Space Science Reviews

, Volume 194, Issue 1–4, pp 97–137 | Cite as

Large-Eddy Simulations of Magnetohydrodynamic Turbulence in Heliophysics and Astrophysics

  • Mark Miesch
  • William Matthaeus
  • Axel Brandenburg
  • Arakel Petrosyan
  • Annick Pouquet
  • Claude Cambon
  • Frank Jenko
  • Dmitri Uzdensky
  • James Stone
  • Steve Tobias
  • Juri Toomre
  • Marco Velli
Article

Abstract

We live in an age in which high-performance computing is transforming the way we do science. Previously intractable problems are now becoming accessible by means of increasingly realistic numerical simulations. One of the most enduring and most challenging of these problems is turbulence. Yet, despite these advances, the extreme parameter regimes encountered in space physics and astrophysics (as in atmospheric and oceanic physics) still preclude direct numerical simulation. Numerical models must take a Large Eddy Simulation (LES) approach, explicitly computing only a fraction of the active dynamical scales. The success of such an approach hinges on how well the model can represent the subgrid-scales (SGS) that are not explicitly resolved. In addition to the parameter regime, heliophysical and astrophysical applications must also face an equally daunting challenge: magnetism. The presence of magnetic fields in a turbulent, electrically conducting fluid flow can dramatically alter the coupling between large and small scales, with potentially profound implications for LES/SGS modeling. In this review article, we summarize the state of the art in LES modeling of turbulent magnetohydrodynamic (MHD) flows. After discussing the nature of MHD turbulence and the small-scale processes that give rise to energy dissipation, plasma heating, and magnetic reconnection, we consider how these processes may best be captured within an LES/SGS framework. We then consider several specific applications in heliophysics and astrophysics, assessing triumphs, challenges, and future directions.

Keywords

Turbulence Magnetohydrodynamics Simulation 

References

  1. O. Agullo, W.-C. Müller, B. Knaepen, D. Carati, Phys. Plasmas 8(7), 3502 (2001) ADSCrossRefGoogle Scholar
  2. A. Alexakis, P.D. Mininni, A. Pouquet, Phys. Rev. E 72, 046301 (2005) MathSciNetADSCrossRefGoogle Scholar
  3. O. Alexandrova, J. Saur, C. Lacombe et al., Phys. Rev. Lett. 103, 165003 (2009) ADSCrossRefGoogle Scholar
  4. J. Ambrosiano, W.H. Matthaeus, M.L. Goldstein, D. Plante, J. Geophys. Res. 93, 14383 (1988) ADSCrossRefGoogle Scholar
  5. J. Baerenzung, H. Politano, Y. Ponty, A. Pouquet, Phys. Rev. E 77, 046303 (2008a) MathSciNetADSCrossRefGoogle Scholar
  6. J. Baerenzung, H. Politano, Y. Ponty, A. Pouquet, Phys. Rev. E 78, 026310 (2008b) MathSciNetADSCrossRefGoogle Scholar
  7. J. Baerenzung, P. Mininni, A. Pouquet, D. Rosenberg, J. Atmos. Sci. 68, 2757 (2011) ADSCrossRefGoogle Scholar
  8. S.A. Balbus, Astrophys. J. 534, 420 (2000) ADSCrossRefGoogle Scholar
  9. S.A. Balbus, Astrophys. J. 562, 909 (2001) ADSCrossRefGoogle Scholar
  10. S.A. Balbus, J.F. Hawley, Astrophys. J. 376, 214 (1991) ADSCrossRefGoogle Scholar
  11. S.A. Balbus, J.F. Hawley, Rev. Mod. Phys. 70, 1 (1998) ADSCrossRefGoogle Scholar
  12. S.D. Bale, P.J. Kellogg, F.S. Mozer, T.S. Horbury, H. Reme, Phys. Rev. Lett. 94, 215002 (2005) ADSCrossRefGoogle Scholar
  13. G.K. Batchelor, The Theory of Homogeneous Turbulence (Cambridge University Press, Cambridge, 1982), 197 pp. MATHGoogle Scholar
  14. J.W. Belcher, L. Davis Jr., J. Geophys. Res. 76, 3534 (1971) ADSCrossRefGoogle Scholar
  15. A.R. Bell, Mon. Not. R. Astron. Soc. 182, 147 (1978) ADSCrossRefGoogle Scholar
  16. F. Bellet, F.S. Godeferd, J.F. Scott, C. Cambon, J. Fluid Mech. 562, 83 (2006) MathSciNetADSCrossRefMATHGoogle Scholar
  17. A. Bhattacharjee, Y.-M. Huang, H. Yang, B. Rogers, Phys. Plasmas 16, 112102 (2009) ADSCrossRefGoogle Scholar
  18. J. Birn, E. Priest, Reconnection of Magnetic Fields (Cambridge University Press, New York, 2007) CrossRefGoogle Scholar
  19. D. Biskamp, Phys. Fluids 29, 1520 (1986) ADSCrossRefMATHGoogle Scholar
