## TL;DR
This article explains the formation of Zen stones, natura...
The melting point goes down with pressure. Therefore, if the stone ...
As can be seen from this phase diagram of pure water, the condition...
The ablation rate of an ice surface can be calculated by considerin...
Just looking at the image of the natural stone below, one notices t...
This is an illustration of the experimental setup for the experimen...
Here's a photo of a Glacier Table
 we have an illustration of this process. If we look at t...
62 PHYSICS TODAY | SEPTEMBER 2022
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Nicolas Taberlet is an associate professor at
the University of Lyon in France. His research
specialties are the physics of granular materials
and pattern formation in ice structures.
A
mong those who live in a cold enough climate, who
has never thrown a large pebble onto the pristine
surface of a frozen lake in the hope of breaking the
ice? In the Siberian winter on Lake Baikal, any at-
tempt is bound to fail, as the ice typically reaches up
to 3 meters thick—enough to support the weight of
an 18- wheeler.
But initial disappointment can turn to amazement: After a
few weeks si ing on the surface, the stone ends up balancing
on a thin pedestal of ice, while the surface around it gradually
vanishes into thin air. The phenomenon is manifest in the for-
mation of Zen stones, shown in the fi gure, so- called because of
their resemblance to stacks of rocks sometimes found balanc-
ing in Japanese Zen gardens.
Sightings are rare, possibly because specifi c meteorological
conditions are required. Not only must the temperature remain
below freezing but the ice surface must remain free of snow for
several consecutive weeks. The climate at Lake Baikal meets
both conditions: The air temperature is below freezing for an
average of fi ve months per year, and precipitation is rare in
winter. Thus, the melting of ice is virtually impossible, and the
region’s exceptionally low humidity mainly causes the ice to
sublimate.
I was struck at how li le explanation exists in the literature
and set out to reproduce the eff ect in the lab.
The umbrella effect
In the case of water, the direct phase transition between the
solid state and a gas occurs at negative temperatures (in Cel-
sius) and in a very dry atmosphere. What’s more, it’s a slow,
endothermic surface process, which therefore requires a con-
stant flux of external energy. Sunlight does the job in nature,
either directly in clear weather or diffusely in overcast condi-
tions. Sublimation causes the ice to vaporize at a rate set by the
temperature, humidity, and amount of sunlight it receives.
From the average winter solar irradiance at the lake and water’s
latent heat of sublimation, I estimate the sublimation rate of an
ice surface at about 2 mm per day.
A pebble placed on the ice blocks that light, however, and
its shade hinders the sublimation beneath it. The rate, nearly
zero underneath, gradually increases with distance from the
center. The stone therefore acts as an umbrella, which protects
the ice from solar irradiance. Known as diff erential ablation,
the process forces the pebble to remain at a constant altitude
on an increasingly taller and narrower foot of ice until it even-
tually falls off . Its lifetime atop the pedestal is roughly the half
width of the stone divided by the ablation rate— about 40 days
for the stone in panel a of the fi gure.
Sublimation is not the only possible factor at play. The melt-
ing temperature of water decreases with applied pressure. And
between 100 MPa and 1 GPa, ice can start melting at tempera-
tures as low as −10 °C. The pressures that Zen stones exert on
the ice remain far below that range, however, and any melting
would only cause the pebble to sink into the ice. Moreover, ice
is known to slowly deform over time— a phenomenon known
as plastic creep— which explains why glaciers can fl ow down
mountains. But that too only causes the stone to sink.
As another possible factor, small wind- driven ice particles
could potentially create mechanical wear. But the smooth sur-
face of the ice pedestals shows no evidence of erosion. And
the typical time required for that ablation process is far longer
than the lifetime of a natural Zen stone.
Stones in the lab
To convince my University of Lyon colleague, Nicolas Plihon,
and myself of the simple sublimation hypothesis, we repro-
duced the phenomenon in a laboratory- scale experimental
setup. We placed an aluminum disk— a proxy for the stone— on
the surface of a block of ice within a commercial lyophilizer, a
device whose temperature, pressure, and humidity favor sub-
limation. The external energy used to sublimate the ice came
not from sunlight but from IR radiation of the walls of the
vacuum chamber, which remained at room temperature.
In the absence of a stone, the ablation is nearly isotropic and
mimics the relative isotropy of natural diff use sunlight in over-
cast weather. And its considerably greater sublimation rate of
typically 8– 10 mm per day allowed us to accelerate the physical
mechanism. Indeed, obtaining Zen stones from actual pebbles
and disks was straightforward.
The fi gure's panel b shows the results we achieved using a
30 mm aluminum disk after 40 hours of sublimation. With the
disk initially placed either on the ice surface or embedded in-
side the ice, the IR forced the ice to sublimate only partially—
the disk’s shade prevented it from vanishing completely.
