the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 11 Feb 2021
Hortonian Overland Flow, Hillslope Morphology and Stream Power I: Spatial Energy Distributions and Steady-state Power Maxima
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 analyzed 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 macro and micro hillslope structures control the free energy balance of Hortonian overland flow. This may ultimately help understanding why these structures have evolved to their present shape. 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 driver of erosion and sediment transport. We then use this framework to analyze 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. Additionally, we provide a first order estimate whether and when rills reduce the overall energy dissipation compared to sheet flow. Finally, we relate accumulated stream power of linear hillslopes to slope angles, closing the loop to Horton's original formulation of erosion force. The analytical analysis of stream power reveals that the common formulation, a function of the depth-discharge product is a reduced version of the more general equations if we neglect changes in velocity and discharge in space. The full equations of stream power result in maximum energy fluxes in space for sinusoidal and exponential hillslope profiles, while linear and negative exponential forms unlimitedly increase these fluxes in the downstream direction. Depending on geometry, rill flow increases or decreases kinetic energy fluxes downslope, effectively counteracting or increasing the dissipation of potential energy. For accumulated power in space for steady state runoff, we find that on linear hillslopes a slope angle of 45° maximizes the conversion of potential energy into dissipation and an angle of 35° maximizes the conversion of potential energy into kinetic energy.
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RC1: 'Comment on hess-2021-79', Anonymous Referee #1, 08 Mar 2021
Traditional analyses of flow rely on the conservation of momentum. In such exchanges, energy is generally lost, and difficult to budget but it is commonly reckoned that losses are larger than energy utilised in runoff and sediment transport.
Here there is no physical analysis of the dissipation terms, and they are deduced only by subtraction of the known terms, although neglecting some energy components, particularly raindrop energy. I do not see any derivations that add to understanding of slope or river processes. Although I am open to the notion of minimising energy dissipation, and recognise that the conventional hydraulic equations do not generally provide closure, this paper does not seem to advance our understanding.
Perhaps I am missing something. Which equation(s) shows the clear benefit of this approach?
Citation: https://doi.org/10.5194/hess-2021-79-RC1 -
AC1: 'Reply on RC1', Samuel Schroers, 09 Mar 2021
We thank the reviewer for his comments on our manuscript.
We agree with the reviewer, that we primarily quantify dissipation during overland flow based on the residual of the free energy balance, and not by comparing different flow laws (e.g. Manning to Darcy Weißbach).
This is appropriate, as free energy is additive. We claim by no means that the usual momentum centered approach to characterize overland flow is not very helpful. We suggest a complementary avenue to explore the role of macroscale controls such as different hillslope width and hillslope forms on the steady state pattern of potential energy and power in overland flow. This reveals the existence of distinct maxima, implying a maximum available force in overland flow e.g. to initial erosion or rill formation. This maximum results from the tradeoff of the downslope increasing mass in overland flow (due to a rising water table) and the decline in geo potential and is sensitive to form and width functions of the hillslopes.
Here, we have tried to separate physical from structural hydraulic losses of energy and focus only on the latter to limit the degrees of freedom. Needless to say, physical roughness may adjust as well but our focus in this study was on macroscale (hillslope form) and microscale (microtopography) structural elements and their influence on runoff power and dissipation.
We also show that the role of rills is not so straightforward as proposed e.g. in Kleidon et a. (2013), who suggest that the related increase in the hydraulic radius implies a reduction in frictional loss per volume stream flow in rivers, which led to the conclusion that rill- or stream flow is generally less dissipative than sheet flow. We showed that this might only be the case if the transition from sheet to rill flow stays within certain limits. Here again formation of structure probably goes hand in hand with adaptation of physical roughness and we agree that this interplay should be investigated in a future study in more detail.
We also agree with the reviewer, that we do not account for the kinetic energy transfer from rainfall splash to the overland flow/sediments. This is certainly a highly interesting topic but beyond the scope of this study as it depends on the drop size distribution, which controls a) mass and velocity of the raindrops and b) the Bond number and thus whether drops can be treated as being elastic or deforming. We do, however, account for the potential energy input due to different effective rainfall rates (or infiltration excess rates), the related tradeoff in potential energy mentioned above. This implies that we also do not account for the role of infiltration explicitly. We leave both for future research.
We thank the reviewer.
