A MODEL FOR THE DYNAMICS OF SANDPILE SURFACES

We propose a new continuum description of the dynamics of sandpile sur faces, which recognizes the existence of two populations of grains: im mobile and rolling. The rolling grains are carried down the slope with a constant drift velocity and have a certain dispersion constant. We introduce a simple bilinear approximation for the interconversion proc ess, which represents both the random sticking of rolling grains (belo w the angle of repose), and the dislodgement of immobile grains by rol ling ones (for greater slopes). We predict that the mean downhill moti on of rolling grains causes surface features to move uphill; shocks ca n arise at large amplitudes. Our equations exhibit a second critical a ngle, larger than the angle of repose, at which the surface of a tilte d immobile sandpile first becomes unstable to an infinitesimal perturb ation. Our model is used to interpret the results of rotating-drum exp eriments. We study the long time behaviour of our equations in the pre sence of noise. For an initially rough surface at the repose angle, wi th no incident flux and an initially constant rolling grain density, t he roughness decays to zero in time with an exponent found from a line arized version of the model. In the presence of spatiotemporal noise, we find that the interconversion nonlinearity is irrelevant, although roughness now becomes large at long times. However, the Kardar-Parisi- Zhang nonlinearity remains relevant. The behaviour of a sandpile with a steady or noisy input of grains at its apex is also briefly consider ed. Finally, we show how our phenomenological description can be deriv ed from a discretized model involving the stochastic motion of individ ual grains.