We use the non-Cartesian, topography-based equations of mass and momentum balance for gravity driven frictional flows of Luca et al. (Math. Mod. Meth. Appl. Sci. 19, 127-171 (2009)) to motivate a study on various approximations of avalanche models for single-phase granular materials. By introducing scaling approximations we develop a hierarchy of model equations which differ by degrees in shallowness, basal curvature, peculiarity of constitutive formulation (non-Newtonian viscous fluids, Savage-Hutter model) and velocity profile parametrization. An interesting result is that differences due to the constitutive behaviour are largely eliminated by scaling approximations. Emphasis is on avalanche flows; however, most equations presented here can be used in the dynamics of other thin films on arbitrary surfaces.

When dealing with geophysical flows across three-dimensional topography or other thin layer flows, for the physical modelling and for computational reasons, it is more convenient to use curvilinear coordinates adapted to the basal solid surface, instead of the Cartesian coordinates. Using such curvilinear coordinates, e.g. introduced by Bouchut and Westdickenberg,3 and the corresponding contravariant components of vector and tensor fields, we derive in full generality the governing equations for the avalanche mass. These are next used to deduce (i) the thin layer equations for arbitrary topography, when the flowing mass is an ideal fluid, and (ii) the thin layer equations corresponding to arbitrary topography and to a viscous fluid that experiences bottom friction, modelled by a viscous sliding law.

This paper presents a three-dimensional, twolayer model for shallow geophysical mass flows, such as debris flows, hydraulic sediment transport, or sub-aquatic turbidity currents down arbitrary natural topographic terrains. The bottom layer is a dense granular fluid which interacts with the stagnant basal topography through an erosion/deposition mechanism. Above this layer is a lighter fluid layer. There is no mass exchange at the layer interface and at the free upper surface, and the materials in both layers are treated as density preserving. The intrinsic modelling equations are written in non-dimensional form and then formulated relative to a topography-adjusted coordinate system. The mass balance equations and momentum balance equations parallel to the bottom topography are depthaveraged over the layers. The emerging governing system of equations is subsequently simplified on the basis of problem-adapted scales, in which a small parameter , the shallowness parameter, plays a central role. The proposed ordering scheme is motivated by an earlier analysis, [1], and depends on the rheological complexities of the stress parameterizations of the two fluids. The ensuing equations are complemented by constitutive assumptions in

No abstract

No abstract