Citation: https://doi.org/10.5194/hess-2021-79-AC1 -
AC2: 'Reply on RC1', Samuel Schroers, 09 Mar 2021
Additionally we would like to add that we do not only find spatially distributed power maxima but further think that the presence of such maxima can help understanding the evolution of macro- and micro structure of hillslopes. To our knowledge a solid quantification of energetic conversion dynamics of surface runoff has not been done and we believe that this new perspective can contribute to a better foundation for future studies.
Citation: https://doi.org/10.5194/hess-2021-79-AC2
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AC1: 'Reply on RC1', Samuel Schroers, 09 Mar 2021
-
RC2: 'Comment on hess-2021-79', Anonymous Referee #2, 18 Apr 2021
General
The paper is a contribution to a growing program concerning possible maximum entropy production phenomena in the Earth system. It shows what happens with potential and kinetic energy in accumulating flow over slopes of different shapes and forms.
One could, as Rev 1 did, argue that such an approach does not really add much to our insight into hillslope processes beyond some trivial findings, such as that the accumulative flow over a hillslope shows different energy distributions than run-on systems such as rivers. Although I do think some clearer argumentation of the advantages or possible new insights is warranted, I would give the authors the advantage of the doubt at this point after a serious re-write.
It would, however, be a good idea to have some empirical support beyond the very general geomorphological observations presented here. One starts with some classical work (Horton, etc) and then neglects three decades with hundreds of hillslope runoff studies to come to a new approach. How does this work compare to that body of scientific literature?
Major remarks
The paper is very long and often one is lost in technical detail that would better be placed in an appendix. Perhaps a matter of taste but the writing is expansive, which further reduces clarity of argument. It reads more like a chapter in a PhD thesis than an article. If this is indeed also to be a chapter in a PhD thesis, I would recommend keeping this version (after some adjustments) as it is more complete, but in an article one would prefer a more succinct narrative that is clear and to the point.
There is a page+ list of variable names, which is good to have, but one is forced to jump back to that list to keep track of the arguments. It is good to show derivations from, more or less, first principles, but not in the main text. One could more or less start with Eq. 16 and show the relation with Horton & Bangold.
A slope that is in steady-state with Peff=50mm/h will soon cease to exist. It is ok to use it as a calculation example but one cannot then conclude (L 510) that only certain configurations are feasible because they have power maxima. The assumption that slopes regularly come into steady-state with even a Peff=5mm/h is more of the problem. This points back a bit at the lack of connection with empirical work.
Finally, a new approach or theoretical ansatz would be much stronger if testable hypotheses are generated. If not, the whole theory becomes untestable and thereby puts itself outside mainstream science. In L520 and further, one comes very close to such a hypothesis (rill initiation) that should be clearly formulated. All the cushioning words in this paragraph make this not a very clear hypothesis and thereby the theory extracts itself from falsifying experiments. Perhaps this is then the main weakness as a grand theory is presented that cannot be tested without the nitty-gritty details that are common in (empirical) hillslope studies.
Minor remarks
As this seems to be part of the larger program started by Kleidon, it is not at all clear how this in the end all fits in that framework. Perhaps this is snowed under the many side remarks made but a clear statement on this, and a much shorter introduction, would improve readability and impact.
If one uses so many variables, it is good to use them consistently. The list suggests there is a difference between Peff and I but it seems from the figures in results and the text that they are the same (l 235).
L 51: Good (but not only) example of why the write-up is not helping towards understanding the core findings. This remark does not contribute to the general findings and arguments. There are important and relevant differences between biological and river networks, as the generalization of West et al., 1997 in Banavar et al., 1999 (Nature 399:130–132) shows. Yes, they are both efficient networks with power-law properties but that is where the similarity ends.
L 194: Why would precipitation be decreasing along a 100 m slope?
L 207 Sheet flow is generally not a fully developed turbulent flow. The exponents in eqs 18&19 are usually not valid for sheet flow. In the conclusion, the authors already hint at the fact that hydraulic details need to be taken into consideration when it comes to rill initiation and the change from laminar-like to full turbulence is likely to have a lot to do with that.
L 299: What is the value of having cases 3 & 4 in Table 2?
L 454: Deduct should probably be deduce.
L 456: Punctuation.
Citation: https://doi.org/10.5194/hess-2021-79-RC2 -
AC3: 'Reply on RC2', Samuel Schroers, 23 Apr 2021
First, we thank the reviewer for her/his time, effort and her/his encouraging comments.
The main idea of our work is indeed, as pointed out by the reviewer, to show that the free energy balance of overland flow on hillslopes as runon-runoff system behaves distinctly different from river systems. Downslope flow accumulation and the decline in geopotential implies downslope increase in potential energy up to a maximum. Honestly, we regard this not as a trivial finding, at least the study of Kleidon et al. (2013) overlooked this difference.
