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von Kármán self-preservation hypothesis for magnetohydrodynamic turbulence and its consequences for universality

Published online by Cambridge University Press:  06 March 2012

Minping Wan ,
Sean Oughton ,
Sergio Servidio  and
William H. Matthaeus
Minping Wan
Affiliation:
Bartol Research Institute and Department of Physics and Astronomy, University of Delaware, Newark, DE 19716, USA
Sean Oughton
Affiliation:
Department of Mathematics, University of Waikato, Hamilton 3240, New Zealand
Sergio Servidio
Affiliation:
Dipartimento di Fisica, Universita’ della Calabria, I-87036 Cosenza, Italy
William H. Matthaeus*
Affiliation:
Bartol Research Institute and Department of Physics and Astronomy, University of Delaware, Newark, DE 19716, USA
*
†Email address for correspondence: whm@udel.edu
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Abstract

We argue that the hypothesis of preservation of shape of dimensionless second- and third-order correlations during decay of incompressible homogeneous magnetohydrodynamic (MHD) turbulence requires, in general, at least two independent similarity length scales. These are associated with the two Elsässer energies. The existence of similarity solutions for the decay of turbulence with varying cross-helicity implies that these length scales cannot remain in proportion, opening the possibility for a wide variety of decay behaviour, in contrast to the simpler classic hydrodynamics case. Although the evolution equations for the second-order correlations lack explicit dependence on either the mean magnetic field or the magnetic helicity, there is inherent implicit dependence on these (and other) quantities through the third-order correlations. The self-similar inertial range, a subclass of the general similarity case, inherits this complexity so that a single universal energy spectral law cannot be anticipated, even though the same pair of third-order laws holds for arbitrary cross-helicity and magnetic helicity. The straightforward notion of universality associated with Kolmogorov theory in hydrodynamics therefore requires careful generalization and reformulation in MHD.

JFM classification

Turbulent Flows: Homogeneous turbulence MHD and Electrohydrodynamics: MHD turbulence Turbulent Flows: Turbulence theory
Type
Papers
Information
Journal of Fluid Mechanics , Volume 697 , 25 April 2012 , pp. 296 - 315
Copyright
Copyright © Cambridge University Press 2012

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References

1. Alexakis, A. 2007 Nonlocal phenomenology for anisotropic magnetohydrodynamic turbulence. Astrophys. J. 667, L93–L96.CrossRefGoogle Scholar
2. Aluie, H. & Eyink, G. L. 2010 Scale locality of magnetohydrodynamic turbulence. Phys. Rev. Lett. 104, 081101.CrossRefGoogle ScholarPubMed
3. Barenblatt, G. I. 1996 Scaling, Self-similarity, and Intermediate Asymptotics. Cambridge University Press.CrossRefGoogle Scholar
4. Batchelor, G. K. 1970 The Theory of Homogeneous Turbulence. Cambridge University Press.Google Scholar
5. Beresnyak, A. 2011 Spectral slope and Kolmogorov constant of MHD turbulence. Phys. Rev. Lett. 106, 075001.CrossRefGoogle ScholarPubMed
6. Beresnyak, A. & Lazarian, A. 2008 Strong imbalanced turbulence. Astrophys. J. 682, 1070–1075.CrossRefGoogle Scholar
