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Mathematical Patterns in Sea Ice Reveal Melt Dynamics


Melt ponds on the surface of Arctic sea ice. Credit: Karen Frey

Some people call Ken Golden the “Indiana Jones” of mathematics due to his frequent excursions to remote, harsh parts of the world. Golden, a professor of mathematics at the University of Utah, studies the dynamics of sea ice, and he regularly goes out into the field to test his hypotheses. He has visited the Arctic and Antarctica 16 times to take ice samples and make observations.

Sea ice has declined rapidly in the past decade; in the summer of 2012, Arctic sea ice was at its lowest recorded levels, below even what climate models predicted. “It’s particularly unsettling that the world’s best climate models have not been able to keep pace with this incredible rate of melting,” Golden said during a March 5 talk entitled Modeling the Melt at the Museum of Mathematics in New York City.

That’s because sea ice is a crucial factor in global climate change. “Sea ice is not only a very sensitive leading indicator of climate change, but it’s also a key player… in Earth’s climate system,” Golden said. White sea ice helps return energy from the sun back into outer space thanks to the high percentage of sunlight it reflects, known as its albedo. As sea ice decreases, dark ocean water absorbs more of that sunlight, contributing to further warming. “One of the most important parameters in climate modeling is the overall albedo of the ice pack,” Golden said.

Despite its importance, modeling the transition of sea ice as it melts and changes from high to low albedo has been difficult. However, in 2012, Golden and his colleagues published a paper showing that as sea ice melts the ponds that form on its surface can in some ways be treated like fractals, mathematical patterns that remain the same or similar at small and large scales.

As isolated simple ponds grow and connect with others, they form larger, more complex ponds; subsets of these ponds show similarities to the larger complex ponds, just as a fractal does. In addition, as ponds get larger and more complex, the total perimeter of the ponds increases much faster than the total area of the ponds, which is also the case with fractals. “Any kind of information you can get about how [melt ponds] evolve, their geometry, their area coverage … are the key parameters that go into assessments of albedo,” Golden says.

Golden has also looked at the microstructure of the sea ice. Specifically, he has modeled “brine inclusions,” or pockets of liquid water within the ice. The formation of these pockets depends on the temperature and crystallographic structure of the ice, which in turn depends on how the sea ice formed. He found that there is a threshold at which these inclusions become connected to one another and form pathways, allowing drainage of melt ponds or percolation of sea water upward from beneath the ice.

For the most common type of sea ice in the Arctic, this threshold occurs when the percentage of liquid brine inclusions in the ice (called the brine volume fraction) is around 5 percent. When the sea ice salinity is at its typical level, about 5 parts per thousand, the 5 percent brine volume fraction occurs at a critical temperature of -5 degrees Celsius. Golden dubbed this “the Rule of Fives” in a 1998 paper in Science.

Although the original Rule of Fives was based on field observations, Golden was later able to confirm the rule through the use of x-ray imaging. By creating 3D models of brine inclusions in ice, Golden and his colleagues showed in a 2007 paper that the connectivity of the brine inclusions does indeed cross a threshold at a brine volume fraction of around 5 percent.

X-ray tomographic image of the brine phase in sea ice. Credit: Hajo Eicken

The Rule of Fives has an important effect on ice albedo, Golden said, because the presence or absence of the pools on top of the ice drastically changes the albedo. And the drainage of the ponds “depends upon how easy it is for fluid to flow through the porous microstructure of the ice,” he said. This permeability of the ice also affects algal communities living in the brine inclusions—when the ice is impermeable, the algae don’t get new nutrients from percolating seawater.

Eventually, Golden thinks that unifying mathematical concepts will connect sea ice across many scales, from tiny inclusions in the ice to large melt ponds. “I kind of smell something universal,” he says.

The Museum of Math hosts monthly talks by mathematicians as part of its ongoing Math Encounters series.

The views expressed are those of the author and are not necessarily those of Scientific American.

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