Disentangle Kinetic From Equilibrium Fractionation Using Primary (δ17O, δ18O, δD) and Secondary (Δ17O, dex) Stable Isotope Parameters on Samples From the Swiss Precipitation Network

Since 1971 water isotope measurements are being conducted by the Climate and Environmental Physics Division at the University of Bern on precipitation, river- and groundwater collected at several places within Switzerland. The water samples were stored in glass flasks for later analyses with improved instrumentation. Conventional isotope ratio measurements on precipitated water from all stations of the network are well correlated as expected. However, Δ17O as well as dex is anticorrelated to these isotope ratio. The combination of these parameters allow to investigate dependencies on temperature, turbulence factor, and humidity of these values as well as to look into the importance and relative contributions of kinetic to equilibrium fractionations. We used published temperature dependent fractionation factors in combination with a simple Rayleigh model approach to investigate the importance of the meteorological parameters on the isotope ratios. A direct comparison of measured and modeled isotope ratios for primary (δ17O, δ18O and (δD) as well as secondary isotope parameters (Δ17O and dex) is shown.


INTRODUCTION
Water isotope ratio measurements are among the first determinations done. The history reaches back to the early days of mass spectrometry (Aston, 1942;Ney and Mann, 1946;Nier and Roberts, 1951;Dibeler, 1954). Applications of mass spectrometry evolved rapidly with several developments on preparation systems of diverse samples den Boer and Borg, 1952;Dostrovsky and Klein, 1952;Friedman and Irsa, 1952;Graff and Rittenberg, 1952;Chinard and Enns, 1953;Dubbs, 1953;Washburn et al., 1953). One important application was the reconstruction of climate conditions based on carbonate oxygen isotopes . Over the many decades it remained an important research topic due to its relevance in many different fields such as hydrology (Aggarwal et al., 2005), meteorology (Yoshimura, 2015), geology (Andrews, 2006), cryosphere (Moser and Stichler, 1980;Masson-Delmotte et al., 2008), biology (Griffiths, 1998;Diefendorf and Freimuth, 2017), chemistry and geochemistry (Bird and Ascough, 2012;Klaus and McDonnell, 2013) and lately of course for applications in the field of climate (Galewsky et al., 2016) and environmental change of the Earth (Cernusak et al., 2016;Allen et al., 2017). Nowadays isotope ratio measurements are standard parameters to be determined in order to characterize the system under investigation, but so-called secondary parameter came into play such as the deuterium excess, d ex defined as d ex δD-8·δ 18 O (Stenni et al., 2010;Aemisegger et al., 2014;Pfahl and Sodemann, 2014;Tanoue and Ichiyanagi, 2016), as well as the Δ 17 O, for its definition see Eq. 4 below, (Schoenemann et al., 2013;Steig et al., 2014;Barkan et al., 2015;Uechi and Uemura, 2019). Both parameters are scaled differences of primary isotope ratios. Δ 17 O, being only dependent on oxygen isotope ratios, is less dependent on temperature compared to d ex , which itself is a combination of hydrogen and oxygen isotope ratios. This fact originates from a close cancellation of the temperature dependence among different isotope ratios of the same element compared to a different isotope ratios of different elements, i.e. d ex . This independence of temperature allows us now to investigate dependencies on other parameters such as the relative humidity linked to kinetic fractionation effects that occur during water vapor diffusion through unsaturated air. Hence, Δ 17 O measurements has the potential to differentiate between kinetic and equilibrium fractionation influence. This opens up a wide range of applications 1) on the hydrological cycle, e.g. humidity conditions at water vapor source locations, influence of re-evaporation on land from lakes and rivers; 2) in biological systems regarding transpiration processes; 3) in paleoclimatology to reconstruct temperature and humidity conditions at the site of precipitation and source of the formed water vapor; 4) in stratosphere-troposphere exchanges and many more.
In this study, we will elaborate on this partitioning of the two fractionations on measurements done on samples from the Swiss network of isotope ratios on precipitation, ISOT (Schotterer et al., 2010). Materials and Methods briefly describes the sites, their characteristics, and the methods of how the isotope ratio measurements are done. Furthermore, it recalls already established equations focusing on Δ 17 O and discusses dependencies of the exponent that relates the two oxygen isotope ratios δ 17 O and δ 18 O, namely on 18 R s , the proportionality constants A between 17 R and 18 R for sample and standard materials (A s , A r ) and Δ 17 O. In the results section, we discuss three theoretical experiments that we compare with corresponding data. These include 1) a two component mixing, 2) mixing of kinetic and equilibrium fractionation and 3) application of the Rayleigh model to data from the ISOT network. The results of these comparisons are discussed in Discussion.

