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.

Surface runoff in rivers and from hillslopes is of key importance to biological, chemical and geomorphological processes. Landscapes, habitats and their functionalities are coupled to the short- and long-term evolution of rainfall–runoff systems. As we live in a changing environment, it has been of major interest to explain the development of runoff systems and how ecological (Zehe et al., 2010; Bejan and Lorente, 2010), chemical (Zhang and Savenije, 2018; Zehe et al., 2013) and geomorphological (Leopold and Langbein, 1962; Kirkby, 1971; Yang, 1971; Kleidon et al., 2013) processes are organized in time and space. Here we focus on the energy balance of surface runoff, particularly at the hillslope scale, using a thermodynamic framework. Typically, the momentum balance of surface runoff and streamflow is strongly dominated by friction, which is usually characterized by the flow laws of Darcy–Weißbach, Manning or Chezy (Nearing et al., 2017). Consequently, hydraulic estimates of flow velocities rely on the semi-empirical parameters of these laws, which in essence express the ability of a system to dissipate free energy via friction into heat and thus to produce entropy (Zehe and Sivapalan, 2009). A thermodynamic perspective appears hence as the natural choice for deeper understanding of how the mass, momentum and energy balances of surface runoff are controlled by and interact with the landscape and how short- and long-term feedbacks determine the co-development of form and functioning of hydrological systems (Paik and Kumar, 2010; Singh, 2003).

Leopold and Langbein (1962) were among the first to introduce thermodynamic principles in landscape evolution. Representing a one-dimensional river profile as a sequence of heat engines with prony brakes (see Fig. 1), they showed that the most likely distribution of potential energy per unit flow along a river's course to the sea follows an exponential function. Their main hypothesis was that streamflow performs the least work or, equivalently, that the production of entropy per flow volume is constant. Yang (1976) extended this principle and termed it minimum stream power and detailed how flow velocity, slope, depth and channel roughness of a stream should adjust to minimize stream power. In his work about optimal stream junction angles, Howard (1990) also assumed that stream power is minimized, while Rodriguez-Iturbe et al. (1992) proposed that optimal channel networks (OCNs) minimize overall energy dissipation. The authors postulated three principles: (1) the principle of minimum energy expenditure in any link of the network, (2) the principle of equal energy expenditure per unit area and (3) the principle of minimum total energy expenditure in the entire network. Subsequent work of these authors (Rodriguez-Iturbe et al., 1994; Ijjasz Vasquez et al., 1993) revealed that application of these principles yielded three-dimensional drainage networks in accordance with Horton's laws of stream number and stream lengths (Smart, 1972).

Hillslope open thermodynamic system with spatial division into a sub-OTS as a two-box open thermodynamic system. Each control volume (sub-OTS) is represented by a prony brake (compare Leopold and Langbein, 1962).

In climate research, Paltrigde (1979) proposed the principle of maximum entropy production. He showed that a simple two-box model allowed for a successful reproduction of the steady-state temperature distribution on Earth, which maximizes entropy production, expressed as the product of the heat flow and the driving temperature difference. Kleidon et al. (2013) argued that maximum entropy production in steady state is equivalent to a maximization of power, which means that the flow extracts free energy at a maximum rate from the driving potential energy gradient. The authors applied the maximum power principle to river systems and proposed that they develop to a state of maximum power in sediment flows: while the driving geopotential gradient is depleted at the maximum rate, the associated sediment export maximizes at the same rate. Furthermore, the authors relate maximum power in the river network to minimum energy expenditure, as minimum dissipation implies that a maximum of potential energy can be converted into kinetic energy of the water and sediment flux.

