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Self-organized criticality revisited: non-local transport by turbulent amplification

Part of: Complex Plasma Phenomena

Published online by Cambridge University Press:  13 November 2015

A. V. Milovanov  and
J. J. Rasmussen
A. V. Milovanov*
Affiliation:
ENEA National Laboratory, Centro Ricerche Frascati, I-00044 Frascati, Rome, Italy Department of Space Plasma Physics, Space Research Institute, Russian Academy of Sciences, 117997 Moscow, Russia
J. J. Rasmussen
Affiliation:
Physics Department, Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark
*
†Email address for correspondence: alexander.milovanov@enea.it
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Abstract

We revise the applications of self-organized criticality (SOC) as a paradigmatic model for tokamak plasma turbulence. The work, presented here, is built around the idea that some systems do not develop a pure critical state associable with SOC, since their dynamical evolution involves as a competing key factor an inverse cascade of the energy in reciprocal space. Then relaxation of slowly increasing stresses will give rise to intermittent bursts of transport in real space and outstanding transport events beyond the range of applicability of the ‘conventional’ SOC. Also, we are concerned with the causes and origins of non-local transport in magnetized plasma, and show that this type of transport occurs naturally in self-consistent strong turbulence via a complexity coupling to the inverse cascade. We expect these coupling phenomena to occur in the parameter range of strong nonlinearity and time scale separation when the Rhines time in the system is small compared with the instability growth time.

Type
Research Article
Information
Journal of Plasma Physics , Volume 81 , Issue 6 , December 2015 , 495810606
Copyright
© Cambridge University Press 2015 