  20. D. Biskamp, Magnetic Reconnection in Plasmas (Cambridge University Press, Cambridge, 2000) CrossRefMATHGoogle Scholar
  21. D. Biskamp, Magnetohydrodynamic Turbulence (Cambridge University Press, Cambridge, 2003), 310 pp. CrossRefMATHGoogle Scholar
  22. G. Bodo, F. Cattaneo, A. Ferrari, A. Mignone, P. Rossi, Astrophys. J. 739, 82 (2011) ADSCrossRefGoogle Scholar
  23. S. Boldyrev, Phys. Rev. Lett. 96, 115002 (2006) ADSCrossRefGoogle Scholar
  24. A. Brandenburg, Astrophys. J. 550, 824 (2001) ADSCrossRefGoogle Scholar
  25. A. Brandenburg, Astrophys. J. 697, 1206 (2009) ADSCrossRefGoogle Scholar
  26. A. Brandenburg, Astrophys. J. 741, 92 (2011) ADSCrossRefGoogle Scholar
  27. A. Brandenburg, Astrophys. J. 791, 12 (2014) ADSCrossRefGoogle Scholar
  28. A. Brandenburg, O. Gressel, S. Jabbari, N. Kleeorin, I. Rogachevskii, Astron. Astrophys. 562, A53 (2014), 15 pp. ADSCrossRefGoogle Scholar
  29. A. Brandenburg, K. Kemel, N. Kleeorin, I. Rogachevskii, Astrophys. J. 749, 179 (2012) ADSCrossRefGoogle Scholar
  30. A. Brandenburg, A. Nordlund, Rep. Prog. Phys. 74, 046901 (2011) ADSCrossRefGoogle Scholar
  31. A. Brandenburg, G.R. Sarson, Phys. Rev. 88, 055003 (2002) ADSGoogle Scholar
  32. A. Brandenburg, K. Subramanian, Phys. Rep. 417, 1 (2005) MathSciNetADSCrossRefGoogle Scholar
  33. A. Brandenburg, D. Sokoloff, K. Subramanian, Space Sci. Rev. 169, 123 (2012) ADSCrossRefGoogle Scholar
  34. B.P. Brown, M.S. Miesch, M.K. Browning, A.S. Brun, J. Toomre, Astrophys. J. 732, 69 (2011) ADSCrossRefGoogle Scholar
  35. C. Cambon, L. Jacquin, J. Fluid Mech. 202, 295 (1989) MathSciNetADSCrossRefMATHGoogle Scholar
  36. C. Cambon, F.S. Godeferd, B. Favier, Incorporating linear dynamics and strong anisotropy in KS. application to diffusion in rotating, stratified, MHD turbulence and to aeroacoustics, in New Approaches in Modeling Multiphase Flows and Dispersion in Turbulence, Fractal Methods and Synthetic Turbulence. ERCOFTAC Series, vol. 18 (Springer, New York, 2012) Google Scholar
  37. P. Cassak, M. Shay, Phys. Plasmas 14, 102114 (2007) ADSCrossRefGoogle Scholar
  38. F. Cattaneo, D.W. Hughes, Phys. Rev. E 54, R4532 (1996) ADSCrossRefGoogle Scholar
  39. B.D.G. Chandran, Astrophys. J. 720, 548 (2010) ADSCrossRefGoogle Scholar
  40. S. Chandrasekhar, Philos. Trans. R. Soc. Lond. Ser. A 242, 557 (1950) MathSciNetADSCrossRefMATHGoogle Scholar
  41. S. Chandrasekhar, Proc. R. Soc. Lond. Ser. A 207, 301 (1951) MathSciNetADSCrossRefGoogle Scholar
  42. P. Charbonneau, Annu. Rev. Astron. Astrophys. 52, 251–290 (2015) ADSCrossRefGoogle Scholar
  43. A.A. Chernyshov, K.V. Karelsky, A.S. Petrosyan, Russ. J. Numer. Anal. Math. Model. 21(1), 1 (2006a) MathSciNetCrossRefMATHGoogle Scholar
  44. A.A. Chernyshov, K.V. Karelsky, A.S. Petrosyan, Phys. Plasmas 13(3), 032304 (2006b) MathSciNetADSCrossRefGoogle Scholar
  45. A.A. Chernyshov, K.V. Karelsky, A.S. Petrosyan, Phys. Plasmas 13(10), 104501 (2006c) MathSciNetADSCrossRefGoogle Scholar
  46. A.A. Chernyshov, K.V. Karelsky, A.S. Petrosyan, Phys. Fluids 19(4) (2007) Google Scholar
  47. A.A. Chernyshov, K.V. Karelsky, A.S. Petrosyan, Phys. Fluids 20(8), 085106 (2008a) ADSCrossRefGoogle Scholar
  48. A.A. Chernyshov, K.V. Karelsky, A.S. Petrosyan, Astrophys. J. 686, 1137 (2008b) ADSCrossRefGoogle Scholar
  49. A.A. Chernyshov, K.V. Karelsky, A.S. Petrosyan, Theor. Comput. Fluid Dyn. 29, 451 (2009) CrossRefGoogle Scholar
  50. A.A. Chernyshov, K.V. Karelsky, A.S. Petrosyan, Phys. Plasmas 17, 102307 (2010) ADSCrossRefGoogle Scholar
  51. A.A. Chernyshov, K.V. Karelsky, A.S. Petrosyan, Flow Turbul. Combust. 89, 563 (2012) CrossRefGoogle Scholar
  52. A.A. Chernyshov, K.V. Karelsky, A.S. Petrosyan, Phys. Usp. 57, 133 (2014) CrossRefGoogle Scholar
  53. J. Cho, E.T. Vishniac, Astrophys. J. 539, 273 (2000) ADSCrossRefGoogle Scholar
  54. J.-P. Chollet, M. Lesieur, J. Atmos. Sci. 38, 2747 (1981) ADSCrossRefGoogle Scholar
  55. S. Corrsin, NACA Res. memo 58B11 (1958) Google Scholar
  56. A. Craya, Contribution à l’analyse de la turbulence associée à des vitesses moyennes. Pub. Sci. et Téch. du Ministère de l’Air (France), No. 345 (1958) Google Scholar
