The Statistics of Supersonic Isothermal Turbulence

By Alexei G. Kritsuk, M. Norman, P. Padoan, Rick Wagner. 2007

471 citations recorded by the source corpus.

We present results of large-scale three-dimensional simulations of supersonic Euler turbulence with the piecewise parabolic method and multiple grid resolutions up to 20483 points. Our numerical experiments describe nonmagnetized 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 5123 points. The simulations allowed us to measure the absolute velocity scaling exponents for the first 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 υ ≡ ρ1/3u. The low-order statistics of υ appear to be invariant with respect to changes in the Mach number. For instance, at Mach 6 the slope of the power spectrum of υ is -1.69, and the exponent of the third-order structure function of υ is unity. We also directly measure the mass dimension of the "fractal" 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.

Subfields: Astronomy and Astrophysics, Computational Mechanics

View Paper: https://www.semanticscholar.org/paper/086ab6eba035dbbe860324790f1ce558120caaa1

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Citations in the corpus

TOWARD A COMPLETE ACCOUNTING OF ENERGY AND MOMENTUM FROM STELLAR FEEDBACK IN GALAXY FORMATION SIMULATIONS (2012, background): “This can be partly due to the fact that natal GMCs are not gravitationally bound (Mac Low & Klessen 2004; Padoan et al. 1997; Li et al. 2004; Kritsuk et al. 2007; Dobbs et al. 2011).”
Numerical simulations of compressively driven interstellar turbulence. I. Isothermal gas (2008, methodology): “However, due to the gradual flattening toward higher wave numbers caused by the bottleneck effect and, possibly, other factors related to PPM (Kritsuk et al. 2007a), the systematic uncertainty of this estimate is much higher thane fitting error margins.”
Physics and Regimes of Supersonic Combustion (2010, background): “Further, due to high helicity, turbulent KE decay is significantly affected by baroclinic and dilatational effects and is faster with respect to the classic Kolmogorov scaling for constantdensity flows [31,32,41,42,53–55].”
ENZO: AN ADAPTIVE MESH REFINEMENT CODE FOR ASTROPHYSICS (2013, background): “…al. 2011), the circumgalactic medium (Hummels et al. 2013), cooling flows (Li & Bryan 2012; Skory et al. 2013), turbulence (Kritsuk & Norman 2004; Kritsuk et al. 2007, 2009a; Collins et al. 2011), the formation of the first stars (Abel et al. 2002; O’Shea & Norman 2007; Turk et al. 2009; Xu et…”
ANALYTICAL THEORY FOR THE INITIAL MASS FUNCTION. II. PROPERTIES OF THE FLOW (2009, methodology): “The dispersion of the local Mach number has been computed in numerical simulations (e.g. Kritsuk et al. 2007, fig.”
yt: A MULTI-CODE ANALYSIS TOOLKIT FOR ASTROPHYSICAL SIMULATION DATA (2010, background): “Examples include two point correlations of galaxies or structure functions, such as the RMS gas velocity structure function used to study the cascade of turbulent energy (Kritsuk et al. 2007).”
Relativistic jet feedback in high-redshift galaxies – I. Dynamics (2016, background): “…D. M., 2009, A&AR, 17, 1 APPENDI X A : PRO BA BI LI TY D E N S I T Y FU N C T I O N S The lognormal distribution has proven to provide excellent description of the density prPDF for simulations of isothermal turbulence (Li, Klessen & Mac Low 2003; Kritsuk et al. 2007; Federrath et al. 2010).”
A comparison between grid and particle methods on the statistics of driven, supersonic, isothermal turbulence (2010, methodology): “…for the clustered and hierarchical nature of star formation (Elmegreen & Falgarone 1996); the measured fractal dimension of interstellar gas (Kritsuk et al. 2007; Federrath et al. 2009, and references therein); the few per- cent efficiency with which gas is converted into stars (Padoan…”