7. Beresnyak, A. & Lazarian, A. 2010 Scaling laws and diffuse locality of balanced and imbalanced magnetohydrodynamic turbulence. Astrophys. J. 722, L110–L113.CrossRefGoogle Scholar
8. Biskamp, D. 1994 Cascade models for magnetohydrodynamic turbulence. Phys. Rev. E 50, 2702.CrossRefGoogle ScholarPubMed
9. Boldyrev, S. 2006 Spectrum of magnetohydrodynamic turbulence. Phys. Rev. Lett. 96, 115002.CrossRefGoogle ScholarPubMed
10. Boldyrev, S., Perez, J. C., Borovsky, J. E. & Podesta, J. J. 2011 Spectral scaling laws in magnetohydrodynamic turbulence simulations and in the solar wind. Astrophys. J. 741, L19.CrossRefGoogle Scholar
11. Bondeson, A. 1985 Cascade properties of shear Alfvén turbulence. Phys. Fluids 28, 2406–2411.CrossRefGoogle Scholar
12. Carbone, V., Sorriso-Valvo, L. & Marino, R. 2009 On the turbulent energy cascade in anisotropic magnetohydrodynamic turbulence. Europhys. Lett. 88, 25001.CrossRefGoogle Scholar
13. Carbone, V. & Veltri, P. 1990 A shell model for anisotropic magnetohydrodynamic turbulence. Geophys. Astrophys. Fluid Dyn. 52, 153–181.CrossRefGoogle Scholar
14. Chandran, B. D. G. 2008 Strong anisotropic MHD turbulence with cross helicity. Astrophys. J. 685, 646–658.CrossRefGoogle Scholar
15. Chandrasekhar, S. 1951a The invariant theory of isotropic turbulence in magneto-hydrodynamics. Proc. R. Soc. Lond. A 204, 435.Google Scholar
16. Chandrasekhar, S. 1951b The invariant theory of isotropic turbulence in magneto-hydrodynamics. II. Proc. R. Soc. Lond. A 207, 301.Google Scholar
17. Cho, J. & Vishniac, E. T. 2000 The anisotropy of magnetohydrodynamic Alfvénic turbulence. Astrophys. J. 539, 273–282.CrossRefGoogle Scholar
18. Davidson, P. A. 2009 The role of angular momentum conservation in homogeneous turbulence. J. Fluid Mech. 632, 329–358.CrossRefGoogle Scholar
19. Davidson, P. A. 2010 On the decay of Saffman turbulence subject to rotation, stratification or an imposed magnetic field. J. Fluid Mech. 663, 268–292.CrossRefGoogle Scholar
20. Dobrowolny, M., Mangeney, A. & Veltri, P. 1980 Fully developed anisotropic hydromagnetic turbulence in interplanetary space. Phys. Rev. Lett. 45, 144–147.CrossRefGoogle Scholar
21. Domaradzki, J. A., Teaca, B. & Carati, D. 2010 Locality properties of the energy flux in magnetohydrodynamic turbulence. Phys. Fluids 22, 051702.CrossRefGoogle Scholar
22. Dryden, H. L. 1943 A review of the statistical theory of turbulence. Q. Appl. Maths 1, 7.CrossRefGoogle Scholar
23. Frisch, U. 1995 Turbulence. Cambridge University Press.CrossRefGoogle Scholar
24. Galtier, S. 2008 von Kármán–Howarth equations for Hall magnetohydrodynamic flows. Phys. Rev. E 77, 015302.CrossRefGoogle ScholarPubMed
25. Galtier, S., Politano, H. & Pouquet, A. 1997 Self-similar energy decay in magnetohydrodynamic turbulence. Phys. Rev. Lett. 79, 2807.CrossRefGoogle Scholar
26. Galtier, S., Zienicke, E., Politano, H. & Pouquet, A. 1999 Parametric investigation of self-similar decay laws in MHD turbulent flows. J. Plasma Phys. 61, 507.CrossRefGoogle Scholar
27. Goldreich, P. & Sridhar, S. 1995 Toward a theory of interstellar turbulence: II. Strong Alfvénic turbulence. Astrophys. J. 438, 763–775.CrossRefGoogle Scholar
28. Grappin, R. 1986 Onset and decay of two-dimensional magnetohydrodynamic turbulence with velocity-magnetic field correlation. Phys. Fluids 29, 2433–2443.CrossRefGoogle Scholar