In line with the reviewer, we think that these spatial maxima in potential energy relate to a transition zone and a related switch in dynamic behavior or at least a local hot spot. Such transitions are the onset of erosion, as the critical shear stress is exceeded, a transition from sheet to rill flow and erosion or the emergence of turbulence. The latter relates however more to the Reynolds number and thus overland flow velocity and depths. While velocity is regarded as constant during steady state conditions, it might change with time when exploring the transient evolution of an overland flow wave running downslope. While this is out of the scope of this work, we plan to investigate this in a follow up study.
We will clarify this main point in the revised manuscript, formulate a related hypothesis about the role of the potential energy maxima and test this using available rainfall simulation experiments at 10 m long stripes, which investigated Hortonian overland flow formation and erosion/ detachment in the Weiherbach catchment (Scherer et al., 2012). We already started to simulate those experiments with a spatially distributed numerical model (runoff and erosion). The output is well suited to explore the emergence of these potential energy maxima and their relation to erosion and emergence of rill flow.
We agree that the manuscript is currently overdoing the idea of reproducibility. We will thus reduce the theoretical background to the minimum necessary amount, and present the remaining details in the appendix. Yet we think it is important to show those, as thermodynamics is not part of the standard hydrological curriculum. The advantage of the concept of free energy, is that we can assess Dissipation as a residual of the energy and the free energy balance.
We agree that a hillslope in steady state with Peff=50mm/h will not stand for a long time. We will clarify that we refer to the steady state phase during a rainfall event. The aforementioned experiments of Scherer et al. (2012) where carried out using 60 mm of rainfall in 1 h. At some sites the runoff coefficient was 80%, implying an effective rainfall of 46 mm in 1 h. While this caused substantial erosion, the hillslope still exists. We furthermore think that steady state overland flow conditions during events might occur less often than assumed, as we implicitly impose them when using steady state approximations of the Navier Stokes equation to simulate overland flow. Only rainfall events with sufficient duration will cause steady state phase, following on a rise and followed by a recession phase. This question is beyond the scope of this work but will be addressed in a follow up study.
For the presentation of the general theory however we think that it is valuable to explore different effective rainfall intensities/ infiltration excess rates in order to highlight that stream power is direct proportional to rainfall intensities. We will, however, better explain that such high intensities will cause substantial, transient erosion and explain that this will rarely occur.
Referring now to the third and last paragraph of the review we thank Rev 2 for the discovered minor errors. Indeed, since we exclude infiltration Peff and I represent the same value. Next, we are not intending to claim that biological and river networks share significant similarities, rather we intend to note that both fields had a very similar motivation in describing networks, that is thermodynamic principals. In L194 refers to the influx of energy caused by precipitation in comparison to the influx of energy by upslope runon Q, we agree that this sentence needs reformulation. L207, we are aware of the problematic by describing sheetflow with channel flow equations, however we think that there is a need to develop more robust concepts of very shallow flow, the parameterization of friction and the resulting energy dynamics. The interesting observation for hillslope surface runoff is its transitional nature and we believe that understanding the underlying energetic dynamics might help separating prevalent flow characteristics (such as laminar and turbulent flow). Cases 3 and 4 of table 2, have been added for comparison. One could argue that they add not additional value but we think it is interesting to highlight that both derivatives, of discharge and and flow velocity contribute (although to a much lesser extent) to dissipation.
Overall, we thank Rev 2 again for his observations and findings. We think the incorporation of an empirical case is a good idea and further agree that readability would benefit from a more concise straight to the point introduction.
Citation: https://doi.org/10.5194/hess-2021-79-AC3
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AC3: 'Reply on RC2', Samuel Schroers, 23 Apr 2021
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RC3: 'Comment on hess-2021-79', Hubert H.G. Savenije, 26 May 2021
Review of hess-2021-79 by Schroers et al.
This is a very interesting paper, investigating a new perspective on the very traditional hydrological process of surface runoff. Horton being maybe the first hydrologist to produce a realistic surface runoff equation, his theory has become so well known that it is considered by most hydrologists as a "physical" fact. Of course, Horton's theory is essentially empirical, as are other steady state equations by e.g. Manning, Chézy and the like. They are all based on a parametrization of energy dissipation. If there exists a physical basis for these empirical equations -- and this is what we can safely assume -- then surely the theory of Entropy production or Maximum Power is a strong candidate for this. So I welcome this paper in this quest, even though the issue is not entirely "solved".