Cosmic-Ray Propagation: Nonlinear Diffusion Parallel and Perpendicular to Mean Magnetic Field (2007, background): “For instance, hydrodynamic turbulence simulations in Kritsuk et al. (2007) showed that only starting with the 10283 numerical cubes the bottleneck effects stop dominating the measured spectral slope.”
Analytical Theory for the Initial Mass Function: CO Clumps and Prestellar Cores (2008, background): “Interestingly enough, the aforementioned favored value for supersonic turbulence, 0:4 (Kritsuk et al. 2007), yields exactly the Salpeter coefficient, dN/dM / M (1þx).”
Inefficient star formation through turbulence, magnetic fields and feedback (2015, background): “…largest scales (Heyer et al. 2006; Brunt et al. 2009) and with a power spectrum, E(k) ∼ k−2, consistent with supersonic, compressible turbulence (Larson 1981; Heyer & Brunt 2004; Roman-Duval et al. 2011) and confirmed by simulations (Kritsuk et al. 2007; Federrath et al. 2010b; Federrath 2013a).”
Turbulent Motions and Shocks Waves in Galaxy Clusters simulated with AMR (2009, methodology): “For completeness, structure functions are also calculated for the density-weighted velocity, u ≡ ρ1/3v, which was introduced by Kritsuk et al.(2007) to study scaling relations for simulated supersonic turbulence.”
Relativistic jet feedback – III. Feedback on gas discs (2018, background): “At higher wavenumbers, the power spectrum may be affected by numerical dissipation and hence the choice of resolution (Kritsuk et al. 2007).”
On the universality of supersonic turbulence (2013, background): “However, the studies by Kritsuk et al. (2007) and Federrath et al. (2010) were limited to 10243 grid cells.”
A NEW JEANS RESOLUTION CRITERION FOR (M)HD SIMULATIONS OF SELF-GRAVITATING GAS: APPLICATION TO MAGNETIC FIELD AMPLIFICATION BY GRAVITY-DRIVEN TURBULENCE (2011, background): “The measured PDFs in Figure 6, however, exhibit significant deviations from the log-normal function due to intermittency (Kritsuk et al. 2007; Schmidt et al. 2009; Federrath et al. 2010b) and the convolution with the radial density distribution inside the Jeans radius of the collapsing Bonnor-Ebert…”
THE FRACTAL DENSITY STRUCTURE IN SUPERSONIC ISOTHERMAL TURBULENCE: SOLENOIDAL VERSUS COMPRESSIVE ENERGY INJECTION (2007, methodology): “The averaging procedure is important to derive meaningful statistics, because all quantities are subject to statistical fluctuations (e.g., Kritsuk et al. 2007).”
THE STAR FORMATION RATE OF TURBULENT MAGNETIZED CLOUDS: COMPARING THEORY, SIMULATIONS, AND OBSERVATIONS (2012, background): “…solenoidal modes are indirectly excited by nonlinear interactions of multiple colliding shock fronts (Vishniac 1994; Sun & Takayama 2003; Kritsuk et al. 2007; Federrath et al. 2010b), by baroclinity, rotation and shear (Del Sordo & Brandenburg 2011), and by viscosity (Mee & Brandenburg…”
On the relationship between molecular hydrogen and carbon monoxide abundances in molecular clouds (2010, background): “Deviations from the lognormal form are apparent in the wings of the PDFs, where one would expect the effects of turbulent intermittency to be most pronounced (Kritsuk et al. 2007; Schmidt et al. 2009; Federrath et al. 2010).”
On the universality of interstellar filaments: theory meets simulations and observations (2015, methodology): “In this regime, numerical simulations find a steeper powerlaw dependence, σv(`) ∝ `1/2 (Kritsuk et al. 2007; Schmidt et al. 2009; Federrath et al. 2010b; Federrath 2013) than in Kolmogorov turbulence. c© 0000 RAS, MNRAS 000, 000–000 This power-law scaling of the turbulence, σv(`) ∝ `α with α > 0,…”