The Savage‐Hutter theory: A system of partial differential equations for avalanche flows of snow, debris, and mud

Abstract

AbstractThe Savage‐Hutter (SH) equations of granular avalanche flows are a hyperbolic system of equations determining the distribution of depth and depth‐averaged velocity components tangential to the sliding bed. We review the equations and point out the geometrical complexities to which these equations have been generalized. Because of the hyperbolicity of the equations, successful numerical modelling is challenging, particularly when large gradients of the physical variables occur, e.g. for a moving front or possibly formed shock waves in avalanche flows if velocities change from supercritical to subcritical e.g. during the deposition. Numerical schemes solving these free surface flows must be able to cope with smooth as well as non‐smooth solutions. In this paper several numerical methods are applied to solve the SH equations and compared, including traditional difference schemes, e.g. central and upstream difference schemes, as well as high‐resolution NOC (Non‐Oscillatory Central Differencing) schemes, in which several second‐order TVD (Total Variation Diminishing) limiters and a third‐order ENO (Essentially Non‐Oscillatory) cell reconstruction scheme are used. Results show that the high‐resolution schemes, particularly the NOC scheme with the Minmod TVD limiter or the van Leer limiter, provide excellent performances. In the SH theory the material response is expressed by only two phenomenological parameters – the internal and the bed friction angles. Parameter investigations show that avalanche flows are much more sensitive against variations of the bed friction angle than that of the internal angle of friction. Effects due to a pressure dependence of the bed friction angle and lateral variations of the basal topography are therefore also numerically examined.