Network Modeling to Study Sea Ice Permeability

By Karthika Swamy Cohen

Sea ice is significant in many ways. It helps keep the polar regions cool. Because of its bright surface, 80 percent of the sunlight striking it is reflected back into space. As the ice melts in warmer weather, parts of the ocean surface are exposed. The ocean surface—being darker—absorbs 90 percent of the sunlight instead of reflecting it. Is heats up the oceans, and temperatures rise further. Sea ice thus helps moderate global climate. Even a mild increase in temperature at the poles can lead to much greater warming over time.

Sea ice is also ecologically important. It harbors a rich variety of flora and fauna, from microorganisms such as algae and bacteria living within its microstructure and crustaceans feeding on its underside to large mammals such as penguins and polar bears that live on it. 

Physically, sea ice is a porous composite. The microstructure of the ice evolves as a function of temperature and salinity. Generally speaking, it mainly comprises a solid matrix of pure ice, with impurities such as brine, salt, and air, among other substances. Brine inclusions in sea ice create brine channels, which allow fluids to transport within ice.

Sea ice algae are among the microorganisms that inhabit the ice. They make up the base of food chain and need nitrogen and carbon in order to survive. They secret extra cellular polymeric substances (EPS), which protect them from the harsh environment; these secretions help shield the organisms from osmotic shock, freezing, and predation. 

At the SIAM Conference on Mathematics of Planet Earth, being held this week in Philadelphia, Pa., Kyle R. Steffen of the University of Utah described a mathematical model for sea ice permeability.

Steffen began by describing sea ice’s permeable features, which allow fluid to transmit through it.  

“Without the ability of sea ice to conduct fluid, algae would not survive,” he said. “The biology is certainly affected by the fluid properties of ice. But how does biology affect the physics?”

In other words, how does the algae affect the fluid permeability of sea ice? 

To get at this answer, Steffen first explained how the secretions from the algae, the EPS, modify the properties of sea ice. Young sea ice with certain types of algal EPS show significant changes in their structure and properties. Sea ice with EPS has a more tortuous microstructure and modified salt retention, showing a net increase in the brine volume fraction and net decrease in fluid permeability. 

Steffen and his group considered a random network model for fluid transport through sea ice with entrained EPS to study EPS-induced changes in sea ice fluid permeability. It is important to study this decrease in permeability, which is significant in physical or ecological process models involving fluid or nutrient transport.

The team found that the cross-sectional areas of the brine inclusions in sea ice are described by a bimodal lognormal distribution. Their model, which describes effective fluid permeability, is based on a random network of pipes with cross-sectional areas chosen from a bimodal distribution.

Each node in the random network is seen to be in balance or flux. The team solved for the pressure at each node and computed the pressure to determine fluid permeability of sea ice.

In mid-nineties, researchers observed that in young sea ice without EPS, brine channels are essentially logarithmically distributed. Steffen and his colleagues used this as a starting point to formulate their analysis. They derived the rigorous upper bound for a general porous medium of a specific volume fraction. This generated a lot of interesting mathematics and the team found that having a lognormal distribution does not appear to be relevant in this case.

They were able to quantify in a rigorous way that the logarithm of the cross-sectional areas seems to be better fit by a bimodal distribution rather than a normal distribution. Hence, they replaced the lognormal distribution with a bimodal-lognormal distribution. Using this new bimodal-lognormal distribution they were able to derive an analogous expression for rigorous upper bound for fluid permeability of sea ice. 

While effective fluid permeability of sea ice is significant in that it affects the properties of sea ice and consequently, polar ecosystems and climate models, the biological and geochemical effects on this parameter and not well understood. The model Steffen described is the first two-dimensional model describing microscale biochemical effects, specifically from the presence of algal secretions, on larger scale sea ice properties. Steffen and his team validated their numerical simulations with experimental observations.

Karthika Swamy Cohen is the managing editor ofSIAM News.