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Mesoscopic Transport Events and the Breakdown of Fick’s Law for Turbulent Fluxes

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Abstract

This paper presents a pedagogical review of the physics of mesoscopic transport events and their role in the breakdown of Fick’s Law for turbulent transport in magnetically confined plasma. It is now clear that the conventional picture of localized turbulence and quasi-linear calculation of fluxes fails to address and account for the phenomenology of tokamak transport. One key issue is the observed departure from the expected gyro-Bohm transport scaling. The causes of this breakdown of Fickian thinking include turbulent avalanching and pulse propagation (turbulence spreading). Both are mesoscopic transport events, and both tend to de-localize the flux–gradient relation. Turbulence spreading is the process of self-scattering and expansion of a slug or other local exciton of turbulence. Spreading is described by theoretically-motivated, phenomenological reaction–diffusion models for the turbulence activity (intensity) field, much in the spirit of Ginzburg–Landau theory. Such models imply that spreading will occur by propagation of intensity fronts. After discussing the basic theory, this paper presents several critical tests of turbulence spreading models using gyrokinetic simulation. Applications include rho-star scaling, penetration of transport barriers and core-edge coupling. Relevant experiment–theory comparisons are addressed, as well. Avalanching refers to a process whereby correlated topplings of nearby localized cells overturn sequentially and drive a burst of transport. Avalanching is a process intrinsic to systems that support a broad range of scales l between a cell size Δ and system size L, i.e. Δ < l < L. Avalanching is also a natural way to produce transport events on scales that exceed the cell size or correlation length. Therefore, the PDF (probability distribution function) of avalanches as a function of l is a crucial quantity, necessary for predicting confinement in a system like ITER, with a very large-scale separation between L and Δ. Avalanching emerged from the theory of selforganized criticality but is a more general phenomenon. The paper traces the intellectual prehistory of avalanching through the advent of self-organized criticality. Special focus is devoted to reduced continuum models of avalanching. The physics of avalanching in confined plasma is discussed in detail, via several multi-faceted comparisons to flux-driven fluid and gyrokinetic simulations. The dominance of bursty, large transport events in the flux is identified. Evidence for avalanching in basic and confinement experiments is summarized. The paper concludes with sections on selected special topics, a discussion of the relation between turbulence spreading and avalanching, and a list of possible future directions. Throughout the paper, an effort is made to set fusion theory and phenomenology in the context of ideas discussed in the broader scientific community.

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References

  1. F. Wagner et al., Phys. Rev. Lett. 49, 1408 (1982).

    ADS  Google Scholar 

  2. P. A. Davidson, J. Turb. 1, N6 (2000).

    ADS  Google Scholar 

  3. P. H. Diamond, S-I. Itoh, K. Itoh and T. S. Hahm, Plasma Phys. Control. Fusion 47, R35 (2005).

    Google Scholar 

  4. Z. B. Guo et al., submitted (2018).

    Google Scholar 

  5. G. I. Barenblatt, Similarity, Self-similarity, and Intermediate Asymptotics (New York and London: Consultant Bureau, 1979).

    MATH  Google Scholar 

  6. L. G. Loitsyansky, Centr. Aero. Hydrodyn. Inst. Moscow, Rep. no. 440(Trans. NACA Tech. Memo. 1079) (1939).

    Google Scholar 

  7. L. D. Landau and E. M. Lifshitz, Fluid Mechanics, 1st ed. (Oxford: Pergamon, 1959), p. 141.

    Google Scholar 

  8. Y. Pomeau, Phys. D: Nonlinear Phenom. 23, 3 (1986).

    ADS  Google Scholar 

  9. Y. Pomeau, C. R. Mecanique 343, 210 (2015).

    ADS  Google Scholar 

  10. G. B. Whitham, Linear and Nonlinear Waves (Wiley- Interscience, New York, 1999).

    Google Scholar 

  11. S. M. Kaye et al., Nucl. Fusion 47, 499 (2007).

    ADS  Google Scholar 

  12. R. Nazikian, K. Shinohara, G. J. Kramer, E. Valeo, K. Hill, T. S. Hahm, G. Rewoldt, S. Ide, Y. Koide, Y. Oyama, H. Shirai and W. Tang, Phys. Rev. Lett. 94, 135002 (2005).

    ADS  Google Scholar 

  13. X. Garbet, L. Laurent, A. Samain and J. Chinardet, Nucl. Fusion 34, 963 (1994).

    ADS  Google Scholar 

  14. T. S. Hahm, P. H. Diamond, Z. Lin, K. Itoh and S-I. Itoh, Plasma Phys. Control. Fusion 46, A323 (2004).

    Google Scholar 

  15. E-J. Kim, P. H. Diamond, M. Malkov, T. S. Hahm, K. Itoh, S-I. Itoh, S. Champeaux, I. Gruzinov, O. Gurcan, C. Holland, M. N. Rosenbluth and A. Smolyakov, Nucl. Fusion 43, 961 (2003).

