DISCRETE AND CONTINUOUS Website: http://AIMsciences.org DYNAMICAL SYSTEMS–SERIES B Volume 3, Number 4, November 2003 pp. 589–599 A MODEL OF GRANULAR FLOWS OVER AN ERODIBLE SURFACE E.B. PITMAN, C.C. NICHITA A.K. PATRA, A.C. BAUER M. BURSIK AND A. WEBB Department of Mathematics, Department of Mechanical and Aerospace Engineering, Department of Geology, University at Buffalo, Buffalo, NY 14260 To Dave Schaeffer, who shows how mathematics can impact science. Abstract. We present a framework for modeling a dry geophysical mass of granular material – a debris or volcanic avalanche or landslide – flowing over an erodible surface. We also describe a computing environment that incorporates topographical data into a parallel, adaptive mesh computational algorithm that solves the model equations. 1. Introduction. Slow moving geophysical mass flows – debris flows, block and ash flows, volcanic avalanches, or landslides – may be initiated by volcanic activity. In these flows, constituent particles are typically centimeter to meter sized, and the flows, sometimes as fast as tens of meters per second, propagate distances of tens of kilometers. As these flows slow, the particle mass sediments out, yielding deposits that can be a hundred meters deep and many kilometers in length [9]. These flows can contain O(10 6 − 10 10 )m 3 or more of material, and many also include a significant amount of water. The range of scales and the complexity of the rheology for geological materials, coupled with the mathematical problem of describing and computing a free surface flow, is a significant challenge. We lack a full understanding of how these mass flows are initiated, but there is a growing understanding of processes governing flow, once that motion has been initiated. This paper describes one effort in modeling and simulating geophysical mass flows. Our efforts try to strike a balance between fidelity to physics on the one hand and mathematical and computational tractability on the other. The starting point for the modeling effort here is the pioneering work of Savage and Hutter [21]. Our contribution is to incorporate into the modeling the effect of erosion on the dynamics of dry, flowing granular materials. In a first approximation we assume that changes in elevation of the basal surface due to erosion are small, and we solve the equations of motion with the bed surface frozen. For the large- scale flows that are our primary interest, this assumption seems justified, but the assumption may not be accurate when comparing model results against small table- top experiments. 1991 Mathematics Subject Classification. 86A60, 86-08, 74H15. Key words and phrases. Geological problems, Computational methods, Numerical approxima- tion of solutions. 589