A MULTIFACETED PROBLEM: Realistically simulating the growth of snowflakes has proved a huge challenge. Above, two examples of faceted snowflake structures.
Image: Barrett/Garcke/Nürnberg
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Windswept from cloud to cloud until they flutter to Earth, snowflakes assume a seemingly endless variety of shapes. Some have the perfect symmetry of a six-pointed star, some are hexagons adorned with hollow columns, whereas others resemble needles, prisms or the branches of a Christmas tree.
Scientists as far back as Johannes Kepler have pondered the mystery of snowflakes: Their formation requires subtle physics that to this day is not well understood. Even a small change in temperature or humidity can radically alter the shape and size of a snowflake, making it notoriously difficult to model these ice crystals on a computer. But after a flurry of attempts by several scientists, a team of mathematicians has for the first time succeeded in simulating a panoply of snowflake shapes using basic conservation laws, such as preserving the number of water molecules in the air.
Harald Garcke of the University of Regensburg in Germany and his colleagues, John Barrett and Robert Nürnberg of Imperial College London, described their findings in an article posted at the physics preprint server, arXiv.org, on February 15. In that sense, Garcke and his collaborators “have done the whole megillah,” says physicist and snowflake maven Ken Libbrecht of the California Institute of Technology. “They have solved a problem that other people have tried and failed to do.”
To model a growing snow crystal on the computer, researchers must accurately simulate how the crystal surface changes with time. The surface is usually approximated by a series of interlocking triangles, but the triangles often deform and collapse in simulations, leading to singularities that bring the simulation to an abrupt halt, Garcke says.
Garcke’s team got around this difficulty by devising a method to describe the curvature and other geometric information about the snowflake surface so that it could be appropriately encoded into a computer. In doing so, the team found a way of avoiding problems other researchers had encountered.
Moreover, they found a new way to model the two main types of snowflake growth simultaneously: faceted growth, in which flat plates, such as hexagons and triangles, dominate the process, and dendritic growth, in which the flakes form treelike branches that themselves beget branches, just as dendrites extend out from nerve cells.
Previous attempts to model snowflakes using a similar approach could not reproduce both growth characteristics. "Our team is the first to do both faceted and dendritic growth, using basic conservation laws and thermodynamics," Garcke says. With the model, Garcke and his colleagues found unexpected aspects of snowflake formation, such as the strong influence of bonds between surface molecules in the crystal. They also found that the speed at which the sharp tips of snowflakes grow is directly proportional to the amount of water vapor in the atmosphere.
Crucially, the team’s approach is based on more realistic physics than past approaches. In their Physical Review E paper from 2009, mathematicians Janko Gravner of University of California, Davis, and David Griffeath of the University of Wisconsin–Madison approximated flake formation using a technique known as cellular automata. Although their work remains a milestone in successfully reproducing the intricate shapes, the method assumed that only neighboring clumps of material, about a micrometer in length, interacted—neglecting processes that occur over a continuum of distance scales.*
The new work by Garcke and his collaborators, Griffeath says, opens an avenue to using powerful mathematical constructs called partial differential equations to study complex snowflake dynamics. Because the differential equations describing the geometry of an evolving snow crystal appear in a similar form in other applications, “we could use our approach for many other problems in which surfaces evolve in time,” Garcke notes. Using variations on their computer model, “we have already computed the shape of red blood cells, soap bubble clusters and the evolution of polycrystalline materials,” he says.
Despite the new advances in modeling snowflake growth, Libbrecht says, the fundamental mystery about snowflakes—how they form in the first place—is still far from solved. To understand how they grow, “we have to marry the mathematics with the physics, and that’s not been done, partly because we don’t know the right physics,” he says.
In a recent paper, Libbrecht, who spent a snowy childhood in North Dakota and has built a machine to make snowflakes in the laboratory, suggested that some of the missing physics may be related to a previously overlooked instability in the changing shape of ice crystals. Libbrecht has urged Garcke to incorporate the proposed instability, which transforms thick prismlike snow crystals into thin plates, in the team’s simulations. Garcke says he and his colleagues are now considering doing so, although he believes other effects may be more important.
