J. Phys. I France 4 (1994) 1383-1410
A model for the dynamics of sandpile surfacesJ.-P. Bouchaud, M. E. Cates, J. Ravi Prakash and S. F. Edwards
Cavendish Laboratory, Madingley Road, Cambridge CB3 OHE, U.K.
(Received 31 March 1994, accepted in final form 30 June 1994)
We propose a new continuum description of the dynamics of sandpile surfaces, which recognizes the existence of two populations of grains: immobile 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 process, which represents both the random sticking of rolling grains (below the angle of repose), and the dislodgement of immobile grains by rolling ones (for greater slopes). We predict that the mean downhill motion of rolling grains causes surface features to move uphill; shocks can arise at large amplitudes. Our equations exhibit a second critical angle, larger than the angle of repose, at which the surface of a tilted immobile sandpile first becomes unstable to an infinitesimal perturbation. Our model is used to interpret the results of rotating-drum experiments. We study the long time behaviour of our equations in the presence of noise. For an initially rough surface at the repose angle, with no incident flux and an initially constant rolling grain density, the roughness decays to zero in time with an exponent found from a linearized version of the model. In the presence of spatiotemporal noise, we find Chat 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 considered. Finally, we show how our phenomenological description can be derived from a discretized model involving the stochastic motion of individual grains.
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