Abstract
Mathematical models of different degrees of complexity, describing the motion of a snow avalanche along a path with given center line and spatially varying width, are formulated and compared. The most complete model integrates the balance equations for mass and momentum over the cross-section and achieves closure through an entrainment function based on shock theory and a modified Voellmy bed friction law where the Coulombic contribution to the bed shear stress is limited by the shear strength of the snow cover. A simplified model results from integrating these balance equations over the (time-dependent) length of the flow and postulating weak similarity of the evolving avalanche shape. On path segments of constant inclination, it can be solved for the flow depth and speed of the front in closed form in terms of the imaginary error function. Finally, the very simplest model assumes constant flow height and length. On an inclined plane, the evolution of flow depth and velocity predicted by the simplified model are close to those from the full model without entrainment and with corresponding parameters, but the simplest model with constant flow depth predicts much higher velocity values. If the friction coefficient is varied in the full model with entrainment, there can be non-monotonous behavior due to the non-linear interplay between entrainment and the limitation on the Coulomb friction.
Highlights
The mathematical modeling of snow avalanche movement is a basic task when estimating and predicting possible hazard parameters for given climatic and geographical conditions ([1], Note 1).There are three main topics arising in the quantitative description of snow avalanches: (1) the physics and mechanics of the snow cover on mountain slopes and its evolution in time, potentially culminating in the release of a snow avalanche; (2) the motion of snow masses on mountain slopes; (3) the impact of moving snow on obstacles and structures
Such a difference appears when we look at the simplification in this simplest point-mass model, which assumes the flow depth H to be constant
The analytical and computational investigations presented here show that the mathematical problem of quantitative description of snow avalanche dynamics is highly nonlinear and that simple explicit solutions are difficult to obtain
Summary
The present paper was originally written in 1996 for the proceedings of a symposium held in Davos, Switzerland, in late 1996, but that volume—and this paper—were never published due to circumstances recounted in Section 1 of the Comment paper [1] written by the editor. In the editor’s opinion, it still contains a host of valuable material that merits publication, but the manuscript needed to be edited to conform to the standards of a refereed journal in 2019. The original text and figures are reproduced in Supplementary Materials Document (SMD) 1. All figures are contained in an appendix, without captions. The most essential ones were inserted into the main text and given captions; SMD 2 contains the remaining figures.
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