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Turbulence Theory

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Numerical Modelling of Astrophysical Turbulence

Part of the book series: SpringerBriefs in Astronomy ((BRIEFSASTRON))

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Abstract

Beginning with the fluid-dynamical equations, theoretical approaches to compressible turbulence are outlined in this chapter. After a brief summary of the main results of the Kolmogorov theory of incompressible turbulence, attempts to treat compressible turbulence with a similar theoretical framework are considered. In the case of isothermal gas, which plays an important role for the theory of star formation, an important aspect is the log-normal distribution of density fluctuations. Scaling relations for a strictly self-similar turbulent cascade are empirically found to break down for higher-order statistics. This is explained by the intermittency of turbulence. A heuristic model that was successfully used to predict scaling properties of incompressible and compressible turbulence is based on a hierarchy of dissipative structure with log-Poisson statistics. This leads to a simple two-parameter model for the relative scaling exponents, which can be determined through experimental measurements or numerical simulations. Apart from the general velocity statistics of compressible turbulence, the density structure of self-gravitating turbulence is an unsolved problem. Crucial questions concern the support of the gas against gravity in the highly non-linear regime and the mass distribution of gravitationally unstable clumps.

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Notes

  1. 1.

    This can be understood as a renormalization of the nominally infinite potential for an infinitely extended mass distribution with positive mean density \(\langle \rho \rangle \) to zero for \(\rho =\langle \rho \rangle \).

  2. 2.

    In the fluid dynamics literature, \(\sigma _{ij}\) often includes the pressure term \(P\delta _{ij}\). In the formulation used here, it is a sperate term.

  3. 3.

    In [12], the symbol \(\mathbf f \) is used for the force density. We keep our definition for consistency in this text.

  4. 4.

    In other contexts, the term clump may have a more specific or different meaning.

References

  1. L. Landau, E. Lifshitz, Fluid Mechanics, Course of Theoretical Physics, vol. 6, 2nd edn. (Pergamon Press, 1987)

    Google Scholar 

  2. A.G. Kritsuk, S.D. Ustyugov, M.L. Norman, P. Padoan, J. Phys. Conf. Ser. 180(1), 012020 (2009). doi:10.1088/1742-6596/180/1/012020

    Google Scholar 

  3. S. Banerjee, S Galtier, Phys. Rev. E 87(1), (2013). doi:10.1103/PhysRevE.87.013019

  4. U. Frisch, Turbulence. The legacy of A.N. Kolmogorov (Cambridge University Press, Cambridge, 1995)

    Google Scholar 

  5. A.N. Kolmogorov, Dokl. Akad. Nauk SSSR 30, 301 (1941)

    Google Scholar 

  6. M.J. Lighthill, in Gas Dynamics of Cosmic Clouds, IAU Symposium, vol. 2 (1955), pp. 1

    Google Scholar 

  7. A.G. Kritsuk, M.L. Norman, P. Padoan, R. Wagner, ApJ 665, 416 (2007). doi:10.1086/519443

    Google Scholar 

  8. Aluie, Physica D 247(1), 54–65 (2013). doi:10.1016/j.physd.2012.12.009

  9. S. Galtier, S. Banerjee, Phys. Rev. Lett 107(13), 134501 (2011). doi:10.1103/PhysRevLett.107.134501

    Article  ADS  Google Scholar 

  10. Kritsuk, Wagner, Norman, J. Fluid Mech. 729, R1 (2013). doi:10.1017/jfm.2013.342

  11. C. Federrath, On the Universality of Compressible Supersonic Turbulence. Submitted to MNRAS (2013)

    Google Scholar 

  12. G. Falkovich, I. Fouxon, Y. Oz, J. Fluid Mech 644, 465 (2010). doi:10.1017/S0022112009993429

    Google Scholar 

  13. R. Wagner, G. Falkovich, A.G. Kritsuk, M.L. Norman, J. Fluid Mech. 713, 482 (2012). doi:10.1017/jfm.2012.470

    Article  MathSciNet  ADS  Google Scholar 

  14. B.G. Elmegreen, J. Scalo, ARA &A 42, 211 (2004). doi:10.1146/annurev.astro.41.011802.094859

  15. M. Mac Low, R.S. Klessen, Rev. Mod. Phys. 76, 125 (2004). doi:10.1103/RevModPhys.76.125

    Article  ADS  Google Scholar 

  16. J. Ballesteros-Paredes, R.S. Klessen, M.M. Mac Low, E. Vazquez-Semadeni, in Protostars and Planets V, ed. by B. Reipurth, D. Jewitt, K. Keil (2007), pp. 63–80

