Linking fractal scaling of river networks to self-organisation and optimality

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 and their watershed co-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.

We found that 18 of the largest streams on Earth have self-organized in a highly similar way despite they spread across nearly all continents, climate zones and various geological settings. Specifically, we show that Horton stream numbers of these rivers exhibit a strongly similar fractal scaling with downstream increasing catchment area. This scaling reflects a step-wise transition of the stream network from a high to a low entropy state by means of channel confluence. By combining this insight with energy balance calculations, we found that the potential energy flux in all these rivers was found to grow with catchment size_, implyingthatthese “river engines” generate power at largely similar rates, creating the necessary degrees of freedom for their downstream self-organization. Based on null model for energy dissipation in the stream network in combination with Lacey’s theory, we show that all these streams perform uniform work along their course, while energy dissipation is minimized at every junction. A minimization of specific energy dissipation per unit discharge yields, furthermore, a theoretical scaling exponent of Horton stream numbers which drops into the error margin of the averaged observed scaling exponents.

Overall, our work reveals that the joint analysis of Horton’s stream laws of stream numbers and areas holds the key to a universal relation for self-organization of river networks towards a functional optimum, minimizing energy dissipation and thus maximizing power in the entire network.