Mishra’s interest in avalanches was sparked by a researcher in the ETH domain: working at the WSL Institute for Snow and Avalanche Research in Davos, Perry Bartelt has developed a software package for calculating the dangers of avalanches and evaluating protective measures. “This avalanche modelling is one of the best in the world, if not the best,” says Mishra. His expertise was in demand in connection with the snow powder: not only can the cloud be immensely destructive, it is also more difficult to model than the core of the avalanche.
At its core, the avalanche consists of a dense accumulation of snow and ice which, when it pitches down the slope, extends in a linear manner to create a slab of snow. With his mathematician’s mind, Mishra understands that the core of an avalanche behaves in a similar fashion to the flow of a river, whereas the dust cloud spreading above it is much more chaotic. From a modelling point of view, it is important to notice that the cloud behaves like a two-phase flow, in which ice dust and air mix.
Turbulences: the enigma of Euler’s equations
In mathematics, there are equations for flows such as this, as well as for pressure and shock waves. Mishra’s particular expertise is to establish the equations necessary for the mathematical modelling of powder cloud avalanches and to write the algorithms so that they can be simulated on the computer. “That’s what I like doing best: creating algorithms to simulate various types of unstable and turbulent flows.”
Like avalanches, these flow equations also have a connection to Switzerland: they date back to the Swiss mathematician Leonhard Euler (1707-1783). The mathematical model that he developed can be used to describe inviscid fluids, in other words the behaviour of many liquids and gases.
The Euler equations are widely used in the natural and engineering sciences because they can be applied to numerous flow phenomena in physics and engineering: using equations similar to those employed to describe avalanches, one can also model phenomena such as tsunamis, hurricanes, aeronautical flows, solar waves and collapsing supernovas.
Despite their dissemination, the Euler equations and their related Navier-Stokes equations, which describe the motion of viscous fluids, remain one of the greatest unsolved mysteries of mathematics. “Even now, we still lack a definitive mathematical theory on how best to model unstable, turbulent flows and calculate them efficiently on the computer”, says Siddhartha Mishra.
Small changes with major impacts
