The sonic scale of interstellar turbulence

This article has been updated

Abstract

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.

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.

from$8.99

All prices are NET prices.

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 http://flash.uchicago.edu/site/flashcode. 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.

References

  1. 1.

    Ferrière, K. M. The interstellar environment of our galaxy. Rev. Mod. Phys. 73, 1031–1066 (2001).

    ADS  Google Scholar 

  2. 2.

    Hennebelle, P. & Falgarone, E. Turbulent molecular clouds. Astron. Astrophys. Rev. 20, 55 (2012).

    ADS  Google Scholar 

  3. 3.

    Mac Low, M.-M. & Klessen, R. S. Control of star formation by supersonic turbulence. Rev. Mod. Phys. 76, 125–194 (2004).

    ADS  Google Scholar 

  4. 4.

    Krumholz, M. R. & McKee, C. F. A general theory of turbulence-regulated star formation, from spirals to ultraluminous infrared galaxies. Astrophys. J. 630, 250–268 (2005).

    ADS  Google Scholar 

  5. 5.

    McKee, C. F. & Ostriker, E. C. Theory of star formation. Annu. Rev. Astron. Astrophys. 45, 565–687 (2007).

    ADS  Google Scholar 

  6. 6.

    Hennebelle, P. & Chabrier, G. Analytical theory for the initial mass function: CO clumps and prestellar cores. Astrophys. J. 684, 395–410 (2008).

    ADS  Google Scholar 

  7. 7.

    Hopkins, P. F. A general theory of turbulent fragmentation. Mon. Not. R. Astron. Soc. 430, 1653–1693 (2013).

    ADS  Google Scholar 

  8. 8.

    Padoan, P. et al. in Protostars and Planets VI (eds Beuther, H. et al.) 77–100 (Univ. Arizona Press, 2014).

  9. 9.

    Federrath, C. & Klessen, R. S. The star formation rate of turbulent magnetized clouds: comparing theory, simulations, and observations. Astrophys. J. 761, 156 (2012).

    ADS  Google Scholar 

  10. 10.

    Larson, R. B. Turbulence and star formation in molecular clouds. Mon. Not. R. Astron. Soc. 194, 809–826 (1981).

    ADS  Google Scholar 

  11. 11.

    Ossenkopf, V. & Mac Low, M.-M. Turbulent velocity structure in molecular clouds. Astron. Astrophys. 390, 307–326 (2002).

    ADS  Google Scholar 

  12. 12.

    Heyer, M. H. & Brunt, C. M. The universality of turbulence in Galactic molecular clouds. Astrophys. J. 615, L45–L48 (2004).

    ADS  Google Scholar 

  13. 13.

    Roman-Duval, J. et al. The turbulence spectrum of molecular clouds in the Galactic Ring Survey: a density-dependent principal component analysis calibration. Astrophys. J. 740, 120 (2011).

    ADS  Google Scholar 

  14. 14.

    André, P. et al. in Protostars and Planets VI (eds Beuther, H. et al.) 27–51 (Univ. Arizona Press, 2014).

  15. 15.

    Federrath, C. On the universality of interstellar filaments: theory meets simulations and observations. Mon. Not. R. Astron. Soc. 457, 375–388 (2016).

    ADS  Google Scholar 

  16. 16.

    Offner, S. S. R. et al. in Protostars and Planets VI (eds Beuther, H. et al.) 53–75 (Univ. Arizona Press, 2014).

  17. 17.

    Federrath, C., Roman-Duval, J., Klessen, R. S., Schmidt, W. & Mac Low, M.-M. Comparing the statistics of interstellar turbulence in simulations and observations. Solenoidal versus compressive turbulence forcing. Astron. Astrophys. 512, A81 (2010).

    ADS  Google Scholar 

  18. 18.

    Konstandin, L., Federrath, C., Klessen, R. S. & Schmidt, W. Statistical properties of supersonic turbulence in the Lagrangian and Eulerian frameworks. J. Fluid Mech. 692, 183–206 (2012).