Sample and Site Selection and Characteristics of Sites Used in This Study
Monthly integrated samples from the ISOT-network form the base for the comparison with the Rayleigh model application.
Measurements have been performed so far for the two time period January 1990 to December 1993 and January 2002 to December 2004 for all stations except Jungfraujoch (JFJ) for which the complete series between January 1983 to December 2011 were measured. We selected seven stations, i.e. Basel, Bern, Meiringen, Guttannen, Grimsel, JFJ, and Locarno on a North to South transect of Switzerland that corresponds simultaneously to an altitude transect from 292 (Basel) to 3,580 (JFJ) meters above sea level. Therefore, we can investigate influences on water isotopes over a mean annual temperature from −7.9°C to +12.4°C. Since the Alps form a natural barrier for wind systems, we also can compare Atlantic to Mediterranean water vapor sources.

Measurements Methods
Measurements that are used in this study have been obtained using a Picarro Cavity Ring Down spectrometer (L-2140-i) (Steig et al., 2014). The measurements have been conducted in 2013 and 2014. Samples were injected at least 6 times from which at least the first two were discarded. Before and after 10 samples standard waters for calibrations are measured, namely Eiswasser and B_SLAP that are tightly linked to an international scale (Schoenemann et al., 2013;Affolter et al., 2015).  or r (reference, standard or an arbitrarily chosen operational parameter by the science community). As discussed in Kaiser (Kaiser, 2008), there are five different types of relationships possible for the isotope composition of a sample and a reference. We do not repeat these evaluations but just recall some of the cases, here. For a detailed understanding, the reader is advised to read the publication of Kaiser. The first case corresponds to the generally assumed case of a sample that follows exactly the same relationship as the standard, i.e. the proportionality constant A s A r A and the exponent λ s λ r λ are the same. Therefore, it follows by division of Eq. 1 for the sample and the reference that 17 R s 17 R r 1 + δ 17 1 + δ 18 λ 18 R s 18 R r λ . (2) As discussed by Angert et al., (Angert et al., 2004), it should be noted that a mass dependent relationship between a substrate and a product is characterized by θ ln( 17 α)/ln (  And hence θ λ.
Since it has been shown that natural processes follow different power laws, i. e different exponents, one had to define an exponent to express deviations from it. The chosen value was 0.528 for λ, corresponding to the value obtained for precipitation samples (Meijer and Li, 1998;Assonov and Brenninkmeijer, 2003). The 17 O excess value, here written as Δ 17 O, is therefore defined as: A more logical definition would have been to use Eq. 5, though the deviations are only important for very high values of Δ' 17 O as shown by the Taylor expansion of Eq. 5, i.e. Eq. 6.
As can be seen by comparing Eqs. 4, 5 there is a slight difference between them in the order of the squared Δ' 17 O, which is generally neglectable and may only become relevant for samples that deviate significantly from the relationship with the exponent 0.528, i.e. stratospheric samples for which λ is close to unity. We can reformulate Eq. 5 such that it has the same mathematical form as the normal delta notation for isotope ratios, yet based on ratios of isotope ratios.
with R' s 17 R s /( 18 R s ) 0.528 and R' r 17 R r /( 18 R r ) 0.528 Δ' 17 O becomes zero when the exponent λ s in Eq. 1 is 0.528. If this is not the case Eq. 7 can be rewritten as Or solved for λ s from Eq. 8 or 1 Evaluating Eq. 9 shows that there is hardly any dependence on 18 R s , but a slight dependence on Δ ′ 17 O documented in the left panel of Figure 1.
However, as shown by Kaiser (Kaiser, 2008) one has to consider not only the value of λ, but also the proportionality constant A which might be different for the sample and reference. Hence Eq. 9 must be replaced by Eq. 10, which is the general dependence of λ on the sample and reference system.
The right panel of Figure 1 clearly documents that the most important dependence by far is the ratio of the proportionality constants. λ s of Eq. 10 becomes 0.528 when Δ ′ 17 O equals zero and A s equals A r , the system that is generally applied. The question, however, remains whether this assumptions that led to this system are indeed valid.

Two Component Mixing
Assuming a two component mixing of two water standards with known values (δ 17 O, δ 18 O, Δ 17 O), then we can write following Eq. 3: The question arises what happens when we look into a mixture of these two waters, i.e. x m times standard water 1 and (1 − x m ) times standard water 2 with x m being a value between 0 and 1. Mathematically, it is clear that the mixed value of the primary delta values (δ 17 O, δ 18 O) are scaling linearly with the mixing value x m . This leads to: And hence The question that arises is whether Eq. 15 is equivalent to Eq. 16?
