Thresholds, floods and the non-uniqueness of the Stream Power River Incision Model parameters

International audienceThe stream power incision model (SPIM) is a cornerstone of quantitative geomorphology. It states that river incision rate is the product of drainage area and channel slope raised to the power exponents m and n, respectively. It is widely used to predict patterns of deformation from channel long profile inversion or to model knickpoint migration and landscape evolution. Numerous studies have attempted to test its applicability with mixed results prompting the question of its validity. In particular, attempts to invert the values of m and n from transient and steady-state case studies have yielded very different values. Hence, no consensus exists on these values except that the ratio m/n should be close to 0.5 to match the observed concavity in rivers. Here, I first present a synthesis of previous work showing that most relationship between channel slope and incision rate are consistent with values of n greater than 1, while knickpoint propagation is more consistent with n=1. This would suggest a non-uniqueness of the SPIM parameters. Reanalysing published incising river datasets and considering the temporal upscaling problem of long-term incision laws, I show that all rivers away from knickpoints or knickzones are in a regime dominated by threshold effects. This implies that an explicit upscaling of flood stochasticity is required to properly derive the standard SPIM and other incision models, and that n should be greater than 1. Threshold effects are expected to be negligible only in some very steep and narrow rivers corresponding to transient knickpoint features where n would be closer to 1. I explore the consequences of the threshold effects using stochastic simulations with dynamic width. I document the existence of composite transient dynamics where knickpoint propagation locally obeys a linear SPIM (n=1) while other parts of the river obey a non-linear SPIM (n>1). This highlights the non-uniqueness of the SPIM parameters along a river and resolves the apparent inconsistency of the values of n observed for steady-state and transient rivers. However the stochastic-threshold SPIM fails to predict the scaling of slope with incision rate at steady-state for cases where width decreases with incision rate. Recent proposed models of dynamic width cannot resolve these deficiencies highlighting the limits of the SPIM and the fact that we are still lacking a long-term river incision model consistent with the large range of field evidence