Theory of shallow ice shelvesM. WEIS, R. GREVE and K. HUTTER AbstractIce shelves consist of two layers, an upper layer of meteoric ice nourished by precipitation and flow from the connected inland ice, and a lower layer of marine ice that is built by the melting and freezing processes at the iceocean 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 iceocean 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 iceshelf 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 threedimensional equations. Its lowestorder approximation  the shallowshelf approximation (SSA)  shows the ice flow to be predominantly horizontal with a velocity field independent of depth, but strongly depthdependent temperature and stress distributions. This zeroth order shallowshelf approximation excludes the treatment of ice rumples, ice rises and the vicinity of the grounding line, but higherorder equations may to within secondorder accuracy in the perturbation parameter accommodate for these more complicated effects. The scaling introduced finally leads to a vertically integrated system of nonlinear partial integrodifferential equations describing the ice flow and evolution equation for temperature and the free surfaces. Continuum Mechanics and Thermodynamics, 11 (1), 1550 (1999). 
