Two-layer models for shallow avalanche flows over arbitrary variable topography

Abstract

This paper presents a three-dimensional, two-layer 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 depth-averaged over the layers. The emerging governing system of equations is subsequently simplified on the basis of problem-adapted scales, in which a small parameter ε, 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 each layer, at the bottom topography and at the layer interface.