2.1 The hillslope as open thermodynamic system

The theory and applications of this paper are an extension to our first publication (Schroers et al., 2022) regarding steady-state dissipation regimes. Therefore, we present here the final equations only and refer to our study for details. In general, we represent the hillslope surface as an open thermodynamic system (OTS) (Kleidon, 2016; Zehe et al., 2013), which exchanges mass, momentum, energy, and entropy with its environment. The boundaries of the system are a subjective choice, depending on the type and objectives of the analysis and are defined here as the hillslope surface without its subsurface soil structure (compare e.g., Zehe et al., 2013), starting at the topographic divide upslope and ending at the drainage channel downslope. Within these boundaries, we set surface runoff into a thermodynamic perspective and apply the first and second law of thermodynamics, which constitute that energy is conserved and entropy of an isolated system can only grow (Kondepui and Prigogine, 1952). We start with the assumption that a hillslope can be defined as a spatially integrated OTS, here denoted by the subscript “HS” (Fig. 1).

Energy dynamics of this OTS black box are therefore driven by a single representative influx of potential energy $J_{\mathrm{HS},\mathrm{in}}^{\mathrm{pe}}\left(t\right)$ in watts, on hillslopes in the form of rainfall, which leads to spatial gradients of geopotential of water. Over a certain flow path distance L_HS, these gradients are then converted into kinetic energy $E_{\mathrm{HS}}^{\mathrm{ke}}\left(t\right)$ in joules (surface runoff) and heat, which is composed of changes in temperature and entropy. These spatial dynamics in time lead to Eq. (1), with net potential energy flow $J_{\mathrm{HS},\mathrm{net}}^{\mathrm{pe}}\left(t\right)$ (watt) (Eq. 3), either increasing potential energy of the system $E_{\mathrm{HS}}^{\mathrm{pe}}\left(t\right)$ or powering the creation of another type of energy P_HS(t) (watt). Equation (2) details how P_HS(t) either leads to the creation of kinetic energy or dissipation D_HS(t). Additionally, net kinetic energy flow $J_{\mathrm{HS},\mathrm{net}}^{\mathrm{ke}}\left(t\right)$ accounts for the net gain or loss of kinetic energy flow of the system. Equations (1) to (4) are a simplification of surface runoff, as we do not consider other types of energy than potential and kinetic energy of water. For the presented applications here however all other energy types can be considered negligible.

In combination, Eqs. (1) and (2) lead to Eq. (4), which relates input and output of energy of a system with the energy stored within the system.

For a rainfall runoff event, the black box OTS (Fig. 1, Eq. 4) of a hillslope surface can be further simplified. We assume that the system receives on its upper end a constant potential energy inflow $J_{\mathrm{HS},\mathrm{in}}^{\mathrm{pe}}\left(t\right)$ and releases a time-dependent energy outflow at the lower end. We assign the lower end a bed level of zero, which makes the specific geopotential of the lower boundary flux only dependent on the water depth. In this case, we regard the potential energy which enters the system to be much larger than the potential energy which leaves the system and therefore also $J_{\mathrm{HS},\mathrm{out}}^{\mathrm{pe}}\left(t\right)$ (watt) to be negligible. The kinetic energy flow at the inflow boundary is also assumed to be zero and temporal gradients, are abbreviated by dot notation (e.g., $\frac{\mathrm{d}{E}_{\mathrm{HS}}^{\mathrm{pe}}}{\mathrm{d}t}={\dot{E}}_{\mathrm{HS}}^{\mathrm{pe}}$) so that we can write the reduced Eq. (4) as

Each of the terms of Eq. (5) shall be derived from integration of spatially distributed hydraulic flow variables. For a detailed calculation of spatially distributed steady-state dynamics, we refer to Schroers et al. (2022), and for the derived transient system, a summary is presented in Appendix A.

From Eq. (5) we deduce that a transient system has several degrees of freedom to increase or decrease dissipation rates (or free energy respectively), whereas a steady-state system can only adjust the outflux of kinetic energy ($J_{\mathrm{out}}^{\mathrm{ke}}$). For the transient case, the influx potential energy can also be converted into potential and kinetic energy, stored within the system itself. For a constant energy influx, power can, e.g., be maximized through minimization of increases in $E_{\mathrm{HS}}^{\mathrm{pe}}$, meaning less influx energy is converted into potential energy and more into kinetic energy. It is therefore possible that a system maximizes power whilst also minimizing dissipation. It is tempting to think that this simplification holds for discrete time steps, but as natural systems are highly transient, it seems more likely that total dissipation in time or a maximum value during a concrete time interval might be optimized. If a system receives a certain amount of energy influx, it is therefore clear that optimization must happen through adjustment of the internal spatial structure which determines temporal derivatives of free energy conversion rates. Previously (Schroers et al., 2022) we assumed the system to be in steady state and analyzed the local adjustment of free energy conversion rates. Dissipation can be minimized by geomorphological adaptations of the hillslope surface optimizing loss of energy per wetted cross section. For a transient event, we integrate over a spatial domain and have an additional degree of freedom, as energy can be stored in time. We therefore expect that the structure of hillslopes is not a result of a steady state but rather an outcome of many transient events (see Wolman and Gerson, 1978), during which free energy gradients are depleted as fast as possible.

2.2 Relative dissipation of surface runoff

As hillslopes vary spatially in vertical as well as horizontal length scales and surface runoff events vary in time, absolute values do not represent relative dynamics of the energy balance and need to be normalized for comparison. Starting with Eq. (5), we first calculate the accumulated dissipation $D_{\mathrm{HS}}^{\mathrm{acc}}$ (joule; Eq. 6) for an event from t=0to t_l and then normalize by the influx of energy $J_{\mathrm{HS},\mathrm{in}}^{\mathrm{acc}}=\underset{t=\mathrm{0}}{\overset{t_l}{\int }}{J}_{\mathrm{HS},\mathrm{in}}^{\mathrm{pe}}\left(t\right)\mathrm{d}t$ which is accumulated at time t_l (Eq. 7).

