SAME SHAKE, VERY DIFFERENT LANDSLIDES

A heap of sand or gravel can sit still on a slope up to a certain steepness, its angle of repose. Below that critical angle, a jolt — an earthquake, for instance — can set grains flowing for a while before they stop. Predicting how much material moves before everything comes to rest matters for changes in landscapes, for landslide hazards, and for handling powders in industry.

Experiments had already shown that slopes close to their limit are easier to set in motion, and that motion then lasts longer and spreads further. A more basic question had not been answered: if you take slopes that are identical in every visible way, and shake them in exactly the same way, how much does the outcome vary?

One push, 128 times

Fumiaki Nakai and Yuto Sasaki, of the University of Osaka, answered it with computer simulations that follow every grain. Their slope holds 2,273 grains, between 1.0 and 1.9 millimetres across, resting on a rough base. To disturb it, they move the base downhill and back once, by one millimetre, in about six hundredths of a second.

At each of 14 angles between 10 and 16.3 degrees, they ran 128 trials. Slope, grains, push: everything identical, except the exact initial arrangement of the grains, as in two real heaps that look the same. They then measured how much material had moved downhill once the slope had settled.

The closer to the edge, the wider the spread

At 12 degrees, the 128 trials end up close to one another. At 16.3 degrees, they differ wildly: some barely move, others carry far more than average.

The critical angle of this pile is about 16.65 degrees. As the slope approaches it:

• the average transport grows roughly in inverse proportion to the remaining distance to that angle: halve the distance, double the average;
• the spread between trials, measured by the standard deviation, the typical gap between one trial and the average, grows faster, as that distance to the power −1.5: halve the distance, and the spread is multiplied by about 2.8;
• so the spread relative to the average keeps rising, and near the critical angle it reaches and can exceed the average itself.

In larger simulated slopes the spread shrinks, roughly as one over the square root of the length, but it rises near the critical angle at every size tested.

Chance in the energy balance

A classic description treats the flowing layer with a constant friction. Gravity pushes it, friction slows it, and it stops when its kinetic energy runs out. This gives the right trend for the average — a transport proportional to one over the distance to the critical angle — but only one answer for given conditions. It cannot produce a spread.

So the authors added random fluctuations to the energy balance: as the grains move, the work done on them jitters at random, and these random kicks add up as more material is carried downhill. The flow stops the first time the energy hits zero, a so-called first-passage problem. The result is a precise distribution of outcomes, called an inverse Gaussian: lopsided, with a long tail towards very large slides. Its spread grows as the distance to the critical angle to the power −1.5, in line with the trend of the simulations. Fitted once to all 14 angles, the model broadly captures the rising average and the stretching tail of large events.

The reason is intuitive: near the critical angle, gravity almost exactly balances friction. The layer is then at the mercy of small random kicks.

What it means for landslides

The simulations describe an idealized pile of grains, not a real mountainside. But they suggest that tiny differences in internal structure, invisible from outside, could make real landslides on similar slopes, under similar shaking, end very differently. Near the stability limit, the authors conclude, assessing a slope requires the full distribution of possible outcomes, and theories of granular flow will need randomness built in.