Triangular and Uphill Avalanches of a Tilted Sandpile

Recent experiments show that an avalanche initiated from a point source propagates downwards by invading a triangular shaped region. The opening angle of this triangle appears to reach 180o for a critical inclination of the pile, beyond which avalanches also propage upwards. We propose a simple interpretation of these observations, based on an extension of a phenomenological model for surface flows. Well-controlled avalanche experiments have recently been performed by Rajchenbach in (narrow) rotating drums [1], and by Daerr and Douady on a thin layer of sand atop a wide inclined plane (of high surface friction) at various angles of inclination [2]. In the latter experiments, an avalanche is triggered locally (e.g. by a probe) and its subsequent evolution is observed. One sees very clearly that below a certain inclination angle, the avalanche rapidly dies out, leaving only a small perturbed region around the initiation point. When the angle exceeds a first critical angle, the avalanche progresses downwards indefinitely; the perturbed region has a triangular shape with its apex at the initiation point. The opening angle of this triangle increases with the inclination and appears to reach 180 degrees at a second critical slope, beyond which the avalanche not only propagates downward, but also upward, progressively invading the whole sample. That avalanches can move upward has also been observed by Rajchenbach [1]. We want to show in this short note that these observations can be qualita2 J.P. BOUCHAUD AND M. E. CATES tively (and perhaps quantitatively) understood within the so-called ‘bcre’ model that was introduced to describe surface flows and avalanches [3]. The starting point of the ‘bcre’ phenomenological description is to recognize that the simultaneous evolution of two physical quantities must be accounted for, namely: • the local ‘height’ of immobile particles, h(~x, t) (which depends both on the horizontal coordinates ~x = (x, y) of the considered surface element and on time t) • the local density of rolling particles R(~x, t), which can be thought of as the thickness of a flowing layer of grains. The presence of two variables, rather than one, crucially affects the “hydrodynamic” behaviour at large lengthand time-scales and, as shown previously [3] can account for the Bagnold hysteresis effect within a simple one-dimensional treatment. (The Bagnold angle is the difference between the “angle of repose” of a surface after an avalanche, and the “maximum angle of stability” at which a new avalanche is initiated.) Variants of the model also allow segregation and stratification [4, 5] or the formation of sand ripples [6] to be understood in a simple way. In what follows, we extend the previous discussion of hysteresis [3] to 2+1 dimensions (the case of a tilted planar surface) and show that the occurence of triangular avalanches can be understood. We also quantify the behaviour of backward propagating avalanche fronts and clarify the physical mechanisms involved in these. In constructing plausible equations governing the time evolution of the two quantities h and R, we considered a regime where the rolling grains quickly reach a constant average velocity v0 along the x direction (reflecting the balance between gravity and inelastic collisions with the immobile bed). We also assumed R(~x, t) to be small enough in order to discard all effects of order R (for example, v0 might depend on R; we return to this point later). Thus we write [3]: