the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 05 Aug 2026
Climate and human organization decouple geometric and functional scaling in river networks
Abstract. Scaling laws are widely regarded as fundamental organizing principles of river networks, however, whether different scaling relationships emerge from common or distinct mechanisms remains unresolved. Using 41,677 rivers across China, we show that geometric and functional scaling exhibit fundamentally different sensitivities to climatic and anthropogenic forcing. The classical Hack’s law relationship between river length and drainage area remains remarkably stable across river hierarchies, climatic gradients, and major basins, indicating a highly conserved geometric organization of river networks. In contrast, runoff-efficiency scaling varies systematically with precipitation, transitioning from strongly negative exponents in arid regions to near-zero values in humid environments. This climatic dependence is associated with hydrological connectivity: river-network structure suppresses runoff efficiency, whereas lake systems enhance the scaling efficiency of runoff generation and transport across basin sizes. Furthermore, administrative fragmentation weakens natural geometric scaling while generating apparent functional scaling relationships absent in intact drainage systems. These findings reveal a fundamental asymmetry in river-network organization: geometric scaling remains highly conserved, whereas functional scaling is strongly shaped by climate, connectivity, and human spatial partitioning. Our results suggest that the organizing principles governing river systems are not fixed properties of natural landscapes, but emergent behaviors continually reshaped by climate variability, hydrological connectivity, and human modification.
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Status: open (until 08 Oct 2026)
- RC1: 'Comment on egusphere-2026-4551', Anonymous Referee #1, 04 Sep 2026 reply
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RC2: 'Comment on egusphere-2026-4551', Anonymous Referee #2, 21 Sep 2026
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General comments:
The manuscript addresses an interesting question and makes use of a large river dataset. However, I have substantial concerns regarding the conceptual framework, the design and interpretation of the administrative-fragmentation experiment, and the interpretation of several key results. These issues affect the main conclusions of the manuscript and, in my view, require substantial reconsideration. Please see my detailed comments below.
Major comments:
1. Although the manuscript presents analyses of both geometric scaling (Hack’s law) and what the authors refer to as functional scaling (runoff-efficiency scaling), the two analyses appear largely juxtaposed rather than conceptually or quantitatively connected. The paper lacks a clear explanation of how these two scaling relationships are expected to interact, and what additional understanding is gained by analyzing them together.
The manuscript showed that Hack’s law is relatively stable across the examined conditions, whereas the exponent of the R/P–area relationship varies with precipitation. However, observing different sensitivities of two different relationships does not by itself demonstrate that geometric and functional scaling are “decoupled.” The manuscript first needs to establish what coupling between these two scaling relationships would mean. For example, is there a theoretical expectation that the Hack exponent h and runoff-efficiency exponent beta should covary? Is there a hypothesized pathway through which network geometry should influence beta? Without establishing such a connection or expectation, it is difficult to interpret the observed differences as evidence of “decoupling.”
Relatedly, Hack’s law characterizes one particular aspect of basin/network geometry-the relationship between river length and drainage area, while the additional network structure metric defined by Eq. 7 characterizes the relative elongation of drainage pathways. Both metrics are derived solely from river length and drainage area and do not characterize other important aspects of network geometry, such as branching patterns and topology. However, the conclusions throughout the manuscript extend from these metrics to much broader statements about “river-network geometry,” “network architecture,” or “drainage structure.” These broader interpretations require substantially more justification or additional analysis.
2. I have substantial concerns about the administrative-fragmentation analysis, both methodologically and conceptually.
The method assumes that both river length and drainage area are equally distributed among all administrative units crossed by a river, such that Ladmin=L/n and Aadmin=A/n. However, administrative boundaries clearly do not partition river length and drainage area into equal fractions. This assumption is spatially unrealistic and can itself affect the resulting scaling relationships. In logarithmic space,
log Ladmin = logL-logn
and
log Aadmin = logA-logn
Thus, the same additional variable, logn, is introduced into both axes of the regression. The resulting change in regression slope may therefore arise mathematically from the imposed transformation itself. A more spatially meaningful analysis would require intersecting the actual river and basin geometries with administrative boundaries and calculating river length and contributing drainage area for the resulting spatial units.
