Morphological controls on surface runoff: an interpretation of steady-state energy patterns, maximum power states and dissipation regimes within a thermodynamic framework

Abstract

Recent research explored an alternative energy-centred perspective on hydrological processes, extending beyond the classical analysis of the catchment's water balance. Particularly, streamflow and the structure of river networks have been analysed in an energy-centred framework, which allows for the incorporation of two additional physical laws: (1) energy is conserved and (2) entropy of an isolated system cannot decrease (first and second law of thermodynamics). This is helpful for understanding the self-organized geometry of river networks and open-catchment systems in general. Here we expand this perspective, by exploring how hillslope topography and the presence of rill networks control the free-energy balance of surface runoff at the hillslope scale. Special emphasis is on the transitions between laminar-, mixed- and turbulent-flow conditions of surface runoff, as they are associated with kinetic energy dissipation as well as with energy transfer to eroded sediments. Starting with a general thermodynamic framework, in a first step we analyse how typical topographic shapes of hillslopes, representing different morphological stages, control the spatial patterns of potential and kinetic energy of surface runoff and energy dissipation along the flow path during steady states. Interestingly, we find that a distinct maximum in potential energy of surface runoff emerges along the flow path, which separates upslope areas of downslope potential energy growth from downslope areas where potential energy declines. A comparison with associated erosion processes indicates that the location of this maximum depends on the relative influence of diffusive and advective flow and erosion processes. In a next step, we use this framework to analyse the energy balance of surface runoff observed during hillslope-scale rainfall simulation experiments, which provide separate measurements of flow velocities for rill and for sheet flow. To this end, we calibrate the physically based hydrological model Catflow, which distributes total surface runoff between a rill and a sheet flow domain, to these experiments and analyse the spatial patterns of potential energy, kinetic energy and dissipation. This reveals again the existence of a maximum of potential energy in surface runoff as well as a connection to the relative contribution of advective and diffusive processes. In the case of a strong rill flow component, the potential energy maximum is located close to the transition zone, where turbulence or at least mixed flow may emerge. Furthermore, the simulations indicate an almost equal partitioning of kinetic energy into the sheet and the rill flow component. When drawing the analogy to an electric circuit, this distribution of power and erosive forces to erode and transport sediment corresponds to a maximum power configuration.