On Hack's law and efficiency of landscapes

Complex evolutionary processes imprint distinct geomorphic signatures on the landscapes, offering valuable insights into the evolution of Earth and other planetary surfaces. In fluvial landscapes, river networks are thought to evolve toward configurations that minimize the energy expenditure required to transport water and sediments. Thus, the evolutionary stage of a landscape can be assessed by its energy expenditure; however, this has not yet been systematically studied using real landscape data. In this study, we quantify the relative energy expenditure of the drainage network in reference to the optimal channel network (OCN) algorithm to gauge the landscape efficiency. Additionally, the evolution of river networks towards optimal patterns would influence the basin morphology, potentially offering insights into the stages of evolution. To examine this, we used Hack's law, a well‐studied power‐law relationship between channel length and basin area, as experimental studies have observed that Hack's exponent () changes with evolution. Our analyses of 25 islands show that and are positively correlated, suggesting that both decrease as landscapes evolve and that drainage basins organize into less elongated and more energy‐efficient configurations. To further verify this relationship, we employed a simplified stream power incision model (SPIM) to simulate landscape evolution under fluvial processes and found that both and decrease over time. Thus, our analyses demonstrate the effectiveness of the numerical indicators, Hack's exponent and energy efficiency, in characterizing fluvial landscape evolution.