This issue directly affects the interpretation of Fig. 4. The reported Hack exponent changes from h=0.573 to 0.473, while beta changes from 0.008 to 0.022. However, for the latter relationship, R^2 changes only from approximately 0.000 to 0.001, indicating essentially no explanatory power in either case.
More fundamentally, even if the administrative fragments were defined using actual spatial boundaries, it remains unclear what physical hypothesis this experiment is intended to test. Administrative partitioning changes the spatial units used to calculate the scaling relationships, but it does not change the underlying river network, flow pathways, runoff generation processes, or hydrological connectivity. The manuscript nevertheless draws the much stronger conclusion that functional scaling is strongly shaped by “climate, connectivity, and human spatial partitioning.” However, the analysis does not demonstrate that human spatial partitioning constitutes a physical control on river-network scaling.
Finally, several important methodological details are missing. The manuscript does not clearly show what administrative boundaries are used or how the administrative units are defined. It is also unclear how runoff efficiency is assigned to the resulting administrative fragments. Are P and R separately calculated for each fragment? How are these values obtained?
3. The analysis of river-level effects is difficult to follow because the manuscript introduces two different scaling metrics without clearly distinguishing how they are used. The Methods state that Hack’s law is fitted separately for different river-level classes and defines a “river-level scaling index,” hi, in Eq. (5). However, hi is mathematically different from the Hack exponent h defined in Eq. (4) and the two are not directly comparable.
It is unclear where the results based on hi are presented or how this index is used in the subsequent analyses. For example, is Fig. 1b based on the river-level scaling index hi, or on Hack exponents h obtained by separately fitting Eq. (4) for each river-level class? The authors should clearly document which metric is used in each analysis and figure, how hi is calculated and interpreted, and why this additional index is needed. Without this information, it is difficult to follow or evaluate the analyses of river-level effects.
4. The manuscript appropriately acknowledges in the Methods that the reported relationships should be interpreted as statistical associations rather than causal estimates. However, the manuscript subsequently makes much stronger mechanistic statements throughout the results and discussions. For example, the negative partial relationship between S and R/P in Fig. 3a is interpreted as evidence that longer transport pathways increase evaporation, infiltration, transmission losses, and flow dispersion. However, the proposed mechanisms are neither directly tested by the analyses nor supported by appropriate references. The same issue occurs in the interpretation of Figs. 2, 4.
5. The manuscript frequently interprets its results in terms of hydrological connectivity, but it is unclear how hydrological connectivity is actually quantified. The two quantities examined in Section 2.5 are the relative elongation of drainage pathways (S), and lake density. Neither is a direct measure of hydrological connectivity. In particular, S is fundamentally a geometric metric derived from river length and drainage area, and the manuscript does not establish why variation in S can be interpreted as variation in hydrological connectivity. Similarly, results involving lake density are frequently generalized into statements about “hydrological connectivity.” However, hydrological connectivity encompasses much more than the abundance of lakes (Wohl et al., 2018).
Reference:
Wohl, E., Brierley, G., Cadol, D., Coulthard, T. J., Covino, T., Fryirs, K. A., Grant, G., Hilton, R. G., Lane, S. N., Magilligan, F. J., Meitzen, K. M., Passalacqua, P., Poeppl, R. E., Rathburn, S. L., and Sklar, L. S. (2019) Connectivity as an emergent property of geomorphic systems. Earth Surf. Process. Landforms, 44: 4–26. https://doi.org/10.1002/esp.4434.
Specific comments:
L83, 90: Why are negative river length and drainage area values present in the original dataset? Similarly, L90-93 suggests that records with non-positive precipitation and runoff also occur. Please report the number/fraction of records removed for each criterion and explain why these physically non-negative quantities contain negative values. This information is important for evaluating the quality of the underlying dataset.
L108: “River level” is repeatedly used throughout the manuscript but is not clearly defined. How are river levels defined?
L161: The study analyzes 41,677 rivers across China, yet neither the main manuscript nor the Supplement provides a map showing their spatial distribution. A map is essential for understanding the dataset and interpreting the analyses.
