The Fractal Scaling Relationship for River Inlets to Lakes

Scaling relationships provide simple rules for understanding complex hydrographic patterns. Globally, river inlet abundance varies among lakes by about three orders of magnitude, but few scaling relationships describe this aspect of lake-river connectivity. In this study, we describe a simple theoretical scaling relationship between lake surface area and river inlet abundance, and test this theory using data from Scandinavia. On average, the number of inlets increases by 67% for each doubling of lake area. However, lakes of vastly different areas can have the same number of inlets with relatively small variations of drainage density, lake shape, or junction angle - characteristics that can often be linked to specific geological processes. Our approach bridges the gap between the detailed understanding of geomorphic processes and large-scale statistical relationships, and engenders predictions about additional patterns including the relationship between lake area and water residence time. Plain Language Summary Lakes and rivers are often connected, but the patterns and consequences of this connectivity are poorly described, because lakes and rivers are studied separately. In this study, we develop and test simple rules that describe lake and river connectivity. Specifically, we focus on predicting the number of river inlets, which varies among lakes from 0 to almost 1,000. The following are the factors controlling inlet abundance: lake area, lake shape, the average distance between rivers, the average angle that rivers intersect lakes, and topographic complexity. We make several predictions based on these rules, including that glacial lakes should have relatively high connectivity with rivers compared to lakes with other geologic origins. We also use these rules to predict the relationship between lake area and the age of water within lakes. Collectively, our results and predictions demonstrate how simple rules like those developed in our study can enrich the understanding of inland waters. Key Points There is a fractal scaling relationship between river inlet abundance and lake area Inlet abundance increases with lake area and shape factor, drainage density, and junction angle Inlet-area scaling rules correctly predict the relationship between water residence time and lake area

Automated summary

Fractal scaling relationship exists between river inlet abundance and lake area. Inlet abundance increases with lake area, shape factor, drainage density, and junction angle. Inlet-area scaling rules accurately predict the relationship between water residence time and lake area.