  57. S. Dalena, A.F. Rappazzo, P. Dmitruk, A. Greco, W.H. Matthaeus, Astrophys. J. 783, 143 (2014) ADSCrossRefGoogle Scholar
  58. W. Daughton, V. Roytershteyn, H. Karimabadi et al., Nat. Phys. 7, 539 (2011a) CrossRefGoogle Scholar
  59. W. Daughton, V. Roytershteyn, H. Karimabadi, L. Yin, B.J. Albright, S.P. Gary, K.J. Bowers, Modern Challenges in Nonlinear Plasma Physics. American Institute of Physics Conference Series, vol. 1320 (2011b), p. 144 Google Scholar
  60. A. Delache, C. Cambon, F.S. Godeferd, Phys. Fluids 26, 025104 (2014) ADSCrossRefGoogle Scholar
  61. F. Del Sordo, G. Guerrero, A. Brandenburg, Mon. Not. R. Astron. Soc. 429, 1686 (2013) ADSCrossRefGoogle Scholar
  62. G. De Stefano, O.V. Vasilyev, J. Comp. Physiol. 238, 240 (2013) ADSCrossRefMATHGoogle Scholar
  63. P. Dmitruk, D.O. Gomez, E.E. DeLuca, Astrophys. J. 505, 974 (1998) ADSCrossRefGoogle Scholar
  64. P. Dmitruk, W.H. Matthaeus, N. Seenu, Astrophys. J. 617, 667 (2004) ADSCrossRefGoogle Scholar
  65. P. Dmitruk, W.H. Matthaeus, Phys. Rev. E 76, 036305 (2007) ADSCrossRefGoogle Scholar
  66. P. Dmitruk, W.H. Matthaeus, Phys. Plasmas 16, 062304 (2009) ADSCrossRefGoogle Scholar
  67. M. Dobrowolny, A. Mangeney, P. Veltri, Phys. Rev. Lett. 45, 144 (1980) MathSciNetADSCrossRefGoogle Scholar
  68. J.F. Drake, M. Swisdak, H. Che, M.A. Shay, Nature 443, 553 (2006) ADSCrossRefGoogle Scholar
  69. G. Einaudi, M. Velli, Phys. Plasmas 6, 4146 (1999) ADSCrossRefGoogle Scholar
  70. G. Erlebacher, M. Hussaini, C. Speziale, T. Zang, J. Fluid Mech. 238, 155 (1992) ADSCrossRefMATHGoogle Scholar
  71. G. Falkovich, Phys. Fluids 6, 1411 (1994) ADSCrossRefMATHGoogle Scholar
  72. B.F.N. Favier, F.S. Godeferd, C. Cambon, A. Delache, W.J.T. Bos, J. Fluid Mech. 681, 434 (2011) MathSciNetCrossRefMATHGoogle Scholar
  73. B.F.N. Favier, F.S. Godeferd, C. Cambon, Geophys. Astrophys. Fluid Dyn. 106, 89 (2012) MathSciNetADSCrossRefGoogle Scholar
  74. C. Federrath, J. Schober, S. Bovino, D.R.G. Schleicher, Astrophys. J. Lett. 797, L19 (2014) ADSCrossRefGoogle Scholar
  75. T. Forbes, E.R. Priest, Magnetic Reconnection: MHD Theory and Applications (Cambridge University Press, New York, 2000). ISBN 978-0521033947 MATHGoogle Scholar
  76. U. Frisch, A. Pouquet, P.-L. Sulem, M. Meneguzzi, J. Mec. Theor. Appl. Suppl. 191–216 (1983) Google Scholar
  77. S. Fromang, J. Papaloizou, Astron. Astrophys. 476, 1113 (2007) ADSCrossRefMATHGoogle Scholar
  78. S. Fromang, J. Stone, Astron. Astrophys. 507, 19 (2009) ADSCrossRefMATHGoogle Scholar
  79. S. Fromang, in Role and Mechanisms of Angular Momentum Transport During the Formation and Early Evolution of Stars, ed. by P. Hennebelle, C. Charbonnel. Series EAS Publ. Ser., vol. 62 (EDP Sciences, Les Ulis, 2013), p. 95 Google Scholar
  80. S. Galtier, S.V. Nazarenko, A.C. Newell, A. Pouquet, J. Plasma Phys. 63, 447 (2000) ADSCrossRefGoogle Scholar
  81. A.V. Glazunov, V.N. Lykossov, Russ. J. Numer. Anal. Math. Model. 18, 279 (2003) MathSciNetCrossRefMATHGoogle Scholar
  82. G.A. Glatzmaier, Annu. Rev. Earth Planet. Sci. 30, 237–257 (2002) ADSCrossRefGoogle Scholar
  83. P. Goldreich, S. Sridhar, Astrophys. J. 438, 763 (1995) ADSCrossRefGoogle Scholar
  84. P. Goldreich, S. Sridhar, Astrophys. J. 485, 680 (1997) ADSCrossRefGoogle Scholar
  85. D.O. Gough, D.R. Moore, E.A. Spiegel, N.O. Weiss, Astrophys. J. 206, 536 (1976) ADSCrossRefGoogle Scholar
  86. R. Grauer, J. Krug, C. Mariani, Phys. Lett. A 195, 335 (1994) ADSCrossRefGoogle Scholar
  87. P. Grete, D.G. Vlaykov, W. Schmidt, D.R.G. Schleicher, C. Federrath, New J. Phys. 17, 023070 (2015) ADSCrossRefGoogle Scholar
  88. F.F. Grinstein, L.G. Margolin, W.J. Rider, Implicit Large Eddy Simulation: Computing Turbulent Fluid Dynamics (Cambridge University Press, Cambridge, 2011), 578 pp. MATHGoogle Scholar
  89. X. Guan, C.F. Gammie, Astrophys. J. 697, 1901 (2009) ADSCrossRefGoogle Scholar
  90. G.W. Hammett, F.W. Perkins, Phys. Rev. Lett. 64, 3019 (1990) ADSCrossRefGoogle Scholar
  91. N.E.L. Haugen, A. Brandenburg, Phys. Rev. E 70, 026405 (2004) ADSCrossRefGoogle Scholar