I have a few observations and questions.
- In an open hydrological system, dissipation and conversion of free energy cannot be limited to a single subsystem. The hydrological system of rainfall-runoff consists of a number of interacting pathways, including surface runoff, open channel flow, subsurface flow, and groundwater flow, each with its own dissipative character. It would be my assumption that these processes are interacting components of the same dissipative structure and cannot be considered in isolation in a thermodynamic framework. In this article, the interaction between the surface runoff system and the other subsurface processes is represented by the exchange term Jpe inf,out . It is a strong assumption that this variable is either constant or independent of other system components, such as subsurface flow and groundwater flow. Maybe the authors can give their view on this aspect.
- The authors try out different hillslope forms (linear, sinusoidal, exponential, negative exponential). From what I have seen in nature, exponential is the dominant form, with linear being a first order approximation. Is there any thermodynamic justification for either one of these shapes?
- I would be curious to know whether or how this theory agrees with field or laboratory experiments, of which many have been documented in the literature. Maybe a thorough empirical analysis is too much for this paper, but some considerations would be welcome.
- I think it is Bagnold and not Bangold (occurring at six times in the text and also in the parameters)
Citation: https://doi.org/10.5194/hess-2021-79-RC3 -
AC4: 'Reply on RC3', Samuel Schroers, 27 May 2021
We sincerely thank Hubert Savenije for his thoughtful and constructive assessment of our manuscript.
First, we agree that energy conversion and dissipation in hydrological systems happens essentially also a) within the partially saturated soil and b) by in-/exfiltration affecting the energy balance. In fact, water infiltration into the soil and subsequent soil water dynamics is associated with conversion and dissipation of potential energy and matric/capillary potential energy. While the former grows with growing soil water content and elevation above ground water, the latter grows non-linearly with declining soil water content (Zehe et al., 2019). Particularly in cohesive soil, capillary potential energy may become very large during dry spells and flow resistances/ frictional dissipation grow non-linearly. Macropores considerably reduce dissipative losses and allow for a faster recharge of dry soil, which implies a faster depletion of matric potential energy gradients (Zehe et al. 2010). The fact that during infiltration gravitational potential energy of soil water grows and the non-linear shape of the retention curve, implies the existence of an infiltration capacity that maximizes dissipation of total free energy in soil water (Zehe et al., 2013). We therefore want to stress, that the main objective of our study was not a complete assessment of dissipation in hydrological systems. Rather, we want to point out that free energy conversions associated with surface runoff on hillslopes differ distinctly from the energetics of stream flow. The latter is associated with downstream potential energy loss while on the hillslope potential energy first builds up in downslope direction until reaching a local maximum and declines afterwards. Note that this implies that frictional laws like Darcy Weißbach are not well defined upslope of the maximum, as dEpot/dx > 0, and it might imply a much stronger erosion upslope as will be shown in the revised manuscript.
The second comment made by Hubert Savenije points to one of the objectives of this study. Kirkby (1971) suggested that commonly observed hillslope forms are related to the dominant erosion processes while Emmett (1970) points out that a sinusoidal hillslope profile is the result of the transition from laminar to turbulent surface runoff. It seems that due to the complexity and amount of interacting processes it is difficult to single out any process alone. Yet all hydrological processes are in its essence the result of energy conversion, dissipating free energy. Therefore we think it is especially interesting considering a thermodynamic perspective for hillslope development. The hillslope profiles in our study have been chosen in line with these preceding investigations and have been analyzed with surface runoff only. Nonetheless our results show that the location of energetic maxima, the magnitude of the energy gradient and the emergence of a power maximum are directly linked to these forms. A conclusion on how complex hillslope profiles in nature relate to their energy conversion processes would require an additional study and was beyond the scope of this paper.
However, in perspective to the third comment made by Hubert Savenije (as well as Rev 2), we agree that the study will benefit from the incorporation of some experiments and a subsequent analysis, applying our theory. Therefore, we have already prepared the results of some field experiments of surface runoff and erosion processes on 12m x 2m plots from the Weiherbach catchment, which we would gladly analyze and test in a revised manuscript.
We greatly appreciate the comments and observations made.
Additional Sources:
Zehe, Erwin; Blume, Theresa; Blöschl, Günter (2010): The principle of 'maximum energy dissipation': a novel thermodynamic perspective on rapid water flow in connected soil structures. In: Philosophical transactions of the Royal Society of London. Series B, Biological sciences 365 (1545), S. 1377–1386. DOI: 10.1098/rstb.2009.0308.
