[This is the fourth of the re-posted articles, from 2009.]

“… by means of these wonderfully delicate and exquisite figures, much may be learned of the history of each crystal and the changes through which it has passed in its journey through cloudland.”
W. A. Bentley, 1898

“… the crystals will in all probability be greatly modified by passing through atmospheric strata varying so greatly in density, temperature, humidity, etc. That they are greatly modified by these flights in the cloud is clearly shown by the interior structure of many of the crystals. ”
W. A. Bentley, 1901

Long ago, Wilson Bentley argued that the various layers (‘strata’) of air that a snow crystal falls through largely determine the appearance of the crystal. The first direct support of this claim came about 30 years later, from laboratory experiments by Ukichiro Nakaya, a Japanese physicist. Now, 100 years later, probably all cloud scientists would agree with Bentley. So, with all the crystal and cloud research that has been done since Bentley and Nakaya, can we determine the history of a crystal if we see its form? Can we finally do what Bentley said we should be able to do?

Hardly. Laboratory experiments have certainly advanced our knowledge of snow, though large, crucial gaps remain and many experiments appear to contradict other experiments. Greater advances have occurred with cloud measurements. For example, researchers recently measured differences in temperature between points less than one centimeter apart in clouds. (Which is not so easy when one is flying at over a hundred meters per second!) Nearly the same sensitivity measurements have been made of cloud humidities and air motions.

Although I was not involved in any of these cloud measurements, the researchers gave me data that I used to estimate the size of the layers that Bentley mentioned. Now, knowing the size of the layers does not allow us to know the exact history of a crystal, but it does help us understand something else dear to Bentley: the wonderful variety of snow crystal forms. As far as I know, nobody else has estimated this size, so I took the liberty of naming it the ‘Bentley length’.

Before describing what this length is, I should clarify two things. One, the length is not fixed: it depends on the cloud conditions, particularly the temperature. Two, these ‘layers’ are not something one can easily discern by looking at a patch of cloud: the size of a layer depends not only on the properties of the cloud, but also on how fast the crystal can respond to changes in the cloud, particularly temperature changes. Specifically, the length is the average distance a crystal falls through the cloudy air before some fluctuation in air temperature or humidity causes the crystal to make a noticeable change in shape.

About this change of shape, one of Nakaya’s great achievements was to demonstrate that snow crystal shape and size is remarkably sensitive to temperature: grow a snow crystal at –11 °C for 30 minutes and it ends up less than half a millimeter across, but grow it at –15 °C, only 4 degrees colder, for the same length of time and the crystal is more than 5 mm across. Using similar data on snow crystals and the above-mentioned data on clouds, I estimated that the Bentley length is about 8 meters at –11 °C, and only 0.8 meters at –15 °C. To a snow crystal, even four degrees is a huge change.

To understand the role of the Bentley length in snow crystal variety, it helps to look into all sources of crystal variety. First, consider an analogy. The uniqueness of snow is sometimes compared to the uniqueness of individuals. That we are all unique is well known to be a result of our genes and our environment. Even identical twins who grow up in the same general environment will, despite having the same genes, develop differently due to small differences in their environment. The genes do not determine the person, but rather determine how the development of the person responds to the environment. Similarly, the structure of the initial embryo of ice, the seed that starts a snow crystal, influences how the crystal will respond to the cloudy environment. Concerning the environment, we can divide the crystal environment into that which would occur if all cloud fluctuations were removed, and the small differences in environment caused by the fluctuations.

For example, consider a crystal that starts at –15 °C the middle of a cloud that everywhere has an updraft speed of 12 cm/s. At first the crystal slowly cools as it rises. But soon it reaches a size at which it falls faster than the updraft and from then on, the crystal slowly, and continuously, warms as it descends towards cloud base at –8 °C. This history would produce a crystal that may look something like that on the left in the illustration below. Conversely, the crystal would look different if it started at –16 °C or some other temperature. But real clouds have fluctuations in the updraft speed, fluctuations in temperature, and fluctuations in other things. These fluctuations also affect the crystal form, just as the small differences in environment affected the twins. One such fluctuation on the right side of the illustration led to a slightly different crystal. So, that which is unique to a given crystal can be said to have originated from a combination of three sources. The third source, the fluctuations, depends on the Bentley length.

To recap, two crystals could be different because they started with different characteristics that affected their final appearance. That’s the first source. The two crystals could instead be different because they started at different temperatures or grew in different clouds. Or, as the third source, the two crystals could have passed through different cloud layers. Bentley also discussed these three sources. He originally assumed that the first one was negligible:

"Of the tens of thousands now filling the air, an infinitesimal proportion fall on this board; nor is there any good reason to doubt that when they started from equal heights on their journey earthward, many of the snow crystals were exactly alike in shape, and size, and probably in density."(ref. 1)

But later, through a “fortunate accident” involving a broken water pipe that flooded a cold room in his house, he decided that crystals are also born with individual characteristics that can affect their appearance

Even with our incomplete knowledge of snow-crystal growth, it is possible to estimate the crystal variety resulting from these sources. Here I define ‘variety’ to mean the number of possible distinguishable crystals that could result from crystals falling through the layers. According to the data, the third source of variety greatly exceeds the second, and the second source greatly exceeds the first. So, the atmospheric ‘layers’, the third source, is likely the main source of crystal individuality. And if one would like to estimate the crystal variety for some set of conditions, one needs to know how many layers a crystal passes through. Here is where we use the Bentley length.