  92. N.E.L. Haugen, A. Brandenburg, Phys. Fluids 18, 075106 (2006) ADSCrossRefGoogle Scholar
  93. C.T. Haynes, D. Burgess, E. Camporeale, Astrophys. J. 783, 38 (2014) ADSCrossRefGoogle Scholar
  94. P. Hellinger, P.M. Trávníček, J. Geophys. Res. Space Phys. 118, 5421 (2013) ADSCrossRefGoogle Scholar
  95. J.R. Herring, Phys. Fluids 17, 859 (1974) ADSCrossRefMATHGoogle Scholar
  96. J.C. Higdon, Astrophys. J. 285, 109 (1984) ADSCrossRefGoogle Scholar
  97. J.V. Hollweg, J. Geophys. Res. 104, 14811 (1999) ADSCrossRefGoogle Scholar
  98. G.G. Howes, W. Dorland, S.C. Cowley et al., Phys. Rev. Lett. 100, 065004 (2008) ADSCrossRefGoogle Scholar
  99. A. Hubbard, A. Brandenburg, Geophys. Astrophys. Fluid Dyn. 104, 577 (2010) MathSciNetADSCrossRefGoogle Scholar
  100. R.S. Hughes, S.P. Gary, J. Wang, Geophys. Res. Lett. 41, 8681 (2014) ADSCrossRefGoogle Scholar
  101. P.S. Iroshnikov, Sov. Astron. 7, 566 (1963) MathSciNetADSGoogle Scholar
  102. A.B. Iskakov, A.A. Schekochihin, S.C. Cowley, J.C. McWilliams, M.R.E. Proctor, Phys. Rev. Lett. 98, 208501 (2007) ADSCrossRefGoogle Scholar
  103. F. Jenko, W. Dorland, M. Kotschenreuther, B.N. Rogers, Phys. Plasmas 7, 1904 (2000) ADSCrossRefGoogle Scholar
  104. J.R. Jokipii, Astrophys. J. 146, 480 (1966) ADSCrossRefGoogle Scholar
  105. C.A. Jones, Annu. Rev. Fluid Mech. 43, 583–614 (2011) ADSCrossRefGoogle Scholar
  106. B.B. Kadomtsev, O.P. Pogutse, Sov. Phys. JETP 38, 283 (1974). See also Zh. Eksp. Teor. Fiz. 65, 575 (1973) ADSGoogle Scholar
  107. Y. Kaneda, T. Ishihara, M. Yokokawa, K. Itakura, A. Uno, Phys. Fluids 15, L2 (2003) CrossRefGoogle Scholar
  108. H. Karimabadi, V. Roytershteyn, M. Wan et al., Phys. Plasmas 20, 012303 (2013) ADSCrossRefGoogle Scholar
  109. T. de Karman, L. Howarth, Proc. R. Soc. Lond. Ser. A 164, 192 (1938) ADSCrossRefGoogle Scholar
  110. A.P. Kazantsev, J. Exp. Theor. Phys. 26, 1031 (1968) ADSGoogle Scholar
  111. P.J. Käpylä, A. Brandenburg, N. Kleeorin, M.J. Mantere, I. Rogachevskii, Mon. Not. R. Astron. Soc. 422, 2465 (2012) ADSCrossRefGoogle Scholar
  112. K. Kemel, A. Brandenburg, N. Kleeorin, D. Mitra, I. Rogachevskii, Sol. Phys. 287, 293 (2013) ADSCrossRefGoogle Scholar
  113. N.I. Kleeorin, I.V. Rogachevskii, A.A. Ruzmaikin, Sov. Astron. Lett. 15(4), 274 (1989) ADSGoogle Scholar
  114. Y. Klimontovich, Phys. Usp. 40, 21 (1997) ADSCrossRefGoogle Scholar
  115. B. Knaepen, P. Moin, Center for Turbulence Research Annual Research Briefs 297 (2003) Google Scholar
  116. A.N. Kolmogorov, Dokl. Akad. Nauk SSSR 30(4), 299 (1941) ADSGoogle Scholar
  117. G. Kowal, A. Lazarian, E. Vishniac, K. Otmianowska-Mazur, Astrophys. J. 70063 (2009) Google Scholar
  118. R.H. Kraichnan, Phys. Fluids 8, 1385 (1965) MathSciNetADSCrossRefGoogle Scholar
  119. R.H. Kraichnan, Phys. Fluids 10, 1417 (1967) ADSCrossRefGoogle Scholar
  120. R.H. Kraichnan, Phys. Rev. 107, 1485 (1958) MathSciNetADSCrossRefGoogle Scholar
  121. R.H. Kraichnan, J. Fluid Mech. 59, 745 (1973) ADSCrossRefMATHGoogle Scholar
  122. F. Krause, K.H. Rädler, Mean-Field Magnetohydrodynamics and Dynamo Theory (Pergamon Press, Oxford, 1980) MATHGoogle Scholar
  123. R.H. Kraichnan, D.C. Montgomery, Phys. Rep. 43, 547 (1980) MathSciNetCrossRefGoogle Scholar
  124. C. Lamriben, P.P. Cortet, F. Moisy, Phys. Rev. Lett. 107, 024503 (2011) ADSCrossRefGoogle Scholar
  125. G. Lapenta, Phys. Rev. Lett. 100, 235001 (2008) ADSCrossRefGoogle Scholar
  126. A. Lazarian, E. Vishniac, Astrophys. J. 517, 700 (1999) ADSCrossRefGoogle Scholar
  127. A. Lazarian, L. Vlahos, G. Kowal, H. Yan, A. Beresnyak, E.M. de Gouveia Dal Pino, Space Sci. Rev. 173, 557 (2012a) ADSCrossRefGoogle Scholar
  128. A. Lazarian, G.L. Eyink, E.T. Vishniac, Phys. Plasmas 19, 012105 (2012b) ADSCrossRefGoogle Scholar
  129. E. Lee, M.E. Brachet, A. Pouquet, P.D. Mininni, D. Rosenberg, Phys. Rev. E 81, 016318 (2010) ADSCrossRefGoogle Scholar
  130. A. Leonard, Adv. Geophys. 18, 237 (1974) ADSCrossRefGoogle Scholar
  131. G. Lesur, P.-Y. Longaretti, Astron. Astrophys. 504, 309 (2009) ADSCrossRefMATHGoogle Scholar
