Energy efficiency in transient surface runoff and sediment fluxes on hillslopes – a concept to quantify the effectiveness of extreme events
Authors: Schroers Samuel , Scherer Ulrike , Zehe Erwin 
Published: July 2023

Abstract. Surface runoff over time shapes the morphology of the landscape. The resulting forms and patterns have been shown to follow distinct rules, which hold throughout almost all terrestrial catchments. Given the complexity and variety of the Earth's runoff processes, those findings have inspired researchers for over a century, and they resulted in many principles and sometimes proclaimed laws to explain the physics that govern the evolution of landforms and river networks. Most of those point to the first and second law of thermodynamics, which describe conservation and dissipation of free energy through fluxes depleting their driving gradients. Here we start with both laws but expand the related principles to explain the co-evolution of surface runoff and hillslope morphology by using measurable hydraulic and hydrological variables. We argue that a release of the frequent assumption of steady states is key, as the maximum work that surface runoff can perform on the sediments relates not only to the surface structure but also to “refueling” of the system with potential energy by rainfall events. To account for both factors, we introduce the concept of relative dissipation, relating frictional energy dissipation to the energy influx, which essentially characterizes energy efficiency of the hillslope when treated as an open, dissipative power engine. Generally, we find that such a hillslope engine is energetically rather inefficient, although the well-known Carnot limit does not apply here, as surface runoff is not driven by temperature differences. Given the transient and intermittent behavior of rainfall runoff, we explore the transient free energy balance with respect to energy efficiency, comparing typical hillslope forms that represent a sequence of morphological stages and dominant erosion processes. In a first part, we simulate three rainfall runoff scenarios by numerically solving the shallow water equations, and we analyze those in terms of relative dissipation. The results suggest that older hillslope forms, where advective soil wash erosion dominates, are less efficient than younger forms which relate to diffusive erosion regimes. In the second part of this study, we use the concept of relative dissipation to analyze two observed rainfall runoff extremes in the small rural Weiherbach catchment. Both flood events are extreme, with estimated return periods of 10 000 years, and produced considerable erosion. Using a previously calibrated, distributed physics-based model, we analyze the free energy balance of surface runoff
simulated for the 169 model hillslopes and determine the work that was
performed on the eroded sediments. This reveals that relative dissipation
is largest on hillslope forms which relate to diffusive soil creep erosion
and lowest for hillslope profiles relating to advective soil wash erosion.
We also find that power in surface runoff and power in the complementary
infiltration flux are during both events almost identical. Moreover, there
is a clear hierarchy of work, which surface runoff expended on the sediments
and relative dissipation between characteristic hillslope clusters. For
hillslope forms that are more energy efficient in producing surface runoff,
on average, a larger share of the free energy of surface runoff performs work
on the sediments (detachment and transport) and vice versa. We thus conclude
that the energy efficiency of overland flow during events does indeed
constrain erosional work and the degree of freedom for morphological changes. We conjecture that hillslope forms and overland dynamics co-evolve,
triggered by an overshoot in power during intermittent rainfall runoff
events, towards a decreasing energy efficiency in overland flow. This
implies a faster depletion of energy gradients during events and a stepwise
downregulation of the available power to trigger further morphological
development.