Though surface runoff on hillslopes is governed by the same physics as streamflow, there are also important differences. Overland flow is an intermittent threshold response to rainfall events (Zehe and Sivapalan, 2009), caused either by infiltration excess (Horton, 1945; Beven, 2004) or saturation excess (Dunne and Black, 1970). Surface runoff flows along a partially saturated soil and may hence either accumulate downslope or re-infiltrate. Downslope re-infiltration implies an export of water mass and thus potential energy into the soil (Zehe et al., 2013), and the related decline in flow depth reduces shear stress, which affects the momentum balance. Overland flow is typically very shallow compared to the roughness elements, which makes the use of the above-mentioned flow laws even more challenging (Phelps, 1975), and it manifests either as diffusive sheet flow or advective flow in rill networks. Due to the transient nature of overland and sediment flows, rill networks are generally transient, but they develop in a self-reinforcing manner (Gómez et al., 2003; Rieke-Zapp and Nearing, 2005; Berger et al., 2010). Micro rills emerge at some critical downstream distance on the hillslope (compare the “belt of no erosion” in Horton, 1945) and continue in parallel for some length before they merge into larger rills (Schumm et al., 1984). Sometimes these rills split apart before converging into larger gullies (Achten et al., 2008; Faulkner, 2008) and finally connecting to a river channel. This transitional emergence of a structured drainage network was firstly stated in Playfair's law (cited in Horton, 1945) and has since then been observed in a variety of studies (Emmett, 1970; Abrahams et al., 1994; Evans and Taylor, 1995). Motivated by the similarity to river networks and surface rill networks, several experimental studies explored whether rill networks grow towards and develop as least-energy structures in accordance with the theory of optimal channel networks (Gómez et al., 2003; Rieke-Zapp and Nearing, 2005; Berger et al., 2010). The studies of Rieke-Zapp and Nearing (2005) and Gómez et al. (2003) revealed that the emergence of rill networks and their development implies indeed a reduction of energy expenditure, which has previously been shown for stream channel networks (Ijjasz Vaquez et al., 1993). In line with these findings, Berkowitz and Zehe (2020) proposed that rill flow reduces the volume specific dissipative energy loss due to a larger hydraulic radius compared to sheet flow, which is equal to smaller rills merging into a larger one, as noted by Parsons et al. (1990).

The possible optimization of river or rill network geometries through the interplay of surface runoff, erosion and deposition of soils/sediments is the first point that motivates an analysis from a thermodynamic perspective. The second point relates to the transition from laminar- to turbulent-flow conditions, which was already corroborated by Emmet (1970) in a set of comprehensive field and laboratory experiments to investigate hydraulics of overland flow. As laminar flow converts more potential energy into kinetic energy per unit volume than turbulent flow, it is of interest whether and how this transition relates to the emergence of rills and their optimization. Parsons et al. (1990) measured the hydraulic properties of overland flow on a semiarid hillslope in Arizona and attributed the observed downslope decrease in the frictional flow resistance to the accumulation of surface flow in fewer but larger rills. This is similar to a transition of inter-rill flow, from here onwards referred to as sheet flow (Dunne and Dietrich, 1980), to rill flow. More recently a concept emerged that upholds a theory of a slope–velocity equilibrium (Govers et al., 2000; Nearing et al., 2005), claiming that physical and therefore hydraulic roughness adapts such that flow velocity is a unique function of the overland flow rate independent of slope.

In the light of this concise selection of studies, we propose that an energy-centred perspective on overland flow on hillslopes might be helpful to better understand the co-evolution of hillslope form and functioning and whether those (and other) hydrological systems evolve towards a meta-stable, energetically optimal configuration (Zehe et al., 2013; Kleidon et al., 2014; Bejan and Lorente, 2010). Following the work of Kleidon (2016), we develop the general thermodynamic framework and explain how surface runoff along rivers and hillslopes fits into this setting (Sect. 2). We argue that despite the similarity of hillslope surface runoff and river runoff, morphological adaptations and the related degree of freedom of both systems manifest at distinctly different scales. Mature river elements are mainly fed by the upstream discharge and local base flow, while hillslope elements receive substantial water masses during runoff events through local rainfall and upslope runon. This causes an interesting trade-off along the overland flow path, where mass grows downslope due to flow accumulation, while geopotential height declines. We hypothesize that these antagonistic effects lead to a peak in potential energy of overland flow at a distinct point on the hillslope. This implies an upslope area, where the potential energy of overland flow is growing due to flow accumulation (though water is flowing downslope) before it starts declining in downslope direction. From a thermodynamic perspective, the ability of surface runoff to perform work increases up to the point of maximum potential energy and is then depleted through a cascade of energy conversion processes. Our second hypothesis is, thus, that this build-up of potential energy occurs under laminar flow conditions with a low degree of freedom for morphological changes, while the location of potential energy maximum coincides with the emergence of turbulent flow and with a maximum degree of freedom for morphological changes, including the emergence of rills.

