Steady state, continuity, and the erosion of layered rocks

Abstract. Considerations of landscape steady state have substantially informed our understanding of the relationships between landscapes, tectonics, climate, and lithology. Topographic steady state, where topography is fixed in time, is a particularly important tool in the interpretation of landscape features, such as bedrock channel profiles, within a context of uplift patterns and rock strength. However, topographic steady state cannot strictly be attained in a landscape with layered rocks with non-vertical contacts. Using a combination of analytical solutions, stream erosion simulations, and full landscape evolution simulations, we show that an assumption of channel continuity, where channel retreat rates in the direction parallel to a contact are equal above and below the contact, provides a more general description of steady state landscapes in layered rocks. Topographic steady state is a special case of the steady state derived from continuity. Contrary to prior work, continuity predicts that channels will be steeper in weaker rocks in the case of subhorizontal rock layers when the stream power erosion exponent n < 1. For subhorizontal layered rocks with different erodibilities, continuity also predicts larger slope contrasts than would be predicted by topographic steady state. Continuity steady state is a type of flux steady state, where uplift is balanced on average by erosion. If uplift rate is constant, continuity steady state is perturbed near base level. These perturbations decay over a length scale that is an inverse function of the contrast in erodibility, such that erodibility contrasts of more than approximately a factor of three lead to rapid decay. Though examples explored here utilze the stream power erosion law, continuity steady state provides a general mathematical tool that can be used to explore the development of landscapes in layered rocks using any erosion model.