Snow Avalanche Simulator (2D)

Will the slab release?

A slab of depth h and density rho sits on a weak layer on a slope of angle theta. The normal stress on the layer is sigma = (rho g h + q) cos^2(theta) and the driving shear stress is tau = (rho g h + q) sin(theta) cos(theta), where q is an extra surface load (a skier, new snow). The layer resists with the Mohr-Coulomb strength c + sigma tan(phi). The factor of safety is FS = (c + sigma tan phi) / tau; the slab fails when FS < 1. With zero cohesion this reduces to FS = tan(phi) / tan(theta), so the critical angle is simply phi. With cohesion the critical tangent solves k t^2 - t + (k + tan phi) = 0 with k = c / (rho g h).

Flow: the Voellmy model

Once released the slab is treated as a moving block of flow depth h_f with du/dt = g sin(theta) - mu g cos(theta) - g u^2 / (xi h_f). The second term is dry Coulomb friction and the third is turbulent drag with coefficient xi (typical values 400-2000 m/s2). On a steady slope the speed approaches u = sqrt(xi h_f (sin theta - mu cos theta)). The track here is a steep start zone, a transition and a gentle run-out zone.

Energy line

If only Coulomb friction acted, friction work is mu g times the horizontal distance, so the block would stop where the straight line from the crown with tan(alpha) = mu meets the profile. Turbulent drag makes the real run-out shorter. The dashed line in the picture shows the angle alpha reached by the simulation.

Honest simplifications

Real avalanches entrain snow, change density, have a width, and depend on snow layers in detail; this is a single-block, constant-depth model with an infinite-slope release criterion, in the family of basic engineering models. It is for learning how slope, strength and friction trade off, not for assessing real slopes.

• Raise the slope angle until the factor of safety crosses 1 on the first graph: the slab releases by itself.
• Compare Dry powder with Deep wet snow: high friction stops the flow much sooner.
• Set the run-out slope to 0 and watch the avalanche spread out and stop on the flat.
• Increase extra surface load on the marginal slope and see the stress bars change.