Building Blocks Of Self -Organized Criticality

Abstract

Why are we having difficulty developing economical nuclear fusion? How can a squirrel cause a statewide power blackout? How do correlations arise in a random complex system? How are these questions related? This thesis addresses these questions through a study of self-organized criticality (SOC). Among the systems that have been proposed as SOC are confined fusion plasmas, the Earth's magnetosphere and earthquake faults. SOC describes how large-scale complex behavior can emerge from small-scale simple interactions. The essence of SOC is that many dynamical systems, regardless of underlying physics, share a common nonlinear mechanism: local gradients grow until exceeding some critical gradient and then relax in events called avalanches. Avalanches range in size from very small to system-wide. Interactions of many avalanches over long times result in robust statistical and dynamical signatures that are surprisingly similar in many different physical systems. Two of the more well-known signatures are power law scaling of probability distribution functions (PDFs) and power spectra. Of particular interest in the literature for approximately a century are 1/f spectra. I studied the SOC running sandpile model and applied the results to confined and space plasmas. My tools were power spectra, PDFs and rescaled range ( R/S) analysis. I found that SOC systems with random external forcing store memory of previous states in their local gradients and can have dynamical correlations over very long time scales regardless of how weak the external forcing is. At time scales much longer than previously thought, the values of the slope of the power spectra, beta and the Hurst exponent, H, are different from the values found for white noise. As forcing changes, beta changes in the range 0.4 <math> <f> &lap;</f> </math> beta &le; 1 but the Hurst exponent remains relatively constant, H &ap; 0.8. The same physics that produces a 1/f spectrum at strong forcing produces a f -0.4 spectrum at weaker forcing. Small amounts of diffusive spreading added to the two dimensional SOC sandpile greatly decreases the frequency and maximum size of large transport events. More diffusion increases the frequency of large events to values much greater than for systems without diffusion.