Avalanches, hydrodynamics, and discharge events in models of sandpiles

Motivated by recent studies of Bak, Tang, and Wiesenfeld [Phys. Rev. Lett. 59, 381 (1987); Phys. Rev. A 38, 364 (1988)], we study self-organized criticality in models of ‘‘running’’ sandpiles. Our analysis reveals rich temporal structures in the flow of sand: at very short time scales, the flow is dominated by single avalanches. These avalanches overlap at intermediate time scales; their interactions lead to 1/f noise in the flow. We show that scaling in this region is a consequence of conservation laws and is exhibited in many examples of driven-diffusion equations for transport. At very long time scales, the sandpiles exhibit system-wide discharge events. These events also obey scaling and are found to be anticorrelated. We derive the ${\mathit{f}}^{1/2}$ mean-field power spectrum for these events and show that a threshold instability of the model, coupled with some stochasticity, is the underlying origin of the long-time anticorrelation.