Mathematical and numerical modelling of ice sheets and glaciers

Tutorial held at the Workshop "Mathematical Approach to Climate Change Impacts", Istituto Nazionale di Alta Matematica Francesco Severi (INdAM), Rome, Italy, 2017.03.15/16.

Summary. Ice sheets (with their attached ice shelves), ice caps and glaciers are active, dynamic components of the climate system of the Earth. As a common feature, these ice bodies show gravity-driven creep flow ("glacial flow"), sustained by the underlying land. This leads to thinning and horizontal spreading, which is essentially compensated by snow accumulation in the higher (interior) areas and melting and calving in the lower (marginal) areas. Any imbalance of this dynamic equilibrium leads to either growing or shrinking ice masses. In this tutorial, we will first review the underlying field theories of continuum mechanics and thermodynamics. Basic concepts are the tensorial measures for stress and strain, and the balance equations for mass, momentum and energy. We will then discuss the material (rheological) properties of polycrystalline ice, which are expressed by Glen's flow law (or variations thereof), Fourier's law of heat conduction and a caloric equation of state. Combining the balance and material equations leads to the field equations for the flow of ice sheets, ice shelves, ice caps and glaciers, which must be complemented by suitable boundary conditions. We will treat a hierarchy of simplifications of the field equations, from the most sophisticated full Stokes formulation to the most simplified shallow-ice and shallow-shelf approximations. Analytical solutions only exist for highly simplified problems, while in general numerical solution techniques must be applied. We will present some examples of numerical solutions for selected problems of ice sheet and glacier flow.