A Plasticity Theory Approach to the Steady-State Shape of a Three-Dimensional Ice Sheet

Abstract

The differential equation determining the elevations of a perfectly plastic three-dimensional steady-state ice sheet is set up. Analytical solutions of the equation are obtained in two simple case, (1) an ice sheet on a horizontal base with an arbitrary edge curve, and (2) an ice sheet on a plane sloping base with a rectilinear ice margin. The solutions are discussed, particularly with reference to the development of ice divides and ice streams. For arbitrary base and ice-margin geometries, solutions are obtained by means of the method of characteristics, which reduces the problem to solving simultaneously three ordinary first-order differential equations. The integration, which is performed by numerical methods, is generally commenced at the ice margin, where the necessary boundary conditions are known. The method has been applied to model the elevation contours and the flow pattern of the central Greenland ice sheet, using the bottom topography revealed by radio echo soundings and the present ice margin geometry. The result is in surprisingly good agreement with our knowledge of the ice-sheet topography and flow pattern, all significant ice divides and ice streams being reproduced. This suggests, that the method can be applied to model the shape and flow pattern of ice sheets under glacial conditions, using information about former ice-margin positions.