${\widehat{D}}_{\mathrm{HS}}$ is dimensionless (J J⁻¹) and represents a thermodynamic descriptor for a spatially defined system which can be analyzed in time for a given rainfall runoff event. In the following, we apply Eq. (7a) for comparison of relative dissipation rates for characteristic hillslope profiles. The energy influx normalization is useful, as it allows a comparison of different transient rainfall runoff events independent of absolute rainfall rates and vertical as well as horizontal hillslope lengths. The second term on the right side of Eq. (7a) can also be termed energy efficiency of overland flow; a larger value leads to less relative dissipation and reversely, a lower value increases ${\widehat{D}}_{\mathrm{HS}}$. Maximum relative dissipation is therefore related to minimum energy efficiency. Additionally we define relative stored energy ${\widehat{E}}_{\mathrm{HS}}=\frac{\underset{t=\mathrm{0}}{\overset{t_l}{\int }}\left({\dot{E}}_{\mathrm{HS}}^{\mathrm{pe}}+{\dot{E}}_{\mathrm{HS}}^{\mathrm{ke}}\right)\mathrm{d}t}{\underset{t=\mathrm{0}}{\overset{t_l}{\int }}{J}_{\mathrm{HS},\mathrm{in}}^{\mathrm{pe}}\mathrm{d}t}$ as well as relative energy flux at the hillslope foot as ${\widehat{J}}_{\mathrm{HS}}=\frac{\underset{t=\mathrm{0}}{\overset{t_l}{\int }}{J}_{\mathrm{HS},\mathrm{out}}^{\mathrm{ke}}\mathrm{d}t}{\underset{t=\mathrm{0}}{\overset{t_l}{\int }}{J}_{\mathrm{HS},\mathrm{in}}^{\mathrm{pe}}\mathrm{d}t}$, leading to a shortened version of Eq. (7a) as follows:

3.1 Erosion process and hillslope form

Quantitative geomorphological modeling is concerned with the development of landforms, given some initial and idealized boundary conditions (e.g., Willgoose et al., 1991; Perron et al., 2009). Typically, the form of a hillslope is modeled by solving partial differential equations of sediment and water mass conservation, coupled by semi-empirical transport laws (Beven, 1996). The parameters of these laws are usually derived from data and reach explicatory value by relating certain parameter combinations to prevalent erosion and transport processes. In its simplest form, sediment transport capacity C is at least dependent on accumulated discharge Q and local gradient S: $C=Q^m\times S^n$. Although the range of (m, n) combinations is broad, we assume the ranges, mentioned by Kirkby (1990; cited in Beven, 1996) to represent the underlying erosion and transport processes (Fig. 2a). With the model provided by Kirkby (1971), the erosion processes of diffusive soil creep, rain splash, soil wash, and advective river transport result in typical 1D hillslope profiles given by Eq. (8) and shown in Fig. 2b. The profiles reflect also the theory from Tarboton et al. (1992) that within a catchment context, hillslope processes can be attributed to convex profiles (more diffusive than advective erosion processes) and channels to concave profiles.

Equation (8) is valid for the transport-limited case and a hillslope with a close to constant width along the flow path, resulting in absolute level along the flow path Z. Here, Z₀ in m is the level at the upslope divide and x_HS in m is the total horizontal hillslope extension

In our previous study we have already shown that convex profiles maximize dissipation of surface runoff per input flux of energy (precipitation) whilst also showing maximum rates of kinetic energy export at the downslope end. This is possible as kinetic energy is on a scale of 1000 times smaller than influx potential energy, and is therefore not significantly affecting the overall energy balance of a hillslope profile. In this context, we extend this steady-state analysis to account for the transient state of surface runoff and analyze maximum power and total work during a full surface runoff event. For simulation of these rainfall runoff events, we implemented a solver of the 1D Saint-Venant equations for viscous flow and analyzed the hydraulic variables on a space–time grid. The events were simulated for soil creep (SC) and soil wash (SW) profiles, as their distributions of geopotential gradient show largest differences.

3.2 Numerical model for transient surface runoff

The simulation of surface runoff on 1D hillslope profiles related to the erosion processes was done by numerical approximation of the system of equations, known as the shallow water equations. In this study, we solve the conservative form of the 1D mass and momentum equations as follows:

where

We applied a finite difference time variation diminishing (TVD) MacCormack scheme, which is presented in Liang et al. (2006). In this study, we adjusted the source term by including the rainfall rate I in m s⁻¹, and we approximated the friction term by the Manning–Strickler equation (Das and Bagheri, 2015) with the Manning coefficient n in m s${}^{-\mathrm{1}/\mathrm{3}}$ instead of the originally proposed Chezy formula. H is the total water column depth in meters, g is the acceleration due to gravity (here 9.81 m s⁻²), and q is the discharge per unit width in m² s⁻¹. β is the correction factor for the non-uniform vertical velocity profile, which has been set to equal 1.0 for a uniform velocity distribution. Due to the influence of the water depth on the friction term, small and zero water depths cause numerical instabilities, and correct wetting–drying algorithms must be applied to ensure stability of the numerical scheme (Liang et al., 2007). We applied similar to Vincent et al. (2001) an algorithm which sets the water depth during each computation time step to a minimum of 10⁻⁵ m and no mass flux (q=0) at these points. The TVD term is included only at the inner computation points, excluding the boundary and the so-called ghost points, which are needed for the calculation of no boundary flux at the hillslope top (solid wall boundary) and the bottom outflow of the accumulated discharge (transmissive wall boundary; see Causon and Mingham, 2010). In the following, we briefly outline the MacCormack scheme (MacCormack, 1969) with the additional TVD term (Liang et al., 2006) for Eq. (10c) as follows:

The superscripts “p” and “c” denote the predictor and corrector steps, while j and i represent the discretization in time and space. It is important to note that the spatial flux term F is discretized backwards in the predictor time step and discretized forward in the corrector time step. The main benefits of this two-stage scheme are that one can solve regions with sharp gradients through the inclusion of the TVD term and that the source term is computationally efficiently treated, whilst maintaining second-order accuracy, in time and space. The complete implementation of the scheme, including transmissive and solid wall boundary conditions, is presented as python script in the supplemental code to this publication.

