# Research

**OVERVIEW:** Ken Golden is an applied mathematician who works on modeling and simulating multiscale systems that arise in geophysics and materials science. He is particularly interested in developing models of sea ice and its role in the climate system, treating sea ice as a complex material exhibiting composite structure on length scales ranging over ten orders of magnitude. Golden brings to bear rigorous mathematics to key problems in the physics and biology of sea ice such as fluid flow through the porous brine microstructure, the evolution of melt ponds on Arctic sea ice, thermal transport and fluid convection, ocean wave propagation in the marginal ice zone, anomolous diffusion and floe trajectories, and the dynamics of microbial communities living in sea ice. He has also worked extensively on the mathematics of sea ice remote sensing and the interaction of electromagnetic waves with composite materials.

A principal theme running through Golden's work is *linkage of scales*: incorporating small scale information into calculations of effective behavior on larger scales, like those relevant in coarse-grained climate models, as well as the inverse problem of reconstructing local characteristics from observed macroscopic behavior. Similar themes run in statistical physics and homogenization theory for composites, which provide approaches and models Golden has often used to study sea ice. His research is helping to advance how sea ice is represented in climate models, and to improve projections of climate change and the response of polar ecosystems.

**THE MATH:** Golden has used a broad array of mathematical techniques to study sea ice and other composites, including lattice and continuum percolation theory, homogenization for elliptic and parabolic partial differential equations, Stieltjes integral representations in one and several complex variables, computation of spectral measures which incorporate composite or flow geometry, Ising and network models, fractal geometry, random matrix theory, advection diffusion processes, and theories of electromagnetic inversion. These studies have often led to more general mathematical investigations and results for systems with applications well beyond sea ice. They include forward and inverse homogenization for matrix-particle composites and polycrystalline materials, critical behavior of transport in composites, integral representations for advection-enhanced diffusion, rigorous bounds on the electromagnetic properties of composites and the fluid permeability of porous media, and an unexpected Anderson transition for classical transport in composites near the percolation threshold. Dr. Golden also maintains research interests in quasiperiodic media and electrorheological fluids.

**FIELD EXPERIMENTS:** Dr. Golden has conducted extensive field experiments on the fluid and electromagnetic transport properties of sea ice in both the Arctic (2007, 2011, 2012, 2013, 2014) and Antarctic (1994, 1999, 2007, 2010, 2012). His field work on the vertical fluid permeability in the Antarctic has been focused recently on determining the percolation threshold for fluid flow in granular ice which comprises a significant proportion of the sea ice produced in the Southern Ocean. In the Arctic the focus has been to help understand the formation and evolution of ponds on top of melting Arctic sea ice. During a 2014 Arctic expedition the experiments (with C. Polashenski) revealed how ponds actually form on the surface of sea ice so permeable that the melt water should drain. Related field measurements of the electrical properties of sea ice are designed to help monitor microstructural transitions and track the state of the sea ice pack in a changing climate. Golden's field experiments are designed to provide data for comparison with theories of transport in sea ice, help guide the development of the models, and lay the groundwork for next generation sensors.

**Review Articles (from Publications):**

47. K. M. Golden, **Climate change and the mathematics of transport in sea ice**, invited article for the *Notices of the American Mathematical Society*, Volume 56, Number 5, pages 562-584 (including issue cover), May 2009. PDF (Additional References, PDF)

60. K. M. Golden, **Mathematics of sea ice**, invited article for *The Princeton Companion to Applied Mathematics*, N. J. Higham (Ed.), M. R. Dennis, P. Glendinning, P. A. Martin, F. Santosa, and J. Tanner (Assoc. Eds.), Princeton University Press, pp. 694-705, September 2015. PDF

**NSF Research Overview Video:**

NSF Discoveries Video: On Golden's Melt Pond (Bringing Mathematics to Sea Ice Research)

**Research Topic Pages (under construction):**

Melt pond evolution

Advection-diffusion processes

Spectral measures

Sea ice remote sensing