Linking energy efficiency of surface runoff, logistic growth and landform changes

This study proposes a free energy centred approach to surface runoff and morphological development of hillslopes and rivers. The starting point is the strong analogy between the functioning of engines on the one hand and watersheds on the other hand. Like an engine converts energy input into motion/kinetic energy, watersheds and hillslopes convert potential energy inputs by rainfall into potential energy and kinetic energy of surface runoff. The latter determines the maximum work water can perform on the sediments. Similar to the energy efficiency of an engine, relating the free energy/work output to the energy input, we define energy efficiencies for each process in the aforementioned cascade of rainfall-runoff formation and sediment transport. The efficiency in generating potential energy of surface runoff depends on precipitation, the runoff coefficient, topography and landforms. While the vast amount of the runoffs potential energy is dissipated, a minute amount sustains the kinetic energy of surface runoff and stream flow, driving erosive changes in watersheds. Here the efficiency depends on the controls of driving and frictional forces: the geo-potential gradient, material roughness and hydraulic radius of the river. We applied this framework to surface runoff at the hillslope scale and the Amazon basin. 

At the hillslope scale we found that typical morphological stages of hillslope forms and related transitions of dominant erosion processes (from soil creep, rain splash, to soil wash) evolve towards a declining energy efficiency in surface runoff. This implies a reduction in power to trigger future landform changes. However, rill and river networks do essentially the opposite. By reducing the specific dissipation, they increase the efficiency in the conversion of potential into kinetic energy of overland and streamflow. In several cases, rill networks were found to even maximize total power of surface runoff in the sheet and rill domains. 

For the Amazon and its tributaries, we found distinct self-similar patterns of stream flow potential energy along each river course. Starting from the source, potential energy in stream flow was growing with downstream distance, up to a maximum value, and exhibited from there an almost linear decline to the river mouth. This implies that the maximum work the river can perform is growing from its source to this maximum, as the rapid growth in the stream flow mass over-weights the steep downstream decline in geopotential. We found the same behavior at the hillslope scale. 

In a third step, we found that for the largest terrestrial river networks on the world that Horton’s laws of stream area and length were close the Feigenbaum constants characterising bifurcations of logistic growth at related deterministic chaos. This suggest parallels between the interplay of growth and mortality of populations, and the interplay of stream power generation and its turbulent dissipation.