Energy efficiency in transient surface runoff and sediment fluxes on hillslopes – a concept to quantify the effectiveness of extreme events

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\u27s 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

SciKit-GStat Uncertainty: A software extension to cope with uncertain geostatistical estimates

This study is focused on an extension of a well established geostatistical software to enable one to effectively and interactively cope with uncertainty in geostatistical applications. The extension includes a rich component library, pre-built interfaces and an online application. We discuss the concept of replacing the empirical variogram with its uncertainty bound. This enables one to acknowledge uncertainties characterizing the underlying geostatistical datasets and typical methodological approaches. This allows for a probabilistic description of the variogram and its parameters at the same time. Our approach enables (1) multiple interpretations of a sample and (2) a multi-model context for geostatistical applications. We focus the sample application on propagating observation uncertainties into manual variogram parametrization and analyze its effects. Using two different datasets, we show how insights on uncertainty can be used to reject variogram models, thus constraining the space of formally equally probable models to tackle the issue of parameter equifinality

Preferential pathways for fluid and solutes in heterogeneous groundwater systems: Self-organization, entropy, work

Patterns of distinct preferential pathways for fluid flow and solute transport are ubiquitous in heterogeneous, saturated and partially saturated porous media. Yet, the underlying reasons for their emergence, and their characterization and quantification, remain enigmatic. Here we analyze simulations of steady-state fluid flow and solute transport in two-dimensional, heterogeneous saturated porous media with a relatively short correlation length. We demonstrate that the downstream concentration of solutes in preferential pathways implies a downstream declining entropy in the transverse distribution of solute transport pathways. This reflects the associated formation and downstream steepening of a concentration gradient transversal to the main flow direction. With an increasing variance of the hydraulic conductivity field, stronger transversal concentration gradients emerge, which is reflected in an even smaller entropy of the transversal distribution of transport pathways. By defining “self-organization” through a reduction in entropy (compared to its maximum), our findings suggest that a higher variance and thus randomness of the hydraulic conductivity coincides with stronger macroscale self-organization of transport pathways. Simulations at lower driving head differences revealed an even stronger self-organization with increasing variance. While these findings appear at first sight striking, they can be explained by recognizing that emergence of spatial self-organization requires, in light of the second law of thermodynamics, that work be performed to establish transversal concentration gradients. The emergence of steeper concentration gradients requires that even more work be performed, with an even higher energy input into an open system. Consistently, we find that the energy input necessary to sustain steady-state fluid flow and tracer transport grows with the variance of the hydraulic conductivity field as well. Solute particles prefer to move through pathways of very high power in the transversal flow component, and these pathways emerge in the vicinity of bottlenecks of low hydraulic conductivity. This is because power depends on the squared spatial head gradient, which is in these simulations largest in regions of low hydraulic conductivity

A Lagrangian model for soil water dynamics during rainfall-driven conditions

Within this study we propose a stochastic approach to simulate soil water dynamics in the unsaturated zone by using a non-linear, space domain random walk of water particles. Soil water is represented by particles of constant mass, which travel according to the Itô form of the Fokker–Planck equation. The model concept builds on established soil physics by estimating the drift velocity and the diffusion term based on the soil water characteristics. A naive random walk, which assumes all water particles to move at the same drift velocity and diffusivity, overestimated depletion of soil moisture gradients compared to a Richards solver. This is because soil water and hence the corresponding water particles in smaller pore size fractions are, due to the non-linear decrease in soil hydraulic conductivity with decreasing soil moisture, much less mobile. After accounting for this subscale variability in particle mobility, the particle model and a Richards solver performed highly similarly during simulated wetting and drying circles in three distinctly different soils. Both models were in very good accordance during rainfall-driven conditions, regardless of the intensity and type of the rainfall forcing and the shape of the initial state. Within subsequent drying cycles the particle model was typically slightly slower in depleting soil moisture gradients than the Richards model. Within a real-world benchmark, the particle model and the Richards solver showed the same deficiencies in matching observed reactions of topsoil moisture to a natural rainfall event. The particle model performance, however, clearly improved after a straightforward implementation of rapid non-equilibrium infiltration, which treats event water as different types of particles, which travel initially in the largest pore fraction at maximum velocity and experience a slow diffusive mixing with the pre-event water particles. The proposed Lagrangian approach is hence a promising, easy-to-implement alternative to the Richards equation for simulating rainfall-driven soil moisture dynamics, which offers straightforward opportunities to account for preferential, non-equilibrium flow

