Abstract
The article is a review of mathematical models of snow avalanches that have been proposed since the middle of the 20th century and are still in use. The main attention is paid to the work of researchers from the Soviet Union and Russia, since many of their works were published only in Russian and are not widely available. Mathematical models of various levels of complexity for avalanches of various types—from dense to powder-snow avalanches—are discussed. Analytical solutions including formulas for the avalanche front speed are described. The results of simulations of the movement of avalanches are given that were used to create avalanche hazard maps. The last part of the article is devoted to constructing models of a new type, in which avalanches are considered as laminar or turbulent flows of non-Newtonian fluids, using the full (not depth-averaged) equations of continuum mechanics. The results of a numerical study of the effect of non-Newtonian rheology and mass entrainment on the avalanche dynamics are presented.
Highlights
Mathematical modeling is an important tool for solving engineering problems related to the protection of people and structures in mountainous areas against snow avalanches
We provide an overview of the mathematical models of snow avalanches that were developed from the middle of the 20th century to the present time
We have presented a review of the studies of dynamics of natural slope flows conducted in USSR and Russia over the last 50 years
Summary
Mathematical modeling is an important tool for solving engineering problems related to the protection of people and structures in mountainous areas against snow avalanches. Development, generalization and use of mass-point or block models for avalanches continue to this day (see, e.g., [19]) After calibration, these models can provide approximate estimates for the velocity and run-out distance of avalanches. In this paper we do not discuss mass-point models in order to save space for the models that explicitly take into account the internal structure of avalanche flows Such models are based on the full (i.e., not averaged over the flow depth) equations of continuum mechanics. Results of the numerical study of the influence of the flow rheological properties and entrainment of the underlying material in laminar and turbulent flows are presented
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