Shortly after the seminal paper "Self-Organized Criticality: An explanation of 1/f noise" by Bak et al. (1987), the idea has been applied to solar physics, in "Avalanches and the Distribution of Solar Flares" by Lu and Hamilton (1991). In the following years, an inspiring cross-fertilization from complexity theory to solar and astrophysics took place, where the SOC concept was initially applied to solar flares, stellar flares, and magnetospheric substorms, and later extended to the radiation belt, the heliosphere, lunar craters, the asteroid belt, the Saturn ring, pulsar glitches, soft X-ray repeaters, blazars, black-hole objects, cosmic rays, and boson clouds. The application of SOC concepts has been performed by numerical cellular automaton simulations, by analytical calculations of statistical (powerlaw-like) distributions based on physical scaling laws, and by observational tests of theoretically predicted size distributions and waiting time distributions. Attempts have been undertaken to import physical models into the numerical SOC toy models, such as the discretization of magneto-hydrodynamics (MHD) processes. The novel applications stimulated also vigorous debates about the discrimination between SOC models, SOC-like, and non-SOC processes, such as phase transitions, turbulence, random-walk diffusion, percolation, branching processes, network theory, chaos theory, fractality, multi-scale, and other complexity phenomena. We review SOC studies from the last 25 years and highlight new trends, open questions, and future challenges, as discussed during two recent ISSI workshops on this theme
Compact optimized stellarators offer novel solutions for confining high-β plasmas and developing magnetic confinement fusion. The three-dimensional plasma shape can be designed to enhance the magnetohydrodynamic (MHD) stability without feedback or nearby conducting structures and provide driftorbit confinement similar to tokamaks. These configurations offer the possibility of combining the steady-state low-recirculating power, external control, and disruption resilience of previous stellarators with the low aspect ratio, high β limit, and good confinement of advanced tokamaks. Quasiaxisymmetric equilibria have been developed for the proposed National Compact Stellarator Experiment (NCSX) with average aspect ratio 4-4.4 and average elongation ∼1.8. Even with bootstrap-current consistent profiles, they are passively stable to the ballooning, kink, vertical, Mercier, and neoclassicaltearing modes for β > 4%, without the need for external feedback or conducting walls. The bootstrap current generates only 1/4 of the magnetic rotational transform at β = 4% (the rest is from the coils); thus the equilibrium is much less non-linear and is more controllable than similar advanced tokamaks. The enhanced stability is a result of 'reversed' global shear, the spatial distribution of local shear, and the large fraction of externally generated transform. Transport simulations show adequate fast-ion confinement and thermal neoclassical transport similar to equivalent tokamaks. Modular coils have been designed which reproduce the physics properties, provide good flux surfaces, and allow flexible variation of the plasma shape to control the predicted MHD stability and transport properties.
The dynamics of relativistic runaway electrons in tokamak plasmas is analyzed using a test particle description that includes acceleration in the toroidal electric field, collisions with the plasma particles, and deceleration due to synchrotron radiation losses. The region of momentum space in which electron runaway takes place is determined. It is found that relativistic and synchrotron radiation effects lead to a critical electric field ER>(kTe/mec2)ED, below which no runaways are generated. In addition, the trajectories of the test electrons in momentum space show a stable equilibrium point that sets a limit on the energy that the runaway electrons can reach. Analytical expressions are given for this energy limit as a function of the toroidal electric field and plasma parameters. The dominant radiative mechanisms limiting the runaway electron energy are identified in the whole range of electric field values.
The dynamics of high energy (up to 20 MeV) runaway electrons in the Frascati Tokamak Upgrade (FTU) is investigated using a gamma-ray spectrometer system which detects photons produced when runaway electrons interact with the plasma facing components [B. Esposito et al., Nucl. Instrum. Methods 476, 522 (2002)]. Runaway electrons are usually generated during the plasma current ramp-up, accelerated to MeV energies, and contained stably during the whole discharge time, which lasts for more than one second. This time is long enough for them to reach the limiting energy that results from the balance between acceleration in the electric field, collisions with the plasma particles and synchrotron radiation losses. The maximum energy inferred from the gamma spectra is shown to be in agreement with the runaway limiting energy predicted by a test particle description of the runaway dynamics [J. R. Martı́n-Solı́s et al., Phys. Plasmas 5, 2370 (1998)]. It is found that the runaway energy behavior during the discharge is determined by the time evolution of the plasma parameters only (mainly E∥, ne and Zeff) and that the synchrotron radiation losses associated with the electron gyromotion around the magnetic field lines can explain the measured limiting runaway energy in FTU.
A new solver for rapidly obtaining magnetohydrodynamic (MHD) equilibria in toroidal systems in the presence of islands and stochastic regions is described. It is based on the Kulsrud-Kruskal MHD energy minimization principle. To carry out this minimization, small displacements are made around a convenient set of curvilinear coordinates obtained from a nearby three-dimensional equilibrium that assumes nested surfaces. Because the changes of the magnetic fields and pressure are small, corresponding to small changes in the initial magnetic and kinetic energies, solutions for the linearized perturbations can be used to rapidly and iteratively find lower energy states with magnetic islands. A physics-based preconditioner is developed to accelerate the convergence of the iterative procedure to obtain an ideal MHD equilibrium with broken magnetic surfaces (islands).
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