Journal of the Korean Physical Society

, Volume 73, Issue 6, pp 747–792 | Cite as

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

  • T. S. Hahm
  • P. H. Diamond
Review Articles
Part of the following topical collections:
  1. JKPS 50th Anniversary Reviews


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.


Self-organization Turbulence spreading Avalanches Entrainment Mesoscopic transport Intermittency, Non-local transport Self-organized criticality Magnetically confined plasma Transport events, Fluctuation front propagation 



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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Copyright information

© The Korean Physical Society 2018

Authors and Affiliations

  1. 1.Department of Nuclear EngineeringSeoul National UniversitySeoulKorea
  2. 2.Center for Astrophysics and Space ScienceUniversity of California San DiegoSan DiegoUSA
  3. 3.Center for Fusion SciencesSouthwestern Institute of PhysicsChengduChina

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