The sonic scale of interstellar turbulence

This article has been updated


Understanding the physics of turbulence is crucial for many applications, including weather, industry and astrophysics. In the interstellar medium1,2, supersonic turbulence plays a crucial role in controlling the gas density and velocity structure, and ultimately the birth of stars3,4,5,6,7,8. Here we present a simulation of interstellar turbulence with a grid resolution of 10,0483 cells that allows us to determine the position and width of the sonic scale (s)—the transition from supersonic to subsonic turbulence. The simulation simultaneously resolves the supersonic and subsonic cascade, with the velocity as a function of scale, v() p, where we measure psup = 0.49 ± 0.01 and psub = 0.39 ± 0.02, respectively. We find that s agrees with the relation \({\ell }_{{\rm{s}}}={\phi }_{{\rm{s}}}{L\,}{{\mathcal{M}}}^{-1/{p}_{\sup }}\), where \({\mathcal{M}}\) is the three-dimensional Mach number, L is either the driving scale of the turbulence or the diameter of a molecular cloud, and ϕs is a dimensionless factor of order unity. If L is the driving scale, we measure \({\phi }_{{\rm{s}}}=0.4{2}_{-0.09}^{+0.12}\), primarily because of the separation between the driving scale and the start of the supersonic cascade. For a supersonic cascade extending beyond the cloud scale, we get \({\phi }_{{\rm{s}}}=0.9{1}_{-0.20}^{+0.25}\). In both cases, ϕs 1, because we find that the supersonic cascade transitions smoothly to the subsonic cascade over a factor of 3 in scale, instead of a sharp transition. Our measurements provide quantitative input for turbulence-regulated models of filament structure and star formation in molecular clouds.

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Fig. 1: Visualization of the gas density distribution of turbulence.
Fig. 2: The sonic scale revealed through structure function analysis.
Fig. 3: Gas density distribution filtered on different scales.

Data availability

Source data are provided with this paper. The simulation raw data (several TB) and the data shown in the Extended Data figures are available from the corresponding author upon reasonable request.

Code availability

Access to the basic simulation code (FLASH) used here can be obtained via The modifications to FLASH implemented here, and the analysis codes used here, are available from the corresponding author upon reasonable request.

Change history

  • 21 January 2021

    In the version of this Letter originally published, Supplementary Video 1 and the Source Data files for Figs. 1–3 and Extended Data Figs. 1–4 were linked to the incorrect documents; this has now been corrected.


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C.F. acknowledges funding provided by the Australian Research Council (Discovery Project DP170100603 and Future Fellowship FT180100495), and the Australia–Germany Joint Research Cooperation Scheme (UA-DAAD). C.F. further acknowledges the Australian Research Council Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), through project number CE170100013. R.S.K. acknowledges support from the German Research Foundation (DFG) via the collaborative research center ‘The Milky Way System’ (SFB 881, Project-ID 138713538, subprojects A1, B1, B2 and B8) as well as support from the Heidelberg cluster of excellence EXC 2181 (Project-ID 390900948) ‘STRUCTURES: A unifying approach to emergent phenomena in the physical world, mathematics, and complex data’ funded by the German Excellence Strategy. R.S.K. also thanks the European Research Council for support via the ERC Advanced Grant ‘STARLIGHT’ (Project ID 339177) and the ERC Synergy Grant ‘ECOGAL’ (Project ID 855130). We further acknowledge high-performance computing resources provided by the Leibniz Rechenzentrum and the Gauss Centre for Supercomputing (grants pr32lo, pr48pi and GCS Large-scale project 10391), the Australian National Computational Infrastructure (grant ek9) in the framework of the National Computational Merit Allocation Scheme and the ANU Merit Allocation Scheme. The simulation software FLASH was in part developed by the DOE-supported Flash Center for Computational Science at the University of Chicago.

Author information




C.F. initiated and led this study, modified the simulation code, ran and analysed the simulations, and drafted the manuscript. R.S.K., L.I. and J.R.B. contributed to data interpretation and to improving the manuscript.

Corresponding author

Correspondence to Christoph Federrath.