The first application of our framework tests hypothesis 1, by exploring how typical shapes of hillslope topography in combination with different width functions control the spatial patterns of potential and kinetic energy of surface runoff and energy dissipation along the flow path during steady states (Sect. 3). As these shapes represent different morphological hillslope stages (Kirkby, 1971), shaped by erosive forces of previous surface runoff events (Rieke-Zapp and Nearing, 2005), we expect differences in the energy balance, including the location of the potential energy maximum. The second application of our framework tests hypothesis 2 (Sect. 4), by analysing the energy balance of surface runoff observed during hillslope scale rainfall simulation experiments in the Weiherbach catchment (Scherer et al., 2012). The experiments provide measurements of eroded sediments and total runoff including sheet and rill flow velocities at the lower end of the irrigated stripes and therefore present an opportunity to explore how rills and rill networks affect the energy balance of surface runoff. For that purpose, we calibrated an extended version of the Catflow model (Zehe et al., 2001), which accounts for the transition from sheet to rill flow, to these experiments and analysed the spatial patterns of potential energy, kinetic energy and dissipation with respect to the transition from laminar to turbulent flow based on simulated flow depths and velocities.

To frame surface runoff processes into a thermodynamic perspective, we define
the surface of a hillslope as an open thermodynamic system (OTS; Kleidon,
2016). In this sense, the hillslope exchanges mass, momentum, energy and
entropy with its environment (Fig. 1). Rainfall adds mass at a certain height and thus free energy in the form of potential energy along the upper system boundary. Mass and free energy leave the system at the lower boundary due to surface runoff or via infiltration as subsurface flow (Zehe et al., 2013). To express energy conservation of surface runoff, we start very generally with the first law of thermodynamics in the following form:

We apply Eq. (1) to balance both potential and kinetic energy of surface
runoff separately and subdivide the hillslope into lateral segments along
the horizontal flow path

Here, we focus on conversion of potential energy into kinetic energy because the former controls the hierarchy of possible energy conversion in surface runoff. We neglect the subsequent kinetic energy transfer to sediments and turbulent velocity fluctuations and refer to

For laminar flow, the downslope accumulation of runoff leads to a steeper
vertical velocity gradient, which might surpass a critical threshold Reynolds number to create turbulent-flow structures (expressed as the relation of inertia to viscous forces). These convert kinetic energy of the mean flow into kinetic energy of small-scale velocity fluctuations and thereby reduce the kinetic energy and thus velocity of the mean flow. Turbulence in turn provides the power and force to detach and lift sediment particles, which also need to be accelerated (in the simplest case) to the mean flow velocity. Both erosion processes feed again on the kinetic energy of the mean flow, while particle detachment also feeds on the kinetic energy of raindrops. In the light of these thoughts, one can expect

Govers et al. (2000) summarize the methods which are still in use today for
estimating how frictional dissipation controls steady-state runoff velocities as a function of roughness, essentially representing the degree of free-energy loss from the mean flow. Most approaches focus on the generalization of a friction coefficient in time and/or space for a given surface area where runoff occurs, which is expressed by a general friction law that relates unit width discharge

Coefficients of general friction law.

Although it is known that friction coefficients on hillslopes vary with the
degree of roughness element inundation (Lawrence, 1997) and sediment
transport concentrations and are transient (Abrahams et al., 1994), mean flow velocities are in practice estimated using constant values. Without additional information about the flow regime and transport process, these
coefficients provide, as explained above, an uncertain estimate of frictional energy dissipation of free energy into heat and related entropy production (Govers et al., 2000). Furthermore, experiments by Govers (1992) for rill flow as well as by Nearing et al. (2017) for sheet flow indicate that friction coefficients vary across the hillslope during steady state. They even seem to be spatially organized, as these studies found that mean runoff velocity can be solely estimated by the runoff rate, independent of topographic slope or rainfall intensities. For the analysis presented in
Sect. 3, we use one of these empirical formulae which was developed by
Nearing et al. (2017) for surface runoff on stony hillslopes:

For the analysis of the rainfall simulation experiments in Sect. 4, the derivation of a similar empirical formula is beyond the data this study has at hand. This implies that absolute values of frictional dissipation rates presented Sect. 4 are uncertain. But they are nevertheless a useful starting point, as our focus lies on their spatial patterns, and the relative differences depend on macroscale properties (measured velocities and runoff rates of rill and sheet flow in this case), which are captured well by these experiments. So even without explicit inclusion of the energy transfers between mean flow and turbulent structures or sediment particles, the analysis of the spatial distribution of potential energy is helpful to understand constraints of runoff and morphological process, as well as the sensitivity to different hillslope forms or the presence of rill networks.

