Supersonic fluct…
Supersonic turbulence generates distributions of shock waves. Here, we analyse the shock waves in three-dimensional numerical simulations of uniformly driven supersonic turbulence, with and without magnetohydrodynamics and self-gravity. We can identify the nature of the turbulence by measuring the distribution of the shock strengths. We find that uniformly driven turbulence possesses a power law distribution of fast shocks with the number of shocks inversely proportional to the square root of the shock jump speed. A tail of high speed shocks steeper than Gaussian results from the random superp…
Understanding the formation of stars in galaxies is central to much of modern astrophysics. For several decades it has been thought that stellar birth is primarily controlled by the interplay between gravity and magnetostatic support, modulated by ambipolar diffusion. Recently, however, both observational and numerical work has begun to suggest that support by supersonic turbulence rather than magnetic fields controls star formation. In this review we outline a new theory of star formation relying on the control by turbulence. We demonstrate that although supersonic turbulence can provide glob…
We performed a series of three-dimensional numerical simulations of supersonic homogeneous Euler turbulence with adaptive mesh refinement (AMR) and effective grid resolution up to 1024^3 zones. Our experiments describe non-magnetized driven supersonic turbulent flows with an isothermal equation of state. Mesh refinement on shocks and shear is implemented to cover dynamically important structures with the highest resolution subgrids and calibrated to match the turbulence statistics obtained from the equivalent uniform grid simulations. We found that at a level of resolution slightly below 512^3…
We study the statistics of density for supersonic turbulence in a medium with magnetic pressure larger than the gaseous pressure. This study is motivated by molecular cloud research. Our simulations exhibit clumpy density structures, which contrast increases with the Mach number. At 10 Machs densities of some clumps are three orders of magnitude higher than the mean density. These clumps give rise to flat and approximately isotropic density spectrum corresponding to the random distribution of clumps in space. We claim that the clumps originate from our random, isotropic turbulence driving. Whe…
Lada et al. (1994) have described a method for studying the distribution of dust in dark clouds using infrared imaging surveys. In particular they show that the method provides some information about the structure of the gas (dust) on scales smaller than their resolution. In the present work we clarify the nature of the information provided by their method. We show that: 1) the 3D density field of the gas is well described by a Log-Normal distribution down to very small scales; 2) the power spectrum and the standard deviation of the 3D density field can be constrained; 3) the origin of such a …
We study the probability distribution function (PDF) of the mass density in simulations of supersonic turbulence with properties appropriate for molecular clouds. For this study we use Athena, a new higher-order Godunov code. We find there are surprisingly similar relationships between the mean of the time-averaged PDF and the turbulent Mach number for driven hydrodynamic and strong-field MHD turbulence. There is, however, a large scatter about these relations, indicating a high level of temporal and spatial variability in the PDF. Thus, the PDF of the mass density is unlikely to be a good mea…
Molecular clouds have broad linewidths suggesting turbulent supersonic motions in the clouds. These motions are usually invoked to explain why molecular clouds take much longer than a free-fall time to form stars. It has classically been thought that supersonic hydrodynamical turbulence would dissipate its energy quickly, but that the introduction of strong magnetic fields could maintain these motions. In a previous paper it has been shown, however, that isothermal, compressible, MHD and hydrodynamical turbulence decay at virtually the same rate, requiring that constant driving occur to mainta…
We study energy dissipation and heating by supersonic MHD turbulence in molecular clouds using Athena, a new higher-order Godunov code. We analyze the dependence of the saturation amplitude, energy dissipation characteristics, power spectra, sonic scaling, and indicators of intermittency in the turbulence on factors such as the magnetic field strength, driving scale, energy injection rate, and numerical resolution. While convergence in the energies is reached at moderate resolutions, we find that the power spectra require much higher resolutions that are difficult to obtain. In a 1024^3 hydro …
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 Lazarian and 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 website. We…
We use unpublished and published VLBI results to investigate the geometry and the statistical properties of the velocity field traced by H2O masers in five galactic regions of star formation -- Sgr B2(M), W49N, W51(MAIN), W51N, and W3(OH). In all sources the angular distribution of the H2O hot spots demonstrates approximate self-similarity (fractality) over almost four orders of magnitude in scale, with the calculated fractal dimension d between (approximately) 0.2 and 1.0. In all sources, the lower order structure functions for the line-of-sight component of the velocity field are satisfactor…
The interstellar medium (ISM) provides a unique laboratory for highly supersonic, driven hydrodynamics turbulence. We present a theory of such turbulence, confirm it by numerical simulations, and use the results to explain observational properties of interstellar molecular clouds, the regions where stars are born.
