Reduced modelling of mountains and ice sheets

Abstract

The motion of rocks that make up mountains and of ice in glaciers and ice sheets can often be treated as a fluid flow. These geophysical systems are, however, very complex in terms of the rheology of their constituents and their interactions with environment. Simple mathematical models are often very helpful to reduce the complexity of these systems to something that can be analysed semi-analytically. In this thesis I present studies of mountain building and the flow of ice sheets that expand our understanding of the complex dynamics of geophysical flows. In chapter 1 I outline how mathematical models may be used to study slow geophysical flows. I also introduce mountains and ice sheets as the two systems to be studied and provide a broad overview of techniques used to model them. In chapter 2 I present a thin-film viscoplastic fluid model of an mountain range where a uniform fluid layer is deformed by a vertical backstop moving at constant speed. I explore the evolution of the system numerically and analytically by deriving a family of self-similar regimes. The model results are tested experimentally using ultrasound gel as the working fluid. I find that the experiments reproduce many features of the theoretically predicted behaviour. Chapter 3 presents a study of a land-terminating ice sheet model on a plastic till. I develop a simple model of subglacial hydrology that allows for meltwater storage and the flow of meltwater inside the till. Steady-state solutions are found and explored by varying the ablation-accumulation forcing and till permeability. I also study the asymptotic behaviour of the ice flow and topography close to the terminus, where a boundary layer can form. In chapter 4 I explore a problem of stability and evolution of a simple land-terminating ice sheet model that is subject to a linear mass balance law that depends on the ice surface height. After finding the steady-state solutions of the model, I discover that they are unstable to perturbations in the snowline height. Next, I look at the evolution of the ice sheet subject to a periodic variation of the snowline height, which represents the seasonal cycle. The thesis is concluded in chapter 5, where I also outline a set of ideas for future work, which range from further modelling and experiments to making comparisons with geophysical observations.