Mach number study of supersonic turbulence: the properties of the density field

Abstract

We model driven, compressible, isothermal, turbulence with Mach numbers ranging from the subsonic (M ≈ 0.65) to the highly supersonic regime (M ≈ 16). The forcing scheme consists both solenoidal (transverse) and compressive (longitudinal) modes in equal parts. We find a relation σ 2 s = b log (1 + b 2 M 2 ) between the Mach number and the standard deviation of the logarithmic density with b = 0.457 ± 0.007. The density spectra follow D(k, M) ∝ k ζ(M) with scaling exponents depending on the Mach number. We find ζ(M) = αM β with a coefficient α that varies slightly with resolution, whereas β changes systematically. We extrapolate to the limit of infinite resolution and find α = −1.91 ± 0.01, β = −0.30 ± 0.03. The dependence of the scaling exponent on the Mach number implies a fractal dimension D = 2 + 0.96M −0.30 . We determine how the scaling parameters depend on the wavenumber and find that the density spectra are slightly curved. This curvature gets more pronounced with increasing Mach number. We propose a physically motivated fitting formula D(k) = D 0 k ζk η by using simple scaling arguments. The fit reproduces the spectral behaviour down to scales k ≈ 80. The density spectrum follows a single power-law η = −0.005 ± 0.01 in the low Mach number regime and the strongest curvature η = −0.04 ± 0.02 for the highest Mach number. These values of η represent a lower limit, as the curvature increases with resolution.

Filaments and striations: anisotropies in observed, supersonic, highly magnetized turbulent clouds

“…The power spectrum allows us to reveal order out of complicated and stochastic structures that are mostly intangible in real space. There are, however, also techniques for directly analysing real-space structures, such as structure functions, topological (Appleby et al 2018;Henderson et al 2019), and fractal methods (Scalo 1990;Elmegreen & Falgarone 1996;Stutzki et al 1998;Kowal et al 2007;Federrath et al 2009;Roman-Duval et al 2010;Donovan Meyer et al 2013;Konstandin et al 2016;Beattie et al 2019b). The power spectrum has a special role in the study of turbulence, since turbulence models rely heavily upon understanding flow characteristics on different length scales in real space, e.g., on the driving scale, in the inertial (for incompressible flows) scaling range (cascade) of turbulence, and on the dissipation scale (Kolmogorov 1941;Burgers 1948).…”

Unbound Young Stellar Systems: Star Formation on the Loose

“…The density contrast in turbulent isothermal gas is also found to scale positively with the turbulence Mach number (e.g., Konstandin et al 2016). Turbulent systems are typically gravitationally unstable (McKee & Ostriker 2007;Hopkins 2013a), with supersonic turbulence driving their density distribution (e.g., Klessen 2011;Hopkins 2012).…”

The distribution of density in supersonic turbulence

“…In addition to the basic functional form of the PDF, which is well-known to accurately match simulations (Hopkins 2013b;Federrath 2013;Konstandin et al 2016), the model's scale Table 1 (see also H13 Fig. 3).…”

“…However, although there are certain wellestablished results-most importantly the density variance-Mach number relation: that the density distribution is approximately lognormal with a variance that increases with Mach number (Passot & Vázquez-Semadeni 1998;Price et al 2011;Padoan & Nordlund 2011;Molina et al 2012;Federrath & Banerjee 2015;Pan et al 2016)-we lack detailed understanding of many important issues. For example, the power spectrum of density and its variation with Mach number is not well understood (Kim & Ryu 2005;Kritsuk et al 2007;Kowal et al 2007;Konstandin et al 2016). Further, an important limitation of the density variance-Mach number relation is that the density can be quite intermittent, viz., it is not distributed log-normally.…”