Overview of the different symbols used in this study.

We come back to the steady-state free-energy balance of surface runoff (Eq. 4), which allows for an estimation of the term

The terms in the first bracket reveal the antagonistic effects of a downslope-growing discharge due to flow accumulation and the decline in topographic elevation on potential energy. As stated in our first hypothesis, we expect that this trade-off leads to a local potential energy maximum. While the existence of such a maximum can in fact already be confirmed by a re-analysis of the experiments of Emmet (1970) (Fig. 2, Sect. 3), the existence of such a maximum is usually not discussed in the case of streamflow. This is because Eq. (7) simplifies in streams to Eq. (8), as kinetic energy fluxes are much smaller than potential energy fluxes, and with increasing discharge the mass balance gets more and more dominated by upstream runon, while precipitation input becomes marginal:

Upper part: digitized results from rainfall simulation experiments
at New Fork River 1 (Emmett, 1970), expressed as normalized potential energy

Rill structures form on event to seasonal timescales due to a fast positive feedback (Rieke-Zapp and Nearing, 2005). On a longer timescale, the redistribution and export of soil material restructures entire topographic hillslope profiles such that typical shapes can be attributed to a dominant erosion process (Kirkby, 1971; Beven, 1996). The latter change in space along the flow path and therefore in close connection to the flow regimes (Shih and Yang, 2009; see Fig. 2). At the upslope divide, erosion is mostly influenced by gravity, resulting in soil creep. With flow accumulation in a downslope direction, the particles eroded by raindrop splash can be transported by surface runoff, until surface runoff becomes turbulent and can erode and transport particles as soil wash. The spatial organization of transition processes (also called threshold processes) can be described by the relative contribution of internal and external forces. Turbulence emerges when gravitational (external) force surpasses a certain threshold in relation to viscous (internal) forces. Similarly, soil wash erosion relates to externally induced bed stress by runoff, while soil creep depends on internal resistance factors of the soil matrix. We therefore propose, as stated in our second hypothesis, that both process transitions are linked through their external forcing, which is attributed to the energy gradient of surface runoff. The distribution of surface runoff energy and its gradient provide therefore insights on erosional as well as flow regimes.

In the following we apply our framework to test our hypotheses on two related temporal and spatial scales. In Sect. 3, we analyse the distribution of energy at the macroscale, representing the hillslope as an open thermodynamic system which adapts morphologically to the distribution of gradients and fluxes on long timescales. To this end we analyse steady-state runoff on typical hillslope profiles that reflect, according to Kirkby (1971), the dominant erosion processes “soil creep”, “rainsplash” and “soil wash”. In Sect. 4 we analyse the energy balance of surface runoff observed during short-term rainfall simulation experiments, where runoff concentrates in rills and distributes energy into a sheet and a rill domain.

In both sections we explore how the transition of flow regime and erosion
processes on hillslopes relate to the distribution of energy and its local
maximum. We want to stress that we speak of laminar flow if there is a clear
dependence between flow Reynolds number of surface runoff and friction
coefficient (Phelps, 1975). For purpose of comparison with earlier studies
of hydraulics of surface runoff (Emmett, 1970; Parsons et al., 1990) we
calculate flow Reynolds number

To clarify and test our hypothesis, we digitized results of rainfall runoff
experiments on hillslope plots from Emmett (1970) and plotted potential
energy

The accumulation of mass along a declining geopotential leads to a maximum of potential energy in space, dividing the flow path into a section where energy is gained (Fig. 2, arrow a) and a section where energy is depleted (Fig. 2, arrow b). In between these two sections (Fig. 2, area highlighted in grey), depletion of potential energy is balanced by the energy influxes of runoff accumulation and rainfall. Volumetric energy

The transition from a laminar into a turbulent-flow regime is indicated by
ranges of critical Reynolds-number