3.3 Averaging in time and space

Depending on the space and time discretization, we can analyze how much of the energy influx by rainfall was converted into free energy of overland flow and how much has dissipated. It is however not trivial to disentangle energy fluxes in space and time and less so to analytically average over both domains to describe the nature of transient energy conversion rates. On the one hand, averaging over the time domain is typically accompanied by setting time derivatives to zero and allows us to analyze the steady-state spatial distribution of energy (Schroers et al., 2022). On the other hand, averaging over the space domain leads to a black box system where we are unaware of the internal spatial distributions and only express the temporal evolution of the system (e.g., Kleidon et al., 2013).

As the partial differential equations of the underlying movement of water (mass and momentum balances) are numerically approximated on a space–time grid, only an average of the energy fluxes in both domains provides an estimate for an entire hillslope and event. In this section, to introduce the reader to the general dynamics of transient surface runoff, we spatially lump the entire hillslope into one OTS which is transient in time. In Sect. 4 of this study, we extend this concept and double average in space as well as in time. Figure 3 shows the space–time grid, where at the computation points (circles) the hydraulic variables H and q are calculated. An exemplary OTS is discretized in space with length dx and temporal conversion dynamics of energy for a time interval with length dt. Spatial derivatives of Eq. (4) ($J_{\mathrm{f},\mathrm{net}}^{\mathrm{pe}}$, $J_{\mathrm{f},\mathrm{net}}^{\mathrm{ke}}$) are averaged in time (dt), and temporal derivatives $\left(\frac{\mathrm{d}{E}_{\mathrm{pe}}}{\mathrm{d}t},\frac{\mathrm{d}{E}_{\mathrm{ke}}}{\mathrm{d}t}\right)$ are averaged in space (dx), leading to a double averaging of power and dissipation of surface runoff (Fig. 3). For calculation of space and time derivatives between computation points i(j) and $i+\mathrm{1}(j+\mathrm{1})$, we apply forward differencing, which reads $\frac{\mathrm{d}f\left(y\right)}{\mathrm{d}y}=\frac{f^{i+\mathrm{1}}\left(y\right)-f^i\left(y\right)}{\mathrm{d}y}$, where f(y) is the averaged variable in time ($\stackrel{\mathrm{̃}}{q}$, $\stackrel{\mathrm{̃}}{H}$) or space ($\stackrel{\mathrm{‾}}{q}$, $\stackrel{\mathrm{‾}}{H}$).

Figure 3Discretization of energy conversion dynamics in space (x) and time (t). q and H are evaluated on nodes (blue circles), energy conversion in time is integral over space (red), [W] in space is integral over time (yellow) [J m⁻¹], and total energy converted is calculated as space–time integral (green) [J].

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Equation (4) (Eq. A8 respectively; see Appendix A) forms the basis for an analysis of surface runoff in space and time. Depending on the system and the rainfall runoff event, we define spatial and temporal boundaries to calculate the total converted energies. For a defined OTS, this allows for calculation of power and dissipation by integration: either for the whole OTS (Fig. 3, red area) in W, for the whole event (Fig. 3, yellow area) in J m⁻¹, or for a specified duration and distance, averaged in time and space (Fig. 3, green area) in J.

3.4 Scenarios and results

To highlight the different transient behaviors of characteristic hillslopes, we compare the hillslope form which is related to advective soil wash erosion (SW) with the one which relates to diffusive soil creep (SC). We ran three simulation scenarios on each hillslope, differing in block rainfall rates (100 nd 50 mm h⁻¹) as well as length of rainfall time interval (120 and 360 s) (Fig. 4; details regarding roughness parameter, dimensions, and spatial discretization can be found in the example of the supplemental code). Based on the calculated hydraulic results, we then proceeded to calculate the transient energy balance averaged in space over the hillslope length. Finally, the residual of the energy balance is interpreted as the total amount of dissipated energy in time and is analyzed relative to the accumulated influx of energy by rainfall (${\widehat{D}}_{\mathrm{HS}}$, see Eq. 7), which allows a thermodynamic description of a temporally transient rainfall runoff event.

Figure 4Block rainfall scenarios and simulated hydrographs for SC- and SW-related 1D hillslope profiles.

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Figure 5Simulated temporal dynamics of spatially lumped (a) stored potential energy, (b) stored kinetic energy, and (c) kinetic energy outflux in watt per meter flow width for SW- and SC-related 1D hillslope profiles.

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3.4.2 Energy conversion dynamics

In the presented transient framework, an influx of energy may either lead to an increase of stored potential energy ${\dot{E}}_{\mathrm{HS}}^{\mathrm{pe}}$, an increase of kinetic energy ${\dot{E}}_{\mathrm{HS}}^{\mathrm{ke}}$, or an increase of the outflux of kinetic energy $J_{\mathrm{HS},\mathrm{out}}^{\mathrm{ke}}$ (Fig. 1). If these energy fluxes are positive, the energy is not dissipated and instead maintained as free energy of surface runoff. ${\dot{E}}_{\mathrm{HS}}^{\mathrm{pe}}$ and ${\dot{E}}_{\mathrm{HS}}^{\mathrm{ke}}$ contribute to the stored energy on the hillslope during the rising limb of the hydrograph, recede to zero when reaching steady state, and dissipate during the falling limb of the hydrograph (${\dot{E}}_{\mathrm{HS}}^{\mathrm{pe}}<\mathrm{0}$, ${\dot{E}}_{\mathrm{HS}}^{\mathrm{ke}}<\mathrm{0}$).

