One-dimensional modelling of rock-ice avalanches: mathematical features, numerical solutions, and strategies to enlarge the hyperbolic range

Rock-ice avalanches are three-phase flows composed of rock, ice, and a liquid. As their occurrences might increase in the future due to climate change, constructing mathematical and numerical models able to simulate these flows could be necessary for good hazard assessment and management in cold mountainous regions. With this aim, in this work, a framework of simplified rock-ice avalanche models is derived from a complete three-phase approach by applying two assumptions. Thanks to these two hypotheses, we obtain five classes of simplified mathematical models that simplify the rock- ice avalanche physics with different levels of approximation. Among these simplified approaches, the mathematical model, which simplifies the flow dynamics to a lesser extent, is a new mathematical model for rock-ice avalanches. For numerical purposes, a detailed analysis of the eigenvalues is performed for the one-dimensional depth-integrated version of the proposed model. Results show that the proposed approach loses hyperbolicity for specific ranges of the flow variables. Due to this feature, numerical modelling is performed by maintaining the numerical solutions in the hyperbolic domain of the flow variables. In this way, we consider the uniformly accelerated flow and the small perturbation of the flow depth and ice concentration as test cases. Additionally, we implement three numerical methods to identify the numerical scheme that can solve the proposed model accurately and to compare the obtained numerical results with those associated with the other simplified rock-ice avalanche approaches. Finally, we apply the linear stability theory to the proposed model to investigate its potential ill-posedness in the ranges of the flow variables where hyperbolicity is lost. Since the proposed approach turns out to be ill-posed, the model regularization is performed by trying to recover its hyperbolicity through a strategy suggested for a two- phase gas-liquid model. Although this strategy can enlarge the hyperbolic flow-variable range, hyperbolicity is still lost for specific flow conditions.