    ADS  Google Scholar 

  16. O. Gurcan, P. H. Diamond, T. S. Hahm and Z. Lin, Phys. Plasmas 12, 032303 (2005).

    ADS  Google Scholar 

  17. H. Sugama and M. Wakatani, J. Phys. Soc. Jpn. 61, 3166 (1992).

    ADS  Google Scholar 

  18. P. H. Diamond, V. B. Lebedev, D. E. Newman, B. A. Carreras, T. S. Hahm, W. M. Tang, G. Rewoldt and K. Avinash, Phys. Rev. Lett. 78, 1472 (1997).

    ADS  Google Scholar 

  19. D. E. Newman, B. A. Carreras, D. Lopez-Bruna, P. H. Diamond and V. B. Lebedev, Phys. Plasmas 5, 938 (1998).

    ADS  Google Scholar 

  20. M. A. Malkov, P. H. Diamond and M. N. Rosenbluth, Phys. Plasmas 8, 5073 (2001).

    ADS  Google Scholar 

  21. D. del-Castillo-Negrete, Phys. Plasmas 13, 082308 (2006).

    ADS  MathSciNet  Google Scholar 

  22. T. S. Hahm, P. H. Diamond, Z. Lin, G. Rewoldt, O. Gurcan and S. Ethier, Phys. Plasmas 12, 090903 (2005).

    Google Scholar 

  23. H. P. Furth, J. Kileen and M. N. Rosenbluth, Phys. Fluids 6, 459 (1963).

    ADS  Google Scholar 

  24. R. D. Sydora, V. K. Decyk and J. M. Dawson, Plasma Phys. Control. Fusion 38, A281 (1996).

    ADS  Google Scholar 

  25. Z. Lin, S. Ethier, T. S. Hahm and W. M. Tang, Phys. Rev. Lett. 88, 195004 (2002).

    ADS  Google Scholar 

  26. V. B. Lebedev and P. H. Diamond, Phys. Plasmas 4, 1087 (1997).

    ADS  MathSciNet  Google Scholar 

  27. S-I. Itoh and K. Itoh, J. Phys. Soc. Japan 69, 408 (2000).

    ADS  Google Scholar 

  28. Y. Sarazin, X. Garbet, Ph. Ghendrih and S. Benkadda, Phys. Plasmas 7, 1085 (2000).

    Google Scholar 

  29. X. Garbet, Y. Sarazin, F. Imbeaux, P. Ghendrih, C. Bourdelle, O. D. Gurcan and P. H. Diamond, Phys. Plasmas 14, 122305 (2007).

    ADS  Google Scholar 

  30. L. Villard, S. J. Allfrey, A. Bottino, M. Brunetti, G. L. Falchetto, V. Grandgirard, R. Hatzky, J. Nuhrenberg, A. G. Peeters, O. Sauter, S. Sorge and J. Vaclavik, Nucl. Fusion 44, 172 (2004).

    ADS  Google Scholar 

  31. L. Chen, R. B. White and F. Zonca, Phys. Rev. Lett. 92, 075004 (2004).

    ADS  Google Scholar 

  32. L. Chen, Z. Lin and R. B. White, Phys. Plasmas 7, 3129 (2000).

    ADS  Google Scholar 

  33. X. Garbet, private communication (2002).

    Google Scholar 

  34. B. B. Kadomtsev, Plasma Turbulence (Academic, New York, 1965).

    Google Scholar 

  35. R. A. Fisher, Ann. Eugenics 7, 353 (1937).

    Google Scholar 

  36. A. Kolmogoroff, I. Petrovsky and N. Piscounoff, Clin. Cancer Res. 1, 1 (1937).

    Google Scholar 

  37. V. Naulin, A. H. Nielsen and J. Juul Rasmussen, Phys. Plasmas 12, 122306 (2005).

    ADS  Google Scholar 

  38. J. Juul Rasmussen, V. Naulin, P. Mantica, J. S. Lonnroth, V. Parail and JET-EFDA Contributors, in Proceedings of the 33rd EPS Conference on Plasma Physics, (Rome, 19–23 June 2006 ECA), Vol. 301, p. 1076.

  39. Z. H. Wang, P. H. Diamond, O. D. Gurcan, X. Garbet and X. G. Wang, Nucl. Fusion 51, 073009 (2011).

    ADS  Google Scholar 

  40. Z. B. Guo and P. H. Diamond, Phys. Plasmas 24, 100705 (2017).

    Google Scholar 

  41. S. Inagaki et al. and the LHD Experiment Group, Nucl. Fusion 53, 113006 (2013).

    ADS  Google Scholar 

  42. R. Heinonen and P. H. Diamond, submitted (2018).

    Google Scholar 

  43. O. D. Gurcan, P. H. Diamond and T. S. Hahm, Phys. Plasmas 13, 052306 (2006).

    ADS  Google Scholar 

  44. R. V. Budny et al., Phys. Plasmas 7, 5038 (2000).

    ADS  Google Scholar 

  45. G. R. McKee et al., Nucl. Fusion 41, 1235 (2001).

    ADS  Google Scholar 

  46. P. Hennequin, R. Sabot, C. Honore, G. T. Hoang, X. Garbet, A. Truc, C. Fenzi and A. Quemeneur, Plasma Phys. Control. Fusion 46, B121 (2004).