Garcke notes that his university town of Regensburg has a special link to snowflake studies. Johannes Kepler, the 17th-century astronomer and mathematician who was the first researcher to write about snowflake formation, died there in 1630 during a brief visit to the city. As Garcke wrote in a recent e-mail about his team’s simulation, he had only to look out the window to contemplate the real thing. After a week of 10-degree Celsius weather, he said, snow had resumed falling.
*Correction (4/27/12): This sentence was edited after publication to rectify an error.
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6 Comments
Add CommentThis probably has no connection, but having read Libbrecht I was thinking about similar things:I made a program that compiles centered figurate numbers. The sum of the centered numbers - or pronic numbers if that is the right term - contain no primes using numbers 8 and 9. I have checked numbers up to 100, and primes up to 100,000, and it seems strange that only these two sequences are free of primes. The fact that figurate numbers 1 to 7 produce primes seemed at first insignificant, given that shapes can only be made
Reply | Report Abuse | Link to thisusing 3 or more numbers, unless supposing a shape using two would be a line, and one being an expanding point! Then it looks like the first seven numbers
are significant, with 8 and 9 being an end, or beginning of another 'sequence'. All the numbers in the 9 sequence have digits that sum to 10, and all 8 pronics are squares, which explains why they can't be primes, but squares of odd numbers in the sequence of odd numbers. All very odd.
I was also struck by the fact that there are 7 crystal systems: from Cubic to Triclinic.
Is this just a arithmetical anomaly or does it have any meaning.
A water molecule has a 105* angle where two hydrogen atoms attach to the oxygen atom. 360* circle, with six facets some over lapping breaks the crystals into one of the strongest possible designs in nature.
Reply | Report Abuse | Link to thisThe recent work described above is an excellent step that goes a long way toward explaining the different morphology encountered in different environments. The one remaining element of snowflake design not covered is their amazing symmetry. Evident especially in the dendritic snowflake is the striking similarity of each dendrite, This symmetry is not limited to the familiar six sided snowflake but is also observed in snowflakes that do not have D6 symmetry. Various explanations for the symmetry have been advanced from vibrations to air vortices. While other explanations for symmetry fail for energetic or resolution reasons there is one explanation that, I believe, satisfies all physical requirements. The snowflake develops a charge structure as water molecules are acquired and the symmetry pattern is the result of the electrostatic forces that cause the acquired water molecules to migrate to their lowest potential energy state which is symmetry.
Reply | Report Abuse | Link to thisThe same symmetry inducing effect can be observed with magnetic particle interactions. One can do the experiment at home using small bar magnets suspended by corks in water. While it is easy to observe with magnetic fields, the operation of the electric fields in snowflake formation is so rapid and at such close ranges that it is not possible to observe the discrete small forces in operation. It should be possible however to add electrostatic forces to the computer models that have successfully reproduced the morphology changes.
We always portray snow flakes in a basic two dimensional growth format and think about symmetry in only one plane. Doe any snow flakes grow symmetric with similar complexity on the other plane as well? If not then why not? What is so special about the two axis growth pattern?
Reply | Report Abuse | Link to thisThis is an excellent comment. It raises some questions, though. First, what about other particles which condense from a vapor phase or aggregate from smaller particles? Soots do not seem to follow this route: they have only a fractal dimension in common, and only over a limited size range. Second, the idea implies that the molecules condensing on the forming snowflake are ionized, or become ions upon landing on the substrate. Where would the energy come from to ionize them?
Reply | Report Abuse | Link to thisKen Libbrecht's Field Guide to Snowflakes puts the symmetry down to the flake being in the same environment from instant to instant.
Reply | Report Abuse | Link to thisConditions change over time, and so do the form of the shapes, but the edges all experience the same conditions at any one time.