Simulations Of Supersonic Turbulence Research Articles

An exact relation for density fluctuations in compressible turbulence

ABSTRACT We derive an exact relation for density fluctuations in statistically stationary compressible turbulence. In weakly compressible turbulence, the relation identifies two contributions, corresponding to the acoustic mode and the pseudo-sound mode, respectively, to the density power spectrum, providing a unifying picture for the origin of density fluctuations in turbulent flows with Mach number ≲ 1. Using numerical simulations of driven turbulence, we verified the validity of the exact relations, and examined the contributions of the acoustic and pseudo-sound terms as a function of the Mach number. For simulations of supersonic turbulence, the exact relations provide a tool to quantify the artificial reduction of the density variance by numerical viscosity. The artificial suppression of density fluctuations increases with increasing Mach number, due to the necessity of applying larger numerical diffusion to stabilize stronger shocks. The exact relation also helps to theoretically establish the relation between the density variance and the density-weighted Mach number, $\langle \delta \rho ^2 \rangle /\bar{\rho }^2 = b^2 \mathcal {M}_{\rho }^2$, in supersonic turbulence, which is of crucial importance for the modelling of star formation. Combining with simulations of supersonic turbulence with solenoidal driving and removing the artificial suppression of density fluctuations by numerical viscosity, our exact relation gives an estimate of b ≃ 0.4 for the b parameter.

Inaccuracy of Spatial Derivatives in Riemann Solver Simulations of Supersonic Turbulence

Abstract We examine the accuracy of spatial derivatives computed from numerical simulations of supersonic turbulence. Two sets of simulations, carried out using a finite-volume code that evolves the hydrodynamic equations with an approximate Riemann solver and a finite-difference code that solves the Navier–Stokes (N–S) equations, are tested against a number of criteria based on the continuity equation, including exact results at statistically steady state. We find that the spatial derivatives in the N–S runs are accurate and satisfy all the criteria. In particular, they satisfy our exact results that, at steady state, the average of the velocity divergence conditioned on the flow density and the conditional average of the advection of density both vanish at all density levels. On the other hand, the Riemann solver simulations fail all the tests that require accurate evaluation of spatial derivatives, resulting in apparent violation of the continuity equation, even if the solver enforces mass conservation. In particular, analysis of the Riemann simulations may lead to the incorrect conclusion that the work tends to preferentially convert kinetic energy into thermal energy, which is inconsistent with the exact result that the energy exchange by work is symmetric in barotropic supersonic turbulence at steady state. The inaccuracy of spatial derivatives is a general problem in the post-processing of simulations of supersonic turbulence with Riemann solvers. Solutions from such simulations must be used with caution in post-processing studies concerning the spatial gradients.

Scaling laws of passive-scalar diffusion in the interstellar medium

Passive scalar mixing (metals, molecules, etc.) in the turbulent interstellar medium (ISM) is critical for abundance patterns of stars and clusters, galaxy and star formation, and cooling from the circumgalactic medium. However, the fundamental scaling laws remain poorly understood in the highly supersonic, magnetized, shearing regime relevant for the ISM. We therefore study the full scaling laws governing passive-scalar transport in idealized simulations of supersonic turbulence. Using simple phenomenological arguments for the variation of diffusivity with scale based on Richardson diffusion, we propose a simple fractional diffusion equation to describe the turbulent advection of an initial passive scalar distribution. These predictions agree well with the measurements from simulations, and vary with turbulent Mach number in the expected manner, remaining valid even in the presence of a large-scale shear flow (e.g. rotation in a galactic disk). The evolution of the scalar distribution is not the same as obtained using simple, constant "effective diffusivity" as in Smagorinsky models, because the scale-dependence of turbulent transport means an initially Gaussian distribution quickly develops highly non-Gaussian tails. We also emphasize that these are mean scalings that only apply to ensemble behaviors (assuming many different, random scalar injection sites): individual Lagrangian "patches" remain coherent (poorly-mixed) and simply advect for a large number of turbulent flow-crossing times.

A simple hybrid finite volume solver for compressible turbulence

SummaryA simple, explicit, hybrid finite volume method for simulating compressible turbulence is developed by combining a fourth‐order central scheme and a shock‐capturing simple low‐dissipation advection upstream splitting method. The total flux on each of the cell faces is computed as a weighted average of central/nondissipative and upwind/dissipative fluxes. The weights are determined using an unphysical oscillation sensor in addition to a more traditional discontinuity sensor used in earlier studies. Shocks are well captured, but overshoots in density are predicted around contact discontinuities that are normal to the flow. The use of the latter sensor effectively prevents these overshoots from generating spurious oscillations that travel away from the contact lines. The efficacy of the proposed method for direct or large‐eddy simulations of supersonic turbulence is established using several canonical test problems. Copyright © 2015 John Wiley & Sons, Ltd.