29. Grappin, R., Frisch, U., Léorat, J. & Pouquet, A. 1982 Alfvénic fluctuations as asymptotic states of MHD turbulence. Astron. Astrophys. 105, 6–14.Google Scholar
30. Grappin, R. & Müller, W.-C. 2010 Scaling and anisotropy in magnetohydrodynamic turbulence in a strong mean magnetic field. Phys. Rev. E 82 (2), 026406.CrossRefGoogle Scholar
31. Grappin, R., Pouquet, A. & Léorat, J. 1983 Dependence of MHD turbulence spectra on the velocity field-magnetic field correlation. Astron. Astrophys. 126, 51–58.Google Scholar
32. Hossain, M., Gray, P. C., Pontius, D. H. Jr., Matthaeus, W. H. & Oughton, S. 1995 Phenomenology for the decay of energy-containing eddies in homogeneous MHD turbulence. Phys. Fluids 7, 2886–2904.CrossRefGoogle Scholar
33. Iroshnikov, R. S. 1963 Turbulence of a conducting fluid in a strong magnetic field. Astron. Zh. 40, 742 (Sov. Astron. 7, 566–571 (1964)).Google Scholar
34. de Kármán, T. & Howarth, L. 1938 On the statistical theory of isotropic turbulence. Proc. R. Soc. Lond. Ser. A 164, 192–215.CrossRefGoogle Scholar
35. von Kármán, T. & Lin, C. C. 1949 On the concept of similarity in the theory of isotropic turbulence. Rev. Mod. Phys. 21, 516.CrossRefGoogle Scholar
36. Kolmogorov, A. N. 1941a Local structure of turbulence in an incompressible viscous fluid at very high Reynolds numbers. Dokl. Akad. Nauk SSSR 30, 301–305 (Reprinted in Proc. R. Soc. Lond. Ser. A 434, 9–13 (1991)).Google Scholar
37. Kolmogorov, A. N. 1941b On degeneration of isotropic turbulence in an incompressible viscous liquid. C.R. Acad. Sci. 31, 538–540.Google Scholar
38. Kolmogorov, A. N. 1941c Dissipation of energy in the locally isotropic turbulence. C.R. Acad. Sci. 32, 16 (Reprinted in Proc. R. Soc. Lond. Ser. A 434, 15–17 (1991)).Google Scholar
39. Kraichnan, R. H. 1965 Inertial-range spectrum of hydromagnetic turbulence. Phys. Fluids 8, 1385–1387.CrossRefGoogle Scholar
40. Kraichnan, R. H. 1973 Helical turbulence and absolute equilibrium. J. Fluid Mech. 59, 745.CrossRefGoogle Scholar
41. Kraichnan, R. H. & Panda, R. 1988 Depression of nonlinearity in decaying isotropic turbulence. Phys. Fluids 31, 2395–2397.CrossRefGoogle Scholar
42. Lee, E., Brachet, M. E., Pouquet, A., Mininni, P. D. & Rosenberg, D. 2010 Lack of universality in decaying magnetohydrodynamic turbulence. Phys. Rev. E 81, 016318.CrossRefGoogle ScholarPubMed
43. Lithwick, Y., Goldreich, P. & Sridhar, S. 2007 Imbalanced strong MHD turbulence. Astrophys. J. 655, 269–274.CrossRefGoogle Scholar
44. MacBride, B. T., Forman, M. A. & Smith, C. W. 2005 Turbulence and third moment of fluctuations: Kolmogorov’s law and its MHD analogues in the solar wind. In Proceedings Solar Wind 11 – Soho 16 Connecting Sun and Heliosphere, vol. SP-592 (ed. Fleck, B., Zurbuchen, T.H. & Lacoste, H. ). pp. 613–616. European Space Agency.Google Scholar
45. Matthaeus, W. H. 1999 Magnetic helicity and homogeneous turbulence models. In Geophysical Monograph 111. Proceedings of Magnetic Helicity in Space and Laboratory Plasmas (ed. Brown, M. R., Canfield, R. C. & Pevtsov, A. A. ). pp. 247–255. AGU.Google Scholar
46. Matthaeus, W. H. & Montgomery, D. 1980 Selective decay hypothesis at high mechanical and magnetic Reynolds numbers. Ann. N. Y. Acad. Sci. 357, 203–222.CrossRefGoogle Scholar
