The paper focuses on climate elasticity of streamflow, which is important to understand how vulnerable river flow is to changes in climate. The authors test if climate elasticity depends on the aridity index by using annual streamflow anomalies from a large data set. This dependency is clearly demonstrated. Further the authors propose a new form to estimate climate elasticity which takes account of the dependency of the climate elasticity to aridity as well as the synchronicity pf precipitation and potential evaporation.

While reading I asked myself if there is sufficient novelty to warrant publication in HESS. Going back to the seminal paper of Sankarasubramian et al. (2001) in WRR one can find quite a similar test if climate elasticity depends on aridity using data from 1200 catchments in the US. The dependency was also found and they also find a significant effect of the synchronicity of precipitation and potential evaporation. Yet in the present paper there is an explicit term on the synchronicity, which was also new to me. So in addition to the even larger data set, there is also a relevant climatic property explicitly addressed.

The second achievement of the present paper is trying to take account for aridity in the elasticity derivation. The proposed functional form seems a bit empirical and ad-hoc. It is validated by a better NSE fitting the annual anomalies than without account for this (Table 4). Here I have a few questions: (a) what is the significance of being better (could be shown by the distribution of NSE across catchments) (b) what would be the goodness of fit using the theoretical elasticity coefficients.

I also want to stress that interannual storage changes affect the annual anomalies of streamflow which cannot directly be estimated by climate variations. Also changes in catchment characteristics like vegetation, water management in a basin will affect annual anomalies. In addition there are also statistical issues like co-variation, trends and non-normal distributions which affect the quality of the empirical derived elasticity coefficients. These issues were not addressed in the present paper, but in the companion paper of the authors earlier this year.

To my mind these issues can lead to larger variation and even the non-physical values of the empirical elasticity coefficients. Therefore before reading the paper I would have rather used the theoretical derived climate elasticity when estimating the potential change of streamflow to changes in climate. Now I might try to use the new formula since I do like the explicit accounting for the synchronicity of precipitation and potential evaporation presented in the paper.

The quality of presentation and the clarity of writing is very high and I enjoyed reading the paper. I would recommend publication in HESS after addressing my remarks and a few minor comments below.

L313: please explain what you mean be mixed results

L358-9: In the discussion there is this sentence: „relationships were developed on catchments with limited interannual memory“ – to me this is a major methodological step (constraining the sample) which was not described in the methods. Please add explicitly, so readers have everything to apply the methods with their own data.

The manuscript titled, “Streamflow elasticity as a function of aridity” empirically validates the relationship between aridity and streamflow elasticity and the proposes a new Budyko type approach to explicitly link elasticity and aridity.

They first compute local elasticities by catchment (multiple regression per basin), then class elasticities for aridity bins (optimising a single triplet of coefficients per aridity class), and finally a regionalized model where coefficients for P and E₀ are smooth functions of aridity, constrained by the shape of the Oldekop Budyko derivative. Performance is measured using mean bounded NSE on streamflow anomalies. The key result is that allowing elasticity to vary with aridity via this parametric form improves performance relative to a single global coefficient set, while keeping elasticities in a constrained “physically plausible” range.

The paper was a pleasure to read. The structure, methodological description, and objectives are all very clearly laid out. I believe that the explicit inclusion of aridity in the proposed method is novel and of interest to the community. My comments are all relatively minor and I would recommend this for publication once they are addressed.

General comments:

I would describe the quantity calculated here as an absolute marginal sensitivity rather than an elasticity, because it is not a proportional measure. The statement asserted throughout, that bounding elasticity between 0 and [-]1, represents a physically realistic response is, in my opinion, misleading. The sensitivities estimated here are bounded due to the specific structure of the analytical models used, and elasticity very often falls outside of these bounds (in the typical way that its calculated, as well as in the empirical estimates here). The bounds are properties of the specific Budyko formulations and their derivatives, not general physical laws. They assume, implicitly, that additional P cannot reduce ET or storage, that additional PET cannot increase Q, and that storage changes do not play a role. These assumptions may not always hold for annual anomalies (e.g. where storage dynamics or snow processes are important). Values outside of that range in Figure 3 are not necessarily physically unrealistic but may rather represent processes which are not well captured by the assumptions of the model. Discussing this, and explicitly explaining why you assume that 0, 1 are realistic physical boundaries would be helpful.

The empirical sensitivities presented in the paper are regression coefficients on interannual anomalies. Have you considered under what conditions the Budyko derivative approaches approximate the empirical sensitivities?  

Elasticity is generally poorly predictable in space (Addor et al., 2018) and while there is a lot of evidence that it relates strongly to the aridity index, the implicit assumption of the regional model is that aridity is the only driver of variation. I still think that what you have done is meritorious, but it would at least be worth discussing that aridity-only regionalisation is a strong simplification.

Minor comments:

In general, I would not exclude the points which are not statistically significant from Figure 3. This can be misleading. Please plot them with e.g. high transparency or at least include the full plot in the SI.

The predictors in used here are, by definition, not independent. Some discussion of collinearity and its impact on coefficient stability would be helpful.

Section 2.3 mentions a “simple grid search algorithm” to calibrate the three coefficients per aridity class, but exact ranges and step sizes aren’t given. For such a low-dimensional problem it would be easy to describe the grid explicitly, improving reproducibility.

Please describe the hydrologic memory filter described in the limitations explicitly in the methods section.

Addor, N., Nearing, G., Prieto, C., Newman, A. J., Le Vine, N., & Clark, M. P. (2018). A Ranking of Hydrological Signatures Based on Their Predictability in Space. Water Resources Research, 54(11), 8792–8812. https://doi.org/10.1029/2018WR022606