Mathematical features of Hibler's model of large-scale sea-ice dynamics

Abstract

Hibler's well established model of large-scale sea-ice dynamics is considered, in which sea-ice, although a granular material made up of floes, is viewed and treated as a medium of continuum mechanics. The ice cover is commonly assumed to be a two-dimensional isotropic compressible fluid of viscous-plastic type which forms a highly stiff constitutive law. The complete model of sea-ice dynamics additionally comprises momentum balance equations for the ice drift velocity and conservation laws for continuity of its mass and compactness. In treating the complete set of equations, the central problem is posed by the sensitive crucial adjustment of the (quasistatic) equilibrium of forces. As a suitable approach to mathematical investigation and numerical treatment, the constitutive equations are reformulated in terms of projection to the convex yield domain in stress space. The mathematical features (existence and uniqueness of a solution) of this strongly nonlinear system of partial differential equations may be clarified by means of general theorems on variational inequalities and equations with monotonic operators in Hilbert spaces. Since the study is ultimately aimed at a numerical method, emphasis is put on the discrete-in-time version of Hibler's problem. As a result, if there is a solution to the problem, it is unique. However, for the ideally plastic constitutive law the existence of a solution is not guaranteed. To ensure a solution, we modified it from perfectly plastic to slightly viscoplastic. On this basis, an appropriate implicit numerical procedure is presented. The nonlinear problem is iterated by a sequence of linearizations.