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Study on Magnetohydrodynamic Turbulence and Its Astrophysical Applications pp 37–83Cite as

Small-Scale Turbulent Dynamo

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Abstract

By following the Kazantsev theory and taking into account both microscopic and turbulent diffusion of magnetic fields, we develop a unified treatment of the kinematic and nonlinear stages of turbulent dynamo and study the dynamo process for a full range of magnetic Prandtl number \(P_m\) and ionization fractions. We find a striking similarity between the dependence of dynamo behavior on \(P_m\) in a conducting fluid and \(\mathcal {R}\) (a function of ionization fraction) in partially ionized gas. In a weakly ionized medium, the kinematic stage is largely extended, including not only exponential growth but a new regime of dynamo characterized by linear-in-time growth of magnetic field strength, and the resulting magnetic energy is much higher than the kinetic energy carried by viscous-scale eddies. Unlike the kinematic stage, the subsequent nonlinear stage is unaffected by microscopic diffusion processes and has a universal linear-in-time growth of magnetic energy with the growth rate as a constant fraction 3 / 38 of the turbulent energy transfer rate, showing a good agreement with earlier numerical results. Applying the analysis to the first stars and galaxies, we find that the kinematic stage is able to generate a field strength only an order of magnitude smaller than the final saturation value. But the generation of large-scale magnetic fields can only be accounted for by the relatively inefficient nonlinear stage and requires longer time than the free-fall time. It suggests that magnetic fields may not have played a dynamically important role during the formation of the first stars. This chapter is based on Xu and Lazarian (ApJ 833:215, 2016, [1]), Xu and Lazarian (New J Phys 19:065005, 2017, [2]).

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Notes

  1. 1.

    Richardson diffusion [39] was initially introduced for hydrodynamic turbulence and is fully consistent with the Kolmogorov theory of turbulence. The explosive separation of magnetic field lines in MHD turbulence conforms to Richardson diffusion, which implies the breakdown of the flux-conservation constraint in MHD turbulence and can be used to recover the Lazarian and Vishniac [38] theory on turbulent reconnection [40, 41].

  2. 2.

    This feature can also be understood from a different perspective called frequency mismatching [21, 59]. For the magnetic fluctuations at scales smaller than the equipartition scale, their Alfvén frequencies \(kV_A\) exceed and mismatch with the turnover rate of the equipartition-scale eddies. As a result, growth of magnetic energy at these scales is no longer possible.

    The numerical testing of the Goldreich and Sridhar [61] model of MHD turbulence was influenced by the simulations that suffer from the bottleneck effect [62]. The recent high-resolution MHD simulations in Beresnyak [63] confirmed the Goldreich and Sridhar [61] scaling.

  3. 3.

    The evolution of \(k_p\) was discussed in e.g. Beresnyak et al. [34] in terms of the change of equipartition scale in turbulent shock precursor dynamo. Our study above provides the analytical derivation from the first principles.

  4. 4.

    As pointed out in Goldreich and Sridhar [69], Lazarian [70], due to the reduction of the viscosity perpendicular to magnetic field, the magnetic field structure formed in the sub-viscous region may not be preserved.

  5. 5.

    In spite of the same formulae for \(\mathcal {E}_\text {cr}\) and \(t_\text {cr}\), the viscosity involved in cases of fully and partially ionized gases are different.

  6. 6.

    A similar conclusion is true for the magnetic field amplification within present-day super-Alfvénic molecular clouds. In such clouds the kinetic energy exceeds the magnetic energy over a broad range of scales. To amplify the magnetic energy up to equipartition on the scale of cloud size, it requires around 6 turbulent crossing times of the cloud (Eq. 2.98), which is longer than the cloud lifetime of \(1{-}2\) crossing times [92].

  7. 7.

    It is worthwhile noticing that even for the folded magnetic fields, Schekochihin et al. [98] claimed that their interaction with the Alfvénic turbulence may lead to unwinding of the folds and further energy transport to larger scales, until eventual saturation with the Alfvénic spectrum of magnetic energy peaking at the outer scale of turbulence.

  8. 8.

    For the stars formed in the magnetized interstellar medium of the first galaxies, we do not rule out the possible magnetic regulation on their formation process.

