First, we would like to thank Keith Beven for his time, effort and insightful comments. We also very much welcome the upload of a commented manuscript and will gladly incorporate his suggestions to clarify the presentation of our approach in a revised manuscript.

As we understand the principal point of criticism made by KB, we agree that the manuscript will benefit from a clearer definition of our theory and a reflection of the main simplifications. We also agree that the results and inferences made apply to landscape forms that are primarily formed by overland flow and the related forces and energy conversions. We will clarify this in the revised manuscript.

Keith Beven’s suggestion to highlight the different behavior of the D_f component of the energy balance for surface runoff with and without sediment load is appreciated. We agree that there is an important difference between viscous dissipation of energy due to friction and energy transfer to the sediment. The former implies that free energy is indeed lost, while the latter means that free energy creates motion of the sediment particles. A water and sediment balance of the rainfall runoff simulation experiments by Gerlinger revealed that suspended sediments may increase the density of the fluid-sediment mix up to 10%. Unfortunately, the available measurements and data are too limited to analyse underlying dynamics and include this into the analysis. In conclusion, we admit that the description of the residual of the energy balance needs some clarification, and it is a good idea to elaborate on the differences of D_f between surface runoff with and without sediment transport. This has direct implications for the inference of Maning’s n from such rainfall simulation experiments.  

We would like to thank KB again for his comment on the Df term (dissipation) and wanted to use this discussion as an opportunity to present a possible extension of the energy scheme which is presented in the current manuscript.

Attached we outline how sediment transport affects the Df term and we distinguish four dissipation regimes, depending on whether sediment is eroded or deposited.If sediment is eroded some energy of the flow will be used for the additionally transported mass, and similarly if sediment is deposited some of the energy of the sediment-water mix becomes available.

This concept could be incorporated as a clarification of the Df term and also be part of the discussion of a revised manuscript.

We thank JD for his comments on our manuscript. The purpose of this study is an evaluation of energy patterns during steady state runoff. While transient floods and runoff are certainly dominant in nature, design of experiments for parameter estimation often departs from a steady state assumption. The results shown in Fig. 10(c) and (d) represent therefore merely the temporal routes of hydraulic variables for reaching a steady state runoff regime and an analysis of the latter, not the routes themselves. However, as mentioned in our previous reply (AC3) to KB’s comment (RC2), we agree that a further study should incorporate temporal patterns.

This study presents therefore just a first step towards an analysis covering distribution of the energy residual in space and time, where we will of course consider JD’s comments on transient dissipation regimes.

We thank Ref.2 for his thoughtful analysis.

First, we would like to address his/her two principal concerns regarding the focus of the study, which we believe are due to the broad range of applicability of the theory. We are thankful for these comments (as also previously made by K. Beven) and we intend to present the theory and its application in a clearer way in a revised version of the manuscript.

We focus in this paper on the analysis of surface runoff from a thermodynamic perspective. Therefore, we depart from the first law of thermodynamics (Eq. 1), which we apply in the subsequent equations for balancing energy fluxes of an open thermodynamic system. As mentioned in the introduction and by the second law of thermodynamics, entropy can only increase. This is constrained by the Carnot limit, which is the maximum amount of work that can be extracted from a heat engine with fixed boundary conditions. For a dissipative system with non-fixed boundaries, which is the case for most natural systems, there exists a trade-off between driving gradient and flux, resulting in a maximum power limit (Kleidon, 2016). We therefore hypothesize that driving gradients and fluxes of surface runoff systems evolve into a state of maximum power and we argue that the spatial organization of structure and dissipative processes is a result of this evolution.

For the surface runoff system of our study this means that the potential energy, which is added by rainfall on different topographic levels needs to be depleted as fast as possible. This can be achieved by maximizing runoff, more specifically, the system maximizes the kinetic energy of the runoff. While doing so, the runoff system spatially organizes in a way that loss of energy is minimized along the flow path, but overall, it maximizes dissipation through faster depletion of the driving gradient of potential energy. This interplay manifests as global maximization of free energy loss in time (dissipation) by local minimization of free energy loss in space (friction).

As the distribution of free energy gradients is key for the resulting fluxes, we show from the results of Emmett’s experiments that measured surface runoff on hillslopes does surprisingly not strictly follow this theory, but results in a maximum of potential energy somewhere along the flow path. This phenomenon stands energetically in contrast to a river system and occurs because downslope mass accumulation over-compensates the declining geopotential. We propose that this switch in the downslope potential energy gradient relates to the transitional character of surface runoff, and we think that there is a critical level of flow accumulation for surface runoff to be classified as a strictly Hortonian surface runoff system, which can be represented by Fig. 1 of the manuscript.

Regarding the second point raised by the Referee, we believe that the process of organization is transient and different adjustments of structure manifest after different time periods and on different scales. On a larger scale much more total work is needed to form typical hillslope profiles, resulting in geological time scales of transient adjustments of hillslope or river profile adjustments until reaching a thermodynamic equilibrium. On a small-scale hillslope plot the same underlying process of maximization of entropy leads much faster to the formation of structure, typically during seasonal if not event time scales. For testing our theory, large scale adjustments would need to be simulated in a laboratory (e.g., Singh et al., 2015) but hillslope plot-scale adjustments can be observed from experiments on real hillslopes. We therefore consider both, sect. 2 and sect. 3 equally important to show different aspects of the same underlying physical process. However, we agree with the Ref. that the current manuscript would benefit from a clearer explanation of the similarities and differences of both sections as well as pointing out the common objectives.

This applies also to the last point raised by the Referee regarding the use of a model, which distributes surface runoff into rill- and sheet flow. As the minimization of energy loss per unit volume in flow path direction results in a structured surface we investigated surface runoff experiments where adjustment of structure by formation of rills was observed. As the resulting rill flow velocities were measured and the sheet flow velocities back calculated from total measured discharge, the model allows an estimation of the spatial distribution of free energy gradients and fluxes. The results could then be used to test our hypothesis that surface runoff on hillslopes evolves into a state of maximum power. Although we agree that the two calibrated experiments are statistically not significant for a definite conclusion, further modelling would imply an additional layer of complexity in this study and was intentionally left for future investigation. We thank Ref.2 for his comments on our study.

References

Singh, A.; Reinhardt, L.; Foufoula-Georgiou, E. (2015): Landscape reorganization under changing climatic forcing: Results from an experimental landscape, Water Resour. Res., 51, 4320–4337, doi:10.1002/2015WR017161.