    ADS  MathSciNet  MATH  Google Scholar 

  19. 19.

    Kolmogorov, A. N. Dissipation of energy in locally isotropic turbulence. Dokl. Akad. Nauk SSSR 32, 16–18 (1941).

    ADS  MathSciNet  MATH  Google Scholar 

  20. 20.

    Burgers, J. M. A mathematical model illustrating the theory of turbulence. Adv. Appl. Mech. 1, 171–199 (1948).

    MathSciNet  Google Scholar 

  21. 21.

    Falgarone, E., Pety, J. & Hily-Blant, P. Intermittency of interstellar turbulence: extreme velocity-shears and CO emission on milliparsec scale. Astron. Astrophys. 507, 355–368 (2009).

    ADS  Google Scholar 

  22. 22.

    Schmidt, W., Federrath, C. & Klessen, R. Is the scaling of supersonic turbulence universal? Phys. Rev. Lett. 101, 194505 (2008).

    ADS  Google Scholar 

  23. 23.

    Squire, J. & Hopkins, P. F. The distribution of density in supersonic turbulence. Mon. Not. R. Astron. Soc. 471, 3753–3767 (2017).

    ADS  Google Scholar 

  24. 24.

    Hopkins, P. F. A model for (non-lognormal) density distributions in isothermal turbulence. Mon. Not. R. Astron. Soc. 430, 1880–1891 (2013).

    ADS  Google Scholar 

  25. 25.

    Federrath, C. On the universality of supersonic turbulence. Mon. Not. R. Astron. Soc. 436, 1245–1257 (2013).

    ADS  Google Scholar 

  26. 26.

    Scalo, J. in Physical Processes in Fragmentation and Star Formation (eds Capuzzo-Dolcetta, R. et al.) 151–176 (Springer, 1990).

  27. 27.

    Sánchez, N., Alfaro, E. J. & Pérez, E. The fractal dimension of projected clouds. Astrophys. J. 625, 849–856 (2005).

    ADS  Google Scholar 

  28. 28.

    Federrath, C., Klessen, R. S. & Schmidt, W. The fractal density structure in supersonic isothermal turbulence: solenoidal versus compressive energy injection. Astrophys. J. 692, 364–374 (2009).

    ADS  Google Scholar 

  29. 29.

    Beattie, J. R., Federrath, C. & Klessen, R. S. The relation between the true and observed fractal dimensions of turbulent clouds. Mon. Not. R. Astron. Soc. 487, 2070–2081 (2019).

    ADS  Google Scholar 

  30. 30.

    Padoan, P., Nordlund, Å & Jones, B. J. T. The universality of the stellar initial mass function. Mon. Not. R. Astron. Soc. 288, 145–152 (1997).

    ADS  Google Scholar 

  31. 31.

    Fryxell, B. et al. FLASH: an adaptive mesh hydrodynamics code for modeling astrophysical thermonuclear flashes. Astrophys. J. Suppl. Ser. 131, 273–334 (2000).

    ADS  Google Scholar 

  32. 32.

    Dubey, A. et al. in Numerical Modeling of Space Plasma Flows (eds Pogorelov, N. V. et al.) 145 (Astronomical Society of the Pacific, 2008).

  33. 33.

    Waagan, K., Federrath, C. & Klingenberg, C. A robust numerical scheme for highly compressible magnetohydrodynamics: nonlinear stability, implementation and tests. J. Comput. Phys. 230, 3331–3351 (2011).

    ADS  MathSciNet  MATH  Google Scholar 

  34. 34.

    Porter, D. H., Pouquet, A. & Woodward, P. R. A numerical study of supersonic turbulence. Theor. Comput. Fluid Dyn. 4, 13–49 (1992).

    MATH  Google Scholar 

  35. 35.

    Kritsuk, A. G., Norman, M. L., Padoan, P. & Wagner, R. The statistics of supersonic isothermal turbulence. Astrophys. J. 665, 416–431 (2007).

    ADS  Google Scholar 

  36. 36.

