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

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 nonlinear relationship. We explore the contribution of rill network geometry observed yield–length scaling Relying an idealized geometry, we formally develop probability functions geometric variables contributing area and In doing so, contribute towards complete probabilistic foundation Hack distribution. Using deterministic empirical functions, then extend hydraulic are related detachment transport. A Monte Carlo simulation samples from different lengths provide estimates yield. results this analysis demonstrate relationship as result geometry. Theory is supported by numerical modeling, wherein surface routed over natural northern Arizona. Numerical routing demonstrates resemble theoretical ones. This work provides unique application Scheidegger settings which, because their finite lengths, in functions. have addressed slopes contributed understanding Hack's law reasoning.