Zehe, Erwin; Loritz, Ralf; Jackisch, Conrad; Westhoff, Martijn; Kleidon, Axel; Blume, Theresa et al. (2019): Energy states of soil water – a thermodynamic perspective on soil water dynamics and storage-controlled streamflow generation in different landscapes. In: Hydrol. Earth Syst. Sci. 23 (2), S. 971–987. DOI: 10.5194/hess-23-971-2019.
Citation: https://doi.org/10.5194/hess-2021-79-AC4
Status: closed
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RC1: 'Comment on hess-2021-79', Anonymous Referee #1, 08 Mar 2021
Traditional analyses of flow rely on the conservation of momentum. In such exchanges, energy is generally lost, and difficult to budget but it is commonly reckoned that losses are larger than energy utilised in runoff and sediment transport.
Here there is no physical analysis of the dissipation terms, and they are deduced only by subtraction of the known terms, although neglecting some energy components, particularly raindrop energy. I do not see any derivations that add to understanding of slope or river processes. Although I am open to the notion of minimising energy dissipation, and recognise that the conventional hydraulic equations do not generally provide closure, this paper does not seem to advance our understanding.
Perhaps I am missing something. Which equation(s) shows the clear benefit of this approach?
Citation: https://doi.org/10.5194/hess-2021-79-RC1 -
AC1: 'Reply on RC1', Samuel Schroers, 09 Mar 2021
We thank the reviewer for his comments on our manuscript.
We agree with the reviewer, that we primarily quantify dissipation during overland flow based on the residual of the free energy balance, and not by comparing different flow laws (e.g. Manning to Darcy Weißbach).
This is appropriate, as free energy is additive. We claim by no means that the usual momentum centered approach to characterize overland flow is not very helpful. We suggest a complementary avenue to explore the role of macroscale controls such as different hillslope width and hillslope forms on the steady state pattern of potential energy and power in overland flow. This reveals the existence of distinct maxima, implying a maximum available force in overland flow e.g. to initial erosion or rill formation. This maximum results from the tradeoff of the downslope increasing mass in overland flow (due to a rising water table) and the decline in geo potential and is sensitive to form and width functions of the hillslopes.
Here, we have tried to separate physical from structural hydraulic losses of energy and focus only on the latter to limit the degrees of freedom. Needless to say, physical roughness may adjust as well but our focus in this study was on macroscale (hillslope form) and microscale (microtopography) structural elements and their influence on runoff power and dissipation.
We also show that the role of rills is not so straightforward as proposed e.g. in Kleidon et a. (2013), who suggest that the related increase in the hydraulic radius implies a reduction in frictional loss per volume stream flow in rivers, which led to the conclusion that rill- or stream flow is generally less dissipative than sheet flow. We showed that this might only be the case if the transition from sheet to rill flow stays within certain limits. Here again formation of structure probably goes hand in hand with adaptation of physical roughness and we agree that this interplay should be investigated in a future study in more detail.
We also agree with the reviewer, that we do not account for the kinetic energy transfer from rainfall splash to the overland flow/sediments. This is certainly a highly interesting topic but beyond the scope of this study as it depends on the drop size distribution, which controls a) mass and velocity of the raindrops and b) the Bond number and thus whether drops can be treated as being elastic or deforming. We do, however, account for the potential energy input due to different effective rainfall rates (or infiltration excess rates), the related tradeoff in potential energy mentioned above. This implies that we also do not account for the role of infiltration explicitly. We leave both for future research.
We thank the reviewer.
Citation: https://doi.org/10.5194/hess-2021-79-AC1 -
AC2: 'Reply on RC1', Samuel Schroers, 09 Mar 2021
Additionally we would like to add that we do not only find spatially distributed power maxima but further think that the presence of such maxima can help understanding the evolution of macro- and micro structure of hillslopes. To our knowledge a solid quantification of energetic conversion dynamics of surface runoff has not been done and we believe that this new perspective can contribute to a better foundation for future studies.
Citation: https://doi.org/10.5194/hess-2021-79-AC2
-
AC1: 'Reply on RC1', Samuel Schroers, 09 Mar 2021
-
RC2: 'Comment on hess-2021-79', Anonymous Referee #2, 18 Apr 2021
General
The paper is a contribution to a growing program concerning possible maximum entropy production phenomena in the Earth system. It shows what happens with potential and kinetic energy in accumulating flow over slopes of different shapes and forms.
