19 Dec 2022
19 Dec 2022
Status: this preprint is currently under review for the journal NPG.

Stieltjes Functions and Spectral Analysis in the Physics of Sea Ice

Kenneth M. Golden, N. Benjamin Murphy, and Elena Cherkaev Kenneth M. Golden et al.
  • Department of Mathematics, University of Utah , 155 S 1400 E RM 233, Salt Lake City, UT 84112-0090

Abstract. Polar sea ice is a critical component of Earth’s climate system. As a material it is a multiscale composite with temperature dependent millimeter-scale brine microstructure, and centimeter-scale polycrystalline microstructure which is largely determined by how the ice was formed. The surface layer of the polar oceans can be viewed as a granular composite of ice floes in a sea water host, with floe sizes ranging from centimeters to tens of kilometers. A principal challenge in modeling sea ice and its role in climate is how to use information on smaller scale structure to find the effective or homogenized properties on larger scales relevant to process studies and coarse-grained climate models. That is, how do you predict macroscopic behavior from microscopic laws, like in statistical mechanics and solid state physics? Also of great interest in climate science is the inverse problem of recovering parameters controlling small scale processes from large scale observations. Motivated by sea ice remote sensing, the analytic continuation method for obtaining rigorous bounds on the homogenized coefficients of two phase composites was applied to the complex permittivity of sea ice, which is a Stieltjes function of the ratio of the permittivities of ice and brine. Integral representations for the effective parameters distill the complexities of the composite microgeometry into the spectral properties of a self-adjoint operator like the Hamiltonian in quantum physics. These techniques have been extended to polycrystalline materials, advection diffusion processes, and ocean waves in the sea ice cover. Here we discuss this powerful approach in homogenization, highlighting the spectral representations and resolvent structure of the fields that are shared by the two component theory and its extensions. Spectral analysis of sea ice structures leads to a random matrix theory picture of percolation processes in composites, establishing parallels to Anderson localization and semiconductor physics, which then provides new insights into the physics of sea ice.

Kenneth M. Golden et al.

Status: open (until 08 Mar 2023)

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Kenneth M. Golden et al.

Kenneth M. Golden et al.


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Short summary
Our paper introduces very powerful methods of finding the effective behavior of complex systems, which can be applied well beyond the initial setting of sea ice. Applications include transport properties of porous and polycrystalline media such as rocks and glacial ice, and advection diffusion processes that arise throughout geophysics. Connections to random matrix theory establish unexpected parallels of these geophysical problems with semiconductor physics and Anderson localization phenomena.