The mysterious balancing stones on frozen lakes
Nicolas Taberlet
During the cold, dry Siberian winter, one can occasionally spot stones perched on impossibly thin
pedestals of ice. How do they get there?
SEPTEMBER 2022 | PHYSICS TODAY 63
Among other results, our experiments confi rmed that the disk’s
thermal properties had li le eff ect. (In some cases, we used
copper disks, whose thermal conductivity and specifi c heat
greatly exceed those of aluminum disks we used in other
cases.) They supported our conclusion that the umbrella eff ect
is the predominant mechanism.
Dip around the pedestal
One interesting difference exists between natural and
laboratory- made Zen stones. In nature a dip always surrounds
the ice foot. But that feature was never encountered in our lab
experiments. Whereas the umbrella effect is clearly responsible
for the pedestal’s formation, a detailed energy balance of the
system reveals second- order phenomena. Like any other ma-
terial, ice and stone emit blackbody IR radiation in a range
whose intensities depend on the temperature and the materi-
al’s emissivity.
In nature, because of sunlight or ambient wind, the ice and
stone are unlikely to remain at the same temperature through-
out the day. And that, in turn, causes an imbalance between
them. More specifi cally, if the stone is a few degrees warmer
than the ice, the IR it radiates into the ice (in addition to that
from sunlight) can exceed that emi ed by the ice itself. The
eff ect becomes important in the later stages of
Zen- stone formation— when a stone sits on a tall and
thin pedestal— as thermal contact is reduced.
Two competing eff ects are therefore at play: the
umbrella eff ect, which protects the ice, and the excess
energy from the stone, which instead accelerates the
sublimation and carves out a cavity in the stone’s vi-
cinity. While the former is responsible for the forma-
tion of the ice pedestal in the early life of a Zen stone,
the la er is responsible for the dip forming around the
ice foot in the later stages.
The excess energy is absent in our experiment be-
cause of three factors: The lyophilizer was operated in
a high vacuum, metal was used for Zen stones, and
such stones were smaller, all of which favor a good
thermal equilibrium between the disk and the ice.
Glacier tables
In addition to Zen stones, other intriguing formations
consisting of a rock resting on a thin pedestal can be
found in nature. In a hoodoo, for instance, a hard
stone protects a tall column of fragile sandstone from
rain and frost- driven erosion. And in a so- called gla-
cier table, a large rock on a low- altitude glacier ends
up on a tall foot of ice. The laer case is akin to the
Zen stones of Lake Baikal because it involves differ-
ential ablation of an ice surface. But the glacier tables’
rock formation, shape, and dynamics are broadly
different.
Glacier tables appear on temperate glaciers where
the ice (remaining at 0 °C) simply melts because of the
warm ambient conditions. Depending on its size and
shape, a rock atop the ice can provide enough thermal
insulation to either hinder the ice from melting (a
process that leads to the formation of the ice pedestal)
or increase the ice's melting rate (a process that leads
to the rock sinking into the ice). A recent study has
shown that the diff erential melting of ice for glacier tables is
predominantly caused by heat exchange with the surrounding
air.
The umbrella eff ect, which controls the formation of Zen
stones, is therefore only a secondary factor for glacier tables.
Conversely, although a material’s thermal properties, such as
conductivity and specifi c heat, are crucial for glacier tables,
they are insignifi cant to the formation of Zen stones. Any
opaque object left on a sublimating ice surface is likely to wind
up atop a narrow foot. Indeed, far from the romantic image
sometimes conjured by a Zen stone, the frozen bodies of de-
ceased penguins in Antarctica can occasionally be found
perched on top of narrow ice pedestals.
Additional resources
‣ N. Mangold, “Ice sublimation as a geomorphic process: A
planetary perspective,” Geomorphology 126, 1 (2011).
‣ N. Taberlet, N. Plihon, “ Sublimation- driven morphogenesis of
Zen stones on ice surfaces,” Proc. Natl. Acad. Sci. USA 118,
e2109107118 (2021).
‣ M. Hénot, N. Plihon, N. Taberlet, “Onset of glacier tables,”
Phys. Rev. Le . 127, 108501 (2021).
PT
a
b
10 cm
20 mm
ZEN STONES in nature and the lab. (a) On Lake Baikal a stone rests on a
narrow ice pedestal. (b) In a laboratory, this 30 mm aluminum disk rests on
a at ice surface after sitting in a lyophilizer for 40 hours. (Adapted from Proc.
Natl. Acad. Sci. USA 118, e2109107118, 2021.)
Here's a photo of a Glacier Table
As can be seen from this phase diagram of pure water, the conditions at Lake Baikal make it very difficult for the ice to melt, whereas sublimation is possible.