    Google Scholar 

  47. Z. Lin, T. S. Hahm, W. W. Lee, W. M. Tang and R. B. White, Science 281, 1835 (1998).

    ADS  Google Scholar 

  48. X. Garbet, Y. Idomura, L. Villard and T. H. Watanabe, Nucl. Fusion 50, 043002 (2010).

    ADS  Google Scholar 

  49. E. A. Frieman and L. Chen, Phys. Fluids 25, 502 (1982).

    ADS  Google Scholar 

  50. T. S. Hahm, Phys. Fluids 31, 2670 (1988).

    ADS  Google Scholar 

  51. S. E. Parker, H. E. Mynick, M. Artun, J. C. Cummings, V. Decyk, J. V. Kepner, W. W. Lee and W. M. Tang, Phys. Plasmas 3, 1959 (1996).

    ADS  Google Scholar 

  52. Y. Kishimoto, T. Tajima, W. Horton, M. J. LeBrun and J. Y. Kim, Phys. Plasmas 3, 1289 (1996).

    ADS  Google Scholar 

  53. W. W. Lee and R. Santoro, Phys. Plasmas 4, 169 (1997).

    ADS  Google Scholar 

  54. Y. Idomura, M. Wakatani and S. Tokuda, Phys. Plasmas 7, 3551 (2000).

    ADS  Google Scholar 

  55. Z. Lin and T. S. Hahm, Phys. Plasmas 11, 1099 (2004).

    ADS  MathSciNet  Google Scholar 

  56. A. M. Dimits et al., Phys. Plasmas 7, 969 (2000).

    ADS  Google Scholar 

  57. B. F. McMillan, X. Lapillonne, S. Brunner, L. Villard, S. Jolliet, A. Bottino, T. Gorler and F. Jenko, Phys. Rev. Lett. 105, 155001 (2010).

    ADS  Google Scholar 

  58. S. Jolliet, A. Bottino, P. Angelino, R. Hatzky, T. M. Tran, B. F. McMillan, O. Sauter, K. Appert, Y. Idomura and L. Villard, Comp. Phys. Comm. 177, 409 (2007).

    ADS  Google Scholar 

  59. T. Gorler, Ph. D. Thesis, Universitat Ulm, 2009.

    Google Scholar 

  60. J. Candy and R. E. Waltz, J. Comput. Phys. 186, 545 (2003).

    ADS  MathSciNet  Google Scholar 

  61. M. Kotschenreuther, G. Rewoldt and W.M. Tang, Comput. Phys. Commun. 88, 128 (1995).

    ADS  Google Scholar 

  62. J. Candy, R. E. Waltz and W. Dorland, Phys. Plasmas 11, L25 (2004).

    Google Scholar 

  63. Z. Lin, S. Ethier, T. S. Hahm and W. M. Tang, Plasma Sci. Technol. 14, 1125 (2012).

    ADS  Google Scholar 

  64. R. E. Waltz and J. Candy, Phys. Plasmas 12, 072303 (2005).

    ADS  Google Scholar 

  65. J. M. Kwon, S. Yi, T. Rhee, P. H. Diamond, K. Miki, T. S. Hahm, J. Y. Kim, O. D. Gurcan and C. McDevitt, Nucl. Fusion 52, 013004 (2012).

    ADS  Google Scholar 

  66. S. Yi, J. M. Kwon, P. H. Diamond and T. S. Hahm, Phys. Plasmas 21, 092509 (2014).

    ADS  Google Scholar 

  67. S. Yi, J. M. Kwon, P. H. Diamond and T. S. Hahm, Nucl. Fusion 55, 092002 (2015).

    ADS  Google Scholar 

  68. P. Mantica et al., Phys. Rev. Lett. 107, 135004 (2011).

    ADS  Google Scholar 

  69. J. W. Hughes, D. A. Mossessian, A. E. Hubbard, B. LaBombard and E. S. Marmar, Phys. Plasmas 9, 3019 (2002).

    ADS  Google Scholar 

  70. J. G. Cordey et al., Plasma Phys. Control. Fusion 36, A267 (1994).

    ADS  Google Scholar 

  71. S. V. Neudatchin, T. Takizuka, H. Shirai, T. Fujita, A. Isayama, Y. Kamada, Y. Koide, T. Suzuki and S. Takeji, Plasma Phys. Control. Fusion 44, A383 (2002).

