the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Morphological controls on Hortonian surface runoff: An interpretation of steady-state energy patterns, maximum power states and dissipation regimes within a thermodynamic framework
- 1Institute of Water Resources and River Basin Management, Karlsruhe Institute of Technology – KIT, Karlsruhe, Germany
- 2Institute for Hydromechanics, Karlsruhe Institute of Technology – KIT, Karlsruhe, Germany
- 3Max-Planck Institute for Biochemistry, Hans Knöll Str. 10, 07745 Jena, Germany
- 4Engler-Bunte-Institut, Water Chemistry and Water Technology – KIT, Karlsruhe, Germany
- 1Institute of Water Resources and River Basin Management, Karlsruhe Institute of Technology – KIT, Karlsruhe, Germany
- 2Institute for Hydromechanics, Karlsruhe Institute of Technology – KIT, Karlsruhe, Germany
- 3Max-Planck Institute for Biochemistry, Hans Knöll Str. 10, 07745 Jena, Germany
- 4Engler-Bunte-Institut, Water Chemistry and Water Technology – KIT, Karlsruhe, Germany
Abstract. Recent developments in hydrology have led to a new perspective on runoff processes, extending beyond the classical mass dynamics of water in a catchment. For instance, stream flow has been analysed in a thermodynamic framework, which allows the incorporation of two additional physical laws and enhances our understanding of catchments as open environmental systems. Related investigations suggested that energetic extremal principles might constrain hydrological processes, because the latter are associated with conversions and dissipation of free energy. Here we expand this thermodynamic perspective by exploring how hillslope structures at the macro- and microscale control the free energy balance of Hortonian overland flow. We put special emphasis on the transitions of surface runoff processes at the hillslope scale, as hillslopes energetically behave distinctly different in comparison to fluvial systems. To this end, we develop a general theory of surface runoff and of the related conversion of geopotential energy gradients into other forms of energy, particularly kinetic energy as the driver of erosion and sediment transport. We then use this framework at a macroscopic scale to analyse how combinations of typical hillslopes profiles and width distributions control the spatial patterns of steady-state stream power and energy dissipation along the flow path. At the microscale, we analyse flow concentration in rills and its influence on the distribution of energy and dissipation in space. Therefore, we develop a new numerical method for the Catflow model, which allows a dynamical separation of Hortonian surface runoff between a rill- and a sheet flow domain. We calibrated the new Catflow-Rill model to rainfall simulation experiments and observed overland flow in the Weiherbach catchment and found evidence that flow accumulation in rills serves as a means to redistribute energy gradients in space, therefore minimizing energy expenditure along the flow path, while also maximizing overall power of the system. Our results indicate that laminar sheet flow and turbulent rill flow on hillslopes develop to a dynamic equilibrium that corresponds to a maximum power state, and that the transition of flow from one domain into the other is marked by an energy maximum in space.
Samuel Schroers et al.
Status: final response (author comments only)
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RC1: 'Comment on hess-2021-479', Keith Beven, 01 Oct 2021
I fully understand that this is a contribution that forms part of an on-going research program to introduce and explore the consequences of thermodynamic principles in hydrological theory. As such the attempt to do this for hillslope runoff is of interest, despite the rather restrictive assumptions under which the analysis is carried out. My main comment is that the discussion should reflect those limitations more, rather than making inferences that are far beyond the results presented.
It would be also useful (I think to this research program as a whole) if some of the concepts could be discussed more clearly. In this case in particular it is the Df component of the energy balance. This is initially the energy dissipation that is the residual of potential and kinetic energy components, and then suddenly becomes equivalent to stream power in a discussion related rivers and sediment transport (even when sediment transport is never actually considered in the analyses). I think a section that discusses how Dfshould be characterised for laminar/transitional/turbulent flows before and after a threshold for sediment mobilisation might be really useful to the reader. As it is, it seems only that a constant Manning n is used to convert to velocity, which is certainly one characterisation but surely might not be considered adequate. Perhaps this suggests a bit more than minor revision.
Further comments are included on the manuscript.
Keith Beven
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AC1: 'Reply on RC1', Samuel Schroers, 04 Oct 2021
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 Df 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 Df between surface runoff with and without sediment transport. This has direct implications for the inference of Maning’s n from such rainfall simulation experiments.
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AC2: 'Reply on RC1', Samuel Schroers, 12 Oct 2021
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.