One could, as Rev 1 did, argue that such an approach does not really add much to our insight into hillslope processes beyond some trivial findings, such as that the accumulative flow over a hillslope shows different energy distributions than run-on systems such as rivers. Although I do think some clearer argumentation of the advantages or possible new insights is warranted, I would give the authors the advantage of the doubt at this point after a serious re-write.
It would, however, be a good idea to have some empirical support beyond the very general geomorphological observations presented here. One starts with some classical work (Horton, etc) and then neglects three decades with hundreds of hillslope runoff studies to come to a new approach. How does this work compare to that body of scientific literature?
Major remarks
The paper is very long and often one is lost in technical detail that would better be placed in an appendix. Perhaps a matter of taste but the writing is expansive, which further reduces clarity of argument. It reads more like a chapter in a PhD thesis than an article. If this is indeed also to be a chapter in a PhD thesis, I would recommend keeping this version (after some adjustments) as it is more complete, but in an article one would prefer a more succinct narrative that is clear and to the point.
There is a page+ list of variable names, which is good to have, but one is forced to jump back to that list to keep track of the arguments. It is good to show derivations from, more or less, first principles, but not in the main text. One could more or less start with Eq. 16 and show the relation with Horton & Bangold.
A slope that is in steady-state with Peff=50mm/h will soon cease to exist. It is ok to use it as a calculation example but one cannot then conclude (L 510) that only certain configurations are feasible because they have power maxima. The assumption that slopes regularly come into steady-state with even a Peff=5mm/h is more of the problem. This points back a bit at the lack of connection with empirical work.
Finally, a new approach or theoretical ansatz would be much stronger if testable hypotheses are generated. If not, the whole theory becomes untestable and thereby puts itself outside mainstream science. In L520 and further, one comes very close to such a hypothesis (rill initiation) that should be clearly formulated. All the cushioning words in this paragraph make this not a very clear hypothesis and thereby the theory extracts itself from falsifying experiments. Perhaps this is then the main weakness as a grand theory is presented that cannot be tested without the nitty-gritty details that are common in (empirical) hillslope studies.
Minor remarks
As this seems to be part of the larger program started by Kleidon, it is not at all clear how this in the end all fits in that framework. Perhaps this is snowed under the many side remarks made but a clear statement on this, and a much shorter introduction, would improve readability and impact.
If one uses so many variables, it is good to use them consistently. The list suggests there is a difference between Peff and I but it seems from the figures in results and the text that they are the same (l 235).
L 51: Good (but not only) example of why the write-up is not helping towards understanding the core findings. This remark does not contribute to the general findings and arguments. There are important and relevant differences between biological and river networks, as the generalization of West et al., 1997 in Banavar et al., 1999 (Nature 399:130–132) shows. Yes, they are both efficient networks with power-law properties but that is where the similarity ends.
L 194: Why would precipitation be decreasing along a 100 m slope?
L 207 Sheet flow is generally not a fully developed turbulent flow. The exponents in eqs 18&19 are usually not valid for sheet flow. In the conclusion, the authors already hint at the fact that hydraulic details need to be taken into consideration when it comes to rill initiation and the change from laminar-like to full turbulence is likely to have a lot to do with that.
L 299: What is the value of having cases 3 & 4 in Table 2?
L 454: Deduct should probably be deduce.
L 456: Punctuation.
Citation: https://doi.org/10.5194/hess-2021-79-RC2 -
AC3: 'Reply on RC2', Samuel Schroers, 23 Apr 2021
First, we thank the reviewer for her/his time, effort and her/his encouraging comments.
The main idea of our work is indeed, as pointed out by the reviewer, to show that the free energy balance of overland flow on hillslopes as runon-runoff system behaves distinctly different from river systems. Downslope flow accumulation and the decline in geopotential implies downslope increase in potential energy up to a maximum. Honestly, we regard this not as a trivial finding, at least the study of Kleidon et al. (2013) overlooked this difference.
In line with the reviewer, we think that these spatial maxima in potential energy relate to a transition zone and a related switch in dynamic behavior or at least a local hot spot. Such transitions are the onset of erosion, as the critical shear stress is exceeded, a transition from sheet to rill flow and erosion or the emergence of turbulence. The latter relates however more to the Reynolds number and thus overland flow velocity and depths. While velocity is regarded as constant during steady state conditions, it might change with time when exploring the transient evolution of an overland flow wave running downslope. While this is out of the scope of this work, we plan to investigate this in a follow up study.