  132. R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems (Cambridge University Press, Cambridge, 2002), 580 pp. CrossRefMATHGoogle Scholar
  133. N.F. Loureiro, A.A. Schekochihin, S.C. Cowley, Phys. Plasmas 14, 100703 (2007) ADSCrossRefGoogle Scholar
  134. N.F. Loureiro, D.A. Uzdensky, A.A. Scheckochihin, S.C. Cowley, T.A. Yousef, Mon. Not. R. Astron. Soc. 399, L146 (2009) ADSCrossRefGoogle Scholar
  135. N.F. Loureiro, R. Samtaney, A.A. Scheckochihin, D.A. Uzdensky, Phys. Plasmas 19, 042303 (2012) ADSCrossRefGoogle Scholar
  136. N.F. Loureiro, A.A. Scheckochihin, D.A. Uzdensky, Phys. Rev. E 87, 013102 (2013) ADSCrossRefGoogle Scholar
  137. S.A. Markovskii, B.J. Vasquez, C.W. Smith, J.V. Hollweg, Astrophys. J. 639, 1177 (2006) ADSCrossRefGoogle Scholar
  138. R. Marino, P.D. Mininni, D. Rosenberg, A. Pouquet, Phys. Rev. E 87, 033016 (2013) ADSCrossRefGoogle Scholar
  139. J. Mason, F. Cattaneo, S. Boldyrev, Phys. Rev. Lett. 97, 255002 (2006) ADSCrossRefGoogle Scholar
  140. W.H. Matthaeus, S.L. Lamkin, Phys. Fluids 28, 303 (1985) ADSCrossRefGoogle Scholar
  141. W.H. Matthaeus, S.L. Lamkin, Phys. Fluids 29, 2513 (1986) ADSCrossRefGoogle Scholar
  142. W.H. Matthaeus, D. Montgomery, Ann. N.Y. Acad. Sci. 357, 203 (1980) ADSCrossRefGoogle Scholar
  143. W.H. Matthaeus, A. Pouquet, P.D. Mininni, P. Dmitruk, B. Breech, Phys. Rev. Lett. 100, 085003 (2008a) ADSCrossRefGoogle Scholar
  144. W.H. Matthaeus, S. Servidio, P. Dmitruk, Phys. Rev. Lett. 101, 149501 (2008b) ADSCrossRefGoogle Scholar
  145. H. Meheut, S. Fromang, G. Lesur, M. Joos, P.-Y. Longaretti, Astron. Astrophys. (2015, submitted). arXiv:1505.05661
  146. M. Meneguzzi, H. Politano, A. Pouquet, M. Zolver, J. Comp. Physiol. 123, 32 (1996) MathSciNetADSCrossRefMATHGoogle Scholar
  147. C. Meneveau, J. Katz, Annu. Rev. Fluid Mech. 32, 1 (2000) MathSciNetADSCrossRefGoogle Scholar
  148. L.J. Milano, W.H. Matthaeus, P. Dmitruk, D.C. Montgomery, Phys. Plasmas 8, 2673 (2001) ADSCrossRefGoogle Scholar
  149. K. Miki, S. Menon, Phys. Plasmas 15, 072306 (2008) ADSCrossRefGoogle Scholar
  150. M.D. Millionshchikov, Izv. Akad. Nauk SSSR, Ser. Geogr. Geofiz. 5(4–5), 433 (1941) Google Scholar
  151. P. Mininni, E. Lee, A. Norton, J. Clyne, New J. Phys. 10, 125007 (2008) ADSCrossRefGoogle Scholar
  152. D. Mitra, S. Candelaresi, P. Chatterjee, R. Tavakol, A. Brandenburg, Astron. Nachr. 331, 130 (2010) ADSCrossRefMATHGoogle Scholar
  153. H.K. Moffatt, Magnetic Field Generation in Electrically Conducting Fluids (Cambridge University Press, Cambridge, 1978), 353 pp. Google Scholar
  154. D.C. Montgomery, L. Turner&, G. Vahala, Phys. Fluids 21, 757 (1978) ADSCrossRefMATHGoogle Scholar
  155. D.C. Montgomery, L. Turner, Phys. Fluids 24, 825 (1981) ADSCrossRefMATHGoogle Scholar
  156. D.C. Montgomery, Phys. Scr. T2/1, 83 (1982) ADSCrossRefGoogle Scholar
  157. P. Morel, A. Bañón Navarro, M. Albrecht-Marc, D. Carati, F. Merz, T. Görler, F. Jenko, Phys. Plasmas 18, 072301 (2011) ADSCrossRefGoogle Scholar
  158. P. Morel, A. Bañón Navarro, M. Albrecht-Marc, D. Carati, F. Merz, T. Görler, F. Jenko, Phys. Plasmas 19, 012311 (2012) ADSCrossRefGoogle Scholar
  159. W.-C. Müller, D. Biskamp, Turbulence and Magnetic Fields in Astrophysics (Springer, New York, 2003) Google Scholar
  160. W.-C. Müller, D. Carati, Phys. Plasmas 9, 824 (2002a) ADSCrossRefGoogle Scholar
  161. W.-C. Müller, D. Carati, Comput. Phys. Commun. 147, 344 (2002b) Google Scholar
  162. W.-C. Müller, S.K. Malapaka, A. Busse, Phys. Rev. E 85, 015302(R) (2012) ADSCrossRefGoogle Scholar
  163. N.J. Nelson, B.P. Brown, A.S. Brun, M.S. Miesch, J. Toomre, Astrophys. J. 739, L38 (2011) ADSCrossRefGoogle Scholar
  164. N.J. Nelson, B.P. Brown, A.S. Brun, M.S. Miesch, J. Toomre, Astrophys. J. 762, 73 (2013) ADSCrossRefGoogle Scholar
  165. A. Nordlund, Astron. Astrophys. 107, 1 (1982) ADSMATHGoogle Scholar
  166. A. Nordlund, Sol. Phys. 100, 209 (1985) ADSCrossRefGoogle Scholar
  167. A. Nordlund, R.F. Stein, M. Asplund, Living Rev. Sol. Phys. 6, 2 (2009). http://www.livingreviews.org/lrsp-2009-2 ADSCrossRefGoogle Scholar
  168. S.A. Orszag, Fluid Dynamics (Les Houches Summer School of Theoretical Physics, Les Houches, 1973), p. 237 Google Scholar
