Self-consistent models of basal entrainment in snow avalanches (Invited)

Observations indicate that the highest erosion rates in snow avalanches usually occur at the very front of the flow. Despite lower erosion rates, basal entrainment may give a similar contribution to the mass balance due to the extended area over which it occurs. For physical reasons, the erosion rate must be determined uniquely by the properties of the snow-cover and the dynamic characteristics of the flow. However, current entrainment models typically contain adjustable parameters whose relation to the snow-cover and flow characteristics remains elusive. In order to shed some light on this question, the dynamics of basal erosion and entrainment is studied in a simplified setting, assuming perfectly brittle behavior of the snow cover: It breaks instantaneously when the shear stress exceeds its shear strength, but no work is expended thereby. A consequence of this assumption is that the shear stress at the top of the snow cover remains locked at this value whenever erosion occurs. Many current avalanche flow models explixcitly or implicitly assume that all shear is concentrated at the interface. It is shown how this assumption together with the brittleness assumption stated above implies a simple explicit formula for the erosion rate as a function of the flow velocity, normal stress at the interface, and shear strength of the snow cover. The velocity dependence of the erosion rate is determined by the form of the assumed friction law. There are no adjustable parameters, yet the model gives erosion rates consistent with observations if the Voellmy friction law is assumed. For flows with non-uniform velocity profile, a quasi-stationary flow configuration is studied, in which the flow height is artificially held constant by skimming off material from the avalanche surface at the same rate as it is entrained from the snow cover. Assuming a value for the erosion rate, the resulting ODE can be solved explicitly for Binghamian fluids and numerically for non-linear rheologies. Self-consistent values of the erosion rate and the flow velocity can be determined by imposing the shear-stress boundary condition at the interface. The velocity profile and erosion rate in non-stationary flow conditions can be approximated by the erosion rate of a corresponding quasi-stationary flow at a different slope angle that reproduces the depth-averaged velocity of the non-stationary flow. This entrainment model is implemented in a quasi-3D (depth-averaged) avalanche model as a precompiled look-up table for the non-dimensionalized erosion rate in terms of non-dimensionalized snow-cover and flow parameters. Selected simulations serve to illustrate the behavior of the model.