The mysterious balancing stones on frozen lakes

Among those who live in a fairly cold climate, who has never thrown a large pebble at the pristine surface of a frozen lake in hopes of breaking the ice? During the Siberian winter on Lake Baikal, any attempt is doomed to failure, as the ice is usually up to 3 meters thick, which is enough to support the weight of an 18-wheeler.

But the initial disappointment can turn into amazement: after a few weeks on the surface, the stone ends up swaying on a thin bed of ice, while the surface around it gradually disappears into thin air. The phenomenon is manifested by the formation of Zen stones, shown in the figure, so called because of their resemblance to piles of stones that are sometimes found in balance in Japanese Zen gardens.

Zen stones in nature and in the lab. (a) On Lake Baikal, a stone rests on a narrow ice base. (b) In a lab, this 30 mm aluminum disc sits on a flat surface of ice after sitting in a freeze dryer for 40 hours. (Adapted from Proc. Natl. Acad. Sci. USA 118, e2109107118, 2021, doi:10.1073/pnas.2109107118.)

Sightings are rare, perhaps because specific weather conditions are required. Not only must the temperature remain below freezing, but the surface of the ice must remain free of snow for several consecutive weeks. The climate of Lake Baikal meets both conditions: the air temperature is below freezing for five months a year on average, and precipitation is scarce in winter. Thus, ice melting is virtually impossible, and the region's exceptionally low humidity primarily causes ice sublimation.

I was struck by the lack of explanations available in the literature and decided to reproduce the effect in the laboratory.

In the case of water, the direct phase transition between the solid state and a gas occurs at negative temperatures (in Celsius) and in a very dry atmosphere. Moreover, it is a slow endothermic surface process, which therefore requires a constant flow of external energy. Sunlight does the work in nature, either directly on a clear day or diffusely on an overcast day. Sublimation causes ice to vaporize at a rate determined by temperature, humidity, and the amount of sunlight it receives. Using the average winter solar irradiance at the lake and the latent heat of water sublimation, I estimate the rate of sublimation of an ice surface to be about 2mm per day.

A pebble placed on the ice, however, blocks the light and its shadow interferes with the sublimation below. The rate, almost zero below, increases gradually with distance from the center. The stone therefore acts as an umbrella, which protects the ice from solar radiation. Known as differential ablation, the process forces the pebble to stay at a constant elevation on an increasingly taller and narrower foot of ice until it eventually falls. Its life at the top of the pedestal is roughly half the width of the stone divided by the ablation rate, or about 40 days for the stone in panel a of the figure.

Sublimation is not the only possible factor at play. The melting temperature of water decreases with applied pressure. And between 100 MPa and 1 GPa, ice can begin to melt at temperatures as low as −10°C. However, the pressures exerted by the Zen stones on the ice remain well below this range and any melting would only cause the pebble to sink into the ice. Additionally, ice is known to deform slowly over time - a phenomenon known as plastic creep - which is why glaciers can flow down mountains. But that too just sinks the stone.

As another possible factor, small wind-driven ice particles could potentially create mechanical wear. But the smooth surface of the ice bases shows no signs of erosion. And the typical time required for this ablation process is far longer than the lifespan of a natural Zen stone.

To convince my colleague from the University of Lyon, Nicolas Plihon, and myself of the simple hypothesis of sublimation, we reproduced the phenomenon...