Spatiotemporal averaging of in-stream solute removal dynamics

The scale dependence of nutrient loads exported from a catchment is a function of complex interactions between hydrologic and biogeochemical processes that modulate the input signals from the hillslope by aggregation and attenuation in a converging river network. Observational data support an empirical inverse relation between the biogeochemical cycling rate constant for nitrate k (T-1) and the stream stage h (L), k = v(f)/h, with v(f), the uptake velocity (LT-1), being constant in space under steady flow conditions. Here we offer a physical explanation for the persistence of this pattern across scales and then extend the analysis to spatiotemporal scaling of k under transient-flow conditions. Inverse k-h dependence arose as an emergent pattern by coupling the mechanistic Transient Storage Model with a network model. Analytical modeling indicated that (1) nitrate processing efficiency increases with increasing variability in the discharge Q and (2) temporal averaging had no effect on the exponent a of the k-h relationship (k = v(f)/h(a)) in catchments with low Q variability, but strong dependence arose in catchments with high variability in Q. Network modeling in domains with low Q variability confirmed that the exponent a was independent of temporal averaging, but v(f) was a function of the averaging timescale. The probability distribution functions for k could be adequately predicted using analytical approaches. Understanding the k-h scaling relationships enables the direct estimation of the variability in nutrient losses due to in-stream reactions without requiring explicit information for spatially distributed network modeling.