| 06 Oct 2025
On the relevance of molecular diffusion for travel time distributions inferred from different water isotopes
Abstract. Water isotopes are a tool of choice for assessing travel time distributions of water and chemical species in soils, aquifers and rivers. However, the question of whether different water isotopes tag the same travel time distributions of the water molecule, or whether the inferred travel time distribution is specific to the chosen water isotope, remains under debate. Here we conjecture that the latter is correct. We state that (a) travel time distributions of water and any tracer reflect the spectrum of fluid velocities and diffusive/dispersive mixing between the flow lines connecting the system in- and outlet, and (b) the self-diffusion coefficients of deuterium, tritium and 18O differ by as much as 10 %. Using particle tracking simulations, we show that these differences do indeed affect the variance of the travel time distribution – as one would expect for well-mixed advective-dispersive transport. Moreover, our simulations suggest that in the case of imperfect mixing, also the average travel time becomes sensitive to the differences in self-diffusion coefficients. We find that when advective trapping occurs in low conductive zones, an isotope with a smaller diffusion coefficient remains there for longer times compared to a substance exhibiting faster diffusion. This implies that for imperfectly mixed transport, average transit times ultimately increase with a decreasing self-diffusion coefficient: deuterium has the longest average travel time, followed by tritium, followed by 18O. Depending on the type of simulated system, we find differences in average travel times ranging from 10 days to more than 2 years. As these differences are in relative terms of order 5–10 %, one could be tempted to erroneously explain them as measurement errors. Our findings suggest instead that these differences are physics based. These differences persist and even grow with increasing space and time scales, rather than being averaged out. We thus conclude that travel time distributions inferred from O-H isotopes of the water molecule are conditioned by the chosen water isotope.
Competing interests: All authors are in the editorial board of HESS
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The manuscript “On the relevance of molecular diffusion for travel time distributions inferred from different water isotopes” by Zehe et al. addresses a so far, at least for studies at larger spatial scales, rarely considered aspect of tracer circulation and its potential effects on water ages and its distributions. Conceived as a modelling study, the experiment is very well designed and quite comprehensive in its extent, analyzing a broad spectrum of potential scenarios. The manuscript is similarly well written, logically structured and, due to clear explanations, easy to follow. Overall, I will be more than glad to eventually see this study published. However, I think the manuscript could be further strengthened by considering the following rather minor points:
(1) Many different model runs with changing structures of the spatial domains (scenarios A-C), tracers (2H, 3H, 18O), hydraulic conductivities, Peclet numbers, etc. were executed. This is very well described and an excellent approach as it analyses many different aspects and influences on the experiment. However, the results of these model runs are reported in somewhat of a rush and with little detail. I think that systematically showing the results of all model runs – at least as figures in the Supplementary Material – and to provide some more detail in the descriptions of the results will benefit the analysis as it will allow the reader to appreciate these different aspects much more.
(2) Related to (1), I also think that the Figures can benefit from being developed a bit more systematically and carefully, so that they are easily readable and consistent in their respective structures across the individual experiments. For example, it is unclear why for scenario A the results of 3H and 18O are not shown in Figure 4. I believe its is important to show the entire sequence of results – even if they do not exhibit major differences between the runs – to allow the reader a complete assessment. Similarly, why are estimates of RThyd only given in Figure 4 for scenario A but not in Figures 5-7 for the other scenarios? Overall, the figures will benefit from a more consistent structure and appearance. In other words, while figure 4 shows mean(T) and RThyd as annotations next to the graphs, and hydraulic conductivities in the legend, the subsequent figures show mean(T) in the legend (omitting RThyd) and definitions of the model runs, i.e. L, Pe, var next to the graphs in the figures. These inconsistencies bring unnecessary “noise” into the manuscript.
In addition, it will be good to increase symbol sizes at least in the legends, as right now the individual model runs are difficult to discern in the individual figures. Also make sure to include (a), (b), (c)… labels for sub-figures. Right now, it is not always immediately clear which subfigure is which. The figures could also benefit from the use of more systematic colour schemes throughout the manuscript.