    ADS  Google Scholar 

  72. B. B. Kadomtsev, Plasma Phys. Control. Fusion 34, 1931 (1992).

    ADS  Google Scholar 

  73. Z. Lin, T. S. Hahm, W. W. Lee, W. M. Tang and P. H. Diamond, Phys. Rev. Lett. 83, 3645 (1999).

    ADS  Google Scholar 

  74. B. LaBombard et al. and the Alcator Group, Nucl. Fusion 44, 1047 (2004).

    ADS  Google Scholar 

  75. G. Rewoldt and W. M. Tang, Phys. Fluids B 2, 318 (1990).

    ADS  Google Scholar 

  76. K. H. Burrell, Phys. Plasmas 4, 1499 (1997).

    ADS  MathSciNet  Google Scholar 

  77. W. X. Wang, T. S. Hahm, W. W. Lee, G. Rewoldt, J. Manickam and W. M. Tang, Phys. Plasmas 14, 072306 (2007).

    ADS  Google Scholar 

  78. T. S. Hahm and K. H. Burrell, Phys. Plasmas 2, 1648 (1995).

    ADS  Google Scholar 

  79. M. Yagi, T. Ueda, S-I. Itoh, M. Azumi, K. Itoh, P. H. Diamond and T. S. Hahm, Plasma Phys. Control. Fusion 48, A409 (2006).

    Google Scholar 

  80. W. Deng and Z. Lin, Phys. Plasmas 16, 102503 (2009).

    ADS  Google Scholar 

  81. C. H. Ma, X. Q. Xu, P. W. Xi and T. Y. Xia, Phys. Plasmas 22, 010702 (2015).

    Google Scholar 

  82. A. Ishizawa and N. Nakajima, Phys. Plasmas 14, 040702 (2007).

    Google Scholar 

  83. A. Ishizawa and N. Nakajima, Nucl. Fusion 49, 055015 (2009).

    ADS  Google Scholar 

  84. E. Poli, A. Bottino and A. G. Peeters, Nucl. Fusion 49, 075010 (2009).

    ADS  Google Scholar 

  85. E. Poli, A. Bottino, W. A. Hornsby, A. G. Peeters, T. Ribeiro, B. D. Scott and M. Siccinio, Plasma Phys. Control. Fusion 52, 124021 (2010).

    ADS  Google Scholar 

  86. K. Ida et al. and LHD Experimental Group, Phys. Rev. Lett. 88, 015002 (2001).

    Google Scholar 

  87. M. J. Choi, J. Kim, J-M. Kwon, H. K. Park, Y. In, W. Lee, K. D. Lee, G. S. Yun, J. Lee, M. Kim, W-H. Ko, J. H. Lee, Y. S. Park, Y-S. Na, N. C. Luhmann Jr. and B. H. Park, Nucl. Fusion 57, 126058 (2017).

    ADS  Google Scholar 

  88. K. Ida, T. Kobayashi, M. Ono, T. E. Evans, G. R. Mc- Kee and M. E. Austin, Phys. Rev. Lett. 120, 245001 (2018).

    ADS  Google Scholar 

  89. M. Jiang, Y. Xu, W. Chen, X. T. Ding, Z. B. Shi, W. L. Zhong, X. Q. Ji, P. W. Shi, J. Q. Li, Z. C. Yang, B. S. Yuan, Y. Liu, Q. W. Yang, M. Xu and HL-2A team, submitted to Phys. Rev. Lett. (2018).

    Google Scholar 

  90. E. J. Synakowski et al., Nucl. Fusion 39, 1733 (1999).

    ADS  Google Scholar 

  91. E. Mazzucato et al., Phys. Rev. Lett. 77, 3145 (1996).

    ADS  Google Scholar 

  92. G. R. McKee, R. J. Fonck, D. K. Gupta, D. J. Schlossberg, M. W. Shafer, R. L. Boivin, W. Solomon, Plasma Fusion Res. 2, S1025 (2007).

