13 Aug 2020

13 Aug 2020

Review status: a revised version of this preprint is currently under review for the journal ESurf.

How Hack distributions of rill networks contribute to nonlinear slope length–soil loss relationships

Tyler H. Doane1, Jon D. Pelletier1, and Mary H. Nichols2 Tyler H. Doane et al.
  • 1Department of Geoscience University of Arizona, 1040 E. 4th St. Tucson, AZ, 85720
  • 2USDA Agriculture Research Service, 2000 E. Allen Rd., Tucson, AZ, 85719

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.

Tyler H. Doane et al.

Status: final response (author comments only)
Status: final response (author comments only)
AC: Author comment | RC: Referee comment | SC: Short comment | EC: Editor comment

Tyler H. Doane et al.

Data sets

Rill Topography - Northern Arizona Jon Pelletier and Tyler Doane

Model code and software

Steady state surface flow Jon Pelletier and Tyler doane

Tyler H. Doane et al.


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Short summary
This paper explores how the geometry of rill networks contributes to observed nonlinear relationships between soil loss and hillslope length. This work develops probability functions of geometrical quantities of the networks and then extends the theory to hydraulic variables by relying on well-known relationships. Theory is complemented by numerical modeling on numerical and natural surfaces. Results suggest that the particular arrangement of rill networks contributes to nonlinear relationships.