References

  1. Xu S, Lazarian A (2016) ApJ 833:215

    Article  ADS  Google Scholar 

  2. Xu S, Lazarian A (2017) New J Phys 19:065005

    Article  Google Scholar 

  3. Reiners A (2012) Living Rev Sol. Phys. 9:1

    Article  ADS  Google Scholar 

  4. Beck R (2012) Space Sci. Rev. 166:215

    Article  ADS  Google Scholar 

  5. Neronov A, Semikoz D, Banafsheh M (2013). arXiv:1305.1450

  6. Bernet ML, Miniati F, Lilly SJ, Kronberg PP, Dessauges-Zavadsky M (2008) Nature 454:302

    Article  ADS  Google Scholar 

  7. Murphy EJ (2009) ApJ 706:482

    Article  ADS  Google Scholar 

  8. Hammond AM, Robishaw T, Gaensler BM (2012). arXiv:1209.1438

  9. Turner MS, Widrow LM (1988) Phys Rev D 37:2743

    Article  ADS  Google Scholar 

  10. Biermann L (1950) Zeitschrift Naturforschung Teil A 5:65

    ADS  Google Scholar 

  11. Lazarian A (1992) A&A 264:326

    ADS  Google Scholar 

  12. Schlickeiser R, Shukla PK (2003) ApJ 599:L57

    Article  ADS  Google Scholar 

  13. Medvedev MV, Silva LO, Fiore M, Fonseca RA, Mori WB (2004) J Korean Astron Soc 37:533

    Article  Google Scholar 

  14. Xu H, O’Shea BW, Collins DC, Norman ML, Li H, Li S (2008) ApJ 688:L57

    Article  ADS  Google Scholar 

  15. Draine BT (2011) Physics of the interstellar and intergalactic medium

    Google Scholar 

  16. Xu S, Lazarian A, Yan H (2015a) ApJ 810:44

    Article  ADS  Google Scholar 

  17. Xu S, Yan H, Lazarian A (2016) ApJ 826:166

    Article  ADS  Google Scholar 

  18. Xu S, Zhang B (2017) ApJ 835:2

    Article  ADS  Google Scholar 

  19. Batchelor GK (1950) R Soc Lond Proc Ser A 201:405

    Article  ADS  Google Scholar 

  20. Kazantsev AP (1968) Sov J Exp Theor Phys 26:1031

    ADS  Google Scholar 

  21. Kulsrud RM, Anderson SW (1992) ApJ 396:606

    Article  ADS  Google Scholar 

  22. Abel T, Bryan GL, Norman ML (2002) Science 295:93

    Article  ADS  Google Scholar 

  23. Greif TH, Johnson JL, Klessen RS, Bromm V (2008) MNRAS 387:1021

    Article  ADS  Google Scholar 

  24. Brandenburg A, Subramanian K (2005) Phys Rep 417:1

    Article  ADS  MathSciNet  Google Scholar 

  25. Ruzmaikin AA, Sokolov DD (1981) Sov Astron Lett 7:388

    ADS  Google Scholar 

  26. Novikov VG, Ruzmaikin AA, Sokoloff DD (1983) Sov Phys JETP 58:527

    Google Scholar 

  27. Subramanian K (1997). arXiv:astro-ph/9708216

  28. Vincenzi D (2001). arXiv:physics/0106090

  29. Schekochihin AA, Boldyrev SA, Kulsrud RM (2002) ApJ 567:828

    Google Scholar 

  30. Boldyrev S, Cattaneo F (2004) Phys. Rev. Lett. 92:144501

    Article  ADS  Google Scholar 

  31. Haugen NE, Brandenburg A, Dobler W (2004) Phys Rev E 70:016308

    Article  ADS  Google Scholar 

  32. Schekochihin AA, Maron JL, Cowley SC, McWilliams JC (2002) ApJ 576:806

    Google Scholar 

  33. Cho J, Vishniac ET, Beresnyak A, Lazarian A, Ryu D (2009) ApJ 693:1449

    Article  ADS  Google Scholar 

  34. Beresnyak A, Jones TW, Lazarian A (2009) ApJ 707:1541

    Article  ADS  Google Scholar 

  35. Beresnyak A (2012) Phys Rev Lett 108:035002

    Article  ADS  Google Scholar 

  36. Beresnyak A, Lazarian A (2015) Lazarian A, de Gouveia Dal Pino EM, Melioli C (eds) Astrophysics and space science library, vol 407, p 163