In this section, we explore how typical hillslope configurations and effective rainfall forcing control runoff accumulation and related energy
conversions. We distinguish three typical hillslope forms, which are related
to a dominant erosion process (Kirkby, 1971). Equation (11) defines the
distribution of geopotential along a representative flow path. The
coefficients

Generally, we found that the trade-off of downslope mass accumulation and
declining geopotential leads to a distinct potential energy maximum, which
has a clear dependence on the slope form, width function and strength of
rainfall forcing (Fig. 4). This implies that the hillslope can be subdivided into three classes of spatial energy dynamics:

Distribution of potential energy

Figure 4a shows that the location of the energetic maximum moves upslope when
changing the width function from divergent (div) to parallel (const) to
convergent (conv). The magnitude of the absolute value of the maximum increases in a similar fashion. The distribution of geopotential from top to bottom clearly influences the location and size of maxima (Fig. 4b). Hillslope profiles which are formed by soil creep (SC) show the maximum of

Similarly, an increasing rainfall infiltration excess intensity

To estimate the relative amount of influx energy that is converted into the
energy balance residual

Figure 5a reveals a distinct pattern of

Spatial distribution of the ratio of

In this section we related the spatial distribution of slope (hillslope form) to the distribution of potential and kinetic energy of surface runoff. As form is also connected to the dominant erosion process, an analysis of energy dissipation provides a link between the erosion process and thermodynamic principles. In a first step we digitized surface runoff experiments by Emmett (1970), and we showed that the distribution of potential energy results in a distinct flow path distance with maximum potential energy. Up to this point, the system net accumulates energy and only undergoes a net loss of energy after this location. The distribution of these zones of energy production and energy depletion seems to be related to the transition of the system from one type of flow regime to another. Magnitude and distribution of energy are relative to a level of null energy at the hillslope end and therefore represent an assumed equilibrium state of the land–water system at the hillslope scale. From a larger perspective, the accumulated discharge at the end of the hillslope can perform work within the context of the whole catchment, which has been discussed previously (compare Rodriguez-Iturbe et al., 1992; Kleidon et al., 2013).

For an analysis of these equilibrium-state hillslopes, we relied on established semi-empirical descriptions of hillslope forms and related erosion processes (Kirkby, 1971), and we assumed that surface runoff on equilibrium hillslopes has dissipated all potential energy at the downslope end (usually the channel bank). The resulting steady-state distribution of potential energy of surface runoff was then calculated by a friction law that was established for stony hillslopes in Arizona (Nearing et al., 2017) but in essence expresses the tendency of a hillslope surface to spatially organize friction as a function of slope and has previously been established with different parameters for rill flow (Govers, 1992). We note that these studies were concerned with surfaces which had little to no vegetation influencing the resistance to erosion of the soil particles, meaning that morphological adaptations were predominantly due to surface runoff. In a similar fashion we did not account for vegetation and infiltration but should mention that these processes would certainly affect the steady-state energy balance and its residual presented here. Therefore, we stress that the presented distribution of potential energy is meant to approximate steady-state runoff on equilibrium hillslopes with respect to frictional adaptation without vegetation and situations with significant infiltration excess runoff.

The resulting distributions reveal that on hillslope forms which relate to
diffusive erosion (SC slope forms),

Energetically, this can be expressed as relative accumulated dissipation per
energy influx

Our assessment is based on an empirical relation between flow velocity and unit discharge and therefore does not provide closure to the energy balance. However, Eq. (6) implicitly incorporates a spatial organization of relative friction (compare Sect. 2.2), which in accordance with our results seems to be supported by thermodynamic theory. Reversely, we show that maximum power and equal energy expenditure per unit discharge for surface runoff on hillslopes should result in friction laws like the ones proposed by Govers (1992) and Nearing et al. (2017). In fact, the proposed slope–velocity equilibrium by Nearing et al. (2005) seems to be a natural outcome of the equal energy expenditure, maximum power and maximum entropy concepts.

Finally, we want to point out that along a similar line of thought, Hooshyar et al. (2020) have recently shown that logarithmic mean elevation profiles of landscapes resemble the logarithmic mean velocity profile in wall-bounded turbulence. The authors concluded that these logarithmic profiles are a consequence of dimensional length-scale independence and therefore apply to different dynamical systems, possibly also to a much smaller hillslope scale. As these profiles were observed at an intermediate region and therefore are spatially transient, we believe they might relate to the transition from energy production to energy depletion proposed here, inspired by the well-known energy cascade of turbulent kinetic energy (Tennekes and Lumley, 1972).

