Sandpiles with Bistable Automata Rules: Towards a Minimal Model of Pedestal Formation and Structure

A sandpile with two stable and two unstable ranges of slopes is presented as a minimal model for the study of $H$-mode pedestal formation and dynamics. Pedestals are observed to form and expand inward with increasing deposition. Transport bifurcation is not critical to pedestal formation, though the pedestal structure obtained with a second, hard stability boundary is qualitatively different from that found in standard sandpiles. Nonperiodic particle ejection events are observed, but do not initiate pedestal collapse. Pedestal formation alters the spectrum of transport avalanches.

Figure 1

Steady-state profiles for deposition rates $N_f=1,2,3,4,5$, and 6. Observe the appearance of a steep gradient as $N_f$ exceeds a critical value.

Figure 2

Widths of the pedestal (triangles) in the simulations with different deposition rates $N_f$.

Figure 3

The space-time pattern of the pile: white cells are stable, while gray and black cells are in first and in second unstable ranges of slopes, respectively. On the left: threshold deposition $N_f=3$, and the pedestal is not formed. On the right: the deposition $N_f=10/3$ exceeds the threshold and the pedestal appears. In this case, the transport through the edge is time intermittent.

Figure 4

Distributions of quiescent time between edge transport events. Left plot: for the marginal case of $N_f=10/3$, when the pedestal just appeared; the fit for quiet time distribution is $e^{[10.5+(\tau -10/3{)}^2/12]}+e^{(9-\tau /6)}$, i.e., Gaussian $+$ exponent. Right plot: when the pedestal is 30–50 cells wide, the main body of the distribution is a power law, and the tail is exponential.

Figure 5

Family of parametric plots of local flux $\Gamma (r)$ versus local slope $Z(r)$ for different depositions $N_f$ from 1 to 8. The “transport” function $\Gamma (Z)$ is multivalued for $\Gamma >3$.

Figure 6

Power spectra of flips for sandpiles in $L$ mode ($N_f\le 3$) and in $H$ mode ($N_f>3$). Note the change of power index from 2.7 to 1.4.

Figure 7

Power spectra of flips in core and in pedestal regions for the case $N_f=6$, when pedestal occupies $\approx 50\%$ of the pile. The power content of flips of the pedestal region is an order of magnitude smaller than that of the core, though the power index seems to be the same in both regions.