    Google Scholar 

  37. Schekochihin AA, Cowley SC, Hammett GW, Maron JL, McWilliams JC (2002) New J Phys 4:84

    Google Scholar 

  38. Lazarian A, Vishniac ET (1999) ApJ 517:700

    Article  ADS  Google Scholar 

  39. Richardson LF (1926) Proc R Soc Lond Ser A 110:709

    Article  ADS  Google Scholar 

  40. Eyink GL (2010) Phys Rev E 82:046314

    Article  ADS  MathSciNet  Google Scholar 

  41. Eyink GL, Lazarian A, Vishniac ET (2011) ApJ 743:51

    Article  ADS  Google Scholar 

  42. Lazarian A (2005) Magnetic fields in the universe: from laboratory and stars to primordial structures. In: de Gouveia dal Pino EM, Lugones G, Lazarian A (eds) American institute of physics conference series, vol 784, pp 42–53

    Google Scholar 

  43. Santos-Lima R, Lazarian A, de Gouveia Dal Pino EM, Cho J (2010) ApJ 714, 442

    Google Scholar 

  44. Li PS, McKee CF, Klein RI (2015) MNRAS, 452, 2500

    Google Scholar 

  45. González-Casanova DF, Lazarian A, Santos-Lima R (2016) ApJ 819:96

    Article  ADS  Google Scholar 

  46. Lazarian A, Esquivel A, Crutcher R (2012) ApJ 757:154

    Google Scholar 

  47. Roberts PH, Glatzmaier GA (2000) Rev Mod Phys 72:1081

    Article  ADS  Google Scholar 

  48. Schober J, Schleicher D, Federrath C, Glover S, Klessen RS, Banerjee R (2012) ApJ 754:99

    Article  ADS  Google Scholar 

  49. Schober J, Schleicher DRG, Klessen RS (2013) A&A 560:A87

    Article  ADS  Google Scholar 

  50. Federrath C, Sur S, Schleicher DRG, Banerjee R, Klessen RS (2011) ApJ 731:62

    Article  ADS  Google Scholar 

  51. Xu H, Li H, Collins DC, Li S, Norman ML (2011) ApJ 739:77

    Article  ADS  Google Scholar 

  52. Zeldovich IB, Ruzmaikin AA, Sokolov DD (eds) (1983) Magnetic fields in astrophysics, vol 3

    Google Scholar 

  53. Kleeorin NI, Ruzmaikin AA, Sokoloff DD (1986) Guyenne TD, Zeleny LM (eds) Plasma astrophysics, vol 251 (ESA Special Publication)

    Google Scholar 

  54. Ruzmaikin A, Sokolov D, Shukurov A (1989) MNRAS 241:1

    Article  ADS  Google Scholar 

  55. Schekochihin A, Cowley S, Maron J, Malyshkin L (2002) Phys Rev E 65:016305

    Google Scholar 

  56. Ott E (1998) Phys Plasmas 5:1636

    Article  ADS  MathSciNet  Google Scholar 

  57. Kinney RM, Chandran B, Cowley S, McWilliams JC (2000) ApJ 545:907

    Article  ADS  Google Scholar 

  58. Biermann L, Schlüter A (1951) Phys Rev 82:863

    Article  ADS  Google Scholar 

  59. Kulsrud RM, Cen R, Ostriker JP, Ryu D (1997) ApJ 480:481

    Article  ADS  Google Scholar 

  60. Subramanian K (1999) Phys Rev Lett 83:2957

    Article  ADS  Google Scholar 

  61. Goldreich P, Sridhar S (1995) ApJ 438:763

    Article  ADS  Google Scholar 

  62. Beresnyak A, Lazarian A (2010) ApJ 722:L110

    Google Scholar 

  63. Beresnyak A (2014) ApJ 784:L20

    Article  ADS  Google Scholar 

  64. Eyink G et al (2013) Nature 497:466

    Google Scholar 

  65. Lazarian A, Eyink G, Vishniac E, Kowal G (2015) Philos Trans R Soc Lond Ser A 373:20140144

    Google Scholar 

  66. Miniati F, Beresnyak A (2015) Nature 523:59

    Article  ADS  Google Scholar 

  67. Spitzer L (1956) Physics of fully ionized gases

    Google Scholar 

  68. Braginskii SI (1965) Rev Plasma Phys 1:205

    ADS  Google Scholar 

  69. Goldreich P, Sridhar S (2006) ApJ 640, L159

    Google Scholar 

  70. Lazarian A (2007) SINS-small ionized and neutral structures in the diffuse interstellar medium. In: Haverkorn M, Goss WM (eds) Astronomical society of the pacific conference series, vol 365, p 324

