For upland ephemeral gullies, gully erosion is strongly related to the formation and migration of cyclic steps. It is necessary to provide insight into the process of cyclic step development to accurately predict the pace of landscape evolution and soil loss. Information on the geometry of cyclic steps in subaerial environments is limited, and, to our knowledge, no model of cyclic step development considers plunge pool erosion. In this study, we analyze the geometric features and controlling factors of erosional cyclic steps through meta-analysis of measured data including new measurements in the Loess Plateau, China. We focus on cyclic step dynamics of fluvial beds controlled by bed shear stress and local plunge pool erosion. We develop a new theory to incorporate plunge pool erosion through adapting existing cyclic step and plunge pool models. Our method agrees with measured data, showing that a larger flow rate leads to larger step length L_d and height H_d and increasing erodibility increases step aspect ratio L_d/H_d. The method is also able to predict how the step length, height, and aspect ratio change with the average channel slope. Our results indicate that plunge pool erosion is an important mechanism of cyclic step evolution. However, plunge pool development alone is not sufficient to explain the wide range of L_d/H_d in the measured data. The posed theory relates to equilibrium conditions and thus cannot consider temporal adjustments in step geometry.

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Cyclic steps are long-wave bedforms that migrate upstream and are bounded by sustained internal hydraulic jumps. Each step has a gentle Froude-subcritical slope in the upstream and a steep slope related to supercritical flow in the downstream (Sun and Fagherazzi, 2003). A hydraulic jump connects two contiguous steps. Cyclic steps can be divided into three categories: purely-erosional cyclic steps, transportational cyclic steps and purely-depositional cyclic steps (Fildani et al., 2006). In purely erosional steps, that usually occur in bedrock or cohesive beds, sediment eroded from the bed is not redeposited again. Transportational steps are formed from a combination of erosion and deposition. In purely-depositional case, that can exist in deep-sea settings, deposition dominates and no sediment entrains from the bed. @en