    Google Scholar 

  93. T. Estrada, C. Hidalgo and T. Happel, Nucl. Fusion 51, 032001 (2011).

    ADS  Google Scholar 

  94. P.H. Diamond et al., presented in 6th Asia-Pacific Transport Working Group Meeting (Seoul, Korea, 2016)

    Google Scholar 

  95. G. K. Zipf, Human Behavior and the Principle of Least Effort (Cambridge, Massachusetts: Addison-Wesley, 1949).

    Google Scholar 

  96. H. E. Hurst, Trans. Am. Soc. Civil Eng. 116, 770 (1951).

    Google Scholar 

  97. H. E. Hurst, Proc. Inst. ivil Eng., Part 1, 519 (1956).

    Google Scholar 

  98. H. E. Hurst, R. P. Black and Y. M. Simaika, Long-Term Storage, and Experimental Study (London, Constable, 1965).

    Google Scholar 

  99. B. B. Mandelbrot, The Fractal Geometry of Nature, (Freeman, New York, 1983).

    Google Scholar 

  100. P. Bak, C. Tang and K. Wiesenfeld, Phys. Rev. Lett. 59, 381 (1987).

    ADS  Google Scholar 

  101. P.H. Diamond et al., presented in 9th Festival de Theorie (Aix-en-Provence, France, 2017)

    Google Scholar 

  102. B. B. Mandelbrot and J. R. Wallis, Water Resources Res. 4, 909 (1968).

    ADS  Google Scholar 

  103. G. Boffetta, A. Mazzino and A. Vulpiani, J. Phys. A 41, 363001 (2008)

    MathSciNet  Google Scholar 

  104. E. W. Montroll and G. H. Weiss, J. Math. Phys. 6, 167 (1965).

    ADS  Google Scholar 

  105. L. P. Kadanoff, S. R. Nagel, L. Wu and S. Zhou, Phys. Rev. A 39, 6524 (1989).

    ADS  Google Scholar 

  106. D. E. Newman, B. A. Carreras, P. H. Diamond and T. S. Hahm, Phys. Plasmas 3, 1858 (1996).

    ADS  Google Scholar 

  107. I. Gruzinov, P. H. Diamond and M. N. Rosenbluth, Phys. Rev. Lett. 89, 255001 (2002).

    ADS  Google Scholar 

  108. I. Gruzinov, P. H. Diamond and M. N. Rosenbluth, Phys. Plasmas 10, 569 (2003).

    ADS  Google Scholar 

  109. T. Hwa and M. Kardar, Phys. Rev. A 45, 7002 (1992).

    ADS  Google Scholar 

  110. P. H. Diamond and T. S. Hahm, Phys. Plasmas 2, 3640 (1995).

    ADS  Google Scholar 

  111. D. Shvarts, U. Alon, D. Ofer, R. L. McCrory and C. P. Verdon, Phys. Plasmas 2, 2465 (1995).

    ADS  Google Scholar 

  112. S. Chandrasekhar, Rev. Mod. Phys. 15, 1 (1943).

    ADS  Google Scholar 

  113. Y. Kosuga, P. H. Diamond, G. Dif-Pradalier and O. D. Gurcan, Phys. Plasmas 21, 055701 (2014).

    ADS  Google Scholar 

  114. Y. Kosuga, P. H. Diamond and O. D. Gurcan, Phys. Rev. Lett. 110, 105002 (2013).

    ADS  Google Scholar 

  115. L. Gil and D. Sornette, Phys. Rev. Lett. 76, 3991 (1996).

    ADS  Google Scholar 

  116. K. Barada, T. L. Rhodes, K. H. Burrell, L. Zeng, L. Bardoczi, Xi Chen, C. M. Muscatello and W. A. Peebles, Phys. Rev. Lett. 120, 135002 (2018).

    Google Scholar 

  117. V. B. Lebedev and P. H. Diamond, Phys. Plasmas 4, 1087 (1997).

    ADS  MathSciNet  Google Scholar 

  118. T. Rhee, J. M. Kwon, P. H. Diamond and W. W. Xiao, Phys. Plasmas 19, 022505 (2012).

    ADS  Google Scholar 

  119. B. A. Carreras, D. Newman, E. Lynch and P. H. Diamond, Phys. Plasmas 3, 2903 (1996).

    ADS  Google Scholar 

  120. X. Garbet and R. E. Waltz, Phys. Plasmas 5, 2836 (1998).

    ADS  Google Scholar 

  121. X. Garbet, Y. Sarazin, P. Beyer, P. Ghendrih, R. E. Waltz, M. Ottaviani and S. Benkadda, Nucl. Fusion 39, 2063 (1999).

    ADS  Google Scholar 

  122. Y. Sarazin and Ph. Ghendrih, Phys. Plasmas 5, 4214 (1998).

    MathSciNet  Google Scholar 

  123. P. Beyer, S. Benkadda, X. Garbet and P. H. Diamond, Phys. Rev. Lett. 85, 4892 (2000).

    ADS  Google Scholar 

  124. E-J. Kim and P. H. Diamond, Phys. Rev. Lett. 22, 225002 (2002).

    Google Scholar 

  125. L. Dan, Private Communications (2000).

    Google Scholar 

  126. Y. Idomura, H. Urano, N. Aiba and S. Tokuda, Nucl. Fusion 49, 065029 (2009).

    ADS  Google Scholar 

  127. V. Grandgirard et al., Plasma Phys. Control. Fusion 49, B173 (2007).

    ADS  Google Scholar 

  128. J. A. Heikkinen, S. J. Janhunen, T. P. Kiviniemi and F. Ogando, J. Comp. Phys. 227, 5582 (2008).

    ADS  Google Scholar 

  129. C. S. Chang, S. Ku, P. H. Diamond, Z. Lin, S. Parker, T. S. Hahm and N. Samatova, Phys. Plasmas 16, 056108 (2009).

    ADS  Google Scholar 

  130. Y. Sarazin et al., Nucl. Fusion 51, 103023 (2011).

    ADS  Google Scholar 

  131. Y. Sarazin, V. Grandgirard, J. Abiteboul, S. Allfrey, X. Garbet, Ph. Ghendrih, G. Latu, A. Strugarek and G. Dif-Pradalier, Nucl. Fusion 50, 054004 (2010).

