2-DIMENSIONAL SPREADING OF A GRANULAR AVALANCHE DOWN AN INCLINED PLANE .1. THEORY

This paper is concerned with the motion of an unconfined finite mass o f a granular material released from rest on an inclined plane. The gra nular mass is treated as a frictional Coulomb-like continuum with a Co ulomb-like basal friction law. Depth averaged equations are deduced fr om the three-dimensional dynamical equations by scaling the equations and imposing the shallowness assumption that the moving piles are long and wide but not deep. Several distinguished limits for small depth t o length and depth to width ratios can be analysed. We develop an appr oximate theory based upon the full dynamical equations parallel to the inclined plane and imposed hydrostatic pressure conditions perpendicu lar to it. The resulting model equations are then applied to construct either yet simpler model equations or else solutions for particular c ases. In a first application the transverse distributions of the veloc ity fields and of the depth profile are prescribed, while representati ve values of these functions (such as the cross sectional averages or maxima) as functions of time and the downhill coordinate are left unsp ecified. For these quantities evolution equations are obtained from a lateral averaging of the vertically averaged equations. In a second ap plication approximate similarity solutions of the spatially two-dimens ional equations are derived. The depth and velocity profiles for the m oving mass are determined in analytical form, and the evolution equati on for the total length and the total width of the pile is integrated numerically. A parameter study illustrates the performance of the mode l.