the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Stieltjes Functions and Spectral Analysis in the Physics of Sea Ice
Kenneth M. Golden
N. Benjamin Murphy
Elena Cherkaev
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: final response (author comments only)
- RC1: 'Comment on npg-2022-17', Anonymous Referee #1, 17 Feb 2023
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RC2: 'Comment on npg-2022-17', Anonymous Referee #2, 28 Feb 2023
The comment was uploaded in the form of a supplement: https://npg.copernicus.org/preprints/npg-2022-17/npg-2022-17-RC2-supplement.pdf
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AC1: 'Comment on npg-2022-17', Kenneth Golden, 25 Apr 2023
We thank the referees for very helpful comments that we believe will significantly improve the manuscript. We respond to each review separately below.
Response to Review #1: We really appreciate this reviewer’s remarks concerning the high level of mathematics and the need to incorporate descriptions of the main results and methods that will be accessible to a broader audience. Adding such descriptions will make the paper more readable, particularly to the intended audience of researchers in the physics of the cryosphere and climate science more broadly. Indeed, in early applications and extensions of the analytic continuation method to sea ice and composites, which rely on the Bergman-Milton integral representation, reviewers for journals in geophysics and materials science made similar requests, which resulted in significantly improving the readability of the paper, and making it accessible to a much broader range of readers. Also, we are more than happy to include the references listed in the review, and are particularly interested to have a close look at the recent papers listed from 2020 and 2022.
Response to Review #2: We also thank reviewer #2 for several incisive comments and for targeting the key mathematical issue at the heart of the analytic continuation approach in any context, which is how the local geometry of the composite or velocity field is encoded into the spectral measure. In particular, this reviewer focuses on the fascinating relationship between connectivity of one of the components, say the brine inclusions in sea ice, and the properties of the spectral measure.
First, to address the noise question concerning Figure 3, we have not computed a measure of noise in this case, although this could be easily done. The noise level is more significant than need be due to resolution issues in the brine images used, where averaging over an ensemble generated through offset sampling can only be done over very small sample sizes. We will likely choose a different sequence of images with much higher resolution that will allow more averaging and less variance.
Concerning the formation of a Dirac delta component at the origin in the spectral variable as the percolation threshold is approached, numerically this is a rather subtle process which has been discussed in some detail in our 2015 Comm. Math. Sci. and 2017 Phys. Rev. Lett. papers. In a 1996 J. Phys.: Cond. Matt. paper by Day and Thorpe they also consider this issue and provide some analytical as well as numerical results that are echoed in our computations of the spectral measure. In the revision we will discuss these very interesting issues in more detail and explain more fully what is accounting for the behavior near the spectral endpoints, and address the excellent points brought up by the reviewer. To help illustrate in our spectral measure calculations the different roles played by the eigenvalues (locations) and the eigenvectors (weights), in the revision we will also display the eigenvalue densities on the spectral interval as the percolation threshold is approached. We believe this will help further explain the spectral measure behavior at the endpoints and address the reviewer’s remarks in this regard.
In a separate paper currently in preparation we have explored in a more general graph-theoretic setting precisely the deep question raised by the reviewer – how does the spectral measure “signal” when a connection is formed. We will briefly mention our results in this general framework in the revised version of the paper in response to the excellent points brought up by the reviewer.
We agree that section 6 may appear to be somewhat disconnected from the rest of the manuscript. We believe, though, that we can easily revise the manuscript there to address the questions raised by the reviewer, which will significantly improve its readability. In fact, we will be including an alternative form for the Stieltjes representation of the effective diffusivity that puts it on a much more equal footing with the classic two component case.
Finally, the suggestion to add a paragraph or two on how the ACM results can contribute to climate models and to mention known limitations of the method in sea ice modeling, is an excellent one. We will certainly be adding a paragraph or two addressing these points in the revision.
Citation: https://doi.org/10.5194/npg-2022-17-AC1
Kenneth M. Golden et al.
Kenneth M. Golden et al.
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