For all simulated scenarios, the total energy which is stored and released is larger for SC than for SW profile forms (Fig. 5a and b). The shortest interval to reach steady state is achieved for SW hillslopes and largest rainfall rates (S1), and contrarily the longest time interval for reaching steady state is related to SC hillslope forms and smallest rainfall rates (S3). Scenario S2 does not reach steady-state runoff and follows the energy dynamics of S1 during the rising limb of the event (both have equal rainfall rates). As however less energy has been stored on the hillslope for S2 than for S1, less energy is dissipated during the falling limb of S2 than of S1. For potential energy, most energy is created at the beginning of the event, with small runoff depths and little to no flow. Most internal kinetic energy ${\dot{E}}_{\mathrm{HS}}^{\mathrm{ke}}$ is produced when flow depths rise (and therefore ${\dot{E}}_{\mathrm{HS}}^{\mathrm{pe}}$ falls) whilst the output of kinetic energy $J_{\mathrm{HS},\mathrm{out}}^{\mathrm{ke}}$ still has not reached its maximum. $J_{\mathrm{HS},\mathrm{out}}^{\mathrm{ke}}$ is linked to the observed runoff at the downslope end of the hillslope profile and is for all three scenarios larger for SC than for SW hillslope forms (Fig. 5c). This export of energy from the system is linked to the internal work from overland flow on the system; the longer it takes for the hillslope system to reach a steady-state value of $J_{\mathrm{HS},\mathrm{out}}^{\mathrm{ke}}$, the more energy is available to perform work on the surface structures. This reflects our notion that certain hillslope morphologies are more likely to experience an overshoot in power and consequently more work which is generated by surface runoff.

Figure 6Computed transient results of (a) absolute dissipation D_HS and (b) relative dissipation ${\widehat{D}}_{\mathrm{HS}}$ for scenarios S1, S2, and S3 on hillslope profiles related to soil wash (SW) and soil creep (SC).

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3.5 Discussion

In this first part of the study, we highlight the connection between surface runoff, dissipation of its free energy, and the evolution of surface morphology. We argue in line with Wolman and Gerson (1978) and Beven (1981) that such events in nature are highly intermittent and transient in time, leading to the question of how this can be interpreted within an optimality context such as has been proposed by many (see Singh, 2003, for an overview). Therefore, we put forward the concept of relative dissipation of free energy or equivalently energy efficiency of surface runoff, which is similar to Carnot's theorem of maximum work which can be extracted from heat flow (Kondepui and Prigogine, 1952). This idea was applied to surface runoff on characteristic 1D hillslope profiles which are related to diffusive soil creep erosion and advective soil wash erosion. Interestingly, our results show that the latter (SW) results in less energy efficiency of surface runoff, or, differently stated, a larger fraction of the provided free energy by rainfall is dissipated than for SC hillslope types (Fig. 6b). This means that there is relatively more energy available for work on the surface of SC profiles (be it in the form of detachment or transport of sediment particles). This reflects the generally accepted theory of the evolution of hillslope profiles (Kirkby, 1971) and river profiles (Leopold and Langbein, 1962) towards concave distributions of geopotential, e.g., a falling energy slope along the flow path. Although we do not specifically account for energy of sediment particles, we derive a simple starting point for a thermodynamic interpretation of erosion regimes and resulting geopotential distributions. The simulated scenarios also hint at the evolution of runoff response. If relative dissipation rates are analyzed on an event scale, our results show that for the same hillslope, shorter but more intense runoff events maximize relative dissipation and minimize energy efficiency.

We stress that these scenarios are only adequate for situations where infiltration is negligible, as a loss of mass affects the transient energy balance. Furthermore, we did not touch small-scale geomorphological adaptations such as rills. We showed in our previous study (Schroers et al., 2022) for steady-state overland flow that rill processes are linked to the distribution of dissipation rates and therefore affect the energy balance. The development of rills is however transient (Rieke-Zapp and Nearing, 2005) and reflects our notion that structural adaptations are a result of an overshoot of power. As a starting point, it is therefore important to understand during which situations such an overshoot is more likely and when transient structural adaptations will occur. The transient, event-based perspective proposed here highlights that larger rainfall rates and shorter rainfall overland flow events lead to larger relative dissipation rates, which is somewhat counterintuitive, as flow velocities and kinetic energy increase as well. The reason for this effect is that larger flow depths increase flow velocities and therefore facilitate during the transient state a faster depletion of the influx of potential energy through rainfall, while relatively less free energy is stored on the hillslope. In terms of energy efficiency of overland flow, this means that long duration, small intensity rainfall overland flow events are most efficient, in contrast to short, high intensity rainfall overland flow events where a larger fraction of the provided free energy dissipates faster. Following this logic, structural patterns on hillslopes should organize over time to decrease efficiency. This means that if we would apply the same event to a hillslope surface twice, the first event will produce smaller relative dissipation rates than the second. Simultaneously, the kinetic energy of surface runoff would increase for the second event as the provided energy gradients are depleted faster. The latter coincides with the theory about minimization of energy expenditure (Rodriguez-Iturbe et al., 1992) as well as experimental results on the plot scale (Rieke-Zapp and Nearing, 2005).