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RC2: 'Reply on AC2', Keith Beven, 12 Oct 2021
Well yes ... it seems a straightforward extension in theory, since again it is not too difficult to write down balance equations. But in practice you have now split your significant unknown term Df into a number of unknown terms (even given the simplifying assumptions of steady state flows, well mixed sediment etc) that hydrologists and geomorphologists have struggled with for over a century (and we STILL resort to Manning's n or some assumed constant fraction of kinetic energy). So I think the critical thing in an revision is to really make it clear how this can help frame the problem of closing that energy balance and having some clearer understanding of how to evaluate the Df term - splitting it into component parts might provide some cursory understanding - but what is needed to allow a proper evaluation (remembering that we are actually interested in transient dynamic flows when we often cannot easily close the mass balance equations in practice)?
Just to add - I certainly do not want to discourage this research programme but by staying in the "theoretical" domain it does not yet seem to be really addressing the most significant question.
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AC3: 'Reply on RC2', Samuel Schroers, 13 Oct 2021
We thank KB for these thoughtful comments. We fully agree that the steady state surface runoff is rarely reached during rainfall runoff events. In fact, we plan to address this point in a follow up study (as explained below). We also share his concerns about the validity of empirical flow laws (in particular Manning's n) in case of very shallow overland flows with flow depths of a few mm. We already stressed that this is particularly delicate upslope of the potential energy maximum, in the current manuscript. Also, we acknowledge that the presented concept does indeed not provide closure of the energy balance. However, we would like to stress that in this study the presentation of the Df term is principally intended to provide a basis for the analysis of the spatial distribution thereof, not it’s magnitude. The question how free energy of flow is dissipated by viscous forces goes beyond the goal of this study, but we agree that this point is important, and that a revision should at least incorporate some perspective.
Yet, solving all these issues is beyond the scope of this study, and more importantly beyond the scope of the field observations at hand. Our work is clearly only a first step, but it underpins that a closer look at the spatial pattern of the steady state energy balance of surface runoff and it’s residual Df is helpful to
- identify distinct differences between hillslopes and rivers,
- single out hot spots, where the flow can perform the largest amount of physical work,
- test thermodynamic concepts (maximum power, least work, etc.) on this dynamical system .
We also like to stress that our work is not purely theoretical. We analyze overland flow measured during sprinkling experiments, which reached steady state conditions according to Scherer et al. (2012). As these experiments provide measured flow velocities in rills and back calculated values of Manning’s n. Hence, we regard these experiments, despite all their short comings (very shallow flow depth and limited validity of all available flow laws) as best suited to test our framework, at least during, albeit rare, steady state conditions. Because the experiments did not provide observed flow depth along the flow path, this approach required a numerical model, as well as expanding the CATFLOW code by a rill domain.
This analysis corroborated the existence of the potential energy maxima and their sensitivity to presence/absence of rills. And we provide first evidence that the transition zone from laminar to mixed flow coincides roughly with the location of the potential energy maximum (when rills are present). We also found that kinetic energy in the sheet and rill domain are equal at the end of the stripe. This is analogue to the maximum power configuration of an electrical circuit and at least an interesting incidence. Of course, two experiments are not sufficient to conclude whether this holds in general.
All In all, we think that our work corroborates that an energetic perspective on overland flow is helpful and has practical implications which should be further explored.
The next logical step is to give up the steady state assumption and work with transient conditions. The mentioned energy residual’s distribution might follow similar thermodynamic concepts in time as in space.
The framework is not restricted to the steady state, but requires numerical simulations of overland flow, ideally with the shallow water equations (as the diffusion wave approach assumes a steady state momentum balance).
Another key challenge is a proper design of experiments to disentangle viscous dissipation and energy transfer to and from the sediments. If this energy transfer is ignored during simulations, this might imply that dissipation is over- or underestimated (see explanation in last reply). And finally, the concept of maximum power i.e., an equal distribution of kinetic energy among rill and sheet flow during steady state, is a testable hypothesis.
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RC3: 'Reply on AC3', Keith Beven, 13 Oct 2021
An excellent reply (and I stress again that I do not want to dicourage this research programme) .... but: you do have to be careful in avoiding circularity of reasoning when coefficients (like Df or its parts) have to be back-caclulated from experimental data (in the same way that it is always possible to get a value for n given observations of velocities or discharges, which is really the only reason why it is still being used as a convenience not requiring much thought). I can see that testing any hypotheses about Df is going to be highly challenging even at small plot scales, while inferences at larger scales (including inferences about minimisation principles) are going to depend very much on assumptions about the distributed and transient nature of Df in model simulations. So I would suggest that you proceed with caution, remembering that rejecting your favoured hypothesis is often more valuable to improving understanding.
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AC6: 'Reply on RC3', Samuel Schroers, 22 Dec 2021
We thank K. Beven for all of his comments as well as his elaboration on doubts regarding circularities.
