A continuum-mechanical formulation for shallow polythermal ice sheets

This paper is concerned with anew theoretical approach to model grounded ice sheets in three dimensions. These are considered as polythermal, i.e. there will be regions with temperatures below the pressure melting point ('cold ice') and regions with temperatures exactly at the pressure melting point ('temperate ice'). In the latter, small quantities of water may occur. Based on previous approaches, an improved theory of polythermal ice sheets is developed, which is founded on continuum-thermodynamic balance relations and jump conditions for mass, momentum and energy. The rheological behaviour is assumed to be that of an incompressible, nonlinear viscous and heat conducting fluid; because of the dependence of viscosity on temperature and on water content, the problem is thermo-mechanically coupled. After presenting analytic solutions for a simple geometry (ice sheet of uniform depth), the theory is subjected to a scaling procedure with the assumptions of a small aspect ratio (ratio between typical vertical dimension and typical horizontal dimension) and a small Froude number. This leads to the introduction of the polythermal shallow-ice approximation (SIA) equations. Finally, as an application of the model to a real problem, a numerically computed steady-state solution for the Greenland Ice Sheet under present climate conditions is presented and compared with the real ice sheet.