This can also be explained with the maximum power principle (Lotka, 1922; Kleidon, 2016), which states that the open thermodynamic system organizes its internal structure to deplete the driving gradients at the maximum rate. In the case of runoff on a hillslope, this would imply that given no other constraints, the hillslope erodes towards a configuration which reacts for the same rainfall event faster with larger runoff rates. The maximum power would be achieved once the runoff approaches the shortest possible runoff response and largest runoff rate. Obviously, this is an extreme case which cannot be achieved in nature, as geology, soil composition, and vegetation constrain the runoff response, but this example helps to understand the evolution of the interaction between runoff and erosion. In the second part of this study, we build on these theoretical results but extend the concept to real-world hillslopes and observed runoff responses in the Weiherbach catchment and analyze whether erosion and the evolution of surface runoff is indeed linked to maximum power of surface runoff.

4.1 The Weiherbach catchment and the flash floods of 1994/1995

The hilly Weiherbach catchment lies in the Kraichgau, which is in the southwest of Germany (Fig. 7). The latter has a size of 3.45 km² and has been a hydrological observatory of for more than 3 decades (Plate and Zehe, 2008). The result is a rich data set with multiple continuous time series of discharge, precipitation, and climate parameters, as well as soil humidity. Furthermore, several measurement campaigns yielded a spatially distributed set of soil hydraulic parameters (Zehe et al., 2001), Manning–Strickler values of the principal land uses as a function of plant growth stage (Gerlinger, 1996), and annual cycles of morphological as well as physiological plant parameters, were used. Sediment concentration measurements at the two discharge measurement stations allowed balancing of total sediment loads (Scherer, 2008). Approximately 90 % of the catchment is agricultural land use, of which the principal plant cultivations are wheat, corn, turnip, and sunflower (see Fig. 7b).

Figure 7The Weiherbach catchment: (a) observed drainage network, surface elevation, and derived hillslopes (see Zehe et al., 2001), and (b) land use patterns during the monitoring period (Scherer, 2008).

The two largest runoff events were recorded on 27 June 1994 and on 13 August 1995. In the following, we will focus on these two events only, and we will therefore refer to them as event 1 and event 2 or by year only (see Table 1). Both events were caused by a convective precipitation event with a return period of 200 years according to the KOSTRA data set (Junghänel et al., 2010). While the event of 1995 could be considered a 10 000-year flood, the 1994 flood peak lies well above the related discharge of 3.3 m³ s⁻¹ (BW-Abfluss) (Blatter et al., 2007). A more detailed analysis of the event runoff generation can be found in Zehe et al. (2005), while for the study at hand, we conclude that the recurrence intervals of peak discharge suffice to consider them effective in terms of landscape formation (Beven, 1981), as corroborated by the considerable amounts of eroded sediments.

Table 1Hydrological variables for extreme events of 1994 and 1995, I_cum is accumulated rainfall, I is average rainfall intensity, QP is measured peak discharge, RC is the calculated runoff coefficient, T_I and T_QP are the return periods of rainfall and flood, and M_sed is the total sediment transport which was measured at the gauge in Menzingen.

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4.2 Model description and calibration

The model we used is an extended version of the physically based model CATFLOW (Maurer, 1997; Zehe et al., 2001), which incorporates a sediment erosion module (Scherer, 2008). In brief, the model subdivides a catchment into several hillslopes and a drainage network, where each hillslope is discretized into a 2D vertical grid. The widths of the elements vary from the top to the foot of the hillslope. For each hillslope, the model simulates the soil water dynamics and solute transport based on the Richards equation in the mixed form as well as a transport equation of the convection diffusion type. The equations are numerically solved using an implicit mass conservative Picard iteration (Celia et al., 1990) and a random walk (particle tracking) scheme. The simulation time step is dynamically adjusted to achieve an optimal change of the simulated soil moisture per time step, which assures fast convergence of the Picard iteration. The hillslope module can simulate infiltration excess runoff, saturation excess runoff, lateral water flow in the subsurface, and return flow. However, in the Weiherbach catchment, only infiltration excess runoff contributes to storm runoff, and lateral flow does not play a role at the event scale. What is important is the redistribution of near surface soil moisture in controlling infiltration and surface runoff. As the portion of the tile drained area in the catchment is smaller than 0.5 %, we did not account for tile drains in the simulation. The here presented setup of the Weiherbach catchment is based on simulations and results from Zehe et al. (2005), who subdivided the catchment into 169 hillslopes in relation to land use and soil patterns (Fig. 7a). The total soil depth represented by the model was 2 m, Manning roughness coefficients for the hillslopes and channels were taken from the mentioned experimental database (Gerlinger, 1996), while relative distribution of macroporosity at the hillslope scale was measured by Zehe (1999). The latter scales the total infiltration capacity during rainfall events in relative terms of the soil hydraulic conductivity, after the soil water content increases field capacity. The model was calibrated by stepwise increasing of macroporosity variability (Zehe et al., 2005) for event 1 and 2 (Table 1), yielding Nash–Sutcliffe model efficiencies of 0.97 (event 1) and 0.98 (event 2) at the downstream gauge in Menzingen (Fig. 8a and b). The main storm runoff generation mechanism for both events is infiltration excess runoff, which is routed in the model on the hillslopes into the channel, both based on the advection-diffusion approximation to the 1D Saint-Venant equations. Individual surface runoff responses of each hillslope $Q_{\mathrm{HS}}^i$ and mean of all hillslopes $Q_{\mathrm{HS}}^m$ for both events can be seen in Fig. 8. For reasons of briefness, we refer to Maurer (1997) or Zehe et al. (2001) and Zehe et al. (2005) for more details on model structure and model equations, as well as the parameters of the river network.

Figure 8Observed precipitation and catchment discharge response, simulated surface runoff at hillslope scale Q_HS, as well as simulated river discharge Q_Menz for (a) the event of 27 June 1994 and (b) the event of 13 August 1995.