    Google Scholar 

  71. Moffatt HK (1961) J Fluid Mech 11:625

    Article  ADS  MathSciNet  Google Scholar 

  72. Cho J, Lazarian A, Vishniac ET (2002) ApJ 566:L49

    Google Scholar 

  73. Cho J, Lazarian A, Vishniac ET (2003) ApJ 595:812

    Google Scholar 

  74. Lazarian A, Vishniac ET, Cho J (2004) ApJ 603:180

    Article  ADS  Google Scholar 

  75. Schekochihin AA, Cowley SC, Taylor SF, Maron JL, McWilliams JC (2004) ApJ 612:276

    Article  ADS  Google Scholar 

  76. Cattaneo F (1997) SCORe’96: solar convection and oscillations and their relationship. In: Pijpers FP, Christensen-Dalsgaard J, Rosenthal CS (eds) Astrophysics and space science library, vol 225, pp 201–222

    Google Scholar 

  77. Cattaneo F (1999) Motions in the solar atmosphere. In: Hanslmeier A., Messerotti M (eds) Astrophysics and space science library, vol 239, pp 119–137

    Google Scholar 

  78. Kulsrud R, Pearce WP (1969) ApJ 156:445

    Article  ADS  Google Scholar 

  79. Shu FH (1992) The physics of astrophysics. Volume II: gas dynamics

    Google Scholar 

  80. Brandenburg A, Lazarian A (2013) Space Sci Rev 178:163

    Article  ADS  Google Scholar 

  81. Lithwick Y, Goldreich P (2001) ApJ 562:279

    Article  ADS  Google Scholar 

  82. Cho J, Lazarian A (2002) Phys Rev Lett 88:245001

    Article  ADS  Google Scholar 

  83. Cho J, Lazarian A (2003) MNRAS 345:325

    Article  ADS  Google Scholar 

  84. Kowal G, Lazarian A (2010) ApJ 720:742

    Google Scholar 

  85. Parker EN (1957) J Geophys Res 62:509

    Article  ADS  Google Scholar 

  86. Sweet PA (1958) Obs. 78:30

    ADS  Google Scholar 

  87. Maron J, Goldreich P (2001) ApJ 554:1175

    Article  ADS  Google Scholar 

  88. Cho J, Lazarian A, Vishniac ET (2002) ApJ 564:291

    Google Scholar 

  89. Schleicher DRG, Banerjee R, Sur S, Arshakian TG, Klessen RS, Beck R, Spaans M (2010) A&A 522:A115

    Article  ADS  Google Scholar 

  90. Draine BT, Roberge WG, Dalgarno A (1983) ApJ 264:485

    Article  ADS  Google Scholar 

  91. Vranjes J, Krstic PS (2013) A&A 554:A22

    Google Scholar 

  92. Elmegreen BG (2000) ApJ 530:277

    Article  ADS  Google Scholar 

  93. Bromm V, Yoshida N (2011) ARA&A 49:373

    Article  ADS  Google Scholar 

  94. Crutcher RM, Wandelt B, Heiles C, Falgarone E, Troland TH (2010) ApJ 725:466

    Article  ADS  Google Scholar 

  95. Kraichnan RH, Nagarajan S (1967) Phys Fluids 10:859

    Article  ADS  Google Scholar 

  96. Sur S, Federrath C, Schleicher DRG, Banerjee R, Klessen RS (2012) MNRAS 423:3148

    Article  ADS  Google Scholar 

  97. Chou H (2001) ApJ 556:1038

    Article  ADS  Google Scholar 

  98. Schekochihin AA, Cowley SC, Hammett GW, Maron JL, McWilliams JC (2002). arXiv:astro-ph/0207151

  99. Murgia M, Govoni F, Feretti L, Giovannini G, Dallacasa D, Fanti R, Taylor GB, Dolag K (2004) A&A 424:429

    Article  ADS  Google Scholar 

  100. Iskakov AB, Schekochihin AA, Cowley SC, McWilliams JC, Proctor MRE (2007) Phys Rev Lett 98:208501

    Article  ADS  Google Scholar 

  101. Schekochihin AA, Iskakov AB, Cowley SC, McWilliams JC, Proctor MRE, Yousef TA (2007) New J Phys 9:300

    Article  Google Scholar 

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  1. University of Wisconsin-Madison, Madison, WI, USA

    Siyao Xu

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Xu, S. (2019). Small-Scale Turbulent Dynamo. In: Study on Magnetohydrodynamic Turbulence and Its Astrophysical Applications. Springer Theses. Springer, Singapore. https://doi.org/10.1007/978-981-13-7515-6_2

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