Glacier longitudinal profiles in regions of active uplift

We combine a simplified equation for ice flow with several proposed glacial erosion laws to solve for longitudinal profiles of glaciers in topographic steady state based on prescribed patterns of rock uplift. The solutions produce realistic looking glaciers and are consistent with previous numerical results. Scaling relationships for the dependence of ice thickness and surface slope on rock uplift and climate (in the form of an imposed ice flux) emerge from the solutions. The general patterns of these dependencies are robust. The ice thickness is dependent upon the ice flux and inversely dependent upon the rock uplift, while the surface slope is inversely dependent upon the ice flux and dependent upon the rock uplift. Different erosion laws lead to only subtle differences in these slope and thickness dependencies. Despite the simplicity of the physics, a first-order match to an actual, over-deepened fjord profile can be obtained. The results provide a theoretical basis for understanding the shape of glacier profiles in tectonically active landscapes, can be used to benchmark numerical models of glacial erosion, and can readily be incorporated directly into simple models of orogen dynamics.