    Google Scholar 

  132. E. Mazzucato and R. Nazikian, Phys. Rev. Lett. 71, 1840 (1993).

    ADS  Google Scholar 

  133. R. J. Fonck, G. Cosby, R. D. Durst, S. F. Paul, N. Bretz, S. Scott, E. Synakowski and G. Taylor, Phys. Rev. Lett. 70, 3736 (1993).

    ADS  Google Scholar 

  134. P. Hennequin et al. and the ASDEX Upgrade Team, in 42nd EPS Conference on Plasma Physics, I1 (2015), p. 102.

  135. G. Dif-Pradalier, G. Hornung, X. Garbet, Ph. Ghendrih, V. Grandgirard, G. Latu and Y. Sarazin, Nucl. Fusion 57, 066026 (2017).

    Google Scholar 

  136. Y. Xiao and Z. Lin, Phys. Rev. Lett. 103, 085004 (2009).

    ADS  Google Scholar 

  137. L. Qi, J. M. Kwon, T. S. Hahm and S. Yi, Nucl. Fusion 57, 124002 (2017).

    ADS  Google Scholar 

  138. S. Ku et al., Nucl. Fusion 52, 063013 (2012).

    ADS  Google Scholar 

  139. P. H. Diamond, C. J. McDevitt, O. D. Gurcan, T. S. Hahm and V. Naulin, Phys. Plasmas 15, 012303 (2008).

    ADS  Google Scholar 

  140. P. H. Diamond, C. J. McDevitt, O. D. Gurcan, T. S. Hahm, W. X. Wang, E. S. Yoon, I. Holod, Z. Lin, V. Naulin and R. Singh, Nucl. Fusion 49, 045002 (2009).

    ADS  Google Scholar 

  141. O. D. Gurcan, P. H. Diamond and T. S. Hahm, Phys. Plasmas 14, 055902 (2007).

    ADS  Google Scholar 

  142. Y. Kosuga, P. H. Diamond and O. D. Gurcan, Phys. Plasmas 17, 102313 (2010).

    ADS  Google Scholar 

  143. J. E. Rice et al., Phys. Rev. Lett. 106, 215001 (2011).

    ADS  Google Scholar 

  144. W. M. Solomon et al., Phys. Plasmas 17, 056108 (2010).

    ADS  Google Scholar 

  145. F. Hariri, V. Naulin, J. Juul Rasmussen, G. S. Xu and N. Yan, Phys. Plasmas 23, 052512 (2016).

    ADS  Google Scholar 

  146. M. Kikuchi and M. Azumi, Rev. Mod. Phys. 84, 1807 (2012).

    ADS  Google Scholar 

  147. K. Imadera, J. Q. Li and Y. Kishimoto, in Proc. 25th Int. Conf. on Fusion Energy (2016), p. TH/P3-3.

    Google Scholar 

  148. Y. Kishimoto, K. Imadera and W. Wang, Private Communications (2017)

    Google Scholar 

  149. W. Wang, Y. Kishimoto and K. Imadera, Private Communications (to be submitted).

  150. K. Ida et al., Nucl. Fusion 55, 013022 (2015).

    ADS  Google Scholar 

  151. B. A. Carreras et al., Phys. Plasmas 5, 3632 (1998).

    ADS  Google Scholar 

  152. P. A. Politzer, Phys. Rev. Lett 84, 1192 (2000).

    ADS  Google Scholar 

  153. P. A. Politzer, M. E. Austin, M. Gilmore, G. R. McKee, T. L. Rhodes, C. X. Yu, E. J. Doyle, T. E. Evans and R. A. Moyere, Phys. Plasmas 9, 1962 (2002).

    ADS  Google Scholar 

  154. M. J. Choi, M. H. Woo, Jae-Min Kwon, S. Ko, Hogun Jhang, H. K. Park, T. S. Hahm, J. Lee, M. Kim, G. S. Yun, arXiv:1806.04947v2 (2018).

  155. S. J. Zweben et al., Phys. Plasmas 9, 1981 (2002).

    ADS  Google Scholar 

  156. J. A. Boedo et al., Phys. Plasmas 10, 1670 (2003).

    ADS  Google Scholar 

  157. S. J. Zweben et al. and the NSTX Team, Nucl. Fusion 44, 134 (2004).

    ADS  Google Scholar 

  158. J. A. Boedo et al. and NSTX Team, Phys. Plasmas 21, 042309 (2014).

    ADS  Google Scholar 

  159. Y. H. Xu, S. Jachmich, R. R. Weynants, A. Huber, B. Unterberg and U. Samm, Phys. Plasmas 11, 5413 (2004).

    ADS  Google Scholar 

  160. B. D. Scott, Contrib. Plasma Phys. 46, 714 (2006).

    ADS  Google Scholar 

  161. P. Manz, T. T. Ribeiro, B. D. Scott, G. Birkenmeier, D. Carralero, G. Fuchert, S. H. Muller, H. W. Muller, U. Stroth and E. Wolfrum, Phys. Plasmas 22, 022308 (2015).