We now explore the spatial distribution of potential energy in sheet and rill overland flow, which was observed during rainfall–runoff experiments carried out in the Weiherbach catchment (Gerlinger, 1997). Therefore, we built an extension to the physical hydrological model Catflow, which allows for the accumulation of flow from sheet flow areas into rills (Catflow-Rill). As these experiments were performed on 12 m plots with a uniform slope, they correspond to the rain-splash-dominated hillslope type, as shown in Fig. 3b.

The Weiherbach catchment is an intensively cultivated catchment which is
almost completely covered with loess up to a depth of 15 m (Scherer et al.,
2012). It is located in the Kraichgau region, northwest of Karlsruhe in
Germany. Because of the hilly landscape, the intensive agricultural use and
the highly erodible loess soils, erosion is a serious environmental problem
in the Kraichgau region. The Weiherbach itself has a catchment area of 6.3 km

Here we analyse 31 rainfall simulation experiments (Gerlinger, 1997; compare data in the Supplement), which were performed to explore formation of overland flow and the erodibility of the loess soils (Scherer et al., 2012). The rainfall simulators were designed to ensure both realistic rainfall intensities and kinetic energies on plots of 2 m by 12 m size. Rainfall intensity of experiments ranged between 34.4 and 62.4 mm h

Classification of rainfall simulation experiments, where green lines
reach steady state during 0.75–1.0 of relative time of rainfall simulation:

Next, we present an extension to the Catflow model (Zehe et al., 2001), accounting for a dynamic link between sheet and rill flow of surface runoff. The model has previously been extended, incorporating water-driven erosion (Scherer, 2008), and has been shown to successfully portray the interplay of overland flow, preferential flow and soil moisture dynamics from the plot to small catchment scales (Graeff et al., 2009; Loritz et al., 2017; Zehe et al., 2005, 2013).

A catchment is represented in Catflow by a set of two-dimensional hillslopes
(length and depth), which may be connected by a river network. Each hillslope is discretized using curvilinear orthogonal coordinates; the third dimension is represented by a variable width. Subsurface water dynamics are described by Richards' equation, which is solved numerically by an implicit mass-conservative Picard iteration scheme. The simulation time step for soil
water dynamics is dynamically adjusted to achieve an optimal change of the
simulated soil moisture, which assures fast convergence of the Picard
iteration. Soil hydraulic properties are usually parameterized using the van Genuchten–Mualem model (Mualem, 1976; van Genuchten, 1980), but other
options are available. Enhanced infiltrability due to activated macropore
flow is conceptualized through enlarging the soil hydraulic conductivity by
a macroporosity factor

Overland flow is simulated in Catflow–Rill with the diffusion wave equation,
which is numerically solved using an explicit upstreaming scheme, a
simplification of the Saint-Venant equations for shallow water flow; for
details of the numerical scheme we refer to Scherer (2008). Flow velocity is
calculated with Manning's equation (Eq. 5). The previous Catflow model
assumes sheet flow only. To incorporate a rill domain that dynamically
interacts with sheet flow, we conceptualize the hillslope surface similar to
the open-book catchment (Wooding, 1965) as an open-book hillslope (Fig. 7).
In this configuration, water may accumulate in a trapezoidal rill of width

Representation of overland flow domains in Catflow–Rill as an open-book hillslope: sheet flow domain (blue area) and rill flow domain (yellow area).

Results of calibrations runs for experiments “lek_2” and “oek2_4”:

From the experimental database, Scherer et al. (2012) created Catflow
simulation setups, which were calibrated to reproduce runoff by adapting the
macroporosity factor to scale infiltration capacity. The hillslopes were
parameterized and initialized using observed data on average topographic
gradient, plant cover, soil hydraulic functions, surface roughness, soil
texture and antecedent soil moisture. The models were driven by a block
rain of the respective intensity and duration of the experiment. From here
onwards, the subscript “sim” relates to the results of the presented calibrated
numerical simulations. Hillslopes were discretized on a 2D numerical grid
with an average lateral distance of 60 cm and vertically increasing
distances, starting with 1 cm at the surface and ending with 5 cm on the soil bottom. This resulted in

