Theory of shallow ice shelves

Ice shelves consist of two layers, an upper layer of meteoric ice nourished by the flow from the connected inland ice and precipitation, and a lower layer of marine ice that is built by the melting and freezing processes at the ice-ocean interface and the accretion of frazil ice from the underlying ocean. The governing thermomechanical equations in the two layers are formulated as are the boundary and transition conditions that apply at the free surface, the material interface between the meteoric and the marine ice and the ice-ocean interface. The equations comprise in the bulk mass balances for the ice and the salt water (in marine ice), momentum balance and energy balance equations, and at the boundaries kinematic equations as well as jump conditions of mass, momentum and energy. The side boundary conditions involve a prescription of the mass flow along the grounding line from the inland ice and a kinematic law describing the mass loss by calving along the floating ice-shelf front. An appropriate scaling, in which the shallowness of the ice shelves is used, gives rise to the development of a perturbation scheme for the solution of the three-dimensional equations. Its lowest-order approximation – the shallow-shelf approximation (SSA) – shows the ice flow to be predominantly horizontal with a velocity field independent of depth, but strongly depth-dependent temperature and stress distributions. This zeroth order shallow-shelf approximation excludes the treatment of ice rumples, ice rises and the vicinity of the grounding line, but higher-order equations may to within second-order accuracy in the perturbation parameter accommodate for these more complicated effects. The scaling introduced finally leads to a vertical integrated system of non-linear partial integro-differential equations describing the ice flow and evolution equation for temperature and the free surfaces.