Self-Organised Criticality

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Self-Organised Criticality (SOC) is a generic concept, applicable to a host of complex systems [1]. A system is said to be in this state when it is at an attractive critical point at which it behaves as in a phase transition (i.e., the spatial and temporal scales are scale-invariant, or nearly so). Note that ordinary phase transitions are not attractive, and maintaining the system near such a phase transition point requires fine-tuning some system parameters. SOC is different in that the system is attracted to the critical point. This situation can only occur in systems that are not in equilibrium, in which fluctuations provide a mechanism for regulating the system and keeping it close to criticality.

In magnetically confined plasmas, this state is thought to be responsible for the global transport phenomena of:

  • profile consistency, which is the observation that profiles tend to have roughly the same shape, regardless of the power and location of the applied heating.[2]
  • the Bohm scaling of confinement in L-mode (scaling of transport with system size) [3], and
  • power degradation, as reflected in global transport scaling laws. The scaling of the plasma energy content with injected power is generally found to be sub-linear, i.e., considerably worse than expected from simple diffusion.

The basic explanation for these phenomena is self-regulation of the profiles by turbulence (see Anomalous transport). [4] The strong temperature and density gradients in fusion-grade plasmas provide free energy that may drive turbulence. The turbulence then enhances transport locally, leading to a local reduction of gradients and a consequential damping of the turbulence amplitude. This feedback could be responsible for keeping the gradients below a critical value. Considered locally, the former is a description of a simple marginal state. But the interaction of such feedback mechanisms at various radial locations would lead to avalanche behaviour and a true (scale-free) self-organised state.

Indeed, there is direct evidence for avalanching behaviour in numerical simulations [5], but experimental evidence is scarce. [6] However, some indirect evidence exists. Typically, such evidence involves the detection of long-range correlations in fluctuations. [7]

Evidence for critical gradients is much more abundant. [8] [9] However, the existence of a critical gradient by itself does not prove the system is in a SOC state.

References

  1. Self-Organised Criticality in the Wikipedia
  2. F. Ryter et al., Plasma Phys. Control. Fusion 43, A323 (2001)
  3. B.A. Carreras, IEEE Trans. Plasma Science 25, 1281 (1997)
  4. D.E. Newman et al., Phys. Plasmas 3, 1858 (1996)
  5. L. García and B.A. Carreras, Phys. Plasmas 12, 092305 (2005)
  6. P.A. Politzer, Phys. Rev. Lett. 84, 1192 (2000)
  7. B.A. Carreras et al., Phys. Plasmas 6, 1885 (1999)
  8. D.R. Baker et al., Phys. Plasmas 8, 4128 (2001)
  9. F. Ryter et al., Plasma Phys. Control. Fusion 43, A323 (2001)