Climatic cycles and potential bias in the relationship between channel steepness and cosmogenic erosion rates

Abstract. We investigate the relationship between river channel steepness, catchment-averaged erosion rates from cosmogenic nuclides, and their potential sensitivity to cyclic precipitation rate. Using a one-dimensional longitudinal river-profile model based on the stream power law, we simulate the temporal evolution of channel steepness and erosion rates under a range of cyclic precipitation regimes. Our analysis explores the roles of sinusoidal precipitation characterized by an amplitude (Pa), mean (Pm), period, and uplift rate. We find that the ratio of precipitation amplitude to mean (Pa / Pm) is the primary control on variations in instantaneous catchment-integrated erosion rates (Ein), erosion rate averaged over cosmogenic nuclide accumulation time (Ecos), and precipitation-modified channel steepness (ks-Q). These oscillations lead to deviations of Ecos and ks-Q from uplift rates and the corresponding channel steepness, and in regions with high Pa / Pm values, the regression between erosion rate and channel steepness becomes unreliable, with extreme phase differences potentially reversing their expected relationship. To address these limitations, we introduce a new erosion rate metric, Cosmogenic-constrained cycle-mean erosion rate (Ecos-Qm), which represents the mean erosion rate over a climatic cycle, derived from cosmogenic nuclide concentrations combined with information on climate cyclicity. Using 1D river profile modelling, combined with ks-Qeff, which is the channel steepness modified by effective precipitation over climatic cycles, we demonstrate that the relationship between Ecos-Qm and ks-Qeff remains stable across sinusoidal, insolation-based and proxy-based precipitation cycles, making them more reliable indicators of long-term erosion or uplift rates. We apply this approach to granite catchments in the Three Rivers region as a natural case study to evaluate the performance of paleo-precipitation-integrated erosion rate and channel steepness metrics, Ecos-Qm and ks-Qeff. The results demonstrate that catchments with higher Pa / Pm values exhibit larger discrepancies between Ecos from Ecos-Qm, and ksn-Q from ksn-Qeff, highlighting the importance of accounting for temporal precipitation variability in interpreting erosion-steepness relationships.