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Figure 9Calculated free energy dynamics for the surface runoff event 1 on 27 June 1994 of changes in (a) potential energy ${\dot{E}}_{\mathrm{HS}}^{\mathrm{pe}}$, (b) kinetic energy ${\dot{E}}_{\mathrm{HS}}^{\mathrm{ke}}$, and (c) energy out flux $J_{\mathrm{HS},\mathrm{out}}^{\mathrm{ke}}$.

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Sediment erosion and transport is modeled using the steady-state sediment continuity equation (Eq. 12). Sediment transport capacity follows an adjusted concept from Meyer and Wischmeier (1969), treating sediment detachment and transport as individual processes. Potential erosion e_pot (kg m⁻² s⁻¹) is simulated in CATFLOW-SED (Scherer et al., 2012) by a semi-empirical approach that bilinearly accounts for detachment by rainfall momentum flux m_r (N m⁻²) as well as overland flow shear stress τ (N m⁻²) (Eq. 11).

The resisting forces acting against detachment are characterized by two empirical parameters: the erosion resistance f_crit (N m⁻²) as well as the erodibility parameter p₁ (–), scaling the growth of the detachment rate in case the attacking forces exceed the threshold f_crit. The parameter p₂ (–) weighs the momentum flux of rainfall against shear stress from overland flow. The empirical parameters were determined for conventionally tilled loess soils using data from rainfall simulation experiments performed in the laboratory (Schmidt, 1996) and at erosion plots in the field (Scherer et al., 2012). Sediment transport is modeled with the approach from Engelund and Hansen (1967) empirically relating a dimensionless transport intensity to dimensionless stream intensity and consequently allowing for a calculation of transport capacity based on hydraulic overland flow conditions. Sedimentation of suspended particles is accounted for depending on Reynolds number and the particle size, characterizing their buoyancy. At each time step CATFLOW-SED then balances sediment transport for each overland flow element based on the stationary form of the sediment continuity equation (Eq. 12).

Here, q_s is sediment mass flow per unit width in kg m⁻¹ s⁻¹, Φ net detachment/sedimentation of sediments from overland flow in kg m⁻² s⁻¹, x length coordinate in meters, and t time step in seconds. For more details on the implementation and model equations, we refer to Scherer et al. (2012) and Scherer (2008). The sediment transport model was able to simulate total erosion for both flash floods with an absolute error of 8 % (Scherer et al., 2012), which is within the error margin of the observations. As previously mentioned, deposition and erosion patterns for individual hillslopes indicate that especially convex-shaped slopes with highly erodible crop types result in high erosion rates (Fig. 7). In the Weiherbach catchment, these slope types are located in the east.

Figure 10Calculated free energy dynamics for the surface runoff event 2 on 13 August 1995 of changes in (a) potential energy ${\dot{E}}_{\mathrm{HS}}^{\mathrm{pe}}$, (b) kinetic energy ${\dot{E}}_{\mathrm{HS}}^{\mathrm{ke}}$, and (c) energy out flux $J_{\mathrm{HS},\mathrm{out}}^{\mathrm{ke}}$.

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Figure 11Temporal dynamics of dissipation D_HS for individual hillslopes and mean of all hillslopes for (a) event 1 and (b) event 2.

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4.3 Transient energy and power

4.3.1 Surface runoff

We estimated for both events the evolution of potential and kinetic energy on each hillslope as well as the kinetic energy export from the hillslope (Eqs. 6 and 7). ${\dot{E}}_{\mathrm{HS}}^{\mathrm{pe}}$ makes up by far the largest portion of free energy at any point in time, while ${\dot{E}}_{\mathrm{HS}}^{\mathrm{ke}}$ and $J_{\mathrm{HS},\mathrm{out}}^{\mathrm{ke}}$ can be considered negligible for the hillslope energy balance (see Fig. 9). For the event in 1994, ${\dot{E}}_{\mathrm{HS}}^{\mathrm{pe}}$ shows three positive and three negative peaks with very limited periods of time independence at roughly 2.5 to 3.3 h (Fig. 9a). For ${\dot{E}}_{\mathrm{HS}}^{\mathrm{pe}}$ as well as ${\dot{E}}_{\mathrm{HS}}^{\mathrm{ke}}$, positive values represent an increase of free energy that is stored on the hillslope and thus an overshoot in power, while negative values indicate that stored free energy is decreasing (Fig. 9a and b). In contrast to the internal free energies, $J_{\mathrm{HS},\mathrm{out}}^{\mathrm{ke}}$ increases on average to a certain level and maintains this flux until the end of the rain event (Fig. 9c). From an external perspective, the system therefore seems to reach steady state but is internally in a transient unsteady state. At this stage, it is interesting to mention that Zehe et al. (2013) quantified the power in soil water fluxes during these events and evaluated their dependency on macroporosity, which resulted in values of 1–2 W m⁻² per hillslope. This translates with a mean hillslope area of 20 000 m² into approximately 2–4×10⁴ W per hillslope, which is of the same scale as the sum of the free energy fluxes ${\dot{E}}_{\mathrm{HS}}^{\mathrm{pe}}$, ${\dot{E}}_{\mathrm{HS}}^{\mathrm{ke}}$, and $J_{\mathrm{HS},\mathrm{out}}^{\mathrm{ke}}$ presented here. Event 2 in 1995 (Fig. 10) shows similar energy dynamics but with lower magnitude and lesser maximum runoff rates. The maximum peak of ${\dot{E}}_{\mathrm{HS}}^{\mathrm{pe}}$ is not mirrored by a negative counterpart (Fig. 10a), indicating that large amounts of stored surface water infiltrates rather than contributes to further surface runoff. Its effect can also be seen from the dynamics of $J_{\mathrm{HS},\mathrm{out}}^{\mathrm{ke}}$ (Fig. 10c), which has on average three peaks with a dip in power between peak one and two, although energy influx from rainfall is maintained almost constant during this period (see Fig. 8b).