L167: The statement that “river morphology differs substantially among levels” needs supporting evidence. What aspects of morphology were quantified, and where are these differences demonstrated?
L171–173: Figure 1c is described as showing “representative river-network structures” corresponding to different river levels. However, it appears to be a conceptual illustration rather than an analysis of actual river networks. How were these structures generated or selected? Without this information, I do not understand how Fig. 1c supports the statement that branching complexity and network architecture differ substantially among levels.
L174–179: The statement that wetter regions “support slightly stronger length–area scaling” appears to overinterpret the observed statistical relationship. What does “stronger scaling” physically mean here?
Figure 1d: It is unclear how Fig. 1d was generated. Are rivers grouped into precipitation bins and a separate Hack exponent fitted within each bin? 1d relate to the analysis shown in Fig. S1?
L180–181: What exactly are meant by “river-network geometry”, “network structure” and “major river basins” here?
L183: The statement that river-level effects are generally small also seems somewhat inconsistent with Fig. 1b, where h appears to increase systematically from river levels 1 through 5.
L190: What is meant by “large-scale reorganization of drainage structure”?
L213: How are “arid” and “humid” regions defined? The classification criteria need to be explicitly stated in the Methods.
L215–220: Again, the physical explanation offered here is speculative. The manuscript attributes strongly negative beta in arid regions to cumulative evaporation, infiltration, and channel transmission losses and attributes near-zero beta in humid regions to stronger hydrological connectivity. However, these processes are not directly evaluated in the presented analysis, nor are relevant references provided to support these specific mechanistic interpretations.
L248: The statement “climatic forcing alone cannot explain how hydrological functioning emerges across river networks” requires justification. What result demonstrates that climatic forcing alone is insufficient?
L250: What exactly does the “network-structure proxy” S represent physically? The manuscript describes it as relative elongation of drainage pathways, but subsequently interprets its effects in terms of hydrological connectivity. The link between these concepts needs to be established.
Tables S2–S3: These tables provide basin names and precipitation values, while noting that basin names refer to river systems rather than individual rivers and may therefore appear repeatedly. In their current form, the tables provide little information about the spatial context of the analyzed rivers and are difficult to interpret. The authors should provide spatial information for individual river records, ideally through the underlying spatial dataset/shapefile or, at minimum, coordinates and a map showing their locations and climatic classifications.
Citation: https://doi.org/10.5194/egusphere-2026-4551-RC2 -
RC3: 'Comment on egusphere-2026-4551', Anonymous Referee #3, 25 Sep 2026
reply
General Comments
I found the topic of this manuscript highly interesting. River-network scaling is a classical geomorphic question that has been studied for more than half a century. The attempt to examine drainage-network organization from the perspective of water availability and runoff efficiency is potentially valuable and recalls previous studies of climatic controls on network geometry, such as tributary junction angles.
However, the central research question is not yet sufficiently focused. Hack’s law, runoff efficiency, hydrological connectivity, lake abundance, and administrative fragmentation are analyzed together, but the theoretical and quantitative connections among them remain unclear. In addition, these variables represent processes operating over very different spatial and temporal scales. Basin geometry develops over geomorphic timescales, whereas precipitation, runoff, lake inventories, and administrative boundaries represent contemporary conditions or observation units. The present analyses do not adequately resolve these scale differences or establish the causal relationships implied throughout the manuscript.
The manuscript is based on a large dataset and addresses a potentially important question. Nevertheless, several central interpretations require fundamental reconsideration. I therefore do not consider the manuscript suitable for publication in its present form.Main Concerns
1. The term “functional scaling” is introduced without sufficient definition or theoretical background. Before concluding that “geometric and functional scaling exhibit fundamentally different sensitivities to climatic and anthropogenic forcing,” the authors need to define what coupling between these scaling relationships would mean. What physical pathway connects drainage geometry to runoff-efficiency scaling?2. I do not agree that Hack’s law can presently be described as “remarkably stable” or as evidence of a “highly conserved geometric organization.” Regional differences in Hack exponents have been used to examine climatic, tectonic, lithological, and geomorphic controls. The authors should define objectively what constitutes stable or conserved scaling.