    ADS  Google Scholar 

  162. B. D. Scott, Phys. Plasmas 12, 082305 (2005).

    ADS  Google Scholar 

  163. K. W. Gentle, R. V. Bravenec, G. Cima, H. Gasquet, G. A. Hallock, P. E. Phillips, D. W. Ross, W. L. Rowan, A. J. Wootton, T. P. Crowley, J. Heard, A. Ouroua, P. M. Schoch and C. Watts, Phys. Plasmas 2, 2292 (1995).

    ADS  Google Scholar 

  164. K. W. Gentle, W. L. Rowan, R. V. Bravenec, G. Cima, T. P. Crowley, H. Gasquet, G. A. Hallock, J. Heard, A. Ouroua, P. E. Phillips, D. W. Ross, P. M. Schoch and C. Watts, Phys. Rev. Lett. 74, 3620 (1995).

    ADS  Google Scholar 

  165. J. D. Callen and M. W. Kissick, Plasma Phys. Control. Fusion 39, B173 (1997).

    Google Scholar 

  166. B. P. van Milligen et al., Nucl. Fusion 42, 787 (2002).

    ADS  Google Scholar 

  167. P. Mantica et al. and JET EFDA Contributors, in Proceedings of the 19th International Conference on Fusion Energy, Lyon, 2002 (IAEA, Vienna, 2002), p. EX/P1- 04.

    Google Scholar 

  168. B. Van Compernolle, G. J. Morales, J. E. Maggs and R. D. Sydora, Phys. Rev. E 91, 031102 (2015).

    ADS  Google Scholar 

  169. G. Dif-Pradalier, P. H. Diamond, V. Grandgirard, Y. Sarazin, J. Abiteboul, X. Garbet, Ph. Ghendrih, A. Strugarek, S. Ku and C. S. Chang, Phys. Rev. E 82, 025401(R) (2010).

    ADS  Google Scholar 

  170. G. M. Zaslavsky, Hamiltonian Chaos and Fractal Dynamics (Oxford University Press, 2005).

    Google Scholar 

  171. T. H. Solomon, E. R. Weeks and H. L. Swinney, Phys. Rev. Lett. 71, 3975 (1993).

    ADS  Google Scholar 

  172. G. M. Zaslavsky, M. Edelman, H. Weitzner, B. Carreras, G. McKee, R. Bravenec and R. Fonck, Phys. Plasmas 7, 3691 (2000).

    ADS  Google Scholar 

  173. G. K. Vallis, Atmospheric and Oceanic Fluid Dynamics (Cambridge University Press, 2006).

    Google Scholar 

  174. G. K. Batchelor, H. K. Moffatt and M. G. Worster, Perspectives in Fluid Dynamics (Cambridge University Press, 2002).

    Google Scholar 

  175. R. B. Wood and M. E. McIntyre, J. Atmospheric Sci. 67, 1261 (2010).

    ADS  Google Scholar 

  176. O. D. Gurcan and P. H. Diamond, J. Phys. A 48, 293001 (2015).

    MathSciNet  Google Scholar 

  177. G. I. Taylor, Philos. Trans. Royal Soc. A 215, 1 (1915).

    ADS  Google Scholar 

  178. D. G. Dritschel and M. E. McIntyre, J. Atmospheric Sci. 65, 855 (2008).

    ADS  Google Scholar 

  179. G. Dif-Pradalier et al., Phys. Rev. Lett. 114, 085004 (2015).

    ADS  Google Scholar 

  180. N. J. Balmforth, S. G. L. Smith and W. R. Young, J. Fluid Mech. 355, 329 (1998).

    ADS  MathSciNet  Google Scholar 

  181. A. Ashourvan and P. H. Diamond, Phys. Rev. E 94, 051202(R) (2016).

    ADS  Google Scholar 

  182. A. Ashourvan and P. H. Diamond, Phys. Plasmas 24, 012305 (2017).

    ADS  Google Scholar 

  183. P. B. Rhines, J. Fluid Mech. 69, 417 (1975).

    ADS  Google Scholar 

  184. W. X. Guo et al., presented in 8th Asia-Pacific Transport Working Group Meeting (Leshan, China, 2018).

    Google Scholar 

  185. G. Dif-Pradalier et al., presented in 8th Asia-Pacific Transport Working Group Meeting (Leshan, China, 2018).

    Google Scholar 

  186. O. D. Gurcan, P. H. Diamond, X. Garbet, V. Berionni, G. Dif-Pradalier, P. Hennequin, P. Morel, Y. Kosuga and L. Vermare, Phys. Plasmas 20, 022307 (2013).