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Peer review information Nature Astronomy thanks Christopher McKee and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Extended data

Extended Data Fig. 1 Sonic scale in simulations with different Mach number and cooling.

a, Same as Fig. 2a, but for models with different Mach number (‘Mach number study’) and without the isothermal approximation (‘Cooling study’), at different numerical resolutions, as indicated in the legend. The vertical lines mark the location of the sonic scale in each of the different models, that is, where the scale-dependent Mach number equals unity (horizontal dotted line). b, Sonic scale measured from the same models in a, as a function of the Mach number \(({\mathcal{M}})\) on the driving scale (L). The solid and dotted lines show Eq. (1) with p = 1/2 and \({\phi }_{{\rm{s}}}=0.4{2}_{-0.09}^{+0.12}\), as measured from the high-resolution (10,0483 grid cells) simulation in the main part of the study. Source data

Extended Data Fig. 2 Statistical convergence study.

Same as Fig. 2a, but for a single time snapshot (at 5 turbulent crossing times), and the structure functions shown here were computed with different numbers of sampling points: 2 × 108, 2 × 109, 2 × 1010, 2 × 1011, and 2 × 1012 pairs. This demonstrates statistical convergence of the 2nd-order structure function on all relevant scales for a sample size of greater than about 1012 pairs per time snapshot. Source data

Extended Data Fig. 3 Velocity power spectra.

Same Fig. 2a, but showing the velocity power spectrum Pv, as a function of wave number k normalised to the driving wave number K = 2π/L. The power spectrum contains the same basic information as the structure function shown in Fig. 2. We find Pv k−1.99±0.02 for the supersonic cascade and Pv k−1.76±0.04 for the subsonic cascade, with the sonic scale in between, consistent with the results derived from Fig. 2. Source data

Extended Data Fig. 4 Numerical resolution study.

Same as Fig. 2a, but for numerical grid resolutions of 5,0243 cells (dashed) and 2,5123 cells (dotted), in addition to our main run with 10,0483 cells (solid line with 1σ uncertainties shown in grey; note that the uncertainties on the dashed and dotted lines are similar to the ones shown on the solid line, but are omitted for clarity). The position of the sonic scale is converged to within 6% of the extrapolated infinite-resolution limit. Measuring the width of the sonic transition range and the scaling in the subsonic regime requires more than about 10,0003 cells, as can be seen by the deviations below the sonic scale in the 5,0243 and 2,5123 resolution cases. Source data

Extended Data Fig. 5 Distribution of interstellar filament widths in observations and theory.

Filament width distribution measured from observations in the IC5146, the Aquila and the Polaris molecular clouds shown as the orange histogram, together with the theoretical prediction based on the sonic scale (blue line; see Supplementary Information for the details of this function). The theoretical prediction is not a fit; instead the peak position is set to x0 = 0.1 parsec and the log-normal standard deviation, σx = 0.105, is determined by the width of the sonic-scale transition measured in Fig. 2. Source data

Supplementary information

Supplementary Information

Supplementary discussion.

Supplementary Video 1

Animation of the three-dimensional projected gas density contrast, starting with a time evolution, then rotating and highlighting the sonic-scale structures, and finally zooming into a dense filament formed in the simulation.

Source data

Source Data Fig. 1

Density contrast slice interpolated to 3,600 × 3,600 pixels, scaled logarithmically in the interval ρ/ρ0 = [0.05, 50].

Source Data Fig. 2

Source data for Fig. 2a (Mach number as a function of scale).

Source Data Fig. 3

Source data for Fig. 3c (density contrast PDFs).

Source Data Extended Data Fig. 1

Source data for Extended Data Fig. 1a (Mach number as a function of scale) and b (sonic scale as a function of large-scale turbulent Mach number).

Source Data Extended Data Fig. 2

Statistical convergence study.

Source Data Extended Data Fig. 3

Velocity power spectrum.

Source Data Extended Data Fig. 4

Numerical resolution study.

Source Data Extended Data Fig. 5

Filament width distribution.

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Federrath, C., Klessen, R.S., Iapichino, L. et al. The sonic scale of interstellar turbulence. Nat Astron (2021).

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