    Schmidt, W., Federrath, C., Hupp, M., Kern, S. & Niemeyer, J. C. Numerical simulations of compressively driven interstellar turbulence. I. Isothermal gas. Astron. Astrophys. 494, 127–145 (2009).

    ADS  Google Scholar 

  37. 37.

    Eswaran, V. & Pope, S. B. An examination of forcing in direct numerical simulations of turbulence. Comput. Fluids 16, 257–278 (1988).

    ADS  MATH  Google Scholar 

  38. 38.

    Haugen, N. E. L., Brandenburg, A. & Mee, A. J. Mach number dependence of the onset of dynamo action. Mon. Not. R. Astron. Soc. 353, 947–952 (2004).

    ADS  Google Scholar 

  39. 39.

    Schekochihin, A. A. et al. Fluctuation dynamo and turbulent induction at low magnetic Prandtl numbers. New J. Phys. 9, 300 (2007).

    ADS  Google Scholar 

  40. 40.

    Federrath, C., Klessen, R. S. & Schmidt, W. The density probability distribution in compressible isothermal turbulence: solenoidal versus compressive forcing. Astrophys. J. 688, L79–L82 (2008).

    ADS  Google Scholar 

  41. 41.

    Frisch, U. Turbulence, the Legacy of A. N. Kolmogorov (Cambridge Univ. Press, 1995).

  42. 42.

    Pope, S. B. Turbulent Flows (Cambridge Univ. Press, 2000).

  43. 43.

    Benzi, R. et al. Intermittency and universality in fully developed inviscid and weakly compressible turbulent flows. Phys. Rev. Lett. 100, 234503 (2008).

    ADS  Google Scholar 

  44. 44.

    Federrath, C. et al. Mach number dependence of turbulent magnetic field amplification: solenoidal versus compressive flows. Phys. Rev. Lett. 107, 114504 (2011).

    ADS  Google Scholar 

  45. 45.

    Federrath, C., Sur, S., Schleicher, D. R. G., Banerjee, R. & Klessen, R. S. A new Jeans resolution criterion for (M)HD simulations of self-gravitating gas: application to magnetic field amplification by gravity-driven turbulence. Astrophys. J. 731, 62 (2011).

    ADS  Google Scholar 

  46. 46.

    Falkovich, G. Bottleneck phenomenon in developed turbulence. Phys. Fluids 6, 1411–1414 (1994).

    ADS  MATH  Google Scholar 

  47. 47.

    Dobler, W., Haugen, N. E., Yousef, T. A. & Brandenburg, A. Bottleneck effect in three-dimensional turbulence simulations. Phys. Rev. E 68, 026304 (2003).

    ADS  Google Scholar 

  48. 48.

    Verma, M. K. & Donzis, D. Energy transfer and bottleneck effect in turbulence. J. Phys. A 40, 4401–4412 (2007).

    ADS  MathSciNet  MATH  Google Scholar 

  49. 49.

    Schmidt, W., Hillebrandt, W. & Niemeyer, J. C. Numerical dissipation and the bottleneck effect in simulations of compressible isotropic turbulence. Comput. Fluids 35, 353–371 (2006).

    MATH  Google Scholar 

  50. 50.

    Koyama, H. & Inutsuka, S.-i. An origin of supersonic motions in interstellar clouds. Astrophys. J. 564, L97–L100 (2002).

    ADS  Google Scholar 

  51. 51.

    Vázquez-Semadeni, E. et al. Molecular cloud evolution. II. From cloud formation to the early stages of star formation in decaying conditions. Astrophys. J. 657, 870–883 (2007).

    ADS  Google Scholar 

  52. 52.

    Körtgen, B., Federrath, C. & Banerjee, R. On the shape and completeness of the column density probability distribution function of molecular clouds. Mon. Not. R. Astron. Soc. 482, 5233–5240 (2019).

    ADS  Google Scholar 

  53. 53.

