Hack distributions of rill networks and nonlinear slope length–soil loss relationships

Authors: DOANE, T.H. - University Of Arizona; PELLETIER, J.D. - University Of Arizona; Nichols, Mary

Submitted to: Earth Surface Dynamics
Publication Type: Peer Reviewed Journal
Publication Acceptance Date: 3/5/2021
Publication Date: 4/16/2021
Citation: Doane, T., Pelletier, J., Nichols, M.H. 2021. Hack distributions of rill networks and nonlinear slope length–soil loss relationships. Earth Surface Dynamics. 9:317-331. https://doi.org/10.5194/esurf-9-317-2021.
DOI: https://doi.org/10.5194/esurf-9-317-2021

Interpretive Summary: Soil erosion on hillslopes is a serious problem, and predicting the amount of soil lost is important for managing eroding lands. The amount of soil that reaches a downslope point, termed sediment yield, is related non-linearly to hillslope length. However, the reasons for the non-linear relationship are not completely understood. We conducted research to understand whether the particular pattern of rills was an important factor in the non-linear relation between sediment yield and hillslope length. We first develop a probabilistic description of contributing area and length as a description of rill networks. Then, we developed probabilistic descriptions of unit stream power, unit shear stress, and sediment concentration. Second, we used a numerical model to route flow down an idealized rill network and a natural rill network to further refine the probabilistic relationships. We used the refined model to simulate erosion and rill development and compared our results with the rill patterns measured on a natural hillslope in Arizona. This research demonstrates that rill patterns are important in explaining the relation between hillslope length and sediment yield.

Technical Abstract: Surface flow on rilled hillslopes tends to produce sediment yields that scale nonlinearly with total hillslope length. The widespread observation lacks a single unifying theory for such a nonlinear relationship. We explore the contribution of rill network geometry to the observed yield-length scaling relationship. Relying on an idealized network geometry, we formally develop probability functions for topological variables of contributing area and rill length. In doing so, we contribute towards a complete probabilistic foundation for the Hack distribution. Using deterministic and empirical functions, we then extend the probability theory to the hydraulic variables that are related to sediment detachment and transport. A Monte Carlo simulation samples hydraulic variables from hillslopes of different lengths to provide estimates of sediment yield. The results of this analysis demonstrate a nonlinear yield-length relationships as a result of the rill network geometry. Theory is supported by numerical modeling wherein surface flow is routed over an idealized numerical surface and a natural one from northern Arizona. Numerical flow routing demonstrates probability functions that resemble the theoretical ones. This work provides a unique application of the Scheidegger network to hillslope settings which, because of their finite lengths, result in unique probability functions. We have addressed sediment yields on rilled slopes and have contributed to an understanding Hack’s law from basic probabilistic reasoning.