Kenneth M. Golden

Bounds for Effective Parameters of Heterogeneous Media by Analytic Continuation

We give a mathematical formulation of a method for obtaining bounds on effective parameters developed by D. Bergman and G. W. Milton. This method, in contrast to others used before, does not rely on a variational principle, but exploits the properties of the effective parameter as an analytic function of the component parameters. The method is at present restricted to two-component media.

Inverse electromagnetic scattering models for sea ice

Inverse scattering algorithms for reconstructing the physical properties of sea ice from scattered electromagnetic field data are presented. The development of these algorithms has advanced the theory of remote sensing, particularly in the microwave region, and has the potential to form the basis for a new generation of techniques for recovering sea ice properties, such as ice thickness, a parameter of geophysical and climatological importance. Moreover, the analysis underlying the algorithms has led to significant advances in the mathematical theory of inverse problems. In particular, the principal results include the following. (1) Inverse algorithms for reconstructing the complex permittivity in the Helmholtz equation in one and higher dimensions, based on layer stripping and nonlinear optimization, have been obtained and successfully applied to a (lossless) laboratory system. In one dimension, causality has been imposed to obtain stability of the solution and layer thicknesses can be obtained from the recovered dielectric profile, or directly from the reflection data through a nonlinear generalization of the Paley-Wiener theorem in Fourier analysis. (2) When the wavelength is much larger than the microstructural scale, the above algorithms reconstruct a profile of the effective complex permittivity of the sea ice, a composite of pure ice with random brine and air inclusions. A theory of inverse homogenization has been developed, which in this quasistatic regime, further inverts the reconstructed permittivities for microstructural information beyond the resolution of the wave. Rigorous bounds on brine volume and inclusion separation for a given value of the effective complex permittivity have been obtained as well as an accurate algorithm for reconstructing the brine volume from a set of values. (3) Inverse algorithms designed to recover sea ice thickness have been developed. A coupled radiative transfer-thermodynamic sea ice inverse model has accurately reconstructed the growth of a thin, artificial sea ice sheet from time-series electromagnetic scattering data.

Inverse bounds for microstructural parameters of composite media derived from complex permittivity measurements

Abstract Bounds on the volume fraction of the constituents in a two-component mixture are derived from measurements of the effective complex permittivity of the mixture, using the analyticity of the effective property. First-order inverse bounds for general anisotropic materials, as well as second-order bounds for isotropic mixtures, are obtained. By exploiting an analytic representation of the effective complex permittivity, the problem of estimating the structural parameters is reduced to a problem of evaluating the moments and support of a measure containing information about the geometrical structure of the material. Rigorous bounds on the volume fraction are found by inverting first- and second-order (Hashin–Shtrikman) forward bounds on the complex permittivity. The inverse bounds are applied to measurements of the effective complex permittivity of sea ice, which is a three-component mixture of ice, brine and air. The sea ice is treated via the two-component theory applied to a mixture of brine and a...

Forward electromagnetic scattering models for sea ice

Recent advances in forward modeling of the electromagnetic scattering properties of sea ice are presented. In particular, the principal results include the following: (1) approximate calculations of electromagnetic scattering from multilayer random media with rough interfaces, based on the distorted Born approximation and radiative transfer (RT) theory; (2) comprehensive theory of the effective complex permittivity of sea ice based on rigorous bounds in the quasi-static case and strong fluctuation theory in the weakly scattering regime; (3) rigorous analysis of the Helmholtz equation and its solutions for idealized sea ice models, which has led in the one dimensional (1D) case to nonlinear generalizations of classical theorems in Fourier analysis. The forward models considered incorporate many detailed features of the sea ice system and compare well with experimental data. The results have advanced the general theory of scattering of electromagnetic waves from complex media as well as homogenization theory, which relates bulk properties of composite media to their microstructural characteristics. Furthermore, the results have direct application to microwave remote sensing and serve as the basis for inverse algorithms for reconstructing the physical properties of sea ice from scattering data.

Bounds on the complex permittivity of a multicomponent material

Abstract Recently D. Bergman introduced a method for obtaining bounds on the effective dielectric constant (or conductivity) of a two-component medium. This method does not rely on a variational principle but instead exploits the properties of the effective parameter as an analytic function of the ratio of the component parameters. Here the method is extended to multicomponent media using techniques of several complex variables. We propose for the first time a series of bounds on the complex dielectric constant of a material of three or more components, as well as rederive the Wiener and Hashin-Shtrikman bounds for real parameters. In addition, we obtain in a simple manner a known infinite sequence of bounds for two-component media.

