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.
Similar content being viewed by others
References
F. Wagner et al., Phys. Rev. Lett. 49, 1408 (1982).
P. A. Davidson, J. Turb. 1, N6 (2000).
P. H. Diamond, S-I. Itoh, K. Itoh and T. S. Hahm, Plasma Phys. Control. Fusion 47, R35 (2005).
Z. B. Guo et al., submitted (2018).
G. I. Barenblatt, Similarity, Self-similarity, and Intermediate Asymptotics (New York and London: Consultant Bureau, 1979).
L. G. Loitsyansky, Centr. Aero. Hydrodyn. Inst. Moscow, Rep. no. 440(Trans. NACA Tech. Memo. 1079) (1939).
L. D. Landau and E. M. Lifshitz, Fluid Mechanics, 1st ed. (Oxford: Pergamon, 1959), p. 141.
Y. Pomeau, Phys. D: Nonlinear Phenom. 23, 3 (1986).
Y. Pomeau, C. R. Mecanique 343, 210 (2015).
G. B. Whitham, Linear and Nonlinear Waves (Wiley- Interscience, New York, 1999).
S. M. Kaye et al., Nucl. Fusion 47, 499 (2007).
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).
X. Garbet, L. Laurent, A. Samain and J. Chinardet, Nucl. Fusion 34, 963 (1994).
T. S. Hahm, P. H. Diamond, Z. Lin, K. Itoh and S-I. Itoh, Plasma Phys. Control. Fusion 46, A323 (2004).
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).
O. Gurcan, P. H. Diamond, T. S. Hahm and Z. Lin, Phys. Plasmas 12, 032303 (2005).
H. Sugama and M. Wakatani, J. Phys. Soc. Jpn. 61, 3166 (1992).
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).
D. E. Newman, B. A. Carreras, D. Lopez-Bruna, P. H. Diamond and V. B. Lebedev, Phys. Plasmas 5, 938 (1998).
M. A. Malkov, P. H. Diamond and M. N. Rosenbluth, Phys. Plasmas 8, 5073 (2001).
D. del-Castillo-Negrete, Phys. Plasmas 13, 082308 (2006).
T. S. Hahm, P. H. Diamond, Z. Lin, G. Rewoldt, O. Gurcan and S. Ethier, Phys. Plasmas 12, 090903 (2005).
H. P. Furth, J. Kileen and M. N. Rosenbluth, Phys. Fluids 6, 459 (1963).
R. D. Sydora, V. K. Decyk and J. M. Dawson, Plasma Phys. Control. Fusion 38, A281 (1996).
Z. Lin, S. Ethier, T. S. Hahm and W. M. Tang, Phys. Rev. Lett. 88, 195004 (2002).
V. B. Lebedev and P. H. Diamond, Phys. Plasmas 4, 1087 (1997).
S-I. Itoh and K. Itoh, J. Phys. Soc. Japan 69, 408 (2000).
Y. Sarazin, X. Garbet, Ph. Ghendrih and S. Benkadda, Phys. Plasmas 7, 1085 (2000).
X. Garbet, Y. Sarazin, F. Imbeaux, P. Ghendrih, C. Bourdelle, O. D. Gurcan and P. H. Diamond, Phys. Plasmas 14, 122305 (2007).
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).
L. Chen, R. B. White and F. Zonca, Phys. Rev. Lett. 92, 075004 (2004).
L. Chen, Z. Lin and R. B. White, Phys. Plasmas 7, 3129 (2000).
X. Garbet, private communication (2002).
B. B. Kadomtsev, Plasma Turbulence (Academic, New York, 1965).
R. A. Fisher, Ann. Eugenics 7, 353 (1937).
A. Kolmogoroff, I. Petrovsky and N. Piscounoff, Clin. Cancer Res. 1, 1 (1937).
V. Naulin, A. H. Nielsen and J. Juul Rasmussen, Phys. Plasmas 12, 122306 (2005).
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.
