Simple Solutions for the Steady-State Longitudinal Profiles of Glacially-Eroded Valleys

Glacial landscapes are among the most spectacular on Earth: many of the steepest and most tectonically active regions, including the Himalaya, Southern Alps, and SE/SC Alaska, are currently undergoing glaciation, and many more regions have been under ice cover in the recent geologic past. The geomorphological processes responsible for carving them have attracted a fair amount of modeling attention in recent years. Models of glacier longitudinal profiles generally use glaciers built by a variety of climate parameters and driven by the physics prescribed by the shallow ice approximation (SIA) or modifications on the SIA including longitudinal smoothing. The erosion rate, in these models, scales with the sliding velocity or the width-averaged discharge, with or without inputs from water pressure variations. In some cases, erosion laws adapted for use on the process scale (meters) have been used over the glacier thickness scale (hundreds of meters) without regard for this inherent scaling. Current studies have grown increasingly complex though enhancements to the glacial flow physics, climate, subglacial hydrology, and erosion laws, with the sensitivity and feedbacks growing similarly complex. Due to many of these complexities, these models are difficult to adapt to a full orogen scale, where glacial erosion is generally treated in a much simpler manner. In comparison, fluvial erosion has been studied from the ground up, starting with simple steady-state models that allow the ability to differentiate the regions where the simplest physical assumptions do not hold and those where they do. These simple models can be used as a source of comparison to both higher-order models and measured profiles. With rivers, these models have led to focused research on stream-power erosion laws for orogen-scale studies, with more process-based research focused on smaller reaches. In order to help provide this type of focus within glaciology, a simple steady-state glacial longitudinal profile model can be developed for different mass balance and uplift conditions. This type of model can be used to better bridge studies of glacier profiles with those of glaciated orogens. It is hoped that this type of model will help glaciologists and geomorphologists develop a better understanding of glacial environments and erosion processes.