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Sea ice plays a critical role in the climate system through its albedo, which constrains light transmission into the upper ocean. In spring and summer, light transmission through sea ice is influenced by its iconic blue melt ponds, which significantly reduce surface albedo. We show that the geometry of surface melt ponds plays an important role in the partitioning of instantaneous solar radiation under sea ice by modeling the three‐dimensional light field under ponded sea ice. We find that aggregate properties of the instantaneous sub‐ice light field, such as the enhancement of available solar energy under bare ice regions, can be described using a new parameter closely related to pond fractal geometry. We then explore the influence of pond geometry on the ecological and thermodynamic sea ice processes that depend on solar radiation.

 

Abstract From quasicrystalline alloys to twisted bilayer graphene, the study of material properties arising from quasiperiodic structure has driven advances in theory and applied science. Here we introduce a class of two-phase composites, structured by deterministic Moiré patterns, and we find that these composites display exotic behavior in their bulk electrical, magnetic, diffusive, thermal, and optical properties. With a slight change in the twist angle, the microstructure goes from periodic to quasiperiodic, and the transport properties switch from those of ordered to randomly disordered materials. This transition is apparent when we distill the relationship between classical transport coefficients and microgeometry into the spectral properties of an operator analogous to the Hamiltonian in quantum physics. We observe this order to disorder transition in terms of band gaps, field localization, and mobility edges analogous to Anderson transitions — even though there are no wave scattering or interference effects at play here. 

The Neumann–Poincaré (NP) operator, a singular integral operator on the boundary of a domain, naturally appears when one solves a conductivity transmission problem via the boundary integral formulation. Recently, a series expression of the NP operator was developed in two dimensions based on geometric function theory [34]. In this paper, we investigate geometric properties of composite materials using this series expansion. In particular, we obtain explicit formulas for the polarisation tensor and the effective conductivity for an inclusion or a periodic array of inclusions of arbitrary shape with extremal conductivity, in terms of the associated exterior conformal mapping. Also, we observe by numerical computations that the spectrum of the NP operator has a monotonic behaviour with respect to the shape deformation of the inclusion. Additionally, we derive inequality relations of the coefficients of the Riemann mapping of an arbitrary Lipschitz domain using the properties of the polarisation tensor corresponding to the domain. 

We adapt the multiple scale method introduced over 40 years ago for the homogenization of periodic structures [1], to the quasiperiodic (cut-and-projection) setting. We make use of partial differential operators (gradient, divergence and curl) acting on periodic functions of m variables in a higher-dimensional space that are projected onto operators acting on quasiperiodic functions in the n-dimensional physical space (m>n). We replace heterogeneous quasiperiodic structures, coined irrational metamaterials in [2], by homogeneous media with anisotropic permittivity and permeability tensors, obtained from the solution of annex problems of electrostatic type in a periodic cell in higher dimensional space. This approach is valid when the wavelength is much larger than the period of the higher dimensional elementary cell.