Brine percolation and the transport properties of sea ice

Abstract Sea ice is distinguished from many other porous composites, such as sandstones or bone, in that its microstructure and bulk material properties can vary dramatically over a small temperature range. For brine-volume fractions below a critical value of about 5%, which corresponds to a critical temperature of about −5°C for salinity of 5 ppt, columnar sea ice is effectively impermeable to fluid transport. For higher brine volumes, the brine phase becomes connected and the sea ice is permeable, allowing transport of brine, sea water, nutrients, biomass and heat through the ice. Over the past several years it has been found that brine transport is fundamental to such processes as sea-ice production through freezing of flooded ice surfaces, the enhancement of thermal and salt fluxes through sea ice, nutrient replenishment for sea-ice algal communities, and to sea-ice remote sensing. Motivated by these observations, recently we have shown how percolation theory can be used to understand the critical behavior of fluid transport in sea ice. We applied a percolation model developed for compressed powders of large polymer particles with much smaller metal particles, which explains the observed behavior of the fluid permeability in the critical temperature regime, as well as Antarctic data on surface flooding and algal growth rates. Moreover, the connectedness properties of the brine phase play a significant role in the microwave signature of sea ice through its effective complex permittivity and surface flooding. Here we review our recent results on brine percolation and its role in understanding the fluid and electromagnetic transport properties of sea ice. We also briefly report on measurements of percolation we made on first-year sea ice during the winter 1999 Mertz Glacier Polynya Experiment.

Bounds for effective parameters of multicomponent media by analytic continuation

Recently D. Bergman introduced a method for obtaining bounds for the effective dielectric constant (or conductivity) of a two-component medium. This method does not rely on a variational principle but instead exploits the properties of the effective parameter as an analytic function of the ratio of the component parameters. We extend the method to multicomponent media using techniques of several complex variables.

Thermal Conduction in Composites

We consider the effective thermal conductivity of two-component isotropic composites and review bounds obtained through analytic continuation of the effective conductivity as a function of the component conductivities. The connection between this conductivity function and Stieltjes functions is emphasized. Many of the well-known bounds on the effective thermal conductivity correspond to bounds on Stieltjes functions and these bounds, in turn, are closely related to Pade approximants.

Bounds on the complex permittivity of sea ice

An analytic method for obtaining bounds on effective properties of composites is applied to the complex permittivity ∈* of sea ice. The sea ice is assumed to be a two-component random medium consisting of pure ice of permittivity ∈1 and brine of permittivity ∈2. The method exploits the properties of ∈* as an analytic function of the ratio ∈1/∈2. Two types of bounds on ∈* are obtained. The first bound R1 is a region in the complex ∈* plane which assumes only that the relative volume fractions p1 and p2 = 1 - p1 of the ice and brine are known. The region R1 is bounded by circular arcs and ∈* for any microgeometry with the given volume fractions must lie inside it. In addition to the volume fractions, the second bound R2 assumes that the sea ice is statistically isotropic within the horizontal plane. The region R2 is again bounded by circular arcs and lies inside R1. Built into the method is a systematic way of obtaining tighter bounds on ∈* by incorporating information about the correlation functions of the brine inclusions. The bounding method developed here, which does not assume any specific geometry for the brine inclusions, offers an alternative to the classical mixing formula approach adopted previously in the study of sea ice. In these mixing formulas, specific assumptions are made about the inclusion geometry, which are simply not satisfied by the sea ice under many conditions. The bounds R1 and R2 are compared with experimental data obtained from artificially grown sea ice at the frequencies 4.8 and 9.5 GHz. Excellent agreement with the data is achieved.

A network model for fluid transport through sea ice

Abstract The flow of liquid through porous sea ice is a fundamental process affecting problems in polar biology, oceanography and geophysics. The geometry and connectedness of the pore microstructure of sea ice determine its fluid permeability, which depends strongly on temperature. Here we analyze a simple pipe network as a basis for modeling fluid flow through the complex porous microstructure, and for numerical approximations of the fluid permeability of sea ice as a function of temperature. For slow flow the fluid system is equivalent to an electrical resistor network, and the network is solved using a fast multi-grid method. The radii of the pipes in the network are chosen randomly from distributions describing measured cross-sectional areas of brine inclusions in sea ice. At this stage, the model reflects only the most general features of brine microstructure and its evolution with temperature. Preliminary results for a basic implementation in two dimensions are presented. They are consistent with theoretical bounds on the vertical fluid permeability of sea ice found recently. Moreover, the results agree roughly with laboratory data for higher porosities. For lower porosities and colder temperatures, the fully connected network of pipes in the model, albeit with smaller radii, overestimates observed values. This finding provides evidence that the brine network becomes more disconnected with lower temperatures, which is consistent with transitional behavior near a percolation threshold.