3. Similar Hack exponents do not necessarily indicate similar drainage geometry. Comparable values of h may result from different combinations of network topology.
4. The use of R/P as a measure of river-network functioning requires stronger justification. This ratio is an integrated catchment water-balance measure, but it does not necessarily represent the efficiency with which a river network collects, transports, or hydrologically connects water. The authors should explain why R/P can be interpreted as river-network functioning and provide a theoretical or literature basis for assuming a power-law relationship between R/P and drainage area.
5. The inferential pathway linking Hack’s law, hydrological connectivity, and runoff efficiency is missing. Similarly, lake density alone does not quantify connectivity because lakes can disconnect flow depending on their position within the network.
6. The manuscript is inconsistent in describing Hack scaling as both climatically responsive and remarkably stable.
7. The 41,677 river records cannot be regarded as independent observations. River reaches and basins within the same drainage network are likely nested and may share upstream area, precipitation, runoff, topography, and geological conditions. The authors should clarify what each river record represents and account for hierarchical dependence and spatial autocorrelation.
8. The administrative-fragmentation experiment does not represent actual human disturbance, water allocation, or management.
9. The lake-density analysis contains a major spatial-scale mismatch. Lake area is aggregated at the provincial level, whereas drainage area and runoff efficiency are analyzed at the individual-river level.
Specific Comments
Equations (4) and (5) are incorrectly formulated.Lines 74–92: The sources, units, spatial resolution, and averaging periods of precipitation and runoff are unclear. Are they basin-averaged observations, station measurements, or modelled values?
Lines 78–81 and 132–140: Please explain exactly how province-level lake area was assigned to individual rivers.
Lines 160–180 and Figure 1d: The climate-related variation in h is not sufficiently reported. Please provide the precipitation bins or windows, sample sizes, the complete range of h, confidence intervals, and variance.
Lines 295–309: The functional exponent changes only from 0.008 to 0.022, while the explanatory power appears to remain effectively zero.
Lines 353–359: Connectivity is described as the mechanism linking climate and functional scaling, but no causal framework is provided.
Citation: https://doi.org/10.5194/egusphere-2026-4551-RC3
Data sets
Compiled river-network and lake datasets for China Weiwei Shao and Haibing Wu https://github.com/LaianLaian/river-scaling-decoupling
Model code and software
Code for reproducing the analyses of geometric and functional scaling in river networks Rui Shao, Jiaqi Li, Weiwei Shao https://github.com/LaianLaian/river-scaling-decoupling
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- 1
This study examines how geometric and functional scaling in Chinese river networks respond to climate, hydrological connectivity, and administrative fragmentation. Using 41,677 river records, the authors claim that Hack’s law is relatively stable across environmental gradients, whereas runoff-efficiency scaling varies strongly with precipitation, lake density, and network structure.
The idea could be interesting, but I identified several major issues in the applied methods that defeat the scientific validity of the manuscript and require reworking. The most important are about the dataset and the whole statistical framework of the analysis. I suggest the authors deeply rework the manuscript, starting from the methods, before resubmitting the paper.
MAJOR COMMENTS
More information about the dataset is required (section 2.1).
- How has it been built, and when was it last updated? What models were used to build and characterize the river network (later on you mention zero or negative L and A, so it seems not to be a DTM-based extraction)? Is there any validation to the resulting L and A values?
- What is the spatial resolution of L and A?
- How can you trust the data if they report negative values of L and A?
- What is the smallest river size it reliably includes? Usually, these datasets miss all the smallest rivers (e.g., with mean flow less than 2 m3/s). This is an inevitable limitation for practical reasons, but we need to know.
- It would be nice to also have a map to accompany this description
- What about P and R? It seems you did not use the ones from the presented dataset (L77), so where did you get them from? If you use long term averages, how do you account for regime changes when new water management structures are built during that period?
I really can't make any sense of the (seemingly arbitrarily) defined river-level scaling index (section 2.3). What does i represent? Also, hi seem to have a similar form to eq. (4) but neglecting the scaling coefficient c. As a consequence, points laying on the line represented by eq. (4) will have a higher hi the lower Ai they have. What would this tell you about climatic influences on geometric scaling?