THE TURBULENT DYNAMO IN HIGHLY COMPRESSIBLE SUPERSONIC PLASMAS

The turbulent dynamo may explain the origin of cosmic magnetism. While the exponential amplification of magnetic fields has been studied for incompressible gases, little is known about dynamo action in highly-compressible, supersonic plasmas, such as the interstellar medium of galaxies and the early Universe. Here we perform the first quantitative comparison of theoretical models of the dynamo growth rate and saturation level with three-dimensional magnetohydrodynamical simulations of supersonic turbulence with grid resolutions of up to 1024^3 cells. We obtain numerical convergence and find that dynamo action occurs for both low and high magnetic Prandtl numbers Pm = nu/eta = 0.1-10 (the ratio of viscous to magnetic dissipation), which had so far only been seen for Pm >= 1 in supersonic turbulence. We measure the critical magnetic Reynolds number, Rm_crit = 129 (+43, -31), showing that the compressible dynamo is almost as efficient as in incompressible gas. Considering the physical conditions of the present and early Universe, we conclude that magnetic fields need to be taken into account during structure formation from the early to the present cosmic ages, because they suppress gas fragmentation and drive powerful jets and outflows, both greatly affecting the initial mass function of stars.

An observational method to measure the relative fractions of solenoidal and compressible modes in interstellar clouds

We introduce a new method for observationally estimating the fraction of momentum density (ρv) power contained in solenoidal modes (for which ∇ · ρv = 0) in molecular clouds. The method is successfully tested with numerical simulations of supersonic turbulence that produce the full range of possible solenoidal/compressible fractions. At present, the method assumes statistical isotropy, and does not account for anisotropies caused by (e.g.) magnetic fields. We also introduce a framework for statistically describing density–velocity correlations in turbulent clouds.

On the universality of supersonic turbulence

Compressible turbulence shapes the structure of the interstellar medium of our Galaxy and likely plays an important role also during structure formation in the early Universe. The density PDF and the power spectrum of such compressible, supersonic turbulence are the key ingredients for theories of star formation. However, both the PDF and the spectrum are still a matter of debate, because theoretical predictions are limited and simulations of supersonic turbulence require enormous resolutions to capture the inertial-range scaling. To advance our limited knowledge of compressible turbulence, we here present and analyse the world's largest simulations of supersonic turbulence. We compare hydrodynamic models with numerical resolutions of 256^3-4096^3 mesh points and with two distinct driving mechanisms, solenoidal (divergence-free) driving and compressive (curl-free) driving. We find convergence of the density PDF, with compressive driving exhibiting a much wider and more intermittent density distribution than solenoidal driving. Analysing the power spectrum of the turbulence, we find a pure velocity scaling close to Burgers turbulence with P(v) k^(-2) for both driving modes in our hydrodynamical simulations with Mach = 17. The spectrum of the density-weighted velocity rho^(1/3)v, however, does not provide the previously suggested universal scaling for supersonic turbulence. We find that the power spectrum P(rho^(1/3)v) scales with wavenumber as k^(-1.74) for solenoidal driving, close to incompressible Kolmogorov turbulence, k^(-5/3), but is significantly steeper with k^(-2.10) for compressive driving. We show that this is consistent with a recent theoretical model for compressible turbulence that predicts P(rho^(1/3)v) k^(-19/9) in the presence of a strong div(v) component as is produced by compressive driving and remains remarkably constant throughout the supersonic turbulent cascade.

THE TURBULENCE SPECTRUM OF MOLECULAR CLOUDS IN THE GALACTIC RING SURVEY: A DENSITY-DEPENDENT PRINCIPAL COMPONENT ANALYSIS CALIBRATION

