IIT, Madras Int J Adv Eng Sci Appl Math (2009) 1:99–121 ORIGINAL PAPER Two-layer models for shallow avalanche flows over arbitrary variable topography I. Luca · K. Hutter · C. Y. Kuo · Y. C. Tai c  Indian Institute of Technology, Madras 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 inter- face and at the free upper surface, and the materials in both layers are treated as density preserving. The intrinsic model- ling 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 ǫ , the shallowness parameter, plays a central role. The pro- posed 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 I. Luca Division of Mechanics, Research Center for Applied Sciences, Academia Sinica, Nangang, Taipei 11529; on leave from Department of Mathematics II, University Politehnica of Bucharest, Splaiul Inde- pendentei 313, 79590 Bucharest, Romania E-mail: iolucaro@yahoo.com K. Hutter Permanent address: Bergstrasse 5, 8044 Z¨ urich, Switzerland C. Y. Kuo Division of Mechanics, Research Center for Applied Sciences, Academia Sinica, Nangang, Taipei 11529 E-mail: cykuo06@gate.sinica.edu.tw Y. C. Tai Department of Civil Engineering, National Chi Nan University, No. 1, University Road, Puli, Nantou 545, Taiwan E-mail: yctai@ncnu.edu.tw each layer, at the bottom topography and at the layer interface. Keywords Avalanche modelling · Two-layer shallow flow · Variable topography · Erosion/deposition rate 1 Introduction Catastrophic debris flows of fluid–solid mixtures which may occur in typhoon or hurricane induced landslides, in fluvial hydraulic currents during strong precipitation events, or in sub-aquatic turbidity currents under slope instabilities, occa- sionally develop into a two-layer flow regime of a relatively dense near bottom debris layer carrying the coarse sediment fraction plus a water layer, which carries the clay and silt fractions in suspension and may be regarded as a slurry. The granular fluid system is nourished by soil mass entrained from the stagnant bottom region, over which the solid–fluid system runs. Both layers may in a first approximation be viewed as one-constituent continuous immiscible fluid like bodies, and hence separated from one another by an inter- face which is material for each layer. Moreover, in a first approximation, the free surface of the upper layer may also be treated as material, thus ignoring the contribution of the precipitation that generally takes place in such events. In general, the eroding soil has a larger density than the dense debris of the lower layer. However, the increase in density by the eroding soil is approximately counter bal- anced by the interstitial fluid that may be exchanged between the two layers. In a single constituent treatment of the mate- rial in the two regions, this water exchange is not modelled, so that the underlying assumption that the two layer materi- als are density preserving and that the interface is material with respect to each layer accounts in a gross fashion for the realistic behavior. The flow of such fluid systems takes place over natural territory, practically “arbitrary” topography, that is best