Dynamical Consistency Constraints on Entrainment and Deposition in Depth-Averaged Models of Snow Avalanches and Other Gravity Mass Flows

Abstract

This paper discusses several issues related to the mathematical and physical formulation of entrainment in depth-averaged models of gravity mass flows (GMF) such as snow avalanches and landslides. (i) The relation between system boundaries, boundary conditions and entrainment-related source terms in the momentum balance and the equation of motionis clarified for different types of GMFs occurring in nature. (ii) Continuous deposition from a GMFrequires that the bed exerts a larger shear stress than the interior of the flow can sustain. This is shown to imply that a GMF cannot accelerate due to deposition, contrary to some claimsin the literature. (iii) Entrainment and deposition differ with respect to the boundary condition for the shear stress at the bed. (iv) The entrainment rate depends onthe instantaneous flow conditions, the bed properties and the GMFrheology and thus cannot be specified independently of the latter. More specific results are obtained by limiting attention to basal erosion and assuming perfectly brittle behavior of the bed, characterized by its shear strength 7.. Forsliding block models, å simple explicit expression for the entrainmentrate is derived in terms of the bed friction law and 7.. For a quasi-stationary, uniform flow of a Bingham fluid over a brittle bed, the erosion rate and the modified velocity and shear-stress profiles are obtained analytically. Although the model cannot directly be applied to GMFs, entrainmentrates consistent with estimates from field measurements are obtained for parameter values typical of snow avalanches. An extension of the theory to non-stationary flows and more general rheologies is proposed, but numerical studies will be needed to assess the accuracy of the approximations involved.