We will clarify this main point in the revised manuscript, formulate a related hypothesis about the role of the potential energy maxima and test this using available rainfall simulation experiments at 10 m long stripes, which investigated Hortonian overland flow formation and erosion/ detachment in the Weiherbach catchment (Scherer et al., 2012). We already started to simulate those experiments with a spatially distributed numerical model (runoff and erosion). The output is well suited to explore the emergence of these potential energy maxima and their relation to erosion and emergence of rill flow.
We agree that the manuscript is currently overdoing the idea of reproducibility. We will thus reduce the theoretical background to the minimum necessary amount, and present the remaining details in the appendix. Yet we think it is important to show those, as thermodynamics is not part of the standard hydrological curriculum. The advantage of the concept of free energy, is that we can assess Dissipation as a residual of the energy and the free energy balance.
We agree that a hillslope in steady state with Peff=50mm/h will not stand for a long time. We will clarify that we refer to the steady state phase during a rainfall event. The aforementioned experiments of Scherer et al. (2012) where carried out using 60 mm of rainfall in 1 h. At some sites the runoff coefficient was 80%, implying an effective rainfall of 46 mm in 1 h. While this caused substantial erosion, the hillslope still exists. We furthermore think that steady state overland flow conditions during events might occur less often than assumed, as we implicitly impose them when using steady state approximations of the Navier Stokes equation to simulate overland flow. Only rainfall events with sufficient duration will cause steady state phase, following on a rise and followed by a recession phase. This question is beyond the scope of this work but will be addressed in a follow up study.
For the presentation of the general theory however we think that it is valuable to explore different effective rainfall intensities/ infiltration excess rates in order to highlight that stream power is direct proportional to rainfall intensities. We will, however, better explain that such high intensities will cause substantial, transient erosion and explain that this will rarely occur.
Referring now to the third and last paragraph of the review we thank Rev 2 for the discovered minor errors. Indeed, since we exclude infiltration Peff and I represent the same value. Next, we are not intending to claim that biological and river networks share significant similarities, rather we intend to note that both fields had a very similar motivation in describing networks, that is thermodynamic principals. In L194 refers to the influx of energy caused by precipitation in comparison to the influx of energy by upslope runon Q, we agree that this sentence needs reformulation. L207, we are aware of the problematic by describing sheetflow with channel flow equations, however we think that there is a need to develop more robust concepts of very shallow flow, the parameterization of friction and the resulting energy dynamics. The interesting observation for hillslope surface runoff is its transitional nature and we believe that understanding the underlying energetic dynamics might help separating prevalent flow characteristics (such as laminar and turbulent flow). Cases 3 and 4 of table 2, have been added for comparison. One could argue that they add not additional value but we think it is interesting to highlight that both derivatives, of discharge and and flow velocity contribute (although to a much lesser extent) to dissipation.
Overall, we thank Rev 2 again for his observations and findings. We think the incorporation of an empirical case is a good idea and further agree that readability would benefit from a more concise straight to the point introduction.
Citation: https://doi.org/10.5194/hess-2021-79-AC3
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AC3: 'Reply on RC2', Samuel Schroers, 23 Apr 2021
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RC3: 'Comment on hess-2021-79', Hubert H.G. Savenije, 26 May 2021
Review of hess-2021-79 by Schroers et al.
This is a very interesting paper, investigating a new perspective on the very traditional hydrological process of surface runoff. Horton being maybe the first hydrologist to produce a realistic surface runoff equation, his theory has become so well known that it is considered by most hydrologists as a "physical" fact. Of course, Horton's theory is essentially empirical, as are other steady state equations by e.g. Manning, Chézy and the like. They are all based on a parametrization of energy dissipation. If there exists a physical basis for these empirical equations -- and this is what we can safely assume -- then surely the theory of Entropy production or Maximum Power is a strong candidate for this. So I welcome this paper in this quest, even though the issue is not entirely "solved".
I have a few observations and questions.
- In an open hydrological system, dissipation and conversion of free energy cannot be limited to a single subsystem. The hydrological system of rainfall-runoff consists of a number of interacting pathways, including surface runoff, open channel flow, subsurface flow, and groundwater flow, each with its own dissipative character. It would be my assumption that these processes are interacting components of the same dissipative structure and cannot be considered in isolation in a thermodynamic framework. In this article, the interaction between the surface runoff system and the other subsurface processes is represented by the exchange term Jpe inf,out . It is a strong assumption that this variable is either constant or independent of other system components, such as subsurface flow and groundwater flow. Maybe the authors can give their view on this aspect.