(3) This is a study that builds exclusively on model experiments. That is fine. However, I think it would benefit the manuscript if a stronger link to experimental studies with real world data is established to allow the reader to place the results into a wider context. While the authors refer to previous studies by Stewart et al. and Rodriguez et al. to frame their work, it is surprising that they do not refer to a recent study by Wang et al. (2023). In that study we found considerable evidence that the considerable differences in 3H and 18O estimates of water ages reported by Stewart et al. (2010), are to a large extent an artefact of the choice of model type used by Stewart et al. (2010) and the cites studies therein. Overall, the study by Wang et al. (2023) found that 18O and 3H result in estimates of water ages that are broadly consistent, with estimates from 18O even showing older (!) ages and thus the opposite of what was reported by Rodriguez et al. and here in the results of this study. Thus, discussion of the results found here in the context of the results reported by Wang et al. (2023) will give the reader a much more complete picture of the current state-of-the-art.
Detailed suggestions:
p.1,l.15: not clear what is meant by “assessing…chemical species…”. Perhaps rephrase
p.2,l.48-50: ok, but the concept was also already known and used before that, e.g. Eriksson (1958), Bolin and Rodhe (1973). Perhaps good to include these references, as well
p.2,l.61-64: not sure if this is a valid generalization. The dependence of water ages on water supply has been known for quite a while (e.g. Nir, 1973) and even explicitly accounted for in time-variable formulations of transfer function approaches (e.g. Niemi, 1977). Please rephrase and include the references.
p.3l.82ff: I think a more precise formulation of the history of the various concepts here would benefit the manuscript. Early studies approached the question in fact with both approaches. While indeed many of them relied on time-invariant or time-variant transfer functions (see e.g. review by McGuire and McDonnell, 2006), many others used methods that are equivalent to the SAS function approach, such as the many studies of hydro-chemical dynamics based on the Birkenes and HBV models (Christophersen and Wright, 1981; Christophersen et al., 1982; Seip et al., 1985; de Groisbois et al., 1988; Hooper et al., 1988; Barnes and Bonell, 1996) and in particular nicely illustrated by Fig. 1 in Bergström et al. (1985). The same is true for Hrachowitz et al. (2013) that is now cited as a transfer function based study. This is factually incorrect. In that study we estimated water ages based on tracking tracers through the systems using “mixing coefficients”, which are functionally the same things as the SAS-approach with piecewise linear age sampling distributions (see Hrachowitz et al., 2016 and Benettin et al., 2022 for more detail).
P3.l.89: probably better to replace Hrachowitz et al. (2010) by Hrachowitz et al. (2021)
p.3,l.103-105: true, but this is likely an artefact due to the choice of model by Stewart et al. (2010, 2021) and the studies cited therein. A detailed comprehensive demonstration thereof can be found in a recent study by Wang et al. (2023), who found broadly equivalent magnitudes of water ages inferred from 3H and 18O. Would be good to rephrase and add the perspectives by Wang et al. (2023)
p.3,l.107: Nice!! This is really excellent.
p.3,l.130ff: please rephrase and add Wang et al. (2023)
p.4,l.137: this is a very useful and clearly described section. One additional thing that I personally would find useful would be to include an explicit description of the difference between self-diffusion and molecular diffusion. After reading this section I am still not sure whether they are the same thing or not and if not, what the difference is.
p.10,figure 3: I do not understand the legend in the bottom row. How can particle numbers Np be negative?
p.11,l.304-306: ok, purely technically seen the mean(T) are larger. But given that the difference is ≤1%, how significant/relevant is it?
p.11,l.326 and elsewhere: the term “average” is somewhat ambiguous as it can refer to any measure of central tendency, i.e. mean, median or mode. Thus, please replace accordingly for clarity.
In addition, it would be good to provide the actual values for the different tracers and show the TTDs – either in Figure 4 or in the supplementary material.
p.12, p.333ff: given that the TTDs of the individual tracers in figures 5-7 are largely indiscernible and plotting on top of each other, it may be good to also show them with log-scale axes to better grasp the differences. In addition, the exclusive focus on mean(T) may conceal other effects. Thus, perhaps it is interesting to also report and discuss differences in young and old quantiles or the medians.
p.13,l.358: how was the RThyd exactly calculated here? Is it based on the flow weighted averages of the upper and lower layers? Does it make a difference whether it is flow weighted?
p.14,l.395-396: ok, but for context it would also be good to mention that this is only ~2%
p.17,l.429-433: also here, reference to the results of Wang et al. (2023) is required to provide a full picture.
Thank you for this interesting contribution!
Best regards,
Markus Hrachowitz
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