Soil hydraulic parameters of Van Genuchten–Mualem model for simulated hillslopes, namely saturated hydraulic conductivity

Results of calibration of flow accumulation to observed rill flow
velocities:

To match the observed flow velocities, we adjusted the flow accumulation
coefficient

Figure 9 shows that calibrated rill flow velocities match the observed values for all 31 experiments well (Fig. 9a). We also note that magnitude of rill flow velocity is correlated to flow accumulation, ranging from the smallest,

Spatial distribution of

In a similar fashion to the comparison of relative dissipation along the typical hillslope profiles in Sect. 3.3, we calculate the kinetic energy export at the hillslope end in relation to the potential energy influx by rainfall and compare the relative contributions of rill flow and sheet flow. However, we can only confidently evaluate this for simulated experiments which can be classified as steady state (for discharge and sediment concentrations; compare Fig. 6) and where

Spatial distribution of

The calibrated Catflow–Rill models also provide an estimate of the spatial
distribution of energy for the rill and the sheet domains. Figure 11a and b
show the spatial distribution of potential energy

By comparing five experiments classified as steady state (compare Fig. 10), we find that

Interestingly, the ratios of kinetic energy in a sheet to rill domain decline downslope, and the gradient of the curve increases (Fig. 12b) when the location and magnitude of

Our approach to model the accumulation of surface runoff by a single rill and the calibration of a flow accumulation parameter resulted in partly good
approximations of observed rill and sheet flow velocities and therefore justifies the presented simplification of surface runoff across two domains.
Although the model uses a single friction coefficient (Manning's

Analysis of relative dissipation of energy per influx energy by rainfall reveals that surface runoff across the rill and sheet domain is related to the existence of a maximum power state. For the analysed experiments, we
distinguished those which reached steady-state discharge and sediment concentrations and calculated the kinetic energy per influx energy that
leaves the plot. For rill flow, it can be shown that kinetic energy export
increases with flow accumulation, while kinetic energy of sheet flow
decreases with growing

For the analysis of flow regime transitions (compare hypothesis 2), we plotted the Reynolds number of rill flow at the flow path distance where potential energy is maximum (compare Fig. 12a). While some

Potential energy in this section is based on a relative calculation of potential energy with the null level of the 12 m plots at the outlet of the Weiherbach catchment, which makes the results (Fig. 12) comparable. We argue that surface runoff on hillslopes in its simplest case can be separated into sheet and rill flow and that the distribution of flow within both domains approaches a maximum power state over time (compare Fig. 10a). At this state dissipation per driving gradient is maximized, while the ratio of kinetic energies approach unity. We found that two of the truly steady-state sites as well as seven other experimental sites cluster in this area. In fact, we see very strong similarities to a maximum power state of an electrical circuit where the load resistance (in the case of surface runoff: the inverse of rill conveyance) has adjusted to meet the source resistance (the inverse of sheet flow conveyance; compare Appendix C). This finding can also be corroborated from Fig. 10a, with the minimum total flux of kinetic energy being related to equal fluxes of kinetic energy (and therefore also equal kinetic energies) across both surface runoff domains.

In this study we linked well-known processes of surface runoff (Shih and Yang, 2009) and erosion (Kirkby, 1971; Beven, 1996) to thermodynamic principles (Kleidon, 2016) and theories derived thereof (Leopold and Langbein, 1962; Rodriguez-Iturbe et al., 1992). The geomorphological development, surface runoff and the dominant erosion process co-evolve. We could show that an approach to account for the energy conversion and dissipation rates is a helpful unifying concept. The core of this concept is the residuals of the observable, free-energy fluxes and particularly their spatial distribution, which is key to evaluating empirical friction laws of surface runoff velocities in a thermodynamic framework. Although we do not provide a full closure of the energy balance of surface runoff, we were able to test and corroborate two hypotheses about the distribution of potential and kinetic energy of surface runoff and the related transition from laminar to turbulent flow, on two related hillslope scales. Hypothesis 1 states that surface runoff systems can be separated into an area of production and an area of depletion of energy. Our second hypothesis relates the typical transitioning of flow (laminar to turbulent) and erosion (diffusive to advective type) regime to these zones.

