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Physicists may have cracked the case of “Zen” stones balanced on ice pedestals

A laboratory reproduction of the Zen stone phenomenon in a lyophilizer.
Enlarge / A laboratory reproduction of the Zen stone phenomenon in a lyophilizer.

Nicolas Taberlet / Nicolas Plihon

Visit the Small Sea of Lake Baikal in Russia during the winter and you’ll likely see an unusual phenomenon: a flat rock balanced on a thin pedestal of ice, akin to stacking Zen stones common to Japanese gardens. The phenomenon is sometimes called a Baikal Zen formation. The typical explanation for how these formations occur is that the rock catches light (and heat) from the Sun and this melts the ice underneath until just a thin pedestal remains to support it. The water under the rock refreezes at night, and it’s been suggested that wind may also be a factor.

Now, two French physicists believe they have solved the mystery of how these structures form, according to a new paper published in the Proceedings of the National Academy of Sciences—and their solution has nothing to do with the thermal conduction of the stone. Rather, they attribute the formation to a phenomenon known as sublimation, whereby snow or ice evaporates directly into vapor without passing through a water phase. Specifically, the shade provided by the stone hinders the sublimation rates of the surrounding ice in its vicinity, while the ice further away sublimates at a faster rate.

Many similar formations occur naturally in nature, such as hoodoos (tall, spindly structures that form over millions of years within sedimentary rock), mushroom rocks or rock pedestals (the base has been eroded by strong dusty winds), and glacier tables (a large stone sitting precariously on top of a narrow pedestal of ice). But the underlying mechanisms by which they form can be very different. 

For instance, as we reported last year, a team of applied mathematicians from New York University studied the so-called “stone forests” common in certain regions of China and Madagascar. These pointed rock formations, like the famed Stone Forest in China’s Yunnan Province, are the result of solids dissolving into liquids in the presence of gravity, which produces natural convective flows.

On the surface, these stone forests look rather similar to “penitentes“: snowy pillars of ice that form in very dry air found high in the Andean glaciers. Charles Darwin described penitentes in 1839 during a March 1835 excursion in which he squeezed his way through snowfields covered in penitentes on the way from Santiago, Chile, to the Argentine city of Mendoza. Physicists have been able to recreate artificial versions of penitentes in the lab. But penitentes and stone forests are actually quite different in terms of the mechanisms involved in their formation. The spikes of a stone forest are carved by flows, which don’t play a big role in the formation of penitentes.

Some physicists have suggested that penitentes form when sunlight evaporates the snow directly into vapor (sublimation). Tiny crests and troughs form, and sunlight gets trapped within them, creating extra heat that carves out even deeper troughs, and those curved surfaces in turn act as a lens, speeding up the sublimation process even more. An alternative proposal adds an additional mechanism to account for the oddly periodic fixed spacing of penitentes: a combination of vapor diffusion and heat transport that produces a steep temperature gradient and, hence, a higher sublimation rate.

Zen stones in nature, in the Small Sea of Lake Baikal (a, b); in the laboratory (c); and in numerical simulations (d). (a) Photograph taken by O. Zima. (b) Photograph taken by A. Yanarev.
Enlarge / Zen stones in nature, in the Small Sea of Lake Baikal (a, b); in the laboratory (c); and in numerical simulations (d). (a) Photograph taken by O. Zima. (b) Photograph taken by A. Yanarev.

Nicolas Taberlet/Nicolas Plihon

In the case of the Baikal Zen stone formations, the process seems similar to the sublimation hypothesis for penitentes, according to co-authors Nicolas Taberlet and Nicolas Plihon of CNRS in Lyon, France. Earlier this month, they published a somewhat related study in Physical Review Letters on the natural formation of glacier tables (a rock supported by a slender column of ice). They were able to produce small-scale artificial glacier tables in a controlled environment, and found two competing effects that control the onset of glacier table formation.

With smaller stone caps with higher thermal conductivity, geometrical amplification of the heat flux causes the cap to sink into the ice. For a larger cap with less thermal conductivity,  a reduction in heat flux arises from the fact that the cap has a higher temperature than the surrounding ice, forming a table.

For this latest study, Taberlet and Plihon wanted to explore the underlying mechanisms behind the natural formation of Baikal Zen structures. “The scarcity of the phenomenon stems from the rarity of thick, flat, snow-free layers of ice, which require long-standing cold and dry weather conditions,” the authors wrote. “Weather records show that melting of the ice is virtually impossible and that, instead, the weather conditions (wind, temperature, and relative humidity) favor sublimation, which has long been known to be characteristic of the Lake Baikal area.”

So the researchers set about trying to reproduce the phenomenon in the laboratory to test their hypothesis. They used metal disks as experimental analogs of the stones, placing the disks on the surface of blocks of ice in a commercial lyophilizer. The instrument freezes material, then reduces the pressure and adds heat, such that the frozen water sublimates. The higher reflectivity of the metal disks compared to stones kept the disks from overheating in the lyophilizer’s chambers.


Both aluminum and copper disks produced the Baikal Zen formations, even though copper has almost twice the thermal conductivity of aluminum. The authors concluded that, therefore, the thermal properties of stone were not a crucial factor in the process. “Far from the stone, the sublimation rate is governed by the diffuse sunlight, while in its vicinity the shade it creates inhibits the sublimation process,” the authors wrote. “We show that the stone only acts as can umbrella whose shade hinders the sublimation, hence protecting the ice underneath, which leads to the formation of the pedestal.”

This was subsequently confirmed by numerical modeling simulations. Taberlet and Plihon also found that the dip, or depression, surrounding the pedestal is the result of far infrared radiation emitted by the stone (or disk) itself, which enhances the overall sublimation rate in its vicinity.

It’s quite different from the process that leads to glacier tables, despite the similar shape of the two formations. In the case of glacier tables, the umbrella effect is only a secondary factor in the underlying mechanism. “Glacier tables appear on low-altitude glaciers when the weather conditions cause the ice to melt instead of to sublimate,” the authors wrote. “They form in warm air while the ice remains at 0 degrees Celsius, whereas Zen stones form in air that is colder than the ice.”

Understanding how these formations occur naturally could help us learn more about other objects in the universe, since ice sublimation has produced penitentes on Pluto and have influenced landscape formation on Mars, Pluto, Ceres, the moons of Jupiter, the moons of Saturn, and several comets. “Indeed, NASA’s Europa Lander project aims to seek biosignatures on Jupiter’s ice-covered moon, on the surface of which differential sublimation may threaten lander stability, and this needs to be fully understood,” the authors concluded.

DOI: PNAS, 2021. 10.1073/pnas.2109107118  (About DOIs).

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