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References

Anderson, P. 2011 More and Different. World Scientific.Google Scholar
Aschwanden, M. J. 2011 Self-Organized Criticality in Astrophysics. The Statistics of Nonlinear Processes in the Universe. Springer.Google Scholar
Aschwanden, M. J. 2013 Theoretical models of SOC systems. In Self-Organized Criticality Systems (ed. Aschwanden, M. J.), chap. 2, pp. 23–72. Open Academic Press GmbH & Co.Google Scholar
Aschwanden, M. J., Crosby, N., Dimitropoulou, M., Georgoulis, M. K., Hergarten, S., Jensen, H. J., McAteer, J., Milovanov, A. V., Mineshige, S., Morales, L. et al. 2014 Twenty-five years of self-organized criticality: solar and astrophysics. Space Sci. Rev. 181 (1–4), (120 pp.) First on-line: 15 July 2014.Google Scholar
Bak, P. & Chen, K. 1989 The physics of fractals. Physica D 38, 5–12.CrossRefGoogle Scholar
Bak, P., Tang, C. & Wiesenfeld, K. 1987 Self-organized criticality: An explanation of $1/f$ noise. Phys. Rev. Lett. 59, 381–384.Google Scholar
Bak, P., Tang, C. & Wiesenfeld, K. 1988 Self-organized criticality. Phys. Rev. A 38, 364–374.Google Scholar
Basu, R., Jessen, T., Naulin, V. & Rasmussen, J. J. 2003 Turbulent flux and the diffusion of passive tracers in electrostatic turbulence. Phys. Plasmas 10 (7), 2696–2703.Google Scholar
Carreras, B. A., Newman, D., Lynch, V. E. & Diamond, P. H. 1996a Self-organized criticality as a paradigm for transport in magnetically confined plasmas. Plasma Phys. Reports 22, 740–753.Google Scholar
Carreras, B. A., Newman, D., Lynch, V. E. & Diamond, P. H. 1996b A model realization of self-organized criticality for plasma confinement. Phys. Plasmas 3, 2903–2911.Google Scholar
del-Castillo-Negrete, D. 2006 Fractional diffusion models of nonlocal transport. Phys. Plasmas 13, 082308.Google Scholar
del-Castillo-Negrete, D., Carreras, B. A. & Lynch, V. E. 2004 Fractional diffusion in plasma turbulence. Phys. Plasmas 11, 3854–3864.CrossRefGoogle Scholar
del-Castillo-Negrete, D., Mantica, P., Naulin, V., Rasmussen, J. J.& JET EFDA contributors 2008 Fractional diffusion models of nonlocal perturbative transport: numerical results and application to JET experiments. Nucl. Fusion 48, 075009.Google Scholar
Chechkin, A. V., Metzler, R., Gonchar, V. Y., Klafter, J. & Tanatarov, L. V. 2003 First passage and arrival time densities for Lévy flights and the failure of the method of images. J. Phys. A: Math. Gen. 36, L537–L544.Google Scholar
Corral, A. & Font-Clos, F. 2013 Criticality and self-organization in branching processes: application to natural hazards. In Self-Organized Criticality Systems (ed. Aschwanden, M. J.), chap. 5, pp. 183–228. Open Academic Press GmbH & Co.Google Scholar
Dendy, R. O & Helander, P. 1997 Sandpiles, silos and tokamak phenomenology: a brief review. Plasma Phys. Control. Fusion 39, 1947–1961.Google Scholar
Diamond, P. H. & Hahm, T. S. 1995 On the dynamics of turbulent transport near marginal stability. Phys. Plasmas 2, 3640–3649.Google Scholar
D’Ippolito, D. A., Myra, J. R. & Zweben, S. J. 2011 Convective transport by intermittent blob-filaments: comparison of theory and experiment. Phys. Plasmas 18, 060501.Google Scholar
Gnedenko, B. V. & Kolmogorov, A. N. 1954 Limit Distributions for Sums of Independent Random Variables. Addison-Wesley.Google Scholar
Hasegawa, A. & Wakatani, M. 1983 Plasma edge turbulence. Phys. Rev. Lett. 50, 682–686.Google Scholar
Horton, W. 1999 Drift waves and transport. Rev. Mod. Phys. 71, 735–778.Google Scholar
Huysmans, G. T. A. 2005 ELMs: MHD instabilities at the transport barrier. Plasma Phys. Control. Fusion 47, B165–B178.Google Scholar
Kadanoff, L. P. 1991 Complex structures from simple systems. Phys. Today 44, 9–11.Google Scholar
Maslov, S. & Zhang, Y. 1996 Self-organized critical directed percolation. Physica A 223, 1–6.CrossRefGoogle Scholar
Metzler, R. & Klafter, J. 2000 The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1–77.CrossRefGoogle Scholar
Metzler, R. & Klafter, J. 2004 The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics. J. Phys. A: Math. Gen. 37, R161–R208.Google Scholar
Milovanov, A. V. 2009 Pseudochaos and low-frequency percolation scaling for turbulent diffusion in magnetized plasma. Phys. Rev. E 79, 046403.Google Scholar
Milovanov, A. V. 2011 Dynamic polarization random walk model and fishbone-like instability for self-organized critical systems. New J. Phys. 13, 043034.Google Scholar
Milovanov, A. V. 2013 Percolation models of self-organized critical phenomena. In Self-Organized Criticality Systems (ed. Aschwanden, M. J.), chap. 4, pp. 103–182. Open Academic Press GmbH & Co., free download: http://www.openacademicpress.de/.Google Scholar
Milovanov, A. V. & Rasmussen, J. J. 2005 Fractional generalization of the Ginzburg–Landau equation: an unconventional approach to critical phenomena in complex media. Phys. Lett. A 337, 75–80.Google Scholar
