Doctor of Philosophy

dissertationSea ice can be viewed as a composite material over multiple scales. On the smallest scale, sea ice is viewed as a two-phase composite of ice and brine. On the mesoscale, one may consider pancake ice and slush as a viscoelastic composite. On the larger scale, one may consider the mix of ice floes and water. With this view, a multitude of mathematical tools may be applied to develop novel models of physical sea ice processes. We model fluid and electrical transport viewing sea ice as a two-phase composite of ice and brine. We may then apply continuum percolation models to study critical behavior which we have experimentally confirmed. These percolation models suggest that the electrical conductivity and fluid permeability follow universal power law behavior as a function of brine volume fraction. We apply the results above for the electrical conductivity of sea ice to develop an inversion algorithm for surface impedance DC tomography. The algorithm retrieves both sea ice thickness and a layered stratigraphy of the sea ice resistivity. This is useful as resistivity carries information about the internal microstructure of the ice. We also apply network models to conductivity of sea ice and use some similar ideas to quantify the horizontal connectivity of melt ponds. On the larger scale, we study the problem of ocean wave dynamics in the marginal ice zone of the Arctic and Antarctic. We adopt the view that the ice and slush may be viewed as a viscoelastic layer atop an inviscid ocean. Models like these produce dispersion relations which describe wave propagation and attenuation into the ice pack. These dispersion relations depend on knowledge of the effective viscoelasticity of the ice/slush mix. This is a difficult parameter to measure in practice. To get around this, we apply homogenization theory to derive bounds on these parameters in the low frequency limit. This is accomplished through the derivation of a Stieltjes integral representation, involving a positive measure of a self-adjoint operator, for the effective elasticity tensor of the ice water composite. We have also developed a simplified wave equation for waves in the ice-water composite