Stieltjes Functions and Spectral Analysis in Sea Ice Physics

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Polar sea ice is a critical component of Earth's climate system. Precipitous losses of Arctic sea ice affect not only the polar marine environment, but can impact, for example, weather patterns, ocean currents, storm tracks, and precipitation amounts in the Northern Hemisphere. As a material, sea ice is a multiscale composite of pure ice with millimeter-scale brine inclusions and centimeter-scale polycrystalline microstructure. Even the ice pack itself is a granular composite of ice floes in a sea water host. A principal challenge in modeling sea ice and its role in climate is how to use information on small scale structure to find the effective or homogenized properties on larger scales relevant to process studies and coarse-grained climate models. Also of interest is the inverse problem of recovering parameters controlling small scale processes from large scale observations, such as in remote sensing. These central issues are addressed in this project by exploiting special mathematical properties of effective parameters that are common to several important problem areas in sea ice modeling. They include fluid and electromagnetic transport through sea ice, diffusion processes like heat flux enhanced by brine convection, and ocean surface waves in the ice pack. Sea ice shares close similarities with other naturally occurring and engineered composites. Our mathematical results on sea ice will give new insights and findings about other composite materials, and vice versa. This research will help to advance how sea ice is represented in climate models and improve projections of the fate of Earth's sea ice packs and the ecosystems they support. This project will also provide support, training, and research opportunities for undergraduate and graduate students. A powerful approach in the mathematical theory of homogenization is the analytic continuation method, which provides integral representations for the effective parameters of composite materials, treated as Stieltjes functions of their parameters. The complexities of composite microgeometries are distilled into the spectral properties of self-adjoint operators, like the Hamiltonian in quantum physics, which become random matrices when the system is discretized. Early applications of the method to sea ice focused on remote sensing and electromagnetic properties, with sea ice treated as a two-phase composite. Extensions of the method to polycrystalline media, advection diffusion processes, and surface waves through the ice pack have yielded only elementary bounds on the effective parameters, based on the coarsest information like the mass of a spectral measure in the integral, which is the brine volume fraction for sea ice as a two-phase composite, or the area fraction of ocean covered by ice floes for surface waves. However, exploiting deep parallels with random matrix theory descriptions of quantum physics brings Anderson transition concepts like field localization, mobility edges, eigenvalue repulsion, and band gaps into homogenization for classical transport in two-phase composites. In the funded work we will explore new insights that this approach can give us in the above problem areas, where the effective parameters are Stieltjes functions. For example, how do sea ice effective properties depend on crystal size and structure, can some ocean waves be localized by the ice pack geometry, and how does the fractal nature of composite geometry influence effective transport and spectral properties. This approach opens many questions of mathematical interest, while at the same time provides novel tools to address questions of critical importance in understanding sea ice processes in the rapidly changing polar marine environment.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.