62 PHYSICS TODAY | SEPTEMBER 2022
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.
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
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
During the cold, dry Siberian winter, one can occasionally spot stones perched on impossibly thin
pedestals of ice. How do they get there?