    Google Scholar 

  17. C.F. McKee, E.C. Ostriker, ARA &A 45, 565 (2007). doi:10.1146/annurev.astro.45.051806.110602

    Article  ADS  Google Scholar 

  18. P. Hennebelle, E. Falgarone, A &A Rev. 20, 55 (2012). doi:10.1007/s00159-012-0055-y

    ADS  Google Scholar 

  19. T. Passot, E. Vázquez-Semadeni, Phys. Rev. E 58, 4501 (1998)

    Article  ADS  Google Scholar 

  20. C. Federrath, R.S. Klessen, W. Schmidt, ApJ 688, L79 (2008). doi:10.1086/595280

    Article  ADS  Google Scholar 

  21. J. Pietarila Graham, R. Cameron, M. Schüssler, ApJ 714, 1606 (2010). doi:10.1088/0004-637X/714/2/1606

    Google Scholar 

  22. A.N. Kolmogorov, J. Fluid Mech. 13, 82 (1962)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  23. Z.S. She, E. Leveque, Phys. Rev. Lett. 72, 336 (1994). doi:10.1103/PhysRevLett.72.336

    Article  ADS  Google Scholar 

  24. B. Dubrulle, Phys. Rev. Lett. 73, 959 (1994). doi:10.1103/PhysRevLett.73.959

    Article  ADS  Google Scholar 

  25. Z.S. She, E.C. Waymire, Phys. Rev. Lett. 74, 262 (1995). doi:10.1103/PhysRevLett.74.262

    Google Scholar 

  26. S. Boldyrev, ApJ 569, 841 (2002)

    Article  ADS  Google Scholar 

  27. S. Boldyrev, A. Nordlund, P. Padoan, ApJ 573, 678 (2002)

    Article  ADS  Google Scholar 

  28. J.H. Jeans, R. Soci. Lond. Philos. Trans. Ser. A 199, 1 (1902)

    Article  ADS  MATH  Google Scholar 

  29. W.B. Bonnor, MNRAS 116, 351 (1956)

    MathSciNet  ADS  Google Scholar 

  30. R. Ebert, Zeitschrift für Astrophysik 37, 217 (1955)

    ADS  MATH  Google Scholar 

  31. S. Chandrasekhar, R. Soc. London Proc. Ser. A 210, 26 (1951)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  32. S. Bonazzola, J. Heyvaerts, E. Falgarone, M. Perault, J.L. Puget, A &A 172, 293 (1987)

    ADS  Google Scholar 

  33. S. Bonazzola, M. Perault, J.L. Puget, J. Heyvaerts, E. Falgarone, J.F. Panis, J. Fluid Mech 245, 1 (1992). doi:10.1017/S0022112092000326

    Article  MathSciNet  ADS  MATH  Google Scholar 

  34. H. Biglari, P.H. Diamond, Physica D Nonlinear Phenomena 37, 206 (1989). doi:10.1016/0167-2789(89)90130-9

    Google Scholar 

  35. P.C. Clark, R.S. Klessen, I.A. Bonnell, MNRAS 379, 57 (2007). doi:10.1111/j.1365-2966.2007.11896.x

  36. P. Padoan, Å. Nordlund, ApJ 576, 870 (2002). doi:10.1086/341790

    Article  ADS  Google Scholar 

  37. P. Padoan, Å. Nordlund, A.G. Kritsuk, M.L. Norman, P.S. Li, ApJ 661, 972 (2007). doi:10.1086/516623

    Article  ADS  Google Scholar 

  38. P. Hennebelle, G. Chabrier, ApJ 684, 395 (2008). doi:10.1086/589916

    Article  ADS  Google Scholar 

  39. W. Schmidt, in Structure Formation, in the Universe, ed. by G. Chabrier. Cambridge Contemporary Astrophysics (Cambridge University Press, Cambridge, 2009), pp. 20–3

    Google Scholar 

  40. W. Schmidt, D.C. Collins, A.G. Kritsuk, MNRAS 431, 3196–3215 (2013). doi:10.1093/mnras/stt399

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  1. Institut für Astrophysik, Georg-August-Universität, Friedrich-Hund-Platz 1, 37077, Göttingen, Germany

    Wolfram Schmidt

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  1. Wolfram Schmidt
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Schmidt, W. (2014). Turbulence Theory. In: Numerical Modelling of Astrophysical Turbulence. SpringerBriefs in Astronomy. Springer, Cham. https://doi.org/10.1007/978-3-319-01475-3_1

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