A finite element formulation for freezing and thawing processes of ice within the framework of the TPM

In recent years the simulation of sea ice evolution, e.g. for the use in climate models, became more important, see for instance Danilov et al. (2015) and the references therein. In many contributions, the ice motion, which goes back to the findings in Hibler III (1979), is investigated. There, a numerical model for the simulation of sea ice circulation and thickness evolution on the basis of an evolution equation is explicitly described. In the present contribution, a coupled finite element method based on the Theory of Porous Media (TPM), see e.g. Bowen (1980) and Bowen (1982), for the direct modeling of phase transition of ice and water is presented. In detail, we investigate the ice deformation, the temperature development and the evolution of energy, enthalpy and mass exchange between the constituents. The main idea is based on a theoretically motivated evolution equation for the phase transition of ice and water, which guarantees the thermodynamical consistency. The resulting finite element is a four-field formulation in terms of ice displacements, liquid pressure, volume fraction of ice and temperature. Here, we make use of a quadratic interpolation of the ice displacements and a linear interpolation for the other degrees of freedom. We present first numerical examples, which examines freezing processes.