The Density Probability Distribution in Compressible Isothermal Turbulence: Solenoidal versus Compressive Forcing

By Christoph Federrath, Ralf Klessen, Wolfram Schmidt. 2008

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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 used in analytical models of star formation is reviewed and a modification of the existing expression is proposed, which takes into account the ratio of solenoidal and compressive modes of the turbulence forcing.

Subfields: Astronomy and Astrophysics, Computational Mechanics

View Paper: https://www.semanticscholar.org/paper/236207254ee2fdb33639dbf5aba90157164c20aa

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Citations in the corpus

Dispersal of molecular clouds by ionizing radiation (2012, background): “Gritschneder et al. (2009) discuss turbulence driving in their simulations of ionising feedback, and report that the energetic ratio of solenoidal to compressible modes (Federrath et al. 2008) is significantly reduced, relative to the fiducial ratio of 2:1.”
THE STAR FORMATION RATE OF TURBULENT MAGNETIZED CLOUDS: COMPARING THEORY, SIMULATIONS, AND OBSERVATIONS (2012, background): “…to the well-known and frequently used hydrodynamic (HD) expression, σ2s = ln ( 1 + b2M2 ) with β →∞ (e.g., Padoan et al. 1997; Passot & Vázquez-Semadeni 1998; Ostriker et al. 2001; Lemaster & Stone 2008; Federrath et al. 2008b; Price et al. 2011) as a necessary condition in the purely HD limit.”
Small-scale dynamo action during the formation of the first stars and galaxies - I. The ideal MHD limit (2010, background): “State of the art numerical simulations of turbulence require a spatial resolution of at least 5123 for a marginally resolved inertial range (Federrath et al. 2008, 2010).”
Phantom: A Smoothed Particle Hydrodynamics and Magnetohydrodynamics Code for Astrophysics (2017, methodology): “Indeed, Phantom was used in the study by Price, Federrath & Brunt (2011) to confirm the relationship between the standard deviation and the Mach number in the form σ2s = ln ( 1 + b2M2 ) , (332) where b = 1/3 for solenoidally driven turbulence, as earlier suggested by Federrath et al. (2008, 2010a).”
Star formation rates and efficiencies in the Galactic Centre (2017, background): “12 The idealized case presented by Federrath et al. (2008), b = 0.33 represents purely isotropic solenoidal turbulence driving.”
A comparison between grid and particle methods on the statistics of driven, supersonic, isothermal turbulence (2010, methodology): “…is well represented by a log-normal distribution (e.g. Padoan et al. 1997; Passot & Vázquez-Semadeni 1998; Klessen 2000; Kritsuk et al. 2007; Federrath et al. 2008; Lemaster & Stone 2008; Federrath et al. 2010), i.e., p(ln ρ)d ln ρ = 1√ 2πσ2 exp [ −1 2 ( ln ρ− ln ρ σ )2 ] d ln ρ.…”
A sub-parsec resolution simulation of the Milky Way: global structure of the interstellar medium and properties of molecular clouds (2013, background): “They can be connected to the width σPDF of the log-normal PDF through σ2PDF = ln ( 1 + b2M2 ) , with b describing the relative importance of solenoidal and compressible modes of turbulence (Federrath et al. 2008, 2010; Molina et al. 2012).”
Probing the evolution of molecular cloud structure: From quiescence to birth (2009, background): “While supersonic turbulence is expected to develop a density PDF close to a log-normal distribution, prominent deviations from that shape are predicted in strongly self-gravitatingsystems (e.g. Klessen 2000; Federrath et al. 2008).”
Numerical simulations of compressively driven interstellar turbulence. I. Isothermal gas (2008, background): “In part, this is the subject of ongoing work, where turbulence simulations with purely solenoidal and compressive forcing are systematically compared for numerical esolutions up to 10243 (Schmidt et al. 2008; Federrath et al. 2008).”
Shocking results without shocks: Subsonic turbulence in smoothed particle hydrodynamics and moving-mesh simulations (2011, methodology): “In any case, if a compressive component was added, we would expect a somewhat broader density PDF for a given Mach number (Federrath et al. 2008).”
The Stellar IMF, Core Mass Function, & The Last-Crossing Distribution (2012, methodology): “…of effective radius R, δ(x |R) is also normally distributed, with a variance S = σ2(R) that can (in principle) be calculated from the power spectrum or simply estimated from the Mach number (see e.g. Passot & Vazquez-Semadeni 1998; Nordlund & Padoan 1999; Federrath et al. 2008; Price et al. 2011).”
Feedback-regulated star formation in molecular clouds and galactic discs (2013, methodology): “The volume-weighted PDF, fV(y), is given by the same expression with the simple replacement µ→ −µ. Federrath et al. (2008) showed that the lognormal parameters depend on both the Mach number M of the turbulence on the driving scale and on the relative importance of solenoidal and compressive…”
Modelling CO formation in the turbulent interstellar medium (2009, background): “…and in these regions there is no simple mapping between gas density and H2 number density, since here the H2 fraction is sensitive not only to the current density and the degree of shielding, but also to the previous dynamical history of the gas (Glover & Mac Low 2007b; Federrath et al. 2008).”
The density variance–Mach number relation in supersonic turbulence – I. Isothermal, magnetized gas (2012, methodology): “These variances are related by (e.g. Federrath et al. 2008b; Price et al. 2011) σ 2s = ln [ 1 + σ 2ρ/ρ0 ] .”
A new class of accurate, mesh-free hydrodynamic simulation methods (2014, methodology): “…length L = 1, density ρ = 1, sound speed cs = 1, and isothermal equation of state γ = 1 is stirred via the usual method in e.g. Schmidt et al. (2008); Federrath et al. (2008); Price & Federrath (2010), where a small range of modes corresponding to wavelengths between 1/2− 1 times the box size…”
THE STAR FORMATION RATE OF SUPERSONIC MAGNETOHYDRODYNAMIC TURBULENCE (2009, background): “…on the distribution of local shock velocity and preshock density and is known to follow a LogNormal pdf (Vazquez-Semadeni 1994; Padoan et al. 1997; Nordlund & Padoan 1999; Ostriker et al. 2001; Li et al. 2004; Kritsuk et al. 2007; Beetz et al. 2008; Lemaster & Stone 2008; Federrath et al. 2008).”
FIRE-2 simulations: physics versus numerics in galaxy formation (2017, background): “…disk, or inject spurious thermal energy/turbulence into 23 Gas which can cool efficiently to T ∼ 10K and has turbulent δv ∼ 10kms−1 should produce an approximately lognormal density distribution with & 10% of the mass exceeding∼ 100× the mean density (see e.g. Federrath et al. 2008). the gas.”