Also the runoff efficiency scaling (section 2.4) seems an arbitrary definition. Did you find it from literature (if so, it needs citations). Or did you choose it yourself? If so, why? Why do you expect runoff efficiency to grow as a power-law of contributing area? R/P is roughly bound between 0 and 1, while a power-law is not by nature. How do you account for that? If this is a model you chose, did you test different ones and compare them? It also feels like you are trying to hide the fact that this is not a good model choice: fig 2c reports RP vs A plots only in a conceptual representation, and fig 3a shows almost 1 order of magnitude of resituals (on the variable RP, which should vary between 0 and 1). Fig 4c and d actually show that there's no real trend in RP vs A. This whole analysis on RP should be completely removed from the manuscript.
Then, in section 2.5 you perform a multiple log regression for R/P, which is a direct extension of the one in 2.4 but you present it as "network structure" analysis instead of climatic. The whole methods need a structural rethinking; right now they are a number of disconnected statistical exercices that don't connect with each other.
You always refer to climate and climatic conditions but really are only looking at precipitation amount. The reality is that similar precipitation amounts can occur in very different climates (thing about seasonal distribution, potential evapotranspiration, rain/snow distribution). Consider either rephrasing the whole manuscript or splitting your rivers by actual climatic areas.
Also I cannot understand how the methods in section 2.6 could make sense. Administrative units are arbitrary regions defined by humans. Why should they relate to any type of scaling between river length and contributing area? For sure you'll find that longer/larger rivers tend to span more administrative units, therefore have a larger n. Also, the fact of partitioning length and area equally among administrative fragments (eq. 10 and 11) does not make any sense. What could you possibly explain from this analysis?
The writing is very dry particularly in the methods section: single lines of text seemingly disconnected from each other, all in a separate line.
The statistical framework used to assess how Hack exponent h varies with precipitation, lake density, and other variables needs reconsideration. The manuscript estimates the h-A and R/P-A scaling separately after stratifying by one variable at a time and then relates the resulting beta values to that same variable. This does not account for correlations among precipitation, lake density, river level, administrative fragmentation, and basin area, so the observed variation cannot be uniquely attributed to the variable used for stratification. A more appropriate approach would be to first define a common statistical model for L as a function of A and all the relevant predictors, including predictor interactions and non-linearity where the hypothesis concerns changes in the scaling exponent, and then assess parameter variation and uncertainty using an appropriate validation/resampling framework. The non-independence introduced by the strongly overlapping moving windows should also be considered.
I am not reviewing the results, discussion and conclusions at this stage as I don't believe they are meaningful given the methods used.
MINOR COMMENTS
I would specify in the title and text that these results refer mostly to China. This is because I have a strong feeling that they are very dependent on the specific dataset that has been used (in terms of data resolution/quality/bias).
- L12: 41677 rivers reaches or measurement points?
- L75: what is river level? Is it Strahler order?
- L77: are L and A local values (e.g. the current river reach) or accumulated upstream?
- L83: what does a river-record represent? A monitoring station, river reach, or what?
- L83-87: how many records in total and how many were removed for each reason? Where they uniformly distributed across China or is there some regional bias? Can we safely assume the remaining data is ok, or is there an inherent bias in P, R, A and L?
- L85: how did you identify artificial channels from the rest?
- L88: what units are R and P? Are they volumes, depths, mean annual flows?
- L88: also, where did you get R and P data from?
- L108: this seems an arbitrary definition with no reason or explanation behind it?
- L112: why not RP instead of R/P?
- L137: moving window framework as in section 2.4, so you fit eq. 6 in sub-groups defined by DL?
- L153: there is no influence of administrative partitioning on hydrological scaling. At most there could be a correlation, but you wouldn't really catch anything useful with the methods you're applying here.
- L155: this is not what a sensitivity analysis is.
- L162: is this a single fit for all available rivers?
- L163: report this range
Figure 1:
- it seems that the very large majority of points have log10(A) < 2.5. This means that the fit is heavily influenced by the larger rivers, therefore h is not representative and R2 and p are artificially inflated.
- how did you generate panel c?
- you also need to report h uncertainty for each point.
- panel a is repeated in figure 4.