A Numerical Study of Plane Ice-Sheet Flow

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Abstract
The plane steady flow of a grounded ice sheet is numerically analysed using the approximate model of Morland or Hutter. In this, the ice behaves as a non-linear viscous fluid with a strongly temperature-dependent rate factor, and ice sheets are assumed to be long and shallow. The climate is assumed to be prescribed via the accumulation/ablation distribution and the surface temperature, both of which are functions of position and unknown height. The rigid base exerts external forcings via the normal heat flow, the geothermal heat, and a given basal sliding condition connecting the tangential velocity, tangential traction, and normal traction. The functional relations are those of Morland (1984) or motivated by his work. We use equations in his notation. The governing equations and boundary conditions in dimensionless form are briefly stated and dimensionless variables are related to their physical counterparts. The thermo-mechanical parabolic boundary-value problem, found to depend on physical scales, constitutive properties, and external forcing functions, has been numerically solved. For reasons of stability, the numerical integration must proceed from the ice divide towards the margin, which requires a special analysis of the ice divide. We present this analysis and then describe the versatility and limitations of the constructed computer code. Results of extensive computations are shown. In particular, we prove that the Morland–Hutter model for ice sheets is only applicable when sliding is sufficiently large (satisfying inequality (30)). In the range of the validity of this inequality, it is then demonstrated that of all physical scaling parameters only a single π-product influences the geometry and the flow within the ice sheet. We analyse the role played by advection, diffusion, and dissipation in the temperature distribution, and discuss the significance of the rheological non-linearities. Variations of the external forcings, such as accumulation/ablation conditions, free surface temperature, and geothermal heat, demonstrate the sensitivity of the ice-sheet geometry to accumulation conditions and the robustness of the flow to variations in the thermal state. We end with a summary of results and a critical review of the model.

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