Z. H. Wang, P. H. Diamond, O. D. Gurcan, X. Garbet and X. G. Wang, Nucl. Fusion 51, 073009 (2011).
Z. B. Guo and P. H. Diamond, Phys. Plasmas 24, 100705 (2017).
S. Inagaki et al. and the LHD Experiment Group, Nucl. Fusion 53, 113006 (2013).
R. Heinonen and P. H. Diamond, submitted (2018).
O. D. Gurcan, P. H. Diamond and T. S. Hahm, Phys. Plasmas 13, 052306 (2006).
R. V. Budny et al., Phys. Plasmas 7, 5038 (2000).
G. R. McKee et al., Nucl. Fusion 41, 1235 (2001).
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).
Z. Lin, T. S. Hahm, W. W. Lee, W. M. Tang and R. B. White, Science 281, 1835 (1998).
X. Garbet, Y. Idomura, L. Villard and T. H. Watanabe, Nucl. Fusion 50, 043002 (2010).
E. A. Frieman and L. Chen, Phys. Fluids 25, 502 (1982).
T. S. Hahm, Phys. Fluids 31, 2670 (1988).
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).
Y. Kishimoto, T. Tajima, W. Horton, M. J. LeBrun and J. Y. Kim, Phys. Plasmas 3, 1289 (1996).
W. W. Lee and R. Santoro, Phys. Plasmas 4, 169 (1997).
Y. Idomura, M. Wakatani and S. Tokuda, Phys. Plasmas 7, 3551 (2000).
Z. Lin and T. S. Hahm, Phys. Plasmas 11, 1099 (2004).
A. M. Dimits et al., Phys. Plasmas 7, 969 (2000).
B. F. McMillan, X. Lapillonne, S. Brunner, L. Villard, S. Jolliet, A. Bottino, T. Gorler and F. Jenko, Phys. Rev. Lett. 105, 155001 (2010).
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).
T. Gorler, Ph. D. Thesis, Universitat Ulm, 2009.
J. Candy and R. E. Waltz, J. Comput. Phys. 186, 545 (2003).
M. Kotschenreuther, G. Rewoldt and W.M. Tang, Comput. Phys. Commun. 88, 128 (1995).
J. Candy, R. E. Waltz and W. Dorland, Phys. Plasmas 11, L25 (2004).
Z. Lin, S. Ethier, T. S. Hahm and W. M. Tang, Plasma Sci. Technol. 14, 1125 (2012).
R. E. Waltz and J. Candy, Phys. Plasmas 12, 072303 (2005).
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).
S. Yi, J. M. Kwon, P. H. Diamond and T. S. Hahm, Phys. Plasmas 21, 092509 (2014).
S. Yi, J. M. Kwon, P. H. Diamond and T. S. Hahm, Nucl. Fusion 55, 092002 (2015).
P. Mantica et al., Phys. Rev. Lett. 107, 135004 (2011).
J. W. Hughes, D. A. Mossessian, A. E. Hubbard, B. LaBombard and E. S. Marmar, Phys. Plasmas 9, 3019 (2002).
J. G. Cordey et al., Plasma Phys. Control. Fusion 36, A267 (1994).
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).
B. B. Kadomtsev, Plasma Phys. Control. Fusion 34, 1931 (1992).
Z. Lin, T. S. Hahm, W. W. Lee, W. M. Tang and P. H. Diamond, Phys. Rev. Lett. 83, 3645 (1999).
B. LaBombard et al. and the Alcator Group, Nucl. Fusion 44, 1047 (2004).
G. Rewoldt and W. M. Tang, Phys. Fluids B 2, 318 (1990).
K. H. Burrell, Phys. Plasmas 4, 1499 (1997).
W. X. Wang, T. S. Hahm, W. W. Lee, G. Rewoldt, J. Manickam and W. M. Tang, Phys. Plasmas 14, 072306 (2007).
T. S. Hahm and K. H. Burrell, Phys. Plasmas 2, 1648 (1995).
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).