Quantification of Evaporation and Drainage Processes in Unsaturated Porous Media Using Magnetic Resonance Imaging

The water loss in packed beds was studied spatially and time‐resolved via magnetic resonance imaging on the pore scale. The packed beds were measured under water‐saturated and unsaturated conditions, while the magnetic resonance method allowed a quantitative differentiation between air, liquid, and solid phases exploring the native contrast of the named phases without additional contrast agents. Beside a qualitative image comparison, subsequent quantitative image processing allowed for a detailed spatially resolved determination of water distribution, the differentiation between water transport processes, and the quantification of liquid clusters in 3‐D. Results are presented for two packed beds that show significant differences in their evaporation and drainage dynamics, which are mainly determined by the physical properties of the packed beds. The water loss of the packed bed of 2–4mmquartz particles reached a level below interpretability after 18.2 hr; meanwhile, a successive decrease of the largest liquid cluster volume from 82.5 to 0.7 mm$^{3}$ was observed. The water content of the packed bed of 2 mm glass spheres was still observable after 70.9 hr. During the experiment, no significant changes in the structure of the liquid clusters were measured. The current work displays the applicability of magnetic resonance imaging for porescale investigations without the addition of contrast agents

Stepping beyond perfectly mixed conditions in soil hydrological modelling using a Lagrangian approach

A recent experiment of Bowers et al. (2020) revealed that diffusive mixing of water isotopes (δ$^2$H and δ$^{18}$O) over a fully saturated soil sample of a few centimetres in length required several days to equilibrate completely. In this study, we present an approach to simulate such time-delayed diffusive mixing processes, on the pore scale, beyond instantaneously and perfectly mixed conditions. The diffusive pore mixing (DIPMI) approach is based on a Lagrangian perspective on water particles moving by diffusion over the pore space of a soil volume and carrying concentrations of solutes or isotopes. The idea of DIPMI is to account for the self-diffusion of water particles across a characteristic length scale of the pore space using pore-size-dependent diffusion coefficients. The model parameters can be derived from the soil-specific water retention curve, and no further calibration is needed. We test our DIPMI approach by simulating diffusive mixing of water isotopes over the pore space of a saturated soil volume using the experimental data of Bowers et al. (2020). Simulation results show the feasibility of the DIPMI approach for reproducing the measured mixing times and concentrations of isotopes at different tensions over the pore space. This result corroborates the finding that diffusive mixing in soils depends on the pore size distribution and the specific soil water retention properties. Additionally, we perform a virtual experiment with the DIPMI approach by simulating mixing and leaching processes of a solute in a vertical, saturated soil column and compare the results against simulations with the common perfect mixing assumption. The results of this virtual experiment reveal that the frequently observed steep rise and long tailing of breakthrough curves, which are typically associated with non-uniform transport in heterogeneous soils, may also occur in homogeneous media as a result of imperfect subscale mixing in a macroscopically homogeneous soil matrix

The role and value of distributed precipitation data in hydrological models

This study investigates the role and value of distributed rainfall for the runoff generation of a mesoscale catchment (20 km$^{2}$). We compare four hydrological model setups and show that a distributed model setup driven by distributed rainfall only improves the model performances during certain periods. These periods are dominated by convective summer storms that are typically characterized by higher spatiotemporal variabilities compared to stratiform precipitation events that dominate rainfall generation in winter. Motivated by these findings, we develop a spatially adaptive model that is capable of dynamically adjusting its spatial structure during model execution. This spatially adaptive model allows the varying relevance of distributed rainfall to be represented within a hydrological model without losing predictive performance compared to a fully distributed model. Our results highlight that spatially adaptive modeling has the potential to reduce computational times as well as improve our understanding of the varying role and value of distributed precipitation data for hydrological models