  169. S. Oughton, E.R. Priest, W.H. Matthaeus, J. Fluid Mech. 280, 95 (1994) ADSCrossRefMATHGoogle Scholar
  170. S. Oughton, K.H. Rädler, W.H. Matthaeus, Phys. Rev. E 56(3), 2875 (1997) MathSciNetADSCrossRefGoogle Scholar
  171. R.V. Ozmidov, Izv., Atmos. Ocean. Phys. 1(8), 853 (1965). Translated into English, Phys. Rev. E 56(3), 2875 (1997) Google Scholar
  172. T.N. Parashar, S. Servidio, B. Breech, M.A. Shay, W.H. Matthaeus, Phys. Plasmas 17, 102304 (2010) ADSCrossRefGoogle Scholar
  173. E.N. Parker, Astrophys. J. 174, 499 (1972) ADSCrossRefGoogle Scholar
  174. E.N. Parker, Cosmical Magnetic Fields (Clarendon Press/Oxford University Press, New York, 1979), 858 pp. Google Scholar
  175. T. Passot, P.L. Sulem, P. Hunana, Phys. Plasmas 19, 082113 (2012) ADSCrossRefGoogle Scholar
  176. R.B. Pelz, V. Yakhot, S.A. Orszag, L. Shtilman, E. Levich, Phys. Rev. Lett. 54, 2505 (1985) ADSCrossRefGoogle Scholar
  177. S. Perri, M.L. Goldstein, J.C. Dorelli, F. Sahraoui, Phys. Rev. Lett. 109, 191101 (2012) ADSCrossRefGoogle Scholar
  178. A. Petrosyan, A. Balogh, M.L. Goldstein, J. Leorat, E. Marsch, K. Petrovay, B. Roberts, R. von Steiger, J.C. Vial, Space Sci. Rev. 156, 135 (2010) ADSCrossRefGoogle Scholar
  179. U. Piomelli, J. Aerosp. Sci. 35, 335 (1999) Google Scholar
  180. H. Politano, A. Pouquet, Phys. Rev. 52(1), 636 (1995) ADSCrossRefGoogle Scholar
  181. S.B. Pope, Turbulent Flows (Cambridge University Press, Cambridge, 2000) CrossRefMATHGoogle Scholar
  182. A. Pouquet, Plasma Astrophysics. Series Lecture Notes in Physics, vol. 468 (1996), p. 163 CrossRefGoogle Scholar
  183. A. Pouquet, J. Baerenzung, J. Graham Pietarila Graham, P. Mininni, H. Politano, Y. Ponty, Modeling of turbulent flows in the presence of magnetic fields or rotation, in TI2009 Conference (Ste Luce), Notes on Numerical Fluid Mechanics and Multidisciplinary Design, vol. 110, ed. by M. Deville, J.-P. Sagaut, T. Hiep (2010), p. 287 Google Scholar
  184. A. Pouquet, U. Frisch, J. Léorat, J. Fluid Mech. 77, 321 (1976) ADSCrossRefMATHGoogle Scholar
  185. E. Priest, C.J. Schrijver, Sol. Phys. 190, 1 (1999) ADSCrossRefGoogle Scholar
  186. F. Pucci, M. Velli, Astrophys. J. Lett. 780, LL19 (2014) ADSCrossRefGoogle Scholar
  187. H. Qin, W.M. Tang, W.W. Lee, Phys. Plasmas 7, 4433 (2000) MathSciNetADSCrossRefGoogle Scholar
  188. E. Quataert, W. Dorland, G.W. Hammett, Astrophys. J. 577, 524 (2002) ADSCrossRefGoogle Scholar
  189. A.F. Rappazzo, M. Velli, G. Einaudi, Astrophys. J. 722, 65 (2010) ADSCrossRefGoogle Scholar
  190. A.F. Rappazzo, W.H. Matthaeus, D. Ruffolo, S. Servidio, M. Velli, Astrophys. J. 758, L14 (2012) ADSCrossRefGoogle Scholar
  191. M.M. Ray, P. Moin, J. Comp. Physiol. 96, 15 (1991) ADSCrossRefGoogle Scholar
  192. M. Rempel, Astrophys. J. 789, 132 (2014), 22 pp. ADSCrossRefGoogle Scholar
  193. H.P. Robertson, Proc. Camb. Philos. Soc. 36, 209 (1940) ADSCrossRefGoogle Scholar
  194. D.C. Robinson, M.G. Rusbridge, Phys. Fluids 14, 2499 (1971) ADSCrossRefGoogle Scholar
  195. P.L. Roe, Annu. Rev. Fluid Mech. 18, 337 (1986) MathSciNetADSCrossRefGoogle Scholar
  196. I. Rogachevskii, N. Kleeorin, Phys. Rev. E 76, 056307 (2007) MathSciNetADSCrossRefGoogle Scholar
  197. G. Rüdiger, L.L. Kitchatinov, A. Brandenburg, Astron. Astrophys. 269, 3 (2011) Google Scholar
  198. P. Sagaut, Large Eddy Simulation for Incompressible Flows: An Introduction. Series on Scientific Computation (Springer, Berlin, 2006), 556 pp. MATHGoogle Scholar
  199. P. Sagaut, C. Cambon, Homogeneous Turbulence Dynamics (Cambridge University Press, New York, 2008) CrossRefMATHGoogle Scholar
  200. R. Samtaney, N.F. Loureiro, D.A. Uzdensky, A.A. Schekochihin, S.C. Cowley, Phys. Rev. Lett. 103, 105004 (2009) ADSCrossRefGoogle Scholar
  201. F. Sahraoui, M.L. Goldstein, P. Robert, Y.V. Khotyaintsev, Phys. Rev. Lett. 102, 231102 (2009) ADSCrossRefGoogle Scholar
  202. F. Sahraoui, M.L. Goldstein, G. Belmont, P. Canu, L. Rezeau, Phys. Rev. Lett. 105, 131101 (2010) ADSCrossRefGoogle Scholar
