Avalanche: snowpack failure dynamics

What is modelled

A dry-snow slab lies on a buried weak layer on a plane slope 30 m long and 20 m wide. The weak layer is a grid of 120 cells. Each cell carries the weight of the slab above it plus the load of any skier, and it fails when the shear stress exceeds its strength. A failing cell passes its load to its neighbours, so a crack can run. If enough of the layer fails and the slope is steep enough for the slab to accelerate, the failed slab slides.

Equations

• Slab weight per unit area: w = ρ g h. On a slope of angle θ it gives shear stress τ = w sinθ and normal stress σ = w cosθ on the weak layer.
• Strength (Mohr-Coulomb): τf = n c + tanφ · σ, with cohesion c, friction angle φ and a random factor n = 1 - heterogeneity x noise for each cell. Stability index S = τf / τ; S < 1 means failure.
• A uniform slab fails at θc = asin(c / (w √(1+tan²φ))) + φ.
• Skier: a vertical load Q on the snow surface adds to the weak layer, by the Boussinesq solution for an elastic half-space, Δσ = 3 Q h³ / (2π (r²+h²)5/2); its shear and normal parts are Q sinθ and Q cosθ times the same kernel.
• Faceting: a large temperature gradient in the pack turns the weak layer into angular crystals; here cohesion falls by up to 60 % as the gradient goes from 5 to 25 C/m.
• Crack: the cells check once every cell length / 30 m/s (a typical crack speed). A cell that fails hands its whole shear load equally to its live neighbours.
• Slide: failed cells accelerate down the slope with a = g (sinθ - μk cosθ) and stop on the flat after v² / (2 μk g).

Checks run in the page

• The failure rule used by the cells finds the same critical angle as the closed formula.
• The Boussinesq load integrated over a disc equals the whole load (1 - (1+(R/h)²)^-3/2).
• A block sliding down the slope and onto the flat reaches the speed and stopping distance of the formulas.
• Load sharing conserves the total shear load (an isolated cell hands its load to the ground).

Honest simplifications

• This is a teaching model, not a forecast. Real slabs fail by an anticrack (collapse of the weak layer) whose spread depends on elastic energy; here load sharing between neighbours stands in for it. Real propagation distances and the size of the triggering load are not predicted.
• The skier load uses a single homogeneous elastic layer; real snow is layered and the load is dynamic (a fall or a jump gives a larger stress than standing).
• The faceting law and the release threshold (35 % of cells failed) are illustrative, not measured. Cohesion of a real weak layer is a few hundred to a few thousand pascals; rates of change with gradient vary strongly.
• The sliding slab is a set of rigid blocks with Coulomb friction; there is no air drag, no entrainment, no turbulent powder cloud and no flow around trees. Velocity is carried unchanged round the toe.
• Trees and boulders are decoration; they are not obstacles.

What to try

• Place one skier on the marginal preset and watch the stability index of the nearby cells fall below 1.
• Raise the temperature gradient: cohesion drops and the whole slope becomes marginal without any skier.
• Set the sliding friction high: the slab fails but may not slide, as sticky snow on a moderate slope.