Generalized scaling of blue metabolism in river basins

Abstract

River networks play a central role in regulating biogeochemical transformations at the catchment scale, yet a general scaling theory linking network structure to integrated metabolic function is still lacking. By positing the fundamental differences between blue (fluvial) and green (terrestrial) metabolisms in river basins, we focus on the former and, inspired by allometric scaling in biology, we treat river networks as model organisms whose effective body size is given by the volume of water stored within the channelized system. Using finite-size scaling theory for space-filling dendritic networks we derive exact relationships linking universal geomorphic exponents, such as Hack’s law, the Hurst exponent, and the probability distribution of the total contributing area to novel scaling laws for river network volume and for a broader class of capacity measures that represent geometric controls on material processing. The theoretical predictions are validated using both optimal channel networks and digital elevation models of real river basins, which display exponent convergence, collapse of rescaled distributions across nested subbasins, and robustness under coarse graining. By linking river network geometry to integrated metabolic capacity, our framework provides a direct, scale-invariant bridge between geomorphic organization and catchment-scale blue metabolism. The resulting allometric relations suggest Kleiber-like scaling between metabolic interfaces and the network equivalent of body size, offering a unifying basis for predicting riverine biogeochemical function across scales and setting the stage for integrated blue-green metabolic theories.