For the two end points, i.e. x m 0 or 1, the two Eqs. 15, 16 are equivalent, which can easily be checked by introducing these x m values into these two equations and considering Eqs. 9, 10. It is slightly more difficult for the other x m values in its given range. It can be checked either purely mathematically but also experimentally. First, mathematically it can be shown, using again a Taylor expansion, that they are different for different δ 18 O values. The difference is highest when x m equals 0.5, i.e. a mixture of 50% of each standard water.
Second, we have also done it experimentally in our lab with three standard waters (Bern_Slap (BS), Bern_DomeC (DC), Meerwasser (MW)) that are different in their Δ 17 O values, and one was different in its δ 18 O values from the other two. We performed mixtures of these standards in 10% steps. For each of the different mixed samples we performed 10 consecutive injections, corresponding to colored dots in Figure 2. We used two additional standards for calibration purposes, namely ST-08 (ST) and Eiswasser (EW). The measurement sequence for series 1 and 2 was setup as follows: Measurement number 1-10 corresponds to MW, 11-20 to ST, 21-30 to EW, 31-50 to BS, 51-60-90% BS +10% DC, 61-70-80% BS + 20% DC, //141-150 to DC, 151-160 to MW, 161-170 to ST, 171-180 to EW and 181-190 BS. A similar setup was made for series 3 and 4 but DC was exchanged with MW.
It clearly documents the dependence of the Δ 17 O on δ 18 O value. Differences shown are in the order of 100 per meg for a difference of 55 per mil for δ 18 O. This is the result of the definition of Δ 17 O based on the logarithmic difference of scaled oxygen isotope ratios according to Eq. 4. In agreement with the theory the maximal difference is reached for a 50%-50% mixture. The results does also not change significantly when using Eq. 5. Differences between the two definitions for Δ 17 O, i.e. Eq. 4 or Eq. 5, do only matter when the relationship between oxygen isotopes deviates significantly from the used power law with exponent 0.528 described in Eq. 1. This is the case for instance for stratospheric samples for which the exponent is near 1 rather than 0.5. For those samples differences between the two definitions can amount up to several 100 per meg.

Mixing of Kinetic and Equilibrium Fractionation
Regarding the water cycle combinations of kinetic and equilibrium fractionations during evaporation, evapotranspiration and condensation are inherent. Therefore, the overall fractionation is a combination of the kinetic, α kin , and equilibrium α eq , fractionation.
The equilibrium fractionation between liquid water and water vapor that we have used here are from Ellhoj (Ellehoj et al., 2013) for T below the freezing point and Horita and Wesolowski (Horita and Wesolowski, 1994) above it. The kinetic fractionation relates to water molecule diffusion in the vapor phase in air and is a function of the prevailing relative humidity, h, D i,air and a turbulence factor n ( Barkan and Luz, 2007).  (Barkan and Luz, 2005) and λ kin 0.518 (Barkan and Luz, 2007). Figure 3 values corresponds to a combination of Eq. 17 for two cases, 1) for h 1 corresponding to the equilibrium fractionation only and 2) the common case of Eq. 17. The balance value x corresponds here to: It corresponds to real situations such as mixing of water vapor that is in equilibrium with the liquid phase (case 1) with water vapor that is additionally exposed to kinetic fractionation (case 2).
There are several dependencies when combining the kinetic and equilibrium fractionation. We first look into the relative humidity and turbulence factor dependence by choosing three values for their respective range, i.e. 0, 0.5 and 1 (Figure 3). Regarding humidity, these values corresponds to completely dry conditions (h 0) with strong kinetic fractionation contribution to the total fractionation, Eqs. 17, 18. For wet conditions (h 1) kinetic fractionation is absent according to Eq. 18, and for 50% humidity (h 0.5) which corresponds to an interplay of kinetic and equilibrium fractionation. Similarly, regarding turbulence, they correspond to completely turbulent conditions (n 0) which corresponds to no diffusion influence in contrast to pure molecular diffusion (n 1) or intermediate, mixed conditions (n 0.5) between pure molecular and turbulent diffusion. The temperature dependence is indirectly shown by the orange dots via the corresponding δ 18 O value dependence. It ranges from −20 to +10 per mil in these plots. A direct temperature dependence is given in Figure 4 and corresponds to an extended temperature range compared to the range of meteoric waters.