  203. A. Salhi, T. Lehner, F. Godeferd, C. Cambon, Phys. Rev. E 85, 026301 (2012) ADSCrossRefGoogle Scholar
  204. A. Salhi, F.G. Jacobitz, K. Schneider, C. Cambon, Phys. Rev. E 89, 013020 (2014) ADSCrossRefGoogle Scholar
  205. A.A. Schekochihin, S.C. Cowley, W. Dorland et al., Astrophys. J. Suppl. 182, 310 (2009) ADSCrossRefGoogle Scholar
  206. W. Schmidt, Living Rev. Comput. Astrophys. (2015, in press). arXiv:1404.2483
  207. J.F. Scott, J. Fluid Mech. 741, 316–349 (2014) MathSciNetADSCrossRefGoogle Scholar
  208. N. Seehafer, Phys. Rev. E 53, 1283 (1996) ADSCrossRefGoogle Scholar
  209. S. Servidio, W.H. Matthaeus, P. Dmitruk, Phys. Rev. Lett. 100, 095005 (2008) ADSCrossRefGoogle Scholar
  210. S. Servidio, W.H. Matthaeus, M.A. Shay, P.A. Cassak, P. Dmitruk, Phys. Rev. Lett. 102, 115003 (2009) ADSCrossRefGoogle Scholar
  211. S. Servidio, W.H.Matthaeus.M.A. Shay, P. Dmitruk, P.A. Cassak, M. Wan, Phys. Plasmas 17, 032315 (2010) ADSCrossRefGoogle Scholar
  212. S. Servidio, F. Valentini, F. Califano, P. Veltri, Phys. Rev. Lett. 108, 045001 (2012) ADSCrossRefGoogle Scholar
  213. S. Servidio, K.T. Osman, F. Valentini, D. Perrone, F. Califano, S. Chapman, W.H. Matthaeus, P. Veltri, Astrophys. J. Lett. 781, LL27 (2014) ADSCrossRefGoogle Scholar
  214. Z.S. She, E. Leveque, Phys. Rev. Lett. 72(3), 336 (1994) ADSCrossRefGoogle Scholar
  215. J.V. Shebalin, W.H. Matthaeus, D. Montgomery, J. Plasma Phys. 29, 525 (1983) ADSCrossRefGoogle Scholar
  216. K.R. Shelnivasan, R.A. Antonia, Annu. Rev. Fluid Mech. 29, 435 (1997) ADSCrossRefGoogle Scholar
  217. D. Smith, S. Ghosh, P. Dmitruk, W.H. Matthaeus, Geophys. Res. Lett. 31, L02805 (2004) ADSGoogle Scholar
  218. E.A. Spiegel, Astron. Astrophys. 9, 323 (1971) ADSCrossRefGoogle Scholar
  219. E.A. Spiegel, Astron. Astrophys. 10, 261 (1972) ADSCrossRefGoogle Scholar
  220. J.E. Stawarz, A. Pouquet, M.E. Brachet, Phys. Rev. E 86, 036307 (2012) ADSCrossRefGoogle Scholar
  221. O. Stawicki, S.P. Gary, H. Li, J. Geophys. Res. 106, 8273 (2001) ADSCrossRefGoogle Scholar
  222. M. Steffen, H.-G. Ludwig, S. Krüß, Astron. Nachr. 123, 371 (1989) Google Scholar
  223. R.F. Stein, A.A. Nordlund, Astrophys. J. 342, L95 (1989) ADSCrossRefGoogle Scholar
  224. R.F. Stein, A.A. Nordlund, Astrophys. J. 499, 914 (1998) ADSCrossRefGoogle Scholar
  225. J.M. Stone, M.L. Norman, Astrophys. J. Suppl. Ser. 80, 753 (1992) ADSCrossRefGoogle Scholar
  226. H.R. Strauss, Phys. Fluids 19, 134 (1976) ADSCrossRefGoogle Scholar
  227. T. Stribling, W.H. Matthaeus, Phys. Fluids B 3, 1848 (1991) ADSCrossRefGoogle Scholar
  228. J.B. Taylor, Phys. Rev. Lett. 33, 1139 (1974) ADSCrossRefGoogle Scholar
  229. J.M. TenBarge, G.G. Howes, W. Dorland, Astrophys. J. 774, 139 (2013) ADSCrossRefGoogle Scholar
  230. A.C. Ting, D. Montgomery, W.H. Matthaeus, Phys. Fluids 29, 3261 (1986) ADSCrossRefGoogle Scholar
  231. S.M. Tobias, F. Cattaneo, S. Boldyrev, MHD dynamos and turbulence, in Ten Chapters in Turbulence, ed. by P. Davidson, Y. Kaneda, K.R. Sreenivasan (Cambridge University Press, Cambridge, 2013), 450 pp Google Scholar
  232. J. Toomre, J.-P. Zahn, J. Latour, E.A. Spiegel, Astrophys. J. 207, 545 (1976) MathSciNetADSCrossRefGoogle Scholar
  233. N.J. Turner, S. Fromang, C. Gammie, H. Klahr, G. Lesur, M. Wardle, X.-N. Bai, in Protostars and Planets VI, ed. by H. Beauther, R. Klessen, C. Dullemond, T. Henning (University of Arizona Press, Tucson, 2014, in press). arXiv:1401.7306
  234. A. Usmanov, M.L. Goldstein, W.H. Matthaeus, Astrophys. J. 788, 43 (2014) ADSCrossRefGoogle Scholar
  235. D.A. Uzdensky, N.F. Loureiro, eprint (2014). arXiv:1411.4295
  236. M. van Noort, L. Rouppe van der Voort, M.G. Löfdahl, Sol. Phys. 228 (2005) Google Scholar
  237. M.K. Verma, Phys. Rep. 401, 229 (2004) MathSciNetADSCrossRefGoogle Scholar
  238. D. Verscharen, E. March, U. Motschmann, J. Müller, Phys. Plasmas 19, 022305 (2012) ADSCrossRefGoogle Scholar
  239. A. Vögler, Shelyag, M. Schüssler, F. Cattaneo, T. Emonet, Astron. Nachr. 429, 335 (2005) Google Scholar
  240. S. Wedemeyer, S. Freytag, B. Steffen, M. Ludwig, H.-G. Holweger, Astron. Nachr. 414, 1121 (2004) Google Scholar
