Balancing Mysterious Stones on Frozen Lakes: Physics Today: Volume 75, Issue 9

Among those who live in a climate cold enough, who have never thrown a large pebble on the pristine surface of a frozen lake in hopes of breaking the ice? In the Siberian winter on Lake Baikal, any attempt is bound to fail, as the ice usually reaches 3 meters in thickness – enough to withstand the weight of 18 wheels.

But the initial disappointment can turn into astonishment: after a few weeks of sitting on the surface, the stone ends up balancing on a thin base of ice, while the surface around it gradually fades into thin air. This phenomenon is manifested in the formation of the zen stones shown in the shapeIt is so called because of its resemblance to the rock piles sometimes found balanced in Japanese Zen gardens.

Observations are rare, possibly due to the need for specific meteorological conditions. Not only must the temperature remain below freezing, but the surface of the ice must remain snow-free for several weeks in a row. The climate in Lake Baikal fulfills two conditions: the air temperature is below freezing an average of five months of the year, and precipitation is rare in winter. Thus, thawing is almost impossible, and the exceptionally low humidity in the region mainly leads to ice sublimation.

I was struck by how little explanation found in the literature and I proceeded to reproduce the effect in vitro.

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

A pebble set on ice blocks nonetheless glows, and its shadow impedes sublimation beneath. The rate, almost zero below it, gradually increases with distance from the center. So the stone acts as an umbrella that protects the ice from solar radiation. The operation, known as differential ablation, forces the pebble to remain at a constant height on an increasingly taller and narrower foot of ice until it eventually falls. The life span of the stone above the base is approximately half the width of the stone divided by the rate of ablation – about 40 days for the stone in panel (A) of the shape.

Sublimation is not the only possible factor in 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 will only cause the pebble to sink into the ice. Moreover, ice is known to deform slowly over time – a phenomenon known as plastic creep – which explains why glaciers flow down mountains. But this also causes the stone to sink.

As another potential factor, small ice particles moved by wind can lead to mechanical erosion. But the smooth surface of the ice bases shows no evidence of erosion. The typical time required for ablation is much longer than the lifespan of a natural zen stone.

To convince my colleague at the University of Lyon, Nicholas Plehoun, and I of the simple sublimation hypothesis, we reproduced this phenomenon in a lab-scale experimental setting. We placed an aluminum disc—a proxy for stone—on the surface of a block of ice inside a commercial lyophilizer, a device whose temperature, pressure, and moisture favor sublimation. The external energy used to sublimate the ice did not come from sunlight but from infrared radiation from the walls of the vacuum chamber, which was kept at room temperature.

In the absence of stone, the extrusion is nearly isotropic and mimics the relative properties of natural sunlight diffused in cloudy weather. The significantly larger sublimation rate, usually 8-10 mm per day, allowed us to speed up the physical mechanism. In fact, obtaining zen stones from actual pebbles and tablets was easy.

The Numbers Panel b shows the results we achieved using a 30 mm aluminum disc after 40 hours of sublimation. With the disc initially either placed on the surface of the ice or embedded within the ice, the infrared rays forced the ice to sublimate only partially—preventing the disc’s shadow from fading out completely. Among other results, our experiments confirmed that the thermal properties of the disc have little effect. (In some cases we have used copper discs, whose thermal conductivity and specific heat greatly exceed the aluminum discs we have used in other cases.) They supported our conclusion that systemic effect is the dominant mechanism.

There is one interesting difference between natural and laboratory zen stones. In nature, snorkeling is always surrounded by icy feet. But this advantage has never been encountered in our laboratory experiments. While the canopy effect is clearly responsible for the formation of the pedestal, the detailed energy balance of the system reveals second-order phenomena. Like any other substance, ice and stone emit blackbody infrared radiation in a range whose intensity depends on the temperature and emission of the substance.

In nature, due to sunlight or ambient wind, ice and stone are unlikely to remain the same temperature all day. This, in turn, leads to an imbalance between them. More specifically, if the stone is a few degrees warmer than the ice, the infrared rays it radiates into the ice (as well as sunlight) can exceed those emitted by the ice itself. The effect becomes significant in later stages of zen stone formation – when the stone sits on a long, thin base – as thermal contact is reduced.

Therefore, two competing effects play a role: the canopy effect that protects the ice, and the excess energy from the stone, which instead speeds up the sublimation process and the formation of a cavity in the vicinity of the stone. While the former is responsible for the formation of the ice base early in the life of the zen stone, the latter is responsible for the formation of dipping around the ice foot in later stages.

Excess energy is absent in our experiment due to three factors: the lyoder was run in a large vacuum, metal was used for the zen stones, and these stones were smaller, all of which favor a good thermal balance between disk and ice.

In addition to Zen stones, other interesting formations consisting of a rock resting on a thin base can be found in nature. In a hood, for example, a solid stone protects a tall column of brittle sandstone from erosion from rain and frost. And in the so-called glacier stream, a large boulder on a low-lying glacier ends on a long glacier. The latter case is similar to the zen stones of Lake Baikal because it involves differential extirpation of an ice surface. But the rock formation, shape and dynamics of glacial streams vary greatly.

Glacier streams appear on temperate glaciers where the ice (remaining at 0°C) simply melts due to warm ambient conditions. Depending on its size and shape, a rock on top of the ice can provide enough thermal insulation to impede snowmelt (a process that leads to the formation of the ice base) or increase the rate of snowmelt (a process that causes a rock to sink into the ice). A recent study showed that the differential ice melting of glacier streams is mostly caused by heat exchange with the surrounding air.

Thus, the canopy effect, which controls the formation of zen stones, is only a minor factor for glacier streams. Conversely, although thermal properties of the material, such as conductivity and specific heat, are essential for glacier streams, they are not important for the formation of zenstones. Any opaque object left on a mounting ice surface is likely to end up over a narrow foot. Indeed, far from the romantic image that a zen stone sometimes conjures, the frozen corpses of deceased penguins can sometimes be found in Antarctica perched atop narrow ice bases.

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