Calculations of Velocity and Temperature in a Polar Glacier using the Finite-Element Method

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Abstract
Numerical methods based on quadrilateral finite elements have been developed for calculating distributions of velocity and temperature in polar ice sheets in which horizontal gradients transverse to the flow direction are negligible. The calculation of the velocity field is based on a variational principle equivalent to the differential equations governing incompressible creeping flow. Glen’s flow law relating effective strain-rateε̇ and shear stressτbyε̇ = (τ/B)nis assumed, with the flow law parameterBvarying from element to element depending on temperature and structure. As boundary conditions, stress may be specified on part of the boundary, in practice usually the upper free surface, and velocity on the rest. For calculation of the steady-state temperature distribution we use Galerkin’s method to develop an integral condition from the differential equations. The calculation includes all contributions from vertical and horizontal conduction and advection and from internal heat generation. Imposed boundary conditions are the temperature distribution on the upper surface and the heat flux elsewhereFor certain simple geometries, the flow calculation has been tested against the analytical solution of Nye (1957), and the temperature calculation against analytical solutions of Robin (1955) and Budd (1969), with excellent results.The programs have been used to calculate velocity and temperature distributions in parts of the Barnes Ice Cap where extensive surface and bore-hole surveys provide information on actual values. The predicted velocities are in good agreement with measured velocities if the flow-law parameterBis assumed to decrease down-glacier from the divide to a point about 2 km above the equilibrium line, and then remain constant nearly to the margin. These variations are consistent with observed and inferred changes in fabric from fine ice with randomc-axis orientations to coarser ice with single- or multiple-maximum fabrics. In the wedge of fine-grained deformed superimposed ice at the margin,Bincreases again.Calculated and measured temperature distributions do not agree well if measured velocities and surface temperatures are used in the model. The measured temperature profiles apparently reflect a recent climatic warming which is not incorporated into the finite-element model. These profiles also appear to be adjusted to a vertical velocity distribution which is more consistent with that required for a steady-state profile than the present vertical velocity distribution.