We here analyse numerical simulations of supersonic, hypersonic and magnetohydrodynamic turbulence that is free to decay. Our goals are to understand the dynamics of the decay and the characteristic properties of the shock waves produced. This will be useful for interpretation of observations of both motions in molecular clouds and sources of non-thermal radiation. We find that decaying hypersonic turbulence possesses an exponential tail of fast shocks and an exponential decay in time, i.e. the number of shocks is proportional to t exp (-ktv) for shock velocity jump v and mean initial wavenumb…
Contradicting results have been reported in the literature with respect to the performance of the numerical techniques employed for the study of supersonic turbulence. We aim at characterising the performance of different particle-based and grid-based techniques on the modelling of decaying supersonic turbulence. Four different grid codes (ENZO, FLASH, TVD, ZEUS) and three different SPH codes (GADGET, PHANTOM, VINE) are compared. We additionally analysed two calculations denoted as PHANTOM A and PHANTOM B using two different implementations of artificial viscosity. Our analysis indicates that …
Modeling the structure of molecular clouds depends on good methods to statistically compare simulations with observations in order to constrain the models. Here we characterize a suite of hydrodynamical and magnetohydrodynamical (MHD) simulations of supersonic turbulence using an averaged wavelet transform, the Delta-variance, that has been successfully used to characterize observations. We find that, independent of numerical resolution and dissipation, the only models that produce scale-free, power-law Delta-variance spectra are those with hypersonic Mach numbers above M ~ 4, while slower sup…
The question of the shape of the density PDF for supersonic turbulence is addressed, using both analytical and numerical methods. For isothermal supersonic turbulence, the PDF is Log-Normal, with a width that scales approximately linearly with the Mach number. For a polytropic equation of state, with an effective gamma smaller than one, the PDF becomes skewed and reminiscent of (but not equal to) a power law on the high density side.
We investigate numerically the combined effects of supersonic turbulence, strong magnetic fields and ambipolar diffusion on cloud evolution leading to star formation. We find that, in clouds that are initially magnetically subcritical, supersonic turbulence can speed up star formation, through enhanced ambipolar diffusion in shocks. The speedup overcomes a major objection to the standard scenario of low-mass star formation involving ambipolar diffusion, since the diffusion time scale at the average density of a molecular cloud is typically longer than the cloud life time. At the same time, the…
We review how supersonic turbulence can both prevent and promote the collapse of molecular clouds into stars. First we show that decaying turbulence cannot significantly delay collapse under conditions typical of molecular clouds, regardless of magnetic field strength so long as the fields are not supporting the cloud magnetohydrostatically. Then we review possible drivers and examine simulations of driven supersonic and trans Alfvénic turbulence, finally including the effects of self-gravity. Our preliminary results show that, although turbulence can support regions against gravitational coll…
We present a direct numerical and analytical study of driven supersonic MHD turbulence that is believed to govern the dynamics of star-forming molecular clouds. We describe statistical properties of the turbulence by measuring the velocity difference structure functions up to the fifth order. In particular, the velocity power spectrum in the inertial range is found to be close to E(k) \~ k^{-1.74}, and the velocity difference scales as <|Δu|>~ L^{0.42}. The results agree well with the Kolmogorov--Burgers analytical model suggested for supersonic turbulence in [astro-ph/0108300]. We then genera…
(Modified) The scaling of velocity fluctuation, dv, as a function of spatial scale L in molecular clouds can be measured from size-linewidth relations, principal component analysis, or line centroid variation. Differing values of the power law index of the scaling relation dv = L^(g3D) in 3D are given by these different methods: the first two give g3D=0.5, while line centroid analysis gives g3D=0. This discrepancy has previously not been fully appreciated, as the variation of projected velocity line centroid fluctuations (dv_{lc} = L^(g2D)) is indeed described, in 2D, by g2D=0.5. However, if p…
We present a detailed computational study of the assembly of protostellar disks and massive stars in molecular clouds with supersonic turbulence. We follow the evolution of large scale filamentary structures in a cluster-forming clump down to protostellar length scales by means of very highly resolved, 3D adaptive mesh refined (AMR) simulations, and show how accretion disks and massive stars form in such environments. We find that an initially elongated cloud core which has a slight spin from oblique shocks collapses first to a filament and later develops a turbulent disk close to the center o…
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 flows 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 first time. The iner…
