On the impact of river network dynamics on the hillslope lengths

The distance to channel represents the length of the hillslope pathway followed by water parcels during their journey to the catchment outlet. The distance to channel distribution of a river basin is a powerful tool in hydrology and it is widely used for computing the discharge produced by an input sequence of rainfall. Lots of empirical and computational studies have been carried out on this topic along the years, which made the computation of the hillslope lengths a well established protocol in geomorphology. In particular, it’s numerically possible to compute the distance to channel function with respect to the channel network just starting from a digital elevation map of the basin. However, to date, the distance to channel distribution has been considered as a static object. Instead, actively flowing river networks display in most settings relevant event-based and seasonal temporal dynamics, thereby implying that the spatial configuration of the active streams may change significantly in time. These channel network dynamics have the ability to reshape the distance to channel function. To date, empirical tools represents the only way to tackle this issue and there’s a lack of theoretical understanding of how stream intermittency impact the shape of the hillslope length distibution. This work aims to develop a general theoretical approach to link the total active network length and the underlying catchment-scale distance to channel distribution. A general framework is introduced first, in which the problem is casted in terms of a partial differential equation (PDE) with variable coefficients. Then, some approximations in the choice of the convective coefficient of the PDE are introduced in order to solve the equation. In particular, both data-inspired and theoretically-based convection terms are discussed, and the corresponding results are compared to the actual solution derived from the analysis of the DTM. Results show a general agreement between the proposed analytical and numerical solutions and the family of distance to channel distributions observed in a set of real-world case studies when the active channel length varies. In particular, as the network length increases, the mean hillslope length decreases and the distance-to-channel distribution becomes monotonically decreasing (i.e. higher probabilities are associated to shorter pathways). This study represents a preliminary step towards the characterization of the hydrologic response of rivers with dynamic intermittent headwaters.