While we agree that there is potential of circularity and that it is highly challenging to describe the spatial pattern of a dissipation term in nature, we think that thermodynamic approaches might hold untapped potential for limiting runoff-erosion processes. In the special case of our study, we made assumptions regarding boundary conditions of a hydrological system, which might only be achievable in laboratory environments. However, describing the thermodynamic principles of a restricted hydrological system in a laboratory would be a first step towards constraining more complex systems in nature. On a brief note, we would like to mention that although the transfer of free energy to bound energy happens on a molecular scale, patterns seem to manifest even at much larger scales. E.g., Hooshyar et al. (2020) show that mean elevation profiles of simulated landscapes resemble the distribution of mean velocity in the viscous boundary layer of turbulent flow. Again, we agree with K. Beven that these findings are partly restricted to a digital idealized environment, but it seems to us that these highly interesting similarities merit further research on a theoretical basis as well as in laboratory setups and finally complex natural systems. For the present study we are grateful for the comments by K. Beven and will highlight these uncertainties and assumptions in a revised version of the manuscript.
References:
Hooshyar, M., Bonetti, S., Singh, A., Foufoula-Georgiou, E., and Porporato, A.: From turbulence to landscapes: Logarithmic mean profiles in bounded complex systems, Physical Review E, 102, 033 107, 2020.
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AC6: 'Reply on RC3', Samuel Schroers, 22 Dec 2021
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AC3: 'Reply on RC2', Samuel Schroers, 13 Oct 2021
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RC2: 'Reply on AC2', Keith Beven, 12 Oct 2021
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AC1: 'Reply on RC1', Samuel Schroers, 04 Oct 2021
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CC1: 'Comment on hess-2021-479', John Ding, 17 Oct 2021
The authors focus the analysis on the rising limbs of overlandflow hydrographs, e.g., Figure 10, Panels (a) and (c) for the sheet and rill flows, and the sheet flow only, respectively. These are essentially the summation or S-curve hydrographs. Not mentioned are their falling limbs after the termination of the effective rainfall, i.e. recession hydrographs.
Because of the absence of the forcing effective rainfall intensity term, Peff, an additional, and simpler, analysis by the authors will likely lead to linearization of the recession hydrograph by an inverse fractional-power transformation of the discharge, 1/Q (1-1/c2). This was derived earlier for a nonlinear storage-discharge function by this writer, e.g., Ding (1974, Eq. 4; second equation after Eq. 11).
For Manning friction law, the flow depth exponent c2 = 5/3 shown in their Table 2.
References
Ding, J.Y., 1974. Variable unit hydrograph. J. Hydrol., 22: 53-69.
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AC4: 'Reply on CC1', Samuel Schroers, 19 Oct 2021
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.
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AC4: 'Reply on CC1', Samuel Schroers, 19 Oct 2021
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RC4: 'Comment on hess-2021-479', Anonymous Referee #2, 16 Dec 2021
In this study, the authors investigated how the hillslope topography controls the overland flow, using thermodynamics metrics. It is a very interesting approach. The manuscript is overall well-written: in particular, the introduction is almost in a complete shape.
Nevertheless, I have a couple of main concerns. I wish the manuscript to be more focused. This is a very theoretical study... and it would not be easy for readers to follow. Clarifying focus would be helpful. I would suggest clarification in two points mainly.
1. If I understand correctly, one of the major motivations is the existence of the peak in the total energy distribution over hillslope shown in Fig 3. Yes, this could be an interesting subject to be investigated. The authors did detailed work in the following section. But after all, I don't really follow why this work and the contents are related to the maximum entropy production principle. Maybe I am misunderstood but I believe readers would be very confused.
2. I feel the contents in sections 3 and 4 are unrelated, and this makes me further confused about the focus of this study.
Below I list minor suggestions.
Title: I don't know whether the word 'Hortonian' here is necessary. This paper does not consider the runoff generation mechanism. The contents here are applicable for any overland flow, and whether the flow is generated through saturation excess or infiltration excess is irrelevant.
L 37: Typo (Dary-> Darcy)
Eq(1): Do you ever use this eq. in this study?
Eq(5) and others: The notation * to express 'multiplication' may be confused with the convolution symbol?
L219: I understand that in Fig3, the volumetric energy and its gradient decrease downstream, but I don't know whether they minimize? How do I know they reach the minimum?
L253: Authors use geopotential to express topography while most works simply use surface elevation profile. This is an interesting approach but I would really wonder what are benefits or reasons?
L352: I don't well understand why authors do this modeling.
Section 4.1: It would be very helpful if the study area is shown as a figure.
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AC5: 'Reply on RC4', Samuel Schroers, 21 Dec 2021
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.
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AC5: 'Reply on RC4', Samuel Schroers, 21 Dec 2021
Samuel Schroers et al.
Samuel Schroers et al.
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