We present numerical studies of compressible, decaying turbulence, with and without magnetic fields, with initial rms Alfven and Mach numbers ranging up to five, and apply the results to the question of the support of star-forming interstellar clouds of molecular gas. We find that, in 1D, magnetized turbulence actually decays faster than unmagnetized turbulence. In all the regimes that we have studied 3D turbulence-super-Alfvenic, supersonic, sub-Alfvenic, and subsonic-the kinetic energy decays as (t-t0)^(-x), with 0.85 < x < 1.2. We compared results from two entirely different algorithms in t…
We present high resolution numerical simulations of compressible magnetohydrodynamic (MHD) turbulence. We concentrate on studies of spectra and anisotropy of velocity and density. We describe a technique of separating different magnetohydrodynamic (MHD) modes (slow, fast and Alfven) and apply it to our simulations. For the mildly supersonic case, velocity and density show anisotropy. However, for the highly supersonic case, we observe steepening of the slow mode velocity spectrum and isotropization of density. Our present studies show that the spectrum of density gets substantially {\it flatte…
In a systematic study, we compare the density statistics in high resolution numerical experiments of supersonic isothermal turbulence, driven by the usually adopted solenoidal (divergence-free) forcing and by compressive (curl-free) forcing. We find that for the same rms Mach number, compressive forcing produces much stronger density enhancements and larger voids compared to solenoidal forcing. Consequently, the Fourier spectra of density fluctuations are significantly steeper. This result is confirmed using the Delta-variance analysis, which yields power-law exponents beta~3.4 for compressive…
The role played by velocity fields in the fragmentation of a cold medium and in the formation of protostars is studied. The velocity field is modeled with a compressible turbulent flow. A supersonic turbulent velocity field can fragment the medium into clumps of mass smaller than a local Jeans' mass, and therefore stabilize the medium against the formation of protostars. Based on this idea, the protostar formation efficiency and the protostar mass distribution are determined as functions of the following ambient parameters: average density $n_{0}$, average temperature $T_{0}$, r.m.s. turbulent…
We employ a generalization of the She & Leveque model to study velocity scaling relations based on our simulations of thermal instability--induced turbulence. Being a by-product of the interstellar phase transition, such multiphase turbulence tends to be more intermittent than compressible isothermal turbulence. Due to radiative cooling, which promotes nonlinear instabilities in supersonic flows, the Hausdorff dimension of the most singular dissipative structures, D, can be as high as 2.3, while in supersonic isothermal turbulence D is limited by shock dissipation to D < 2. We also show that s…
Aim: To study turbulence in the Orion Molecular Cloud (OMC1) by comparing observed and simulated characteristics of the gas motions. Method: Using a dataset of vibrationally excited H2 emission in OMC1 containing radial velocity and brightness which covers scales from 70AU to 30000AU, we present the transversal structure functions and the scaling of the structure functions with their order. These are compared with the predictions of two-dimensional projections of simulations of supersonic hydrodynamic turbulence. Results: The structure functions of OMC1 are not well represented by power laws, …
The observational record of turbulence within the molecular gas phase of the interstellar medium is summarized. We briefly review the analysis methods used to recover the velocity structure function from spectroscopic imaging and the application of these tools on sets of cloud data. These studies identify a near-invariant velocity structure function that is independent of local the environment and star formation activity. Such universality accounts for the cloud-to-cloud scaling law between the global line-width and size of molecular clouds found by Larson (1981) and constrains the degree to w…
Turbulence is a crucial component of dynamics of astrophysical fluids dynamics, including those of ISM, clusters of galaxies and circumstellar regions. Doppler shifted spectral lines provide a unique source of information on turbulent velocities. We discuss Velocity-Channel Analysis (VCA) and its offspring Velocity Coordinate Spectrum (VCS) that are based on the analytical description of the spectral line statistics. Those techniques are well suited for studies of supersonic turbulence. We stress that a great advantage of VCS is that it does not necessary require good spatial resolution. Addre…
We compute the structure function scaling of a 2MASS extinction map of the Taurus molecular cloud complex. The scaling exponents of the structure functions of the extinction map follow the Boldyrev's velocity structure function scaling of supersonic turbulence. This confirms our previous result based on a spectral map of 13CO J=1-0 covering the same region and suggests that supersonic turbulence is important in the fragmentation of this star--forming cloud.
