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 l
ayer of marine ice that is built by the melting and freezing processes at t
he ice-ocean interface and the accretion of frazil ice from the underlying
ocean. The governing thermomechanical equations in the two layers are formu
lated 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 ju
mp conditions of mass, momentum and energy. The side boundary conditions in
volve a prescription of the mass flow along the grounding line from the inl
and ice and a kinematic law describing the mass loss by calving along the f
loating ice-shelf front. An appropriate scaling, in which the shallowness o
f the ice shelves is used, gives rise to the development of a perturbation
scheme for the solution of the three-dimensional equations, Its lowest-orde
r approximation - the shallow-shelf approximation (SSA) - shows the ice flo
w to be predominantly horizontal with a velocity field independent of depth
, but strongly depth-dependent temperature and stress distributions. This z
eroth order shallow-shelf approximation excludes the treatment of ice rumpl
es, ice rises and the vicinity of the grounding line, but higher-order equa
tions may to within second-order accuracy in the perturbation parameter acc
ommodate for these more complicated effects. The scaling introduced finally
leads to a vertical integrated system of non-linear partial integrodiffere
ntial equations describing the ice flow and evolution equation for temperat
ure and the Free surfaces.