Using the energy influx $J_{\mathrm{HS}}^{\mathrm{in}}$, we calculated D_HS for each hillslope (Eq. 5), event, and as average of all profiles (Fig. 11). D_HS is very dynamic and is for both events unsteady, with a global maximum occurring at the beginning of an event and followed by one or more subsequent smaller local maxima. We also note that the spread of D_HS between individual hillslopes is large, especially at the points in time of maxima.

Figure 12(a) Simulated erosion and (b) approximated expended energy on erosion per hillslope for the 1994 event.

4.4 Energy efficiency of characteristic hillslope forms

The calculations of transient energy and power for both calibrated rainfall runoff events provide an estimate of energy efficiency of overland flow for each hillslope in the Weiherbach catchment. These energy efficiencies are linked to the geomorphological development stage of each hillslope, facilitating an interpretation of geomorphology within the energy balance of surface runoff. To this end, we cluster the hillslopes into groups, representing the typical hillslope profile groups SW, RS, and SC, as introduced in Sect. 3.1 and detailed below.

4.4.1 Clustering hillslope forms

To cluster the 169 hillslope profiles, each one is normalized in its vertical and horizontal length and then plotted as a single point into a 3D space, consisting of the following axes: (1) mean vertical height, (2) percentage length of negative curvature, and (3) horizontal length coordinate of maximum slope. The same procedure is applied to the normalized characteristic hillslope profiles SW, RS, and SC from Sect. 3.1, forming cluster centroids. This allows clustering of model hillslopes according to their minimum Euclidian distance in the parameter space and resulted in 27 hillslopes being classified as SC type, 129 profiles as RS type, and 13 as belonging to SW (Fig. 13a). This confirms the perception that most erosion can be attributed to a combined impact of kinetic energy by rain splash plus shear stress of overland flow accumulation. The classification also showed that 27 hillslopes that can be related to soil creep lie mostly in the eastern part of the Weiherbach catchment (Fig. 13b), where highest erosion rates were simulated. In the next section, we do not only confirm this general erosion pattern, but also show that highest erosion rates coincide with lowest relative dissipation rates and therefore maximum work, which overland flow performed on the sediments.

Figure 13Classification of Weiherbach hillslope profiles into forms related to soil creep (blue), rain splash (white), and soil wash (red): (a) normalized profiles and (b) spatial distribution.

Figure 14Clusters of geomorphological hillslope types (SC, RS, SW) and (a) dissipated energy D_HS as well as (b) relative dissipation ${\widehat{D}}_{\mathrm{HS}}$ for runoff events 1 and 2.

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4.4.2 Relative dissipation patterns and energy efficiency of surface runoff

For both events we plot the hillslope clusters for calculated total dissipated energy as well relative dissipated energy (Fig. 14). In both cases we find distinct differences between SC, RS and SW hillslope types. In absolute terms, more energy is dissipated for both events on SC profiles than RS and SW types, while SW types show lowest dissipated energy levels. Contrarily, relative dissipated energy is highest for SW hillslope types and lowest for SC classified profiles. ${\widehat{D}}_{\mathrm{HS}}$ values ranging from 91 % to 99 % indicate that almost all energy has been dissipated or has been transferred to the sediments at the end of the rainfall event at t_end=5 h.

SC hillslopes receive larger quantities of energy influxes through rainfall but in comparison to SW profiles dissipate a smaller portion of this energy. Both events show similar total dissipated energy levels, which is due to very similar total rainfall volumes. Figure 14b however shows that although total energy influx and dissipation is similar, relative dissipation is larger for the 1995 event than for the 1994 event. This difference arises from the larger surface runoff rates of the latter (due to less infiltration (see Zehe et al., 2005) at its peak up to three times larger, see Fig. 8), leading to more kinetic energy of surface runoff at the outlet.

Similarly, we compare relative free energies ${\widehat{E}}_{\mathrm{HS}}$ and relative outflux energies ${\widehat{J}}_{\mathrm{HS},\mathrm{out}}$ of the three hillslope types (Fig. 15).

Figure 15Clusters of geomorphological hillslope types (SC, RS, SW) and maximum (a) relative stored free energy ${\widehat{E}}_{\mathrm{HS}}$ as well as (b) relative free energy flux ${\widehat{J}}_{\mathrm{HS},\mathrm{out}}$ at the hillslope foot for runoff events 1 and 2.

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Figure 15a shows the maximum values of transient relative free energy that is not dissipated during each surface runoff event for all simulated hillslopes (see Eq. 7b). The results indicate a tendency of SW and RS profiles to lead to less relative free energy in comparison to SC hillslope profiles. Relative free energy ${\widehat{E}}_{\mathrm{HS}}^{\max }$ mirrors ${\widehat{D}}_{\mathrm{HS}}$, highlighting the connection between maximum free energy that is stored in time on the hillslope and total dissipated free energy over the whole event.

Figure 16Simulated surface runoff coefficient RC_HS vs. eroded sediment for each hillslope of (a) event in 1994 and (b) event in 1995 for each hillslope and hillslope cluster (SC, RS, SW).

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Compared with each other, the 1994 event generates larger relative kinetic and potential energy fluxes than the 1995 event, with less total runoff volume. ${\widehat{E}}_{\mathrm{HS}}$ of the 1994 event is therefore much larger than during the 1995 event.

Free energy during a transient event consists of the stored potential and kinetic energy as well as the energy outflux at the hillslope end. An analysis of the latter (Fig. 15b) reveals that there is only a small difference between the three hillslope types and between events. This means that for the analyzed events hillslope geomorphology seems not to be imprinted in kinetic energy export at the hillslope outlet.