    Mandal, A., Federrath, C. & Körtgen, B. Molecular cloud formation by compression of magnetized turbulent gas subjected to radiative cooling. Mon. Not. R. Astron. Soc. 493, 3098–3113 (2020).

    ADS  Google Scholar 

  54. 54.

    Glover, S. C. O., Federrath, C., Mac Low, M.-M. & Klessen, R. S. Modelling CO formation in the turbulent interstellar medium. Mon. Not. R. Astron. Soc. 404, 2–29 (2010).

    ADS  Google Scholar 

  55. 55.

    Arzoumanian, D. et al. Characterizing interstellar filaments with Herschel in IC 5146. Astron. Astrophys. 529, L6 (2011).

    ADS  Google Scholar 

  56. 56.

    Arzoumanian, D., Shimajiri, Y., Inutsuka, S.-i, Inoue, T. & Tachihara, K. Molecular filament formation and filament-cloud interaction: hints from Nobeyama 45 m telescope observations. Publ. Astron. Soc. Jpn 70, 96 (2018).

    ADS  Google Scholar 

  57. 57.

    Vázquez-Semadeni, E., Ryu, D., Passot, T., González, R. F. & Gazol, A. Molecular cloud evolution. I. Molecular cloud and thin cold neutral medium sheet formation. Astrophys. J. 643, 245–259 (2006).

    ADS  Google Scholar 

  58. 58.

    Heitsch, F., Hartmann, L. W., Slyz, A. D., Devriendt, J. E. G. & Burkert, A. Cooling, gravity, and geometry: flow-driven massive core formation. Astrophys. J. 674, 316–328 (2008).

    ADS  Google Scholar 

  59. 59.

    Banerjee, R., Vázquez-Semadeni, E., Hennebelle, P. & Klessen, R. S. Clump morphology and evolution in MHD simulations of molecular cloud formation. Mon. Not. R. Astron. Soc. 398, 1082–1092 (2009).

    ADS  Google Scholar 

  60. 60.

    Audit, E. & Hennebelle, P. On the structure of the turbulent interstellar clouds. Influence of the equation of state on the dynamics of 3D compressible flows. Astron. Astrophys. 511, A76 (2010).

    ADS  Google Scholar 

  61. 61.

    Zamora-Aviles, M., Vazquez-Semadeni, E., Koertgen, B., Banerjee, R. & Hartmann, L. Magnetic suppression of turbulence and the star formation activity of molecular clouds. Mon. Not. R. Astron. Soc. 474, 4824–4836 (2018).

    ADS  Google Scholar 

  62. 62.

    Padoan, P., Pan, L., Haugbølle, T. & Nordlund, Å Supernova driving. I. The origin of molecular cloud turbulence. Astrophys. J. 822, 11 (2016).

    ADS  Google Scholar 

  63. 63.

    Pan, L., Padoan, P., Haugbølle, T. & Nordlund, Å Supernova driving. II. Compressive ratio in molecular-cloud turbulence. Astrophys. J. 825, 30 (2016).

    ADS  Google Scholar 

  64. 64.

    Körtgen, B., Federrath, C. & Banerjee, R. The driving of turbulence in simulations of molecular cloud formation and evolution. Mon. Not. R. Astron. Soc. 472, 2496–2503 (2017).

    ADS  Google Scholar 

  65. 65.

    Schneider, N. et al. What determines the density structure of molecular clouds? A case study of Orion B with Herschel. Astrophys. J. 766, L17 (2013).

    ADS  Google Scholar 

  66. 66.

    Solomon, P. M., Rivolo, A. R., Barrett, J. & Yahil, A. Mass, luminosity, and line width relations of Galactic molecular clouds. Astrophys. J. 319, 730–741 (1987).

    ADS  Google Scholar 

Download references

Acknowledgements

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

Affiliations

Authors

Contributions

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.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

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

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Federrath, C., Klessen, R.S., Iapichino, L. et al. The sonic scale of interstellar turbulence. Nat Astron (2021). https://doi.org/10.1038/s41550-020-01282-z

Download citation

Further reading

Search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing