The relationship between mean hillslope length and drainage density: from the Horton equation to dynamic river networks

This study investigates the concept of mean hillslope length, defined as the average distance from all points within a river basin to the nearest channel. This measure is essential for understanding key hydro-morphological processes, such as water flow, erosion, and ecosystem dynamics. In the literature, hillslope length and the associated drainage density are often linked through the widely accepted Horton relationship, which suggests that the mean hillslope length is half the reciprocal of the drainage density. Although the Horton equation was derived using an idealized V-shaped valley geometry, it has been successfully applied in many practical settings to capture the relationship between hillslope length and the extent of the channel network. In this study, we propose a novel analytical framework that is used to derive a closed-form expression for mean hillslope length based on the total length of the channel network. This approach is applicable to both individual watersheds with dynamic stream networks and across diverse catchments, and it includes the Horton relationship as an asymptotic case (i.e., when river networks are sufficiently long). The method was tested on data from 15 river catchments across Europe and the USA, showing strong performance in all cases. Our analysis demonstrates that, while the Horton model is extremely accurate for high drainage densities, it cannot be applied in cases where the total channel network length is smaller than the square root of the catchment area, as it significantly overestimates the mean hillslope length in such circumstances. Our new approach offers a more accurate and reliable alternative to the traditional Horton formula while remaining easy to use. The proposed method is particularly valuable in cases involving short or dry river networks, such as dynamic headwater catchments or non-perennial rivers. This research provides a novel method for estimating mean hillslope lengths, enhancing the understanding of key hydrological processes in river basins for both scientists and engineers.