Furthermore, one can consider air masses with water vapor of different origin to be mixed. In this case an additional parameter comes into play, i.e. the balance between these air masses and corresponding water vapor contents and isotope compositions. The simplest case is a mixture of two air masses which can be characterized by a balance value x in the range of 0-1 as shown above. Assuming that one water vapor is in complete equilibrium (h 1) and the other experienced kinetic fractionation (h 0 or 0.5) in addition with an intermediate turbulence factor n 0.6 and additionally for different δ 18 O values leads to the dependence of Δ 17 O on x shown in Figure 4. Similarly the direct influence of temperature is given in Figure 4 for n 0.6 and dry, intermediate and humid conditions h (0, 0.5, 1).
In order to shed light on the relevance of their weighting functions we can have a look at data obtained on selected sites within the Swiss network for precipitation (ISOT), see Materials and Methods. To further investigate these fractionation behavior under real conditions, Disentangling Kinetic from Equilibrium Fractionation, we use our measurements on seven stations.

Disentangling Kinetic From Equilibrium Fractionation
Fractionation occurs at the source where the water vapor is produced, on its path to the site and at the site itself.
Generally, these fractionations are combinations of kinetic associated with diffusion of water molecules in the vapor phase in air and equilibrium fractionations associated with phase changes. Additionally, as discussed above air masses of different origins containing different vapor water contents with potentially different isotope ratios could mix. A way of summarizing these different fractionations in a simplified approach is often done by the so-called Rayleigh model (Kendall and Caldwell, 1998) (Figure 5). It relates the isotope composition of precipitation to the corresponding source composition and mathematically it reads: where f corresponds to the remaining water content that can be estimated based on the prevailing water saturation pressures that are themselves dependent on temperatures at the source and the site. R i s and R i 0 are the isotope ratios of liquid water at the site of precipitation and the source of water vapor production, respectively. R i 0 corresponds to the isotope composition of the ocean water and will be taken as a reference here R i 0 R i r . All fractionation factors show the dependences as given in Eqs. 17, 18. In our calculations we use a simplification of the fractionation along the path by using a weighted mean of the fractionations at the source and the site.
This indirectly takes into account a re-evaporation mechanisms at the site where the precipitation takes place since it includes also the turbulence factor as well as the humidity factor at the site.
In order to investigate the validity and robustness of this simplified method, we compared the data derived values with the Rayleigh model estimates. For that we estimated the expected f values for seven stations in Switzerland. The characteristics of these stations are given in Table 1 and the results of our estimates are listed in Table 2.
In a more sophisticated approach one can use source conditions that are based on back-trajectory analyses for each individual station. This would be particularly important for the Locarno station since it is located south of the Alps whereas all other are in the center or north of the Alps. Furthermore, the fractionation associated with condensation based on the actual meteorological parameters including values along vertical falling path of precipitation.
Here, we used the same source conditions for all stations. The condition of the water vapor source was set to a mean annual temperature of 24°C with an amplitude (max-min) of 4°C and a relative humidity value of 94.7%. The site conditions are corresponding to mean annual temperatures and their FIGURE 4 | Dependence of Δ 17 O on temperature for a turbulence factor n of 0.6, x 1 and for different relative humidity values of 1, 0.5, and 0 (upper, middle and lower left panels). Orange dashed line for fractionation values between ice and vapor (Ellehoj et al., 2013) and blue dashed line for fractionation values between liquid and vapor water phase (Horita and Wesolowski, 1994). Dependence of Δ 17 O on the balance factor x for a turbulence factor n of 0.6 for different relative humidity values of 1, 0.5, and 0 (upper, middle and lower right panels). The dots correspond to δ 18 O values from −20 to 10 per mil in steps of 2.
Frontiers in Earth Science | www.frontiersin.org March 2021 | Volume 9 | Article 598061 corresponding amplitudes obtained from measured data from MeteoSwiss ( Table 2) as well as a relative humidity value of 94%. The turbulence factor has been fixed for these calculations at 0.88 for the humidity source location and at 1 at the site of precipitation. Based on these values, fractionation factors were calculated following Eqs. 17, 18.