47. Matthaeus, W. H. & Montgomery, D. 1984 Dynamic alignment and selective decay in MHD. In Statistical Physics and Chaos in Fusion Plasmas (ed. Horton, C. W. Jr. & Reichl, L. E. ), pp. 285–291. Wiley.Google Scholar
48. Matthaeus, W. H., Oughton, S., Pontius, D. & Zhou, Y. 1994 Evolution of energy containing turbulent eddies in the solar wind. J. Geophys. Res. 99, 19267–19287.CrossRefGoogle Scholar
49. Matthaeus, W. H., Pouquet, A., Mininni, P. D., Dmitruk, P. & Breech, B. 2008 Rapid alignment of velocity and magnetic field in magnetohydrodynamic turbulence. Phys. Rev. Lett. 100, 085003.CrossRefGoogle ScholarPubMed
50. Matthaeus, W. H. & Smith, C. 1981 Structure of correlation tensors in homogeneous anisotropic turbulence. Phys. Rev. A 24, 2135–2144.CrossRefGoogle Scholar
51. Matthaeus, W. H. & Zhou, Y. 1989 Extended inertial range phenomenology of magnetohydrodynamic turbulence. Phys. Fluids B 1, 1929–1931.CrossRefGoogle Scholar
52. Milano, L. J., Matthaeus, W. H., Dmitruk, P. & Montgomery, D. C. 2001 Local anisotropy in incompressible magnetohydrodynamic turbulence. Phys. Plasmas 8, 2673–2681.CrossRefGoogle Scholar
53. Mininni, P. D. 2011 Scale interactions in magnetohydrodynamic turbulence. Ann. Rev. Fluid Mech. 43 (1), 377–397.CrossRefGoogle Scholar
54. Mininni, P., Alexakis, A. & Pouquet, A. 2005 Shell-to-shell energy transfer in magnetohydrodynamics. II. Kinematic dynamo. Phys. Rev. E 72, 046302.CrossRefGoogle ScholarPubMed
55. Montgomery, D. C. 1982 Major disruption, inverse cascades, and the Strauss equations. Phys. Scr. T2/1, 83–88.CrossRefGoogle Scholar
56. Montgomery, D., Turner, L. & Vahala, G. 1979 Most probable states in magnetohydrodynamics. J. Plasma Phys. 21, 239.CrossRefGoogle Scholar
57. Obukhov, A. M. 1941 On the energy distribution in the spectrum of a turbulent flow. Dokl. Akad. Nauk SSSR 32, 22–24 (C.R. (Dokl.) Acad. Sci. URSS 32, 19 (1963)).Google Scholar
58. Oughton, S., Priest, E. R. & Matthaeus, W. H. 1994 The influence of a mean magnetic field on three-dimensional MHD turbulence. J. Fluid Mech. 280, 95–117.CrossRefGoogle Scholar
59. Perez, J. C. & Boldyrev, S. 2009 Role of cross-helicity in magnetohydrodynamic turbulence. Phys. Rev. Lett. 102, 025003.CrossRefGoogle ScholarPubMed
60. Podesta, J. J. 2008 Laws for third-order moments in homogeneous anisotropic incompressible magnetohydrodynamic turbulence. J. Fluid Mech. 609, 171–194.CrossRefGoogle Scholar
61. Podesta, J. J., Forman, M. A. & Smith, C. W. 2007 Anisotropic form of third-order moments and relationship to the cascade rate in axisymmetric magnetohydrodynamic turbulence. Phys. Plasmas 14 (9), 092305.CrossRefGoogle Scholar
62. Politano, H., Gomez, T. & Pouquet, A. 2003 von Kármán–Howarth relationship for helical magnetohydrodynamic flows. Phys. Rev. E 68 (2), 026315.CrossRefGoogle ScholarPubMed
63. Politano, H. & Pouquet, A. 1998a Dynamical length scales for turbulent magnetized flows. Geophys. Res. Lett. 25, 273–276.CrossRefGoogle Scholar
64. Politano, H. & Pouquet, A. 1998b von Kármán–Howarth equation for magnetohydrodynamics and its consequences on third-order longitudinal structure and correlation functions. Phys. Rev. E 57, R21.CrossRefGoogle Scholar
65. Politano, H., Pouquet, A. & Sulem, P. L. 1995 Current and vorticity dynamics in three-dimensional magnetohydrodynamic turbulence. Phys. Plasmas 2, 2931.CrossRefGoogle Scholar