  241. F. Waleffe, Phys. Fluids A 4, 350 (1992) MathSciNetADSCrossRefMATHGoogle Scholar
  242. M. Wan, S. Oughton, S. Servidio, W.H. Matthaeus, Phys. Plasmas 17, 082308 (2010) ADSCrossRefGoogle Scholar
  243. M. Wan, W.H. Matthaeus, H. Karimabadi et al., Phys. Rev. Lett. 109, 195001 (2012a) ADSCrossRefGoogle Scholar
  244. M. Wan, S. Oughton, S. Servidio, W.H. Matthaeus, J. Fluid Mech. 697, 296 (2012b) MathSciNetADSCrossRefMATHGoogle Scholar
  245. M. Wan, W.H. Matthaeus, S. Servidio, S. Oughton, Phys. Plasmas 20, 042307 (2013) ADSCrossRefGoogle Scholar
  246. M. Wan, A.F. Rappazzo, W.H. Matthaeus, S. Servidio, S. Oughton, Astrophys. J. 797, 63 (2014) ADSCrossRefGoogle Scholar
  247. P.R. Woodward, D.H. Porter, S. Anderson, T. Fuchs, in Numerical Modeling of Space Plasma Flows, ed. by N.V. Pogorelov, G.P. Zank. ASP Conf. Ser., vol. 359 (2006a), pp. 97–110 Google Scholar
  248. P.R. Woodward, D.H. Porter, S. Anderson, T. Fuchs, F. Herwig, J. Phys. Conf. Ser. 46, 370–384 (2006b) MathSciNetADSCrossRefGoogle Scholar
  249. P. Wu, S. Perri, K. Osman et al., Astrophys. J. Lett. 763, LL30 (2013a) ADSCrossRefGoogle Scholar
  250. P. Wu, M. Wan, W.H. Matthaeus, M.A. Shay, M. Swisdak, Phys. Rev. Lett. 111, 121105 (2013b) ADSCrossRefGoogle Scholar
  251. J.C. Wyngaard, J. Atmos. Sci. 61, 1816 (2004) ADSCrossRefGoogle Scholar
  252. M. Yamada, R. Kulsrud, H. Ji, Rev. Mod. Phys. 82, 603 (2010) ADSCrossRefGoogle Scholar
  253. N. Yokoi, A. Yoshizawa, Phys. Fluids A 5, 464 (1993) MathSciNetADSCrossRefMATHGoogle Scholar
  254. N. Yokoi, R. Rubinstein, A. Yoshizawa, F. Hamba, J. Turbul. 9, N37 (2008) MathSciNetADSCrossRefGoogle Scholar
  255. N. Yokoi, K. Higashimori, M. Hoshino, Phys. Plasmas 20, 122310 (2013) ADSCrossRefGoogle Scholar
  256. N. Yokoi, Geophys. Astrophys. Fluid Dyn. 107, 114 (2013) MathSciNetADSCrossRefGoogle Scholar
  257. A. Yoshizawa, Phys. Fluids 30, 1089 (1987) ADSCrossRefMATHGoogle Scholar
  258. T.A. Zang, R.B. Dahlburg, J.P. Dahlburg, Phys. Fluids A 4(1), 127 (1992) ADSCrossRefMATHGoogle Scholar
  259. G.P. Zank, W.H. Matthaeus, J. Plasma Phys. 48, 85 (1992) ADSCrossRefGoogle Scholar
  260. O. Zeman, Phys. Fluids 6, 3221 (1994) ADSCrossRefMATHGoogle Scholar
  261. V. Zhdankin, D.A. Uzdensky, J.C. Perez, S. Boldyrev, Astrophys. J. 771, 124 (2013) ADSCrossRefGoogle Scholar
  262. V. Zhdankin, D.A. Uzdensky, S. Boldyrev, Phys. Rev. Lett. 114, 065002 (2015) ADSCrossRefGoogle Scholar
  263. Y. Zhou, G. Vahala, J. Plasma Phys. 45, 239 (1991) ADSCrossRefGoogle Scholar
  264. Y. Zhou, W.H. Matthaeus, P. Dmitruk, Rev. Mod. Phys. 76, 1015 (2004) ADSCrossRefGoogle Scholar
  265. S.J. Zweben, C.R. Menyuk, R.J. Taylor, Phys. Rev. Lett. 42, 1720 (1979) ADSCrossRefGoogle Scholar
  266. E.G. Zweibel, M. Yamada, Annu. Rev. Astron. Astrophys. 47, 291 (2009) ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  • Mark Miesch
    • 1
  • William Matthaeus
    • 2
  • Axel Brandenburg
    • 3
  • Arakel Petrosyan
    • 4
    • 5
  • Annick Pouquet
    • 6
    • 7
  • Claude Cambon
    • 8
  • Frank Jenko
    • 9
  • Dmitri Uzdensky
    • 10
  • James Stone
    • 11
  • Steve Tobias
    • 12
  • Juri Toomre
    • 13
  • Marco Velli
    • 14
  1. 1.HAO/NCARBoulderUSA
  2. 2.Dept. of Physics and AstronomyUniversity of DelawareNewarkUSA
  3. 3.NORDITAKTH Royal Inst. Tech. and Stockholm Univ.StockholmSweden
  4. 4.Space Research Inst.Russian Academy of SciencesMoscowRussia
  5. 5.Moscow Institute of Physics and Technology (State University)DolgoprudnyRussia
  6. 6.NCARBoulderUSA
  7. 7.Dept. of Applied Math.Univ. of ColoradoBoulderUSA
  8. 8.Laboratoire de Mécanique des Fluides et d’AcoustiqueÉcole Centrale de LyonEcully cedexFrance
  9. 9.Dept. of Physics and AstronomyUCLALos AngelesUSA
  10. 10.CIPS, Physics Dept., UCB-390Univ. ColoradoBoulderUSA
  11. 11.Dept. of Astrophysical SciencesPrinceton Univ.PrincetonUSA
  12. 12.Dept. of Applied MathematicsUniversity of LeedsLeedsUK
  13. 13.JILA and Dept. of Astrophysical and Planetary SciencesUniv. of ColoradoBoulderUSA
  14. 14.Jet Propulsion LaboratoryCalifornia Inst. Tech.PasadenaUSA

Personalised recommendations