## TL;DR
This article explains the formation of Zen stones, natural structures where a stone on an ice surface ends up balanced on a narrow ice pedestal. The article provides a physical explanation for their formation, showing that slow surface sublimation is responsible for the differential ablation, and that the stone acts as an umbrella that inhibits sublimation and protects the ice underneath. Laboratory experiments using metal disks ruled out thermal conduction as an influence on the process, and numerical simulations showed the influence of stone shape on the ice foot formation. The stone's far-infrared black-body irradiance leads to a depression surrounding the pedestal.
This is an illustration of the experimental setup for the experiment: a metal disk (of radius 15 mm) representing the stone is placed at the surface of a block of ice sublimating in the chamber of a commercial lyophilizer (Materials and Methods). The external energy required to sublimate the ice originates from the IR black-body radiation of the outer walls of the vacuum chamber, which remains at room temperature. The ablation is then isotropic (in the absence of a stone) and mimics the relative isotropy of natural diffuse sunlight in overcast weather, with a sublimation rate of typically 8 to 10 mm/d.
Just looking at the image of the natural stone below, one notices the the stone is much thicker on one side -- and thus much more heavy. Any umbrella effect would be nearly independent of the thickness of the stone.
At the same time, the stem is so thin, that it cannot support any bending forces, with other words the stem must be exactly located under the center of mass of the stone.
These two facts together mean that whatever causes the stem to form must cause the sublimation to go slower with the pressure applied.
The melting point goes down with pressure. Therefore, if the stone is leaning towards one side, the pressure there will increase and thus will the melting... IF this term dominates the stone would topple over.
In Fig. a) we have an illustration of this process. If we look at the typical energy profiles of diffuse sunlight (blue) and of far-infrared radiation FIR (orange) we can see that the irradiance of the sun is 100 times larger than the black-body emission of the stone, but the values of the integrated energies are comparable (359 W/m2 for the solar energy vs. 259 W/m2 from the stone at 260 K).
Additionally, there is a significant difference in the range of wavelengths between sunlight and FIR: 300 to 2,500 nm for sunlight compared to 5 to 50 μm for FIR. This distinction has critical consequences, as the extinction rate (i.e., the imaginary part of the refraction index) varies by 10 orders of magnitude (Figure d). Likewise, the ice absorption length is highly dependent on wavelength (Figure d), with 10 km for 400-nm radiation, which renders the ice layer almost transparent, in contrast to 10-5 m for 10-μm radiation, where energy is absorbed within a 10-μm-thick layer, promoting surface sublimation of the ice. It is important to note that the data in Figure d pertain to pure ice; while multiple scattering in white ice may alter the exact extinction rate values, the discrepancy remains vast, and the conclusions are still valid. This immense difference indicates that even if FIR irradiance is 100 times lower than diffuse sunlight, it can locally and significantly boost the ablation rate. This clarifies why the depression reflects the stone's shape (Figure e).
The ablation rate of an ice surface can be calculated by considering the energy balance at the surface. The primary energy source contributing to ice ablation is solar irradiance, which provides energy for melting and sublimation. The ablation rate can be estimated using the following formula:
$$
\text{Ablation rate (AR)} = \frac{(\text{Solar irradiance (SI)} \times \text{Absorbed fraction (AF)})}{(\text{Latent heat of sublimation (LHS)} \times \text{Ice density} (\rho))}
$$
Here,
- AR: Ablation rate (in meters per second or similar units)
- SI: Solar irradiance (in Watts per square meter, W/m²)
- AF: Absorbed fraction (unitless) - it represents the fraction of incoming solar radiation that is absorbed by the ice surface. This depends on the albedo (reflectivity) of the ice surface. A typical value for ice is about 0.9, meaning 90% of the incoming solar radiation is absorbed.
- LHS: Latent heat of sublimation (in Joules per kilogram, J/kg) - it is the amount of energy required to change a given mass of ice directly from solid to vapor without going through the liquid phase. The latent heat of sublimation for water is approximately 2.83 x $10^6$ J/kg.
- ρ: Ice density (in kilograms per cubic meter, kg/m³) - the density of ice is about 900 to 920 kg/m³.
This formula provides a simplified estimation of the ablation rate and doesn't account for all factors influencing the ice ablation process, such as air temperature, wind speed, or long wave radiation from the atmosphere.
At the latitude of Lake Baikal, in February or March, the irradiance reaches from 400 to 500 W/$m^2$ for typically 6 to 8 h. Assuming an emissivity of 0.95, the ice itself (at 260 K) radiates 246 W/$m^2$ back into the atmosphere resulting in a net energy input of 60W/$m^2$ per day. Substituting for the values at Lake Baikal:
$$
AR = \frac{60}{2.83 \times 10^6 \times 900} = 2.355 \times 10^{-8} m/s = 2 \ mm/day
$$