Features of glacial valley profiles simply explained

[ 1] Glacial occupation of alpine valleys results in a distinct signature in the long-valley profile, including steepening of the profile in the headwaters, flattening at lower elevations, and a step in the profile at the convergence of headwater tributaries. We present analytic results for glacial erosion patterns by making the following assumptions: ( 1) the initial profile is linear, ( 2) the width of the valley is uniform, ( 3) the annual mass balance varies linearly with elevation, ( 4) the glacier at any time is quasi-steady, ( 5) the erosion rate is proportional to ice discharge per unit valley width, and ( 6) glacial erosion rates far exceed fluvial erosion rates. A steady glacier under these conditions would erode a parabolic divot in the longitudinal valley profile, with its maximum depth coinciding with the down-valley position of the equilibrium line altitude (ELA). The calculated flattening of the valley floor down valley of the ELA and the steepening of it up valley captures the essence of the glacial signature. When a reasonable probability distribution of ELAs is allowed, the predicted erosion peaks at 30 - 40% of the down-valley distance to the glacial limit, and the pattern merges smoothly with the steeper fluvial profile downstream of the glacial limit. Profiles of mass balance that are capped at a maximum value produce shorter glaciers and a slight asymmetry in the expected erosion pattern. Only when we mimic the tributaries of glacial headwaters by specifying a valley width distribution do we obtain the upper step observed in many valley profiles.