- The authors try out different hillslope forms (linear, sinusoidal, exponential, negative exponential). From what I have seen in nature, exponential is the dominant form, with linear being a first order approximation. Is there any thermodynamic justification for either one of these shapes?
- I would be curious to know whether or how this theory agrees with field or laboratory experiments, of which many have been documented in the literature. Maybe a thorough empirical analysis is too much for this paper, but some considerations would be welcome.
- I think it is Bagnold and not Bangold (occurring at six times in the text and also in the parameters)
Citation: https://doi.org/10.5194/hess-2021-79-RC3 -
AC4: 'Reply on RC3', Samuel Schroers, 27 May 2021
We sincerely thank Hubert Savenije for his thoughtful and constructive assessment of our manuscript.
First, we agree that energy conversion and dissipation in hydrological systems happens essentially also a) within the partially saturated soil and b) by in-/exfiltration affecting the energy balance. In fact, water infiltration into the soil and subsequent soil water dynamics is associated with conversion and dissipation of potential energy and matric/capillary potential energy. While the former grows with growing soil water content and elevation above ground water, the latter grows non-linearly with declining soil water content (Zehe et al., 2019). Particularly in cohesive soil, capillary potential energy may become very large during dry spells and flow resistances/ frictional dissipation grow non-linearly. Macropores considerably reduce dissipative losses and allow for a faster recharge of dry soil, which implies a faster depletion of matric potential energy gradients (Zehe et al. 2010). The fact that during infiltration gravitational potential energy of soil water grows and the non-linear shape of the retention curve, implies the existence of an infiltration capacity that maximizes dissipation of total free energy in soil water (Zehe et al., 2013). We therefore want to stress, that the main objective of our study was not a complete assessment of dissipation in hydrological systems. Rather, we want to point out that free energy conversions associated with surface runoff on hillslopes differ distinctly from the energetics of stream flow. The latter is associated with downstream potential energy loss while on the hillslope potential energy first builds up in downslope direction until reaching a local maximum and declines afterwards. Note that this implies that frictional laws like Darcy Weißbach are not well defined upslope of the maximum, as dEpot/dx > 0, and it might imply a much stronger erosion upslope as will be shown in the revised manuscript.
The second comment made by Hubert Savenije points to one of the objectives of this study. Kirkby (1971) suggested that commonly observed hillslope forms are related to the dominant erosion processes while Emmett (1970) points out that a sinusoidal hillslope profile is the result of the transition from laminar to turbulent surface runoff. It seems that due to the complexity and amount of interacting processes it is difficult to single out any process alone. Yet all hydrological processes are in its essence the result of energy conversion, dissipating free energy. Therefore we think it is especially interesting considering a thermodynamic perspective for hillslope development. The hillslope profiles in our study have been chosen in line with these preceding investigations and have been analyzed with surface runoff only. Nonetheless our results show that the location of energetic maxima, the magnitude of the energy gradient and the emergence of a power maximum are directly linked to these forms. A conclusion on how complex hillslope profiles in nature relate to their energy conversion processes would require an additional study and was beyond the scope of this paper.
However, in perspective to the third comment made by Hubert Savenije (as well as Rev 2), we agree that the study will benefit from the incorporation of some experiments and a subsequent analysis, applying our theory. Therefore, we have already prepared the results of some field experiments of surface runoff and erosion processes on 12m x 2m plots from the Weiherbach catchment, which we would gladly analyze and test in a revised manuscript.
We greatly appreciate the comments and observations made.
Additional Sources:
Zehe, Erwin; Blume, Theresa; Blöschl, Günter (2010): The principle of 'maximum energy dissipation': a novel thermodynamic perspective on rapid water flow in connected soil structures. In: Philosophical transactions of the Royal Society of London. Series B, Biological sciences 365 (1545), S. 1377–1386. DOI: 10.1098/rstb.2009.0308.
Zehe, Erwin; Loritz, Ralf; Jackisch, Conrad; Westhoff, Martijn; Kleidon, Axel; Blume, Theresa et al. (2019): Energy states of soil water – a thermodynamic perspective on soil water dynamics and storage-controlled streamflow generation in different landscapes. In: Hydrol. Earth Syst. Sci. 23 (2), S. 971–987. DOI: 10.5194/hess-23-971-2019.
Citation: https://doi.org/10.5194/hess-2021-79-AC4
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