Mathematicians have re-created the intricate patterns of ice formation, a breakthrough that could lead to new models of red blood cells, soap bubbles and other surfaces that evolve over time
March 16, 2012?|
A MULTIFACETED PROBLEM: Realistically simulating the growth of snowflakes has proved a huge challenge. Above, two examples of faceted snowflake structures. Image: Barrett/Garcke/N?rnberg
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 N?rnberg 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 molecules interacted?neglecting processes that occur over a continuum of distance scales.