W. Deng and Z. Lin, Phys. Plasmas 16, 102503 (2009).
C. H. Ma, X. Q. Xu, P. W. Xi and T. Y. Xia, Phys. Plasmas 22, 010702 (2015).
A. Ishizawa and N. Nakajima, Phys. Plasmas 14, 040702 (2007).
A. Ishizawa and N. Nakajima, Nucl. Fusion 49, 055015 (2009).
E. Poli, A. Bottino and A. G. Peeters, Nucl. Fusion 49, 075010 (2009).
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).
K. Ida et al. and LHD Experimental Group, Phys. Rev. Lett. 88, 015002 (2001).
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).
K. Ida, T. Kobayashi, M. Ono, T. E. Evans, G. R. Mc- Kee and M. E. Austin, Phys. Rev. Lett. 120, 245001 (2018).
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).
E. J. Synakowski et al., Nucl. Fusion 39, 1733 (1999).
E. Mazzucato et al., Phys. Rev. Lett. 77, 3145 (1996).
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).
T. Estrada, C. Hidalgo and T. Happel, Nucl. Fusion 51, 032001 (2011).
P.H. Diamond et al., presented in 6th Asia-Pacific Transport Working Group Meeting (Seoul, Korea, 2016)
G. K. Zipf, Human Behavior and the Principle of Least Effort (Cambridge, Massachusetts: Addison-Wesley, 1949).
H. E. Hurst, Trans. Am. Soc. Civil Eng. 116, 770 (1951).
H. E. Hurst, Proc. Inst. ivil Eng., Part 1, 519 (1956).
H. E. Hurst, R. P. Black and Y. M. Simaika, Long-Term Storage, and Experimental Study (London, Constable, 1965).
B. B. Mandelbrot, The Fractal Geometry of Nature, (Freeman, New York, 1983).
P. Bak, C. Tang and K. Wiesenfeld, Phys. Rev. Lett. 59, 381 (1987).
P.H. Diamond et al., presented in 9th Festival de Theorie (Aix-en-Provence, France, 2017)
B. B. Mandelbrot and J. R. Wallis, Water Resources Res. 4, 909 (1968).
G. Boffetta, A. Mazzino and A. Vulpiani, J. Phys. A 41, 363001 (2008)
E. W. Montroll and G. H. Weiss, J. Math. Phys. 6, 167 (1965).
L. P. Kadanoff, S. R. Nagel, L. Wu and S. Zhou, Phys. Rev. A 39, 6524 (1989).
D. E. Newman, B. A. Carreras, P. H. Diamond and T. S. Hahm, Phys. Plasmas 3, 1858 (1996).
I. Gruzinov, P. H. Diamond and M. N. Rosenbluth, Phys. Rev. Lett. 89, 255001 (2002).
I. Gruzinov, P. H. Diamond and M. N. Rosenbluth, Phys. Plasmas 10, 569 (2003).
T. Hwa and M. Kardar, Phys. Rev. A 45, 7002 (1992).
P. H. Diamond and T. S. Hahm, Phys. Plasmas 2, 3640 (1995).
D. Shvarts, U. Alon, D. Ofer, R. L. McCrory and C. P. Verdon, Phys. Plasmas 2, 2465 (1995).
S. Chandrasekhar, Rev. Mod. Phys. 15, 1 (1943).
Y. Kosuga, P. H. Diamond, G. Dif-Pradalier and O. D. Gurcan, Phys. Plasmas 21, 055701 (2014).
Y. Kosuga, P. H. Diamond and O. D. Gurcan, Phys. Rev. Lett. 110, 105002 (2013).
L. Gil and D. Sornette, Phys. Rev. Lett. 76, 3991 (1996).
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).
