Incisional cyclic steps of permanent form in mixed bedrock-alluvial rivers

Most bedrock river channels have a relatively thin, discontinuous cover of alluvium and are thus termed mixed bedrock-alluvial channels. Such channels often show a series of steps formed at relatively regular intervals. This bed form is the bedrock equivalent of cyclic steps formed on beds composed of cohesive soil in gullies. In this paper, we perform a full nonlinear analysis for the case of cyclic steps in mixed bedrock-alluvial channels to explain the formation of these steps. We employ the shallow water equations in conjunction with equations describing the process of bedrock incision. As a model of bedrock incision, we employ the recently introduced Macro-Roughness Saltation Abrasion Alluviation model, which allows direct interaction between alluvial and bedrock morphodynamics. The analysis is greatly simplified by making the quasi-steady assumption that alluvial processes occur much faster than bedrock erosional processes. From our analysis, we obtain the conditions for the formation of cyclic steps in bedrock, as well as the longitudinal profiles of bed elevation, water surface elevation, and areal fraction of alluvial cover. It is found from the analysis that when the sediment supply is small relative to the transport capacity, cyclic steps form only on slopes with very high gradients. The analysis indicates that the shape of a step formed on bedrock is characterized by a relatively short upstream portion with an adverse slope and a long, almost planar downstream portion with a constant slope.