In line with our first hypothesis, we showed that hillslopes as mass-accumulating systems are characterized by a distinct energetic behaviour: the trade-off between downslope mass gain and geopotential loss along a runoff flow path leads to a maximum of potential energy. We found that the location and magnitude of this maximum are a function of hillslope form and accumulated surface runoff. Specifically, we analysed the influence of typical hillslope macro-topographical profiles with a fixed accumulated runoff on the spatial pattern of overland flow energy. We found that hillslope forms which relate to diffusive erosion processes (soil creep SC) have an energetic maximum located farther downslope than hillslope profiles related to advective erosion (soil wash SW). One might therefore be inclined to relate maximum dissipation rates to the former hillslope type SC as for our example more energy is depleted on a shorter flow path. However, in relative terms we see that SW forms have much larger dissipation rates than RS or SC forms, implying that dissipation is increased and even maximized as relative dissipation per unit flow path is close to unity. At the same time, SW forms also increase kinetic energy per influx energy, a criterion proposed by various authors for maximization of power (Kleidon et al., 2013) as well as maximum entropy production (Leopold and Langbein, 1962).

Referring to our second hypothesis, we interpret these findings as results of the transition of the dominant energy conversion process of surface runoff. Hereby we present a theory why laminar flow regime should be related to sheet flow and mixed/turbulent flow is related to concentrated flow in rills and channels. For the second application of this study, we create an extension to the numerical model Catflow, which allows for an adjustment of flow accumulation, by separating runoff into sheet and rill flow and dynamically linking both one dimensional flow domains. The calibration to observed rill and sheet flow velocities from rainfall simulation experiments in the Weiherbach catchments revealed distinct flow accumulation coefficients, which clearly relate to the distribution of kinetic energy of and the relative contribution to surface runoff from both domains. In fact, we showed that maximum relative dissipation rates are achieved when kinetic energy exports from both domains are equal. This can be interpreted as a maximum power state with minimum production of total kinetic energy, and related experiments therefore result in minimum sediment concentrations.

For those experiments that reached an energetic steady state, our simulations show that the build-up of potential energy on hillslopes is likely to occur under laminar flow conditions, while a decrease in potential energy along the flow path seems to be related to concentrated rill flow, with

Our final comment is aimed at the common picture of runoff as a fixed volume of water losing energy by friction (e.g. Bagnold, 1966). We think that we have shown that this picture should be revised because a loss of mean flow energy does not necessarily imply an equal increase in production of heat but can also be translated into velocity fluctuations of turbulence or lift and accelerate sediment particles. All this affects real dissipation rates and needs to be considered if one ever attempts to depart from empirical friction laws of channel flow for estimation of surface runoff on hillslopes.

For each OTS

Figures B1 to B4 are based on values derived from measurements
(Manning's

Manning's

Calibrated flow accumulation

Calibrated flow accumulation

Slope of experiment plots vs. rill flow velocity.

Flow on hillslope equivalent to current in circuit.

Additional symbols.

Therefore, channel conveyance is the inverse of the resistance of the channel to transport flow.

If water is mainly falling on the sheet flow area and flows therefore first onto the
sheet flow area with

Total power in the rill is then

Equation (C2) becomes maximum if the term “

The derivative (Eq. C3) becomes zero if

The model code used is available upon request.

The used dataset (Gerlinger, 1997) has been published by the KIT and is accessible through its library. A summarized file is available on request.

The supplement related to this article is available online at:

SSc conceptualized and implemented the Catflow–Rill extension, conducted the analysis and wrote the paper. OE conceptualized and supervised the hydraulic concepts. AK reviewed and edited the thermodynamic concepts. US provided the original Catflow setups and commented on surface runoff dynamics. JW contributed to paper writing and Catflow modelling. EZ oversaw the study and theory development as a mentor.

At least one of the (co-)authors is a member of the editorial board of

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

We would like to thank the reviewers Keith Beven and Hubert Savenije for their valuable comments. Especially the discussion with Keith Beven was very appreciated and led to a substantial improvement of this paper.

The article processing charges for this open-access publication were covered by the Karlsruhe Institute of Technology (KIT).

This paper was edited by Roger Moussa and reviewed by Keith Beven, Hubert H. G. Savenije, and two anonymous referees.