The validity of river steepness as a proxy for erosion rates

River steepness (K_s) is a crucial geomorphic metric used to characterize river slope, normalized by local river drainage area. According to many erosion laws, K_s should be proportional to erosion rate, offering insights into the dynamic processes of landscape evolution. This relationship is complicated by spatial and temporal variations in precipitation rate. To address this issue, modern precipitation-corrected river steepness (K_sp) has been adopted and used as a proxy for erosion rates. However, the utilization of modern precipitation rates may not be entirely suitable to assess erosion rates over the timescales at which river profiles form, spanning thousands or even millions of years, due to temporal changes in precipitation rate, including those driven by glacial-interglacial cycles. To test the viability of river steepness as a proxy for erosion rates under conditions of time-dependent precipitation, we develop a 1-D longitudinal river profile model incorporating periodic precipitation fluctuations and apply this analysis to river profiles of the Three Rivers Region (TRR) in the Southeastern Tibetan Plateau. 

This model calculates K_s and K_sp as well as instantaneous erosion rates(E_in) and erosion rates as measured by time-averaging cosmogenic isotope concentrations (E_cos) on river profiles subjected to Milankovic-cycle precipitation fluctuations. Based on this model, we propose a new metric, mean precipitation-corrected river steepness (K_spm), which is corrected by both local river drainage area and mean precipitation over glacial-interglacial cycles. We find that the precipitation oscillation introduces scale-dependent effects on K_sp, E_in, and E_cos with the variation in E_cos being smaller than in E_in. K_s is largely unaffected by cyclic changes in precipitation but is dependent on mean precipitation levels. In contrast, K_spm remains constant despite fluctuations in precipitation and is not dependent on mean precipitation rates. Hence, K_spm emerges as a preferable indicator to correct precipitation dependence on river steepness. There remains a bias in the measured erosion rates that is dependent on the phase of the imposed precipitation rate.

We examine the three kinds of river steepness to the rivers in the TRR and compare the steepnesses of the trunk rivers and the major tributaries. The south-north trends in tributary basin-averaged river steepness and trunk river steepness are generally similar in pattern, but the tributaries are steeper than the corresponding trunks unless an unusually large concavity parameter is selected for the analysis. The steepest segments of normalized K_sp and normalized K_spm are located further south than those of normalized K_s. This observation suggests an influence of precipitation on river steepness, pointing out the potential bias on measured K_s. For the same trunk segments, the variation of the corresponding tributary basin-averaged normalized K_spm is smaller than that of normalized K_sp, which suggests that some noise in the tributary steepness is a consequence of glacial-interglacial precipitation variation, which can be removed through the use of K_spm. We propose that normalized K_spm is a better metric for long-term erosion rates, but more erosion data are still needed to confirm the use of these metrics as proxies for erosion rates.