A quasi-analytical ice-sheet model for climate studies

A simple quasi-analytical model is developed to study the response of ice-sheets to climate change. The model is axi-symmetrical and rests on a bed with a constant slope. The mechanics are highly parameterised. The climatic conditions are represented by the altitude of the runoff line. Above the runoff line the accumulation rate is constant (but may depend on, for instance, the ice-sheet size), below the runoff line the balance gradient is constant. The ice-sheet may extend into the sea and can respond to changes in sea level. At the grounding line the ice velocity is assumed to be proportional to the water depth. For this set-up an explicit expression for the total mass budget of the ice-sheet is derived. To illustrate the properties and possibilities of the model, equilibrium states are analysed and the response to periodic forcing is studied as well. The coupling of mass balance and surface elevation of the ice-sheet leads to nonlinear behaviour and branching of the equilibrium solutions. The qualitative behaviour of the system is that of the cusp catastrophe. Nonlinear effects are more pronounced when the slope of the bed is smaller. A case is discussed in which two ice-sheets are coupled by making the altitude of the runoff line dependent on the total area of the two ice-sheets. On two continents, having a slightly different glaciation threshold, periodic forcing of the altitude of the runoff line is imposed. It is shown that in such a situation variations on a long time scale (two to three times the period of the forcing) are introduced. Finally the model is forced by the GISP δ18O record for the last 120 000 years. For an appropriate choice of parameters the model simulates well the waxing and waning of the Laurentide, Fennoscandian and Barentsz ice-sheets.