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Monday, April 13, 2009

Kenneth M. Golden, Climate Change and the Mathematics of Transport in Sea Ice

Notices of the AMS, May 2009,Vol. 56, No. 5, pp. 562-585.

Climate Change and the Mathematics of Transport in Sea Ice

by Kenneth M. Golden, University of Utah, Salt Lake City, UT, U.S.A. e-mail:

As the boundary layer between the ocean and atmosphere in the polar regions, sea ice is a critical component of the global climate system. As temperatures on Earth have warmed, the Arctic sea ice pack in particular has exhibited a dramatic decline in its summer extent. Indeed, the polar sea ice packs are harbingers of climate change. Predicting what may happen over the next ten, fifty, or one hundred years requires extensive modeling of critical sea ice processes and the role that sea ice plays in global climate. Currently, large-scale climate models in general do not realistically treat a number of sea ice processes that can significantly affect predictions. Moreover, monitoring the state of Earth’s sea ice packs, in particular their thickness distributions, provides key data on the impact of global warming. Mathematics is currently playing an important role in addressing these fundamental issues and will likely play an even greater role in the future. Here we give a brief bullet-by-bullet summary of the contents of this

Question 1. What is the role of polar sea ice in global climate?

• Sea ice forms the boundary between the polar oceans and the atmosphere, and mediates the exchange of heat, gases, and momentum between them. The polar sea ice packs help regulate Earth’s climate and are acute indicators of climate change.
• While sea water absorbs most incident solar radiation, sea ice reflects most ofit. Earth’s sea ice packs act as solar heat reflectors, as well as ocean insulators, keeping significant heat from escaping to the atmosphere.
• Sea ice is a porous composite of pure ice with liquid brine inclusions. The flow of fluids through sea ice mediates processes important to climate, from melt pond evolution, which controls the reflectance of the sea ice packs, to upward percolation of sea water, which floods the surface and
then freezes, forming snow-ice, an increasingly important component in the sea ice system. Brine flow though sea ice can also enhance heat exchange.
• Sea ice hosts algal and bacterial communities which support life in the polar oceans. Nutrient replenishment processes are controlled by fluid transport through the microstructure.
• The state of the polar sea ice packs—their extent and thickness—provides important
information in monitoring climate change.

Question 2. What is the role of mathematics in modeling transport in sea ice?
• Various techniques within the field of homogenization are used to derive macroscopic
information about transport in sea ice from partial information about its microstructure.
• Variational formulations for the trapping constant and mean survival time for diffusion processes which interact with the pore boundaries are used to obtain rigorous bounds on the fluid permeability tensor for sea ice, based on general microstructural information.
• Fluid transport in sea ice exhibits a permeable/impermeable transition at a critical brine volume fraction of about 5%, which controls geophysical and biological processes.
X-ray computed tomography and mapping of the pore microstructure onto random graphs are used to demonstrate that the brine phase of sea ice undergoes a transition in connectedness at this brine volume fraction. Percolation theory is used to theoretically predict the transition
and to mathematically characterize the thermal evolution of the fluid pores and their connectedness.
• Lattice and continuum percolation models are used to predict critical behavior of the fluid permeability in sea ice, with a universal exponent describing the behavior above the percolation threshold, and critical path analysis yields the scaling factor.
• Hierarchicalmodels developed for porous rocks are used to predict the fluid permeability of sea ice over the entire range of brine porosities.
• A random pipe network, which is equivalent to a random resistor network and is solved using fast multigrid methods, is used to numerically simulate fluid flow through sea ice.
• Methods of complex analysis and functional analysis are used to obtain rigorous bounds on the effective complex permittivity of sea ice, the key parameter characterizing its electromagnetic behavior and the response of sea ice in remote sensing applications, such as monitoring sea ice thickness.
• Inverse methods yield microstructural information fromcomplex permittivity data, paving the way for electromagnetic monitoring of fluid and thermal transport in sea ice.
• We have made measurements of fluid and electrical transport in sea ice on a 2007 Antarctic expedition in order to validate our models and investigate new phenomena.
• Large-scale sea ice and climate models currently do not generally incorporate the key processes our work on transport describes.
These global models have grid sizes on the order of many kilometers. Future mathematical challenges include quantification of how local transport properties on the scale of individual floes influence pack behavior on much larger scales relevant to global models.

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