Turbulence plays a major role in the formation and evolution of molecular clouds. The problem is that turbulent velocities are convolved with the density of an observed region. To correct for this convolution, we investigate the relation between the turbulence spectrum of model clouds, and the statistics of their synthetic observations obtained from Principal Component Analysis (PCA). We apply PCA to spectral maps generated from simulated density and velocity fields, obtained from hydrodynamic simulations of supersonic turbulence, and from fractional Brownian motion fields with varying velocity, density spectra, and density dispersion. We examine the dependence of the slope of the PCA structure function, alpha_PCA, on intermittency, on the turbulence velocity (beta_v) and density (beta_n) spectral indexes, and on density dispersion. We find that PCA is insensitive to beta_n and to the log-density dispersion sigma_s, provided sigma_s < 2. For sigma_s>2, alpha_PCA increases with sigma_s due to the intermittent sampling of the velocity field by the density field. The PCA calibration also depends on intermittency. We derive a PCA calibration based on fBms with sigma_s<2 and apply it to 367 CO spectral maps of molecular clouds in the Galactic Ring Survey. The average slope of the PCA structure function, <alpha_PCA>=0.62\pm0.2, is consistent with the hydrodynamic simulations and leads to a turbulence velocity exponent <beta_v>=2.06\pm0.6 for a non-intermittent, low density dispersion flow. Accounting for intermittency and density dispersion, the coincidence between the PCA slope of the GRS clouds and the hydrodynamic simulations suggests beta_v~1.9, consistent with both Burgers and compressible intermittent turbulence.

Dissipative Structures in Supersonic Turbulence

We show that density-weighted moments of the dissipation rate, epsilonl, averaged over a scale l, in supersonic turbulence can be successfully explained by the She and Lévêque model [Phys. Rev. Lett. 72, 336 (1994)]. A general method is developed to measure the two parameters of the model, gamma and d, based directly on their physical interpretations as the scaling exponent of the dissipation rate in the most intermittent structures (gamma) and the dimension of the structures (d). We find that the best-fit parameters (gamma=0.71 and d=1.90) derived from the epsilonl scalings in a simulation of supersonic turbulence at Mach 6 agree with their direct measurements, confirming the validity of the model in supersonic turbulence.

Density Probability Distribution Functions in Supersonic Hydrodynamic and MHD Turbulence

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 measure of magnetic field strength. We also find that the PDF of decaying MHD turbulence deviates from the mean-Mach relation found in the driven case. This implies that the instantaneous Mach number alone is not enough to determine the statistical properties of turbulence that is out of equilibrium. The scatter about the mean-Mach relation for driven turbulence, along with the large departure of decaying turbulence PDFs from those of driven turbulence, may illuminate one factor contributing to the large observed cloud-to-cloud variation in the star formation rate per solar mass.

Supersonic turbulence as an agent of structure formation in space

AbstractSupersonic turbulence is a main agent for structure formation in space, from stellar winds to wind‐blown bubbles, supernovae remnants, ‐ray bursts, and star‐formation in molecular clouds. Dissipation is mostly due to shocks. Under isothermal conditions, the density follows a log‐normal distribution whose variance grows with Mach number. The exact scaling is under debate. We focus on problems with not strictly periodic boundary conditions and present 2D and 3D numerical simulation of supersonic turbulence in plan‐parallel slabs. From dimensional analysis we derive scaling laws for the mean quantities under isothermal conditions. Simulations including spatially extended radiative cooling layers reveal a complex interplay between turbulence and RM‐ and RT‐unstable interfaces. Turbulent Mach numbers here are significantly reduced compared to the isothermal case and are of order 1. However, unstable interfaces between the different thermal component of the gas are equally able to produce structure. (© 2008 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

The Turbulent Shock Origin of Proto–Stellar Cores

The fragmentation of molecular clouds (MC) into proto-stellar cores is a central aspect of the process of star formation. Because of the turbulent nature of supersonic motions in MCs, it has been suggested that dense structures such as filaments and clumps are formed by shocks in a turbulent flow. In this work we present strong evidence in favor of the turbulent origin of the fragmentation of MCs. The most generic result of turbulent fragmentation is that dense postshock gas traces a gas component with a smaller velocity dispersion than lower density gas, since shocks correspond to regions of converging flows, where the kinetic energy of the turbulent motion is dissipated. Using synthetic maps of spectra of molecular transitions, computed from the results of numerical simulations of supersonic turbulence, we show that the dependence of velocity dispersion on gas density generates an observable relation between the rms velocity centroid and the integrated intensity (column density), σ(V0)-I, which is indeed found in the observational data. The comparison between the theoretical model (maps of synthetic 13CO spectra) with 13CO maps from the Perseus, Rosette, and Taurus MC complexes shows excellent agreement in the σ(V0)-I relation. The σ(V0)-I relation of different observational maps with the same total rms velocity are remarkably similar, which is a strong indication of their origin from a very general property of the fluid equations, such as the turbulent fragmentation process.