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25 Years of Self-organized Criticality: Space and Laboratory Plasmas

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Abstract

Studies of complexity in extended dissipative dynamical systems, in nature and in laboratory, require multiple approaches and the framework of self-organized criticality (SOC) has been used extensively in the studies of such nonequilibrium systems. Plasmas are inherently nonlinear and many ubiquitous features such as multiscale behavior, intermittency and turbulence have been analyzed using SOC concepts. The role of SOC in advancing our understanding of space and laboratory plasmas as nonequilibrium systems is reviewed in this article. The main emphasis is on how SOC and related approaches have provided new insights and models of nonequilibrium plasma phenomena. Among the natural plasmas the magnetosphere, driven by the solar wind, is a prominent example and extensive data from ground-based and space-borne instruments have been used to study phenomena of direct relevance to space weather, viz. geomagnetic storms and substorms. During geomagnetically active periods the magnetosphere is far from equilibrium, due to its internal dynamics and being driven by the turbulent solar wind, and substorms are prominent features of the complex driven system. Studies using solar wind and magnetospheric data have shown both global and multiscale features of substorms. While the global behavior exhibits system-wide changes, the multiscale behavior shows scaling features. Along with the studies based on observational data, analogue models of the magnetosphere have advanced the understanding of space plasmas as well as the role of SOC in natural systems. In laboratory systems, SOC has been used in modeling the plasma behavior in fusion experiments, mainly in tokamaks and stellarators. Tokamaks are the dominant plasma confinement system and modeling based on SOC have provided a complementary approach to the understanding of plasma behavior under fusion conditions. These studies have provided insights into key features of toroidally confined plasmas, e.g., the existence of critical temperature gradients above which the transport rates increase drastically. The SOC models address the transport properties from a more general approach, compared to those based on turbulence arising from specific plasma instabilities, and provide a better framework for modeling features such as superdiffusion. The studies of space and laboratory plasmas as nonequilibrium systems have been motivated by features such as scaling and critical behavior, and have provided new insights by highlighting the properties that are common with other systems.

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Notes

  1. The Hurst exponent \(H\) is related to the spectral index \(\beta\) via the Berry formula \(\beta= 2H +1\). \(H=1/2\) corresponds to white noise, with \(\beta= 0\); \(H < 0\) to fractional Brownian noise; \(H=0\) to flicker noise; \(0 < H < 1\) to fractional Brownian motion.

  2. Indeed neoclassical transport is classical transport including the effects of toroidal geometry. The transport is still driven by Coulomb collisions—no anomalous transport due to plasma microscopic instabilities is included. The collisional character of the transport implies that there is a characteristic scale posed by the dynamics. So this scale will be always there in the thermodynamic (large-system) limit at odds with the multi-scale features characterizing SOC. The net result is that SOC is to be expected in regimes where Coulomb collisions lose their importance, which is somehow the case of rarefied thermonuclear plasmas.

  3. The inverse spectral energy transfer is inherent to fluid-turbulence systems in two dimensions (2D), including the ubiquitous drift-wave turbulence. In 2D turbulence energy cascades from small to large scales because the phenomenon of vortex stretching is forbidden.

  4. The idea that activity in one region can stimulate activity in another region, particularly in a nonlinear context, is in fact very general and as such must occur in many applications. Here we mention the processes of stimulated galaxy formation discussed by Schulman and Seiden (1986), who used this to model the hierarchical structure in the distribution of galaxies with power law correlations.

  5. Because of amplification, we expect a steeper drop-off in the energy spectrum of the coupled SOC-turbulence system as compared to the inertial range of the fluid (drift-wave) turbulence. In this connection, we should stress that the avalanching transport is triggered by the explicit radial dependence in the profiles, and, when account is taken for the inverse cascade of the energy, by boundary feedbacks, so that the assumptions of constant energy transfer and of infiniteness of the system, resulting in the fluid-like \(-5/3\) behavior, do not really apply here.

  6. The Fick paradigm states that the internal fluxes are described by a set of local transport coefficients—diffusivities or conductivities—related to the local thermodynamic forces which induce the fluxes through Fick’s law. Models based on these assumptions are referred to as local models.

  7. The Russian word “roy” means a “swarm” in English.

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Acknowledgements

The author team acknowledges the hospitality and partial support for two workshops on “Self-Organized Criticality and Turbulence” at the International Space Science Institute (ISSI) in Bern, Switzerland, during October 15–19, 2012, and September 16–20, 2013. Discussions with V.P. Budaev, J. Chen, G. Dif-Pradalier, A. Iomin, P.K. Kaw, V. Krishnamurthy, K. Papadopoulos, J.J. Rasmussen, A. Sen, X. Shao, M.I. Sitnov, A.Y. Ukhorskiy, D. Vassiliadis, T. Veeramani, L.M. Zelenyi, and F. Zonca are gratefully acknowledged. The research at the University of Maryland was supported by NSF grants AGS-1036473 and IIP-1338634.

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Authors and Affiliations

  1. Department of Astronomy, University of Maryland, College Park, MD, 20740, USA

    A. Surjalal Sharma

  2. Solar and Astrophysics Laboratory (LMSAL), Lockheed Martin, Palo Alto, CA, 94304, USA

    Markus J. Aschwanden

  3. Belgian Institute for Space Aeronomy, Ringlaan-3-Avenue Circulaire, 1180, Brussels, Belgium

    Norma B. Crosby

  4. NASA Goddard Space Flight Center, Code 671.0, Greenbelt, MD, 20771, USA

    Alexander J. Klimas

  5. ENEA National Laboratory, Centro Ricerche Frascati, 00044, Frascati, Rome, Italy

    Alexander V. Milovanov

  6. Space Research Institute, Russian Academy of Sciences, Profsoyuznaya str. 84/32, 117997, Moscow, Russia

    Alexander V. Milovanov

  7. Space Science and Technology Branch, Canadian Space Agency, 6767 Route de l’Aeroport, Saint Hubert, Quebec, J3Y8Y9, Canada

    Laura Morales

  8. Dept. Fisica, Universidad Carlos III de Madrid, Avda. de la Universidad 30, 28911, Leganes, Madrid, Spain

    Raul Sanchez

  9. Catholic University of America at NASA Goddard Space Flight Center, Code 671, Greenbelt, MD, 20771, USA

    Vadim Uritsky

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  1. A. Surjalal Sharma
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  2. Markus J. Aschwanden
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Correspondence to A. Surjalal Sharma.

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Sharma, A.S., Aschwanden, M.J., Crosby, N.B. et al. 25 Years of Self-organized Criticality: Space and Laboratory Plasmas. Space Sci Rev 198, 167–216 (2016). https://doi.org/10.1007/s11214-015-0225-0

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