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 (P_a), mean (P_m), period, and uplift rate. We find that the ratio of precipitation amplitude to mean (P_a / P_m) is the primary control on variations in instantaneous catchment-integrated erosion rates (E_in), erosion rate averaged over cosmogenic nuclide accumulation time (E_cos), and precipitation-modified channel steepness (k_s-Q). These oscillations lead to deviations of E_cos and k_s-Q from uplift rates and the corresponding channel steepness, and in regions with high P_a / P_m 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 (E_cos-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 k_s-Qeff, which is the channel steepness modified by effective precipitation over climatic cycles, we demonstrate that the relationship between E_cos-Qm and k_s-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, E_cos-Qm and k_s-Qeff. The results demonstrate that catchments with higher P_a / P_m values exhibit larger discrepancies between E_cos from E_cos-Qm, and k_sn-Q from k_sn-Qeff, highlighting the importance of accounting for temporal precipitation variability in interpreting erosion-steepness relationships.