    ADS  Google Scholar 

  187. M. R. Flynn, A. R. Kasimov, J. C. Nave, R. R. Rosales and B. Seibold, Phys. Rev. E 79, 056113 (2009).

    ADS  MathSciNet  Google Scholar 

  188. J. B. Taylor, Rev. Mod. Phys. 58, 741 (1986).

    ADS  Google Scholar 

  189. S. Galtier, Introduction to Modern Magnetohydrodynamics (Cambridge University Press, 2016).

    Google Scholar 

  190. U. Frisch, M. Lesieur and P. L. Sulem, Phys. Rev. Lett. 37, 895 (1976).

    ADS  Google Scholar 

  191. A. Pouquet, U. Frisch and J. Leorat, J. Fluid Mech. 77, 321 (1976).

    ADS  Google Scholar 

  192. A. H. Boozer, J. Plasma Phys. 35, 133 (1986).

    ADS  Google Scholar 

  193. P. H. Diamond and M. Malkov, Phys. Plasmas 10, 2322 (2003).

    ADS  MathSciNet  Google Scholar 

  194. J. A. Holmes, B. A. Carreras, P. H. Diamond and V. E. Lynch, Phys. Fluids 31, 1166 (1988).

    ADS  Google Scholar 

  195. I. H. Hutchinson, M. Malacarne, P. Noonan and D. Brotherton-Ratcliffe, Nucl. Fusion 24, 59 (1984).

    Google Scholar 

  196. R. Lorenzini et al. and RFX-mod team and collaborators, Nat. Phys. 5, 570 (2009).

    Google Scholar 

  197. F. Zonca, S. Briguglio, L. Chen, G. Fogaccia, T. S. Hahm, A. V. Milovanov and G. Vlad, Plasma Phys. Control. Fusion 48, B15 (2006).

    ADS  Google Scholar 

  198. Z. Guo, L. Chen and F. Zonca, Phys. Rev. Lett. 103, 055002 (2009).

    ADS  Google Scholar 

  199. F. Zonca, L. Chen, S. Briguglio, G. Fogaccia, A. V. Milovanov, Z Qiu, G. Vlad and X. Wang, Plasma Phys. Control. Fusion 57, 014024 (2015).

    ADS  Google Scholar 

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Acknowledgments

We would like to acknowledge useful discussions with A. Ashourvan, S. Cappello, B. A. Carreras, L. Chen, M. J. Choi, B. Compernolle, P. Davidson, G. Dif-Pradalier, X. T. Ding, X. Fan, X. Garbet, N. Goldenfeld, D. Guo, W. X. Guo, Z. B. Guo, O. D. Gurcan, R. Hajjar, R. Heinonen, P. Hennequin, C. Hidalgo, R. Hong, D. W. Hughes, T. Hwa, K. Ida, S. Inagaki, K. Itoh, S-I. Itoh, H. G. Jhang, R. Ke, S. Keating, E-J. Kim, S. S. Kim, Y. Kosuga, S. Ku, J. M. Kwon, J. C. Li, Z. Lin, T. Long, R. Ma, V. Naulin, D. E. Newman, Y. Pomeau, T. Rhee, Y. Sarazin, B. D. Scott, Z. B. Shi, H. J. Sun, R. D. Sydora, K. Thompson, L. Villard, L. Wang, W. X. Wang, Z. H. Wang, W. Xiao, Y. Xu, M. Yagi, W. R. Young, S. Yi, Y. Zhang and F. Zonca. We have also benefitted from the Festival de Theorie 2003, 2005 and 2017 where many of the subjects addressed in this review were discussed. We would also like to thank Mr. G. J. Choi for his dedicated work in preparing this manuscript.

This work was supported by the Ministry of Science, ICT and Future Planning of the Republic of Korea under the Korean ITER project contract, and National R&D Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (No. 2014M1A7A1A03045368), by the U.S. Department of Energy, Office of Science, Office of Fusion Energy Sciences under Award Number DEFG02- 04ER54738, and by the Center for Fusion Science, Southwest Institute of Physics, China. PD thanks SWIP for hospitality during the completion of a portion of this work.

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  1. Department of Nuclear Engineering, Seoul National University, Seoul, 08826, Korea

    T. S. Hahm

  2. Center for Astrophysics and Space Science, University of California San Diego, San Diego, 92093, USA

    P. H. Diamond

  3. Center for Fusion Sciences, Southwestern Institute of Physics, Chengdu, 610041, China

    P. H. Diamond

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Hahm, T.S., Diamond, P.H. Mesoscopic Transport Events and the Breakdown of Fick’s Law for Turbulent Fluxes. J. Korean Phys. Soc. 73, 747–792 (2018). https://doi.org/10.3938/jkps.73.747

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