To see just how much the Bentley-length source dominates snow crystal variety, we apply a little mathematics to the example in the illustration. According to experiments on ice-crystal growth rates and fall speeds, by the time the crystal falls below cloudbase, it would have fallen through about 1850 meters of air and grown to about 7.6 millimeters across. (Yes, this is a big crystal.)

With a Bentley length of 0.8 meters, which is appropriate for the crystal’s starting temperature, the number of layers the crystal passed through would be the 1850 meters divided by the 0.8-meter length, and assuming that half of the layers produced changes to the crystal, the crystal experienced 1156 changes. One example of a change is shown in the illustration. However, the Bentley length depends on temperature, and by the time the crystal gets to cloud base, the length is more than four times larger. If we instead use the average Bentley length for the crystal’s journey, the crystal experienced 468 changes. This might seem like a small number, but we have not yet calculated the variety.

When the crystal passed through a given layer, the growth could have sped up or it could have slowed down. In the case of the illustration, the colder region caused the growth to speed up, and this produced the narrower tip and sidebranches. If instead, the fluctuation had been a region of warmer air, the growth would have slowed and the crystal tip would have widened. However, the Bentley length tells us only how many changes occurred during the crystal’s fall to earth; it does not tell us where (or when) each change occurred. Indeed, each change could have occurred at any one of many stages in the crystal’s history. The larger a crystal is, or the closer we choose to inspect the crystal features, the greater the number of stages. Assuming that we can use a good-quality microscope to examine the crystal to 1/1000 of a millimeter, the number of distinct stages for our example is about 3800. At this point, we have everything needed to calculate the variety.

Each of the 468 changes were one of two, equally likely types, and could have occurred in any one of the 3800 stages of the crystal’s growth. The resulting number of possibilities is a staggeringly large number: a one followed by 768 zeros or 10^768. (For math-inclined readers, this is “3800 choose 468 times 2 to the 468th power.) To get an idea of the immense size of this number, consider that the number of atoms in the entire universe has been estimated to be a 1 followed by only 70 zeros or 10^70. The number of snow crystals that have ever formed on Earth is much smaller - 1 followed by only about 35 zeros or 10^35. (See estimate in ref. 4.) If every crystal that grows under such cloud conditions has equal chance to be any one of these 10^768 possible crystal forms, we can be sure that no two of them will be alike. Of course, Bentley intuitively knew this result.

But it turns out that we cannot be so sure of uniqueness if we instead examined crystals that were born much closer to the bottom of the cloud. Crystals born near cloud base pass through fewer Bentley lengths, particularly so if the temperature is much warmer than –15 °C. In fact, one can argue that there probably have been two such crystals that appear the same. More cases and details in refs. 2 and 3.

The Bentley length depends on fundamental properties of snow crystals and clouds. The values I used here were calculated from a few sets of cloud measurements and one set of snow-crystal growth measurements. As we learn more about both, the values will undoubtedly be refined. But they are still fundamental. The famous physicists Neils Bohr, Arthur Compton, and Max Planck have atomic length scales named after them, lengths learned by most students of atomic physics. Karl Schwarzschild has a size named after him, a size in black-hole physics that depends on the mass of the star, and Peter Debye has a material-dependent length named after him. Here's a possible one for Wilson A. Bentley. The Bentley length depends on temperature, but nevertheless represents a well-defined property of nature.

Will meteorology and physics students in the future learn about the Bentley length? Probably not, though it is nice to honor the man with a scientific term.

--JN

References and notes

1) W. A. Bentley “Forty years’ study of snow crystals” Monthly Weather Review, Nov. 1924, page 532. He studied the crystals in the room, assuming they all started their growth under the same conditions. However, the editor pointed out in the article that the conditions throughout the room were not necessarily the same. Also, some of the crystals Bentley examined may have been born on the glass slide upon which he was photographing. The form of such crystals can be affected by the glass surface.

2) More details are in J. Nelson “Origin of diversity in falling snow” Atmos. Chem. Phys., 8 (2008), pages 5669-5682. Can be accessed at
https://acp.copernicus.org/articles/8/5669/2008/acp-8-5669-2008.pdf

3) A more careful calculation, in the above article, results in a slightly smaller number. But this difference does not affect the main points given here.

4) See the last page of C. Knight and N. Knight “Snow crystals”, Scientific American 228 (1973), pages 100-107. Another estimate of this number is in Frank, F. C.: Snow crystals, Contemp. Phys., 23, 3–22, 1982.