Tracing stream flow in confluent rivers – a journey from chaos to order

The beauty of river networks has continuously inspired science to elucidate their self-similarity and the underlying organizing principles. In his pioneering work, Robert Horton postulated several laws explaining the scaling of stream networks, which are today widely accepted in fluvial geomorphology. Another avenue to explain the nature of river networks acknowledges that landforms in general and rivers in particular have been shaped by the physical work of surface runoff in the past. Several studies proposed thus that river networks evolve towards energetically optimal steady states, minimizing total dissipation or energy expenditure in the entire network. Here we reconcile both research avenues, by linking Horton’s stream laws with the theories of river hydraulics and of non-linear, dissipative dynamic systems.

By analyzing the confluence rates of 18 of the largest rivers in the world, we found a universal relation between Horton’s laws of stream numbers and the logistic growth model introduced by Bob May. The confluence ratios converge for Strahler orders smaller than 5 regardless of the climate and physiographic setting to the first Feigenbaum constant, characterizing the route of the logistic growth model into deterministic Chaos. Using the concept of entropy we show furthermore that the transition of the classical logistic growth model from determinism to Chaos corresponds to a step-wise transition from a minimum to a maximum entropy state. A Lagrangian perspective tracing the pathways of surface runoff from the watershed downslope into the first order streams and further downstream, reveals that the entropy of the flow path distribution exhibits continuous downstream decline as well. Consistently with the requirement that a downstream decline in entropy requires a downstream increase in free energy, we found that the potential energy flux in rivers does indeed generally increase with downstream distance up to a Strahler order of 4-5. This is because the downstream accumulation of flowing water mass outweighs the decline in topographic elevation. 

We finally show that the growth and mortality in the logistic population model are the equivalents to power generation and energy dissipation in the stream. Assuming bank full discharge and using Lacey’s equations we found that the free energy per stream obeys at confluence points a logistic equation as well. We  conclude that Horton’s law of stream numbers is a manifestation of the gradual downstream transition of the flow path density from Chaos, seen as state of minimum predictability and thus maximum entropy, to perfect Order, which is mediated by a maximization of energy efficiency at every confluence point.