Supersonic turbulence fragments the interstellar medium into dense sheets, filaments, cores and large low density voids. The turbulence is driven on large scales, probably predominantly by supernovae. The scaling properties of supersonic turbulence are well described by a new analytical theory. The distribution of proto-stellar core masses depends primarily on the power spectrum of the turbulent flow, and on the jump conditions for near isothermal shocks in a magnetized gas. For the predicted velocity power spectrum index beta=1.74, consistent with results of numerical experiments of supersoni…
Turbulence is essential for understanding the structure and dynamics of molecular clouds and star-forming regions. There is a need for adequate tools to describe and characterize the properties of turbulent flows. One-point probability distribution functions (pdf's) of dynamical variables have been suggested as appropriate statistical measures and applied to several observed molecular clouds. However, the interpretation of these data requires comparison with numerical simulations. To address this issue, SPH simulations of driven and decaying, supersonic, turbulent flows with and without self-g…
Turbulence is a crucial component of dynamics of astrophysical fluids dynamics, including those of ISM, clusters of galaxies and circumstellar regions. Doppler shifted spectral lines provide a unique source of information on turbulent velocities. We discuss Velocity-Channel Analysis (VCA) and its offspring Velocity Coordinate Spectrum (VCS) that are based on the analytical description of the spectral line statistics. Those techniques are well suited for studies of supersonic turbulence. We stress that a great advantage of VCS is that it does not necessary require good spatial resolution. Addre…
Colliding hypersonic flows play a decisive role in many astrophysical objects. In this paper, we look at the idealized model of a 2D plane parallel isothermal slab (CDL) and at symmetric settings, where both flows have equal parameters. We performed a set of high-resolution simulations with upwind Mach numbers, 5 < M_u < 90. We find that the CDL is irregularly shaped and has a patchy and filamentary interior. The size of these structures increases with l_cdl, the extension of the CDL. On average, but not at each moment, the solution is about self-similar and depends only on M_u. We find the ro…
We investigate diffusion in supersonic, turbulent, compressible flows. Supersonic turbulence can be characterized as network of interacting shocks. We consider flows with different rms Mach numbers and where energy necessary to maintain dynamical equilibrium is inserted at different spatial scales. We find that turbulent transport exhibits super-diffusive behavior due to induced bulk motions. In a comoving reference frame, however, diffusion behaves normal and can be described by mixing length theory extended into the supersonic regime.
The probability density function (PDF) of the gas density in turbulent supersonic flows is investigated with high-resolution numerical simulations. In a systematic study, we compare the density statistics of compressible turbulence driven by the usually adopted solenoidal forcing (divergence-free) and by compressive forcing (curl-free). Our results are in agreement with studies using solenoidal forcing. However, compressive forcing yields a significantly broader density distribution with standard deviation ~3 times larger at the same rms Mach number. The standard deviation-Mach number relation…