Milovanov, A. V. & Rasmussen, J. J. 2014 A mixed SOC-turbulence model for nonlocal transport and Lévy-fractional Fokker–Planck equation. Phys. Lett. A 378, 1492–1500.CrossRefGoogle Scholar
Milovanov, A. V. & Iomin, A. 2012 Localization–delocalization transition on a separatrix system of nonlinear Schrödinger equation with disorder. Europhys. Lett. 100, 10006.Google Scholar
Milovanov, A. V. & Iomin, A. 2014 Topological approximation of the nonlinear Anderson model. Phys. Rev. E 89, 062921.Google Scholar
Milovanov, A. V. & Iomin, A. 2015 Topology of delocalization in the nonlinear Anderson model and anomalous diffusion on finite clusters. Discontinuity, Nonlinearity, Complexity 4 (2), 151–162.Google Scholar
Montroll, E. W. & Shlesinger, M. F. 1982 On $1/f$ noise and other distributions with long tails. Proc. Natl Acad. Sci. USA 79, 3380–3383.Google Scholar
Naulin, V. 2002 Aspects of flow generation and saturation in drift-wave turbulence. New J. Phys. 4, 28.1–28.18.Google Scholar
Naulin, V., Garcia, O. E., Nielsen, A. H. & Rasmussen, J. J. 2004 Statistical properties of transport in plasma turbulence. Phys. Lett. A 321, 355–365.CrossRefGoogle Scholar
Naulin, V., Nielsen, A. H. & Rasmussen, J. J. 1999 Dispersion of ideal particles in a two-dimensional model of electrostatic turbulence. Phys. Plasmas 6, 4575–4585.Google Scholar
Newman, D. E., Carreras, B. A., Diamond, P. H. & Hahm, T. S. 1996 The dynamics of marginality and self-organized criticality as a paradigm turbulent transport. Phys. Plasmas 3 (5), 1858–1866.Google Scholar
Nicolis, G. & Nicolis, C. 2007 Foundations of Complex Systems. World Scientific.Google Scholar
Politzer, P. A. 2000 Observation of avalanchelike phenomena in a magnetically confined plasma. Phys. Rev. Lett. 84, 1192–1195.Google Scholar
Politzer, P. A., Austin, M. E., Gilmore, M., McKee, G. R., Rhodes, T. L., Yu, C. X., Doyle, E. J., Evans, T. E. & Moyere, R. A. 2002 Characterization of avalanche-like events in a confined plasma. Phys. Plasmas 9, 1962–1969.Google Scholar
Reuss, J.-D. & Misguich, J. H. 1996 Low-frequency percolation scaling for particle diffusion in electrostatic turbulence. Phys. Rev. E 54, 1857–1869.Google Scholar
Rhines, P. B. 1975 Waves and turbulence on a beta-plane. J. Fluid Mech. 69, 417–443.CrossRefGoogle Scholar
Rhodes, T. L., Moyer, R. A., Groebner, R., Doyle, E. J., Lehmer, R., Peebles, W. A. & Retting, C. L. 1999 Experimental evidence for self-organized criticality in tokamak plasma turbulence. Phys. Lett. A 253, 181–186.Google Scholar
Samko, S. G., Kilbas, A. A. & Marichev, O. I. 1993 Fractional Integrals and Derivatives. Theory and Applications. Gordon and Breach.Google Scholar
Sanchez, R., Newman, D. E. & Carreras, B. A. 2001 Mixed SOC diffusive dynamics as a paradigm for transport in fusion devices. Nucl. Fusion 41, 247–256.Google Scholar
Schulman, L. S. & Seiden, P. E. 1986 Hierarchical structure in the distribution of galaxies. Astrophys. J. 311, 1–5.CrossRefGoogle Scholar
Sharma, A. S., Aschwanden, M. J., Crosby, N., Klimas, A. J., Milovanov, A. V., Morales, L., Sanchez, R. & Uritsky, V. 2015 Twenty-five years of self-organized criticality: space and laboratory plasmas. Space Sci. Rev. (accepted), in the production queue.Google Scholar
Sibani, P. & Jensen, H. J. 2013 Stochastic Dynamics of Complex Systems. Imperial College Press.Google Scholar
Sokolov, I. M., Klafter, J. & Blumen, A. 2002 Fractional kinetics. Phys. Today 55 (11), 48–54.Google Scholar
Sornette, D. 1992 Critical phase transitions made self-organized: a dynamical system feedback mechanism for self-organized criticality. J. Phys. I France 2, 2065–2073.Google Scholar
Sornette, D. & Ouillon, G. 2012 Dragon kings: mechanisms, statistical methods and empirical evidence. Eur. Phys. J. Special Topics 205, 1–26.Google Scholar
Sparre Andersen, E. 1953 On the fluctuations of sums of random variables. Math. Scand. 1, 263–285.Google Scholar
Tang, C. & Bak, P. 1988 Critical exponents and scaling relations for self-organized critical phenomena. Phys. Rev. Lett. 60, 2347–2350.Google Scholar
Xu, G. S., Naulin, V., Fundamenski, W., Hidalgo, C., Alonso, J. A., Silva, C., Gonçalves, B., Nielsen, A. H., Rasmussen, J. J., Krasheninnikov, S. I. et al. & JET EFDA Contributors 2009 Blob/hole formation and zonal-flow generation in the edge plasma of the JET tokamak. Nucl. Fusion 49, 092002.Google Scholar
Xu, G. S., Naulin, V., Fundamenski, W., Rasmussen, J. J., Nielsen, A. H. & Wan, B. N. 2010 Intermittent convective transport carried by propagating electromagnetic filamentary structures in nonuniformly magnetized plasma. Phys. Plasmas 17, 022501.Google Scholar
Zhang, Y.-C. 1989 Scaling theory of self-organized criticality. Phys. Rev. Lett. 63, 470–473.Google Scholar