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Depth-averaged models in the geosciences: An overview and a new model for the simulation of gravity-driven pyroclastic currents.
de' Michieli Vitturi M., Esposti Ongaro T., Lari G., Aravena Ponce A., Neri A.
Nonlinear shallow-water equations were firstly derived in 1871 by A.J.C. de Saint-Venant in its unidimensional form. Despite their age, they are still widely used for the simplicity of their form and the number of fields to which they apply. In the geosciences, they are employed to describe tsunamis, debris flow, snow avalanches, lava flows and other wave propagation phenomena where the horizontal scale of the flow is much greater than the depth of the fluid. We present here a modified 2D shallow water model designed to model pyroclastic avalanches, focusing on the importance of a proper numerical discretization to guarantee the positivity of the flow thickness and the correct coupling between gravitational and frictional forces. Pyroclastic avalanches (PA) are gravity-driven pyroclastic currents, typically confined within steep volcanic slopes, whose dynamics are controlled by the balance between the longitudinal component of gravity and the frictional stress, and by their interaction with the topography. Physical and numerical modelling is a key tool for studying their behaviour and evaluating their hazards, but it is particularly challenging because of the complex (and relatively poorly understood) rheology of the basal, concentrated granular flow and its interplay with the substrate and the overriding, relatively dilute, turbulent ash cloud. In the present implementation, the model solves the basic depth-averaged mass and momentum equations for a single-fluid mixture having constant density and specific rheological laws (Voellmy-Salm or constant stress). Model equations are written in conservative form in a geographic (absolute) coordinate system, thus avoiding the need of metric corrections ($e.g.$ curvature) on complex topographies. On steep slopes, non-hydrostatic formulation of the vertical momentum equation can also be implemented to provide accurate modelling of avalanches with non-negligible vertical acceleration. Furthermore, an implicit treatment of the non-linear friction terms has been implemented, in order to avoid the problems typically associated with operator-splitting or predictor-corrector algorithms in the regime of flow stoppage. We present here some first benchmark tests to show the model capabilities and an application to the simulation of PA generated by the partial collapse of scoria craters cones at Mt. Etna volcano (Italy).