The perfect river - an analysis of three transcendental properties of river networks

Rivers and their drainage networks have been shown to follow distinct patterns of organization in space and time. These patterns manifest on different scales and are related to structural equilibria which have previously been linked to energetic optimality concepts. In this study I have analyzed three different properties which are based on the work of Horton, Leopold and Langbein and Stolum. Here I hypothesize that all three aspects are indeed related to the physics-based concept of minimum energy expenditure. Additionally, they have in common that they represent dimensionless properties of stream networks, representing the water sediment dynamics in phase space. In this study I show how the confluence rate of river networks is related to the Feigenbaum numbers δ and α, how the decline of geopotential along the flow path is related to Euler’s number e, and how the meandering process is related to the number π.

For the largest rivers on earth I found that these transcendental numbers can indeed be identified, although on distinct scales. On average, the Feigenbaum numbers can be found for converging flow in the upper stream network, Eulers number e relates to geopotential decline in the medium reaches, and strong meandering can be found in the lower parts of the stream. This pattern can be found for each of the considered river networks, indicating a general principle, valid across different scales and climates. Although the state strongly oscillates, it is astonishing that the average state of the analyzed river networks can be described within lower single digits percentage error to the mathematical constants.

I interpret this finding as the dynamics of water and sediment to be attracted to states in phase space which can be described by geometric forms that directly relate to the mathematical constants of δ, α, e, and π.