An avalanche is a downhill slide of a large mass, usually of snow, ice, or rock debris prompted by a small initial disturbance. Avalanches, along with landslides, are one of the major natural disasters that still present significant danger for people in the mountains. On average, 25 people die in avalanches every winter in Switzerland alone. Dozens of people were killed on September 23, 2002, in a gigantic avalanche in Northern Ossetia, Russia, when a 150 m thick chunk of the Kolka Glacier broke off and triggered an avalanche of ice and debris that slid some 25 km along Karmadon gorge. In 1999, some 3000 avalanches occurred in the Swiss Alps.
Avalanches vary widely in size, from minor slides to large movements of snow reaching a volume of 105 m3 and a weight of 30,000 tons. The speed of the downhill snow movement can reach 100 m/s. There are two main types of avalanches-loose avalanche and slab avalanche: depending on the physical properties of snow. Soft dry snow typically produces loose avalanches that form a wedge downward from the starting point, mainly determined by the physical properties of the granular material. In wet or icy conditions, on the other hand, a whole slab of solid dense snow may slide down. The initiation of the second type occurs as a fracture line at the top of the slab. The study of real avalanches and landslides is mostly an empirical science that is traditionally a part of geophysics and draws from the physics of snow, ice, and soil. Semi-empirical computer codes have been developed for prediction of avalanches dependent on the weather conditions (snowfall, wind, temperature profiles) and topography.
Figure 1. Only several layers of mustard seeds are involved in the rolling motion inside the avalanche: moving grains are smeared out in this long-exposure photograph. Reproduced with permission from Jaeger et al. (1998).
More fundamental aspects of avalanche dynamics have been studied in controlled laboratory experiments with dry or wet granular piles, or sandpiles. Granular slope can be characterized by two angles of repose—the static angle of repose θs which is the maximum angle at which the granular slope can remain static, and the dynamic angle of repose θd, or a minimum angle at which the granular flow down the slope can persist. Typically, in dry granular media, the difference between static and dynamic angles of repose is about 2–5°, for smooth glass beads θs≈25°, θd≈23°. Avalanches may occur in the bistable regime when the slope angle satisfies θd<θ<θs. The bistability is explained by the need to dilate the granular material for it to enter flowing regime (Bagnold’s dilatancy).
An avalanche can be initiated by a small localized fluctuation from which the fluidized region expands downhill and sometimes also uphill, while the sand always slides downhill. An avalanche in a deep sandpile usually involves a narrow layer near the surface (see Figure 1). Avalanches have also been studied in finite-depth granular layers on inclined planes. The two-dimensional structure of a developing avalanche depends on the thickness of the granular layer and the slope angle. For thin layers and small angles, wedge-shaped avalanches are formed similar to the loose snow avalanches (Figure 2a). In thicker layers and at higher inclination angles, avalanches have a balloon-type shape that expands both down- and uphill (Figure 2b).
The kinematics of the fluidized layer in one dimension can be described by a set of hydraulic equations for the local thickness R(x, t) of the layer of rolling particles flowing over a sandpile of immobile particles with variable profile h(x, t) (BCRE model, after Bouchaud et al., (1994)),
∂tR=−υ∂xR+Γ(R, h)+(diffusive terms),
(1)
∂th=−Γ(R, h)+(diffusive terms),
(2)
Figure 2. Structure of the avalanche in a thin (4 grain diameters) layer of glass beads: (a) wedge-shaped avalanche for θ=31.5°; (b) balloon-shaped avalanche propagating both up- and downhill for θ=32.5°. Reprinted by permission from Nature (Daerr & Douady, 1999). Copyright (1999) Macmillan Publishers Ltd.
where Γ is the entrainment flux of immobile particles into the rolling layer and the downhill transport velocity υ is assumed constant. Γ becomes positive when the local slope becomes steeper than the static repose angle θs, and in the simplest case, Γ=γR(∂xh− tan θs). This model allows for a complete analytical treatment.
A more sophisticated continuum theory of granular avalanches is based on the fluid dynamics (NavierStokes) equations coupled with a phenomenological description of the first-order phase transition from a static to a fluidized state driven by the local shear stress (Aranson & Tsimring, 2001). The local phase state is described by the local order parameter ρ that is controlled by a Ginzburg-Landau-type equation with bistable free energy F(ρ, δ):
(3)
The control parameter δ in this equation depends on the ratio of shear to normal stress. This theory can describe a variety of “partially fluidized” granular flows, including avalanches in sandpiles. In a “shallow-water” approximation, it yields the BCRE-type equations for the local slope and the thickness of the rolling layer.
The wide distribution of scales in real avalanches led Bak et al. (1988) to propose a “sandpile cellular automaton” (SeeSandpile model) as a paradigm model for self-organized criticality (SOC), the phenomenon that occurs in slowly driven nonequilibrium spatially extended systems when they asymptotically reach a critical state characterized by a power-law distribution of event sizes. The BTW model is remarkably simple, yet it exhibits a highly nontrivial behavior. The sandpile is formed on a lattice by dropping “grains” on a random site from above, one at a time. “Grains” form stacks of integer height at each lattice site. After each grain dropping the sandpile is allowed to relax. Relaxation occurs when the slope (a difference in heights of two adjacent stacks) reaches a critical value (“angle of repose”) and the grain hops to a lower stack. This may prompt a series of subsequent hops and so trigger an avalanche. The size of the avalanche is determined by the number of grains set into motion by adding a single grain to a sandpile. In the asymptotic regime in a large system, the avalanche size distribution becomes scale-invariant, with α≈1.5.
The relevance of this model and its generalizations to real avalanches is still a matter of debate. The sandpile model is defined via a single repose angle, and so its asymptotic behavior has the properties of the critical state for a second-order phase transition. Real sandpiles are characterized by two angles of repose and thus exhibit features of the first-order phase transition. Experiments with avalanches in slowly rotating drums do not confirm the scale-invariant distribution of avalanches. However, in such experiments, the internal structures of the sandpile (the force chains) are constantly changing in the process of rotation. In other experiments with large monodispersed glass beads dropped on a conical sandpile, SOC with α≈1.5 was observed. The characteristics of the size distribution depend on the geometry of the sandpile and the physical and geometrical properties of grains. SOC was also observed in the avalanche statistics in a three-dimensional pile of long rice; however, a smaller scaling exponent α≈1.2 was measured for the avalanche size distribution.
An avalanche in a pile of sand has been used as a metaphor in many other physical phenomena including the avalanche diodes, vortices in type-II superconductors, Barkhausen effect in ferro-magnetics, 1/f noise, and.
with α≈1.5. 