These findings imply that during a surface runoff event, the largest differences between hillslope types can be observed in the pattern of free energy components along the flow paths and not locally, e.g., at the hillslope end. These results differ from our previous analysis of steady-state runoff, where SW hillslope types increased the relative kinetic energy outflux in comparison to RS and SC profiles (Schroers et al., 2022). As the latter did not account for infiltration processes, we hypothesize that distributed infiltration in the catchment levels out these differences.

Figure 17Relative dissipated energy ${\widehat{D}}_{\mathrm{HS}}$ vs. total expended energy for sediment erosion e_HS of (a) the event in 1994 and (b) the event in 1995 for each hillslope and hillslope cluster (SC, RS, SW).

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4.5 Discussion

In this second part of the study, we have explored a range of concepts to connect runoff generation process, erosional regimes, and geomorphological evolution of hillslopes in a thermodynamic framework. We put the focus on the analysis of two extreme rainfall runoff events, which were observed in the Weiherbach catchment. This certainly raises the question how representative these events are, given their rare occurrence. We argue however, in line with Wolman and Gerson (1978) and also Beven (1981), that only certain events contribute to effective landscape formation. Those events must be extraordinary, as an overshoot in power is needed to exceed a threshold and trigger significant erosion and structure formation (Zehe and Sivapalan, 2009). Our analysis of the surface runoff during these two extreme events clearly shows that driving downward, dissipative cascade of energy conversions from potential energy to kinetic energy and work on the sediment should be seen within a transient framework, as neither the mass nor the momentum balance during overland flow events is at steady state. We found that the resulting power of surface runoff is of the same order as power of water infiltration into the soil via macropores (Zehe et al., 2013). This might imply that surface and subsurface flow co-evolve into a maximum power state where dissipation and power are equally distributed between complementary domains or more precisely flow paths (Schroers et al., 2022).

We then connected the energy balance and energy efficiency of surface runoff events to the geomorphological forms of the derived hillslope systems. While this rests on the assumption that the delimited hillslopes represent homogeneous dynamics, we are confident that this is the case, as those are defined by topography as well as land use, the main controls of infiltration rate, and surface runoff (Zehe et al., 2001). Most hillslopes were classified as profiles relating to rain splash erosion and only few as soil creep or soil wash profile. We find a clear hierarchy relating relative dissipation and thus energy efficiency to erosion rates. ${\widehat{D}}_{\mathrm{HS}}$ is largest on SW then RS and smallest on SC profiles, indicating that SW profiles are conserving the least percentage of the energy influx by rainfall, while SC profiles are most efficient in generating power in surface runoff. The energy efficiency of overland flow $\mathrm{1}-{\widehat{D}}_{\mathrm{HS}}$ therefore constrains the effectiveness of a rainfall runoff event to change land forms and trigger landscape evolution (Wolman and Miller, 1960). A larger value indicates that more potential energy is conserved as free energy, which implies that overland flow acts with larger average forces and can perform more work on the surface materials (overshoot in power for structure formation). The 1995 event on average resulted in larger ${\widehat{D}}_{\mathrm{HS}}$ values than the 1994 event, which explains the higher erosion rates of the latter (see Table 1: $M_{\mathrm{sed}}^{\mathrm{1994}}=\mathrm{1800}$ t vs. $M_{\mathrm{sed}}^{\mathrm{1995}}=\mathrm{500}$ t). Importantly, as accumulated rainfall amounts are almost equal for both events (see Table 1: $I_{\mathrm{cum}}^{\mathrm{1994}}=\mathrm{78.3}$ mm vs. $I_{\mathrm{cum}}^{\mathrm{1995}}=\mathrm{73.2}$ mm), this difference does not relates to differences in energy influxes by rainfall. This is indicated by almost equal absolute dissipated energy (Fig. 14a) and can also not be deduced from kinetic energy fluxes at the hillslope foot (Fig. 15b). The difference between both events arises from storage rates of free energies within the hillslope systems in the form of potential and kinetic energies. Importantly, energy storage and therefore effectiveness of a surface runoff event relate to transient conditions. Although we found that runoff coefficients and total erosion amounts were not correlated, largest erosion rates were found for hillslopes with a medium RC_HS. This is somewhat surprising, as one would think that highest RC_HS values would also result in largest erosion rates. However, our results give evidence that larger RC_HS values are related to hillslope profiles which are closer to a dynamic equilibrium, store less free energy, and therefore produce less erosion. The maximum work surface runoff can perform on the sediments relates to the potential flux in overland flow and thus on runoff and the specific geopotential gradient (Schroers et al., 2022). As the concept of relative dissipation captures both, we found a strong relation between mean ${\widehat{D}}_{\mathrm{HS}}$ and the average work/free energy expended on sediments e_HS (detachment and transport) for the three analyzed hillslope classes (Fig. 17). Clearly most work on the eroded sediments was performed on SC and RS and only very little on SW hillslopes. In terms of efficiency, we find that SC profiles are on average more efficient in power generation of surface runoff ($\mathrm{1}-{\widehat{D}}_{\mathrm{HS}}$ is larger), which implies that more work can be performed on sediments (e_HS is larger), while SW profiles are less efficient ($\mathrm{1}-{\widehat{D}}_{\mathrm{HS}}$ is smaller and e_HS is smaller).

This finding is in line with a general pattern, characterizing the co-evolution of surface runoff dynamics and erosion and hillslope geomorphology, which holds for various climatological as well as geological settings (Perron et al., 2009). More generally, the evolution of the hillslope system towards less energy efficiency is consistent with the idea of maximization of dissipation and therefore entropy production (Leopold and Langbein, 1962).