V. B. Lebedev and P. H. Diamond, Phys. Plasmas 4, 1087 (1997).
T. Rhee, J. M. Kwon, P. H. Diamond and W. W. Xiao, Phys. Plasmas 19, 022505 (2012).
B. A. Carreras, D. Newman, E. Lynch and P. H. Diamond, Phys. Plasmas 3, 2903 (1996).
X. Garbet and R. E. Waltz, Phys. Plasmas 5, 2836 (1998).
X. Garbet, Y. Sarazin, P. Beyer, P. Ghendrih, R. E. Waltz, M. Ottaviani and S. Benkadda, Nucl. Fusion 39, 2063 (1999).
Y. Sarazin and Ph. Ghendrih, Phys. Plasmas 5, 4214 (1998).
P. Beyer, S. Benkadda, X. Garbet and P. H. Diamond, Phys. Rev. Lett. 85, 4892 (2000).
E-J. Kim and P. H. Diamond, Phys. Rev. Lett. 22, 225002 (2002).
L. Dan, Private Communications (2000).
Y. Idomura, H. Urano, N. Aiba and S. Tokuda, Nucl. Fusion 49, 065029 (2009).
V. Grandgirard et al., Plasma Phys. Control. Fusion 49, B173 (2007).
J. A. Heikkinen, S. J. Janhunen, T. P. Kiviniemi and F. Ogando, J. Comp. Phys. 227, 5582 (2008).
C. S. Chang, S. Ku, P. H. Diamond, Z. Lin, S. Parker, T. S. Hahm and N. Samatova, Phys. Plasmas 16, 056108 (2009).
Y. Sarazin et al., Nucl. Fusion 51, 103023 (2011).
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).
E. Mazzucato and R. Nazikian, Phys. Rev. Lett. 71, 1840 (1993).
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).
P. Hennequin et al. and the ASDEX Upgrade Team, in 42nd EPS Conference on Plasma Physics, I1 (2015), p. 102.
G. Dif-Pradalier, G. Hornung, X. Garbet, Ph. Ghendrih, V. Grandgirard, G. Latu and Y. Sarazin, Nucl. Fusion 57, 066026 (2017).
Y. Xiao and Z. Lin, Phys. Rev. Lett. 103, 085004 (2009).
L. Qi, J. M. Kwon, T. S. Hahm and S. Yi, Nucl. Fusion 57, 124002 (2017).
S. Ku et al., Nucl. Fusion 52, 063013 (2012).
P. H. Diamond, C. J. McDevitt, O. D. Gurcan, T. S. Hahm and V. Naulin, Phys. Plasmas 15, 012303 (2008).
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).
O. D. Gurcan, P. H. Diamond and T. S. Hahm, Phys. Plasmas 14, 055902 (2007).
Y. Kosuga, P. H. Diamond and O. D. Gurcan, Phys. Plasmas 17, 102313 (2010).
J. E. Rice et al., Phys. Rev. Lett. 106, 215001 (2011).
W. M. Solomon et al., Phys. Plasmas 17, 056108 (2010).
F. Hariri, V. Naulin, J. Juul Rasmussen, G. S. Xu and N. Yan, Phys. Plasmas 23, 052512 (2016).
M. Kikuchi and M. Azumi, Rev. Mod. Phys. 84, 1807 (2012).
K. Imadera, J. Q. Li and Y. Kishimoto, in Proc. 25th Int. Conf. on Fusion Energy (2016), p. TH/P3-3.
Y. Kishimoto, K. Imadera and W. Wang, Private Communications (2017)
W. Wang, Y. Kishimoto and K. Imadera, Private Communications (to be submitted).
K. Ida et al., Nucl. Fusion 55, 013022 (2015).
B. A. Carreras et al., Phys. Plasmas 5, 3632 (1998).
P. A. Politzer, Phys. Rev. Lett 84, 1192 (2000).
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).
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).
S. J. Zweben et al., Phys. Plasmas 9, 1981 (2002).
J. A. Boedo et al., Phys. Plasmas 10, 1670 (2003).
S. J. Zweben et al. and the NSTX Team, Nucl. Fusion 44, 134 (2004).
J. A. Boedo et al. and NSTX Team, Phys. Plasmas 21, 042309 (2014).
Y. H. Xu, S. Jachmich, R. R. Weynants, A. Huber, B. Unterberg and U. Samm, Phys. Plasmas 11, 5413 (2004).
B. D. Scott, Contrib. Plasma Phys. 46, 714 (2006).
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).