66. Pouquet, A., Brachet, M.-E., Lee, E., Mininni, P., Rosenberg, D. & Uritsky, V. 2011 Lack of universality in MHD turbulence, and the possible emergence of a new paradigm? In Astrophysical Dynamics: From Stars to Galaxies, Proceedings IAU Symposium S271 , vol. 6. pp. 304–316. Cambridge University Press.Google Scholar
67. Pouquet, A., Frisch, U. & Léorat, J. 1976 Strong MHD helical turbulence and the nonlinear dynamo effect. J. Fluid Mech. 77, 321–354.CrossRefGoogle Scholar
68. Pouquet, A., Meneguzzi, M. & Frisch, U. 1986 Growth of correlations in magnetohydrodynamic turbulence. Phys. Rev. A 33, 4266–4275.CrossRefGoogle ScholarPubMed
69. Robertson, H. P. 1940 The invariant theory of isotropic turbulence. Proc. Camb. Phil. Soc. 36, 209.CrossRefGoogle Scholar
70. Robinson, D. C. & Rusbridge, M. G. 1971 Structure of turbulence in the zeta plasma. Phys. Fluids 14, 2499–2511.CrossRefGoogle Scholar
71. Schekochihin, A. A., Cowley, S. C., Hammett, G. W., Maron, J. L. & McWilliams, J. C. 2002 A model of nonlinear evolution and saturation of the turbulent MHD dynamo. New J. Phys. 4, 84.1–84.22.CrossRefGoogle Scholar
72. Schekochihin, A. A., Cowley, S. C. & Yousef, T. A. 2008 MHD turbulence: nonlocal, anisotropic, nonuniversal? In Computational Physics and New Perspectives in Turbulence (ed. Kaneda, Y. ), vol. 4. pp. 347–354. Springer.Google Scholar
73. Servidio, S., Matthaeus, W. H. & Dmitruk, P. 2008 Depression of nonlinearity in decaying isotropic MHD turbulence. Phys. Rev. Lett. 100, 095005.CrossRefGoogle ScholarPubMed
74. Shebalin, J. V., Matthaeus, W. H. & Montgomery, D. 1983 Anisotropy in MHD turbulence due to a mean magnetic field. J. Plasma Phys. 29, 525–547.CrossRefGoogle Scholar
75. Smith, C. W. 1981 The structure of axisymmetric turbulence. PhD thesis, College of William and Mary, Williamsburg, VA 23185.Google Scholar
76. Sorriso-Valvo, L., Marino, R., Carbone, V., Noullez, A., Lepreti, F., Veltri, P., Bruno, R., Bavassano, B. & Pietropaolo, E. 2007 Observation of inertial energy cascade in interplanetary space plasma. Phys. Rev. Lett. 99, 115001.CrossRefGoogle ScholarPubMed
77. Speziale, C. G. & Bernard, P. S. 1992 The energy decay in self-preserving isotropic turbulence revisited. J. Fluid Mech. 241, 645–667.CrossRefGoogle Scholar
78. Stribling, T. & Matthaeus, W. H. 1991 Relaxation processes in a low order three-dimensional magnetohydrodynamics model. Phys. Fluids B 3, 1848.CrossRefGoogle Scholar
79. Tennekes, H. & Lumley, J. L. 1972 A First Course in Turbulence. MIT.CrossRefGoogle Scholar
80. Ting, A. C., Matthaeus, W. H. & Montgomery, D. 1986 Turbulent relaxation processes in magnetohydrodynamics. Phys. Fluids 29, 3261.CrossRefGoogle Scholar
81. Tobias, S. M. & Cattaneo, F. 2008 Limited role of spectra in dynamo theory: Coherent versus random dynamos. Phys. Rev. Lett. 101, 125003.CrossRefGoogle ScholarPubMed
82. Wan, M., Servidio, S., Oughton, S. & Matthaeus, W. H. 2009 The third-order law for increments in magnetohydrodynamic turbulence with constant shear. Phys. Plasmas 16.CrossRefGoogle Scholar
83. Zank, G. P. & Matthaeus, W. H. 1992 The equations of reduced magnetohydrodynamics. J. Plasma Phys. 48, 85–100.CrossRefGoogle Scholar
84. Zhou, Y. & Oughton, S. 2011 Nonlocality and the critical Reynolds numbers of the minimum state magnetohydrodynamic turbulence. Phys. Plasmas 18, 072304.CrossRefGoogle Scholar