B. D. Scott, Phys. Plasmas 12, 082305 (2005).
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).
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).
J. D. Callen and M. W. Kissick, Plasma Phys. Control. Fusion 39, B173 (1997).
B. P. van Milligen et al., Nucl. Fusion 42, 787 (2002).
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.
B. Van Compernolle, G. J. Morales, J. E. Maggs and R. D. Sydora, Phys. Rev. E 91, 031102 (2015).
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).
G. M. Zaslavsky, Hamiltonian Chaos and Fractal Dynamics (Oxford University Press, 2005).
T. H. Solomon, E. R. Weeks and H. L. Swinney, Phys. Rev. Lett. 71, 3975 (1993).
G. M. Zaslavsky, M. Edelman, H. Weitzner, B. Carreras, G. McKee, R. Bravenec and R. Fonck, Phys. Plasmas 7, 3691 (2000).
G. K. Vallis, Atmospheric and Oceanic Fluid Dynamics (Cambridge University Press, 2006).
G. K. Batchelor, H. K. Moffatt and M. G. Worster, Perspectives in Fluid Dynamics (Cambridge University Press, 2002).
R. B. Wood and M. E. McIntyre, J. Atmospheric Sci. 67, 1261 (2010).
O. D. Gurcan and P. H. Diamond, J. Phys. A 48, 293001 (2015).
G. I. Taylor, Philos. Trans. Royal Soc. A 215, 1 (1915).
D. G. Dritschel and M. E. McIntyre, J. Atmospheric Sci. 65, 855 (2008).
G. Dif-Pradalier et al., Phys. Rev. Lett. 114, 085004 (2015).
N. J. Balmforth, S. G. L. Smith and W. R. Young, J. Fluid Mech. 355, 329 (1998).
A. Ashourvan and P. H. Diamond, Phys. Rev. E 94, 051202(R) (2016).
A. Ashourvan and P. H. Diamond, Phys. Plasmas 24, 012305 (2017).
P. B. Rhines, J. Fluid Mech. 69, 417 (1975).
W. X. Guo et al., presented in 8th Asia-Pacific Transport Working Group Meeting (Leshan, China, 2018).
G. Dif-Pradalier et al., presented in 8th Asia-Pacific Transport Working Group Meeting (Leshan, China, 2018).
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).
M. R. Flynn, A. R. Kasimov, J. C. Nave, R. R. Rosales and B. Seibold, Phys. Rev. E 79, 056113 (2009).
J. B. Taylor, Rev. Mod. Phys. 58, 741 (1986).
S. Galtier, Introduction to Modern Magnetohydrodynamics (Cambridge University Press, 2016).
U. Frisch, M. Lesieur and P. L. Sulem, Phys. Rev. Lett. 37, 895 (1976).
A. Pouquet, U. Frisch and J. Leorat, J. Fluid Mech. 77, 321 (1976).
A. H. Boozer, J. Plasma Phys. 35, 133 (1986).
P. H. Diamond and M. Malkov, Phys. Plasmas 10, 2322 (2003).
J. A. Holmes, B. A. Carreras, P. H. Diamond and V. E. Lynch, Phys. Fluids 31, 1166 (1988).
I. H. Hutchinson, M. Malacarne, P. Noonan and D. Brotherton-Ratcliffe, Nucl. Fusion 24, 59 (1984).
R. Lorenzini et al. and RFX-mod team and collaborators, Nat. Phys. 5, 570 (2009).
F. Zonca, S. Briguglio, L. Chen, G. Fogaccia, T. S. Hahm, A. V. Milovanov and G. Vlad, Plasma Phys. Control. Fusion 48, B15 (2006).
Z. Guo, L. Chen and F. Zonca, Phys. Rev. Lett. 103, 055002 (2009).
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).
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
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
Received:
Published:
Issue Date:
DOI: https://doi.org/10.3938/jkps.73.747