We used two different approaches an isobaric case with f values between 0.11 and 0.429 and a dry adiabatic case with values between 0.162 and 0.493. Dry adiabatic conditions lead to higher remaining water vapor contents. The ranges are characterized by the temperature dependence on altitude as expected and an observed Δ 17 O altitude dependence ( Figure 6). These values where R 18 s is measured, the fractionation factors can be calculated from the above-mentioned references and the prevailing humidity and turbulence factor n (see discussion below), λ s corresponds to the slope when regressing ln (R 17 s /R 17 r ) as y-axis to ln (R 18 s /R 18 r ) as x-axis, A s and A r corresponds to the proportionality constants for the sample and the reference, respectively. A r is given whereas A s can also be calculated from the intercept b of the above regression and by using Eq. 1, it corresponds to Furthermore, we optimized the parameter n, h for the source and site, and the balance of contribution between these, i.e. y, by minimizing the sum of squared differences of measured and modeled monthly means as well as squared differences between monthly measured and modeled mean monthly amplitudes of the individual station's precipitation isotope signature. The optimization was done with an iterative Generalized Reduced Gradient solver approach using non-linear engine. We performed calculations with multiple initial conditions to enhance the chance to find the global minimum. This led to the values for n, h, and y given in Table 3. The comparison between the data and modeled values are given in Figures 7-13.

DISCUSSION
The expected dependence of Δ 17 O on δ 18 O based on the definition of Δ 17 O was proven by a two water standard mixing experiment. The primary delta value scale linearly with the mixing ratio whereas the Δ 17 O is dependent on the mixing ratio as well as on the δ 18 O. However, it also shows that if the δ 18 O is equal or similar, Δ 17 O scales linearly with the mixing ratio. We intentionally performed these experiments to show that one has to pay attention when mixing two waters in order to come up with a range of standards for high precision 17 O measurements. δ 17 O and δ 18 O will be linearly scaling with the mixing ratio of these waters but not necessarily the mixed Δ 17 O. This is of particular interest since one can derive Δ 17 O with higher precision than the δ 17 O due to cancellation of variations in both oxygen isotope ratios.
Dependencies of Δ 17 O on relative humidity, turbulence factor, temperature as shown in Figure 3, 4 have also been investigated previously experimentally as well as theoretically (Landais et al., 2010;Gibson et al., 2016;Surma et al., 2018;Uechi and Uemura, 2019). Regarding relative humidity, different results were obtained from correlation analysis of Δ 17 O with primary isotopes (δ 17 O and δ 18 O), i.e. either a positive (Uechi and Uemura, 2019) or a negative dependence (Landais et al., 2010), yet in significantly different environments (high and low relative humidity conditions). In both studies convincing theoretical considerations are given, how the measured Δ 17 O values can be explained in which the relative humidity plays a key role. In tropical and subtropical regions Δ 17 O values is a measure for the relative humidity of the oceanic moisture source region (Uechi and Uemura, 2019) whereas over land the relative humidity among other is a driver for re-evaporation processes as discussed in Landais et al., (Landais et al., 2010). These two studies exemplarily show how different the influences on the water stable isotope composition can be, particularly when strongly divergent relative humidity conditions are compared. Additionally to the humidity condition the amount effect comes into play for which the slope between the primary isotopes (δ 17 O and δ 18 O) as discussed in (Uechi and Uemura, 2019) are critical for the interpretation of Δ 17 O values shown in (Landais et al., 2010). Regarding our measurements of the Swiss precipitation FIGURE 7 | Comparison of monthly mean measured (red) and modeled (for the isobaric case (orange) and for the dry adiabatic case (blue)) isotope ratios for Basel. The dashed red line corresponds to the assumed source temperature.
FIGURE 8 | Comparison of monthly mean measured (red) and modeled (for the isobaric case (orange) and for the dry adiabatic case (blue)) isotope ratios for Locarno. The dashed red line corresponds to the assumed source temperature.
Frontiers in Earth Science | www.frontiersin.org March 2021 | Volume 9 | Article 598061 FIGURE 9 | Comparison of monthly mean measured (red) and modeled (for the isobaric case (orange) and for the dry adiabatic case (blue)) isotope ratios for Bern. The dashed red line corresponds to the assumed source temperature.
FIGURE 10 | Comparison of monthly mean measured (red) and modeled (for the isobaric case (orange) and for the dry adiabatic case (blue)) isotope ratios for Meiringen. The dashed red line corresponds to the assumed source temperature.
FIGURE 11 | Comparison of measured (red) and modeled (for the isobaric case (orange) and for the dry adiabatic case (blue)) isotope ratios for Guttannen. The dashed red line corresponds to the assumed source temperature.
Frontiers in Earth Science | www.frontiersin.org March 2021 | Volume 9 | Article 598061 sites, measured relative humidity exhibits intermediate to high observed values in the range of 50%-95%. Therefore, we would not expect a strong effect of re-evaporation, however we cannot exclude it for specific individual rain events during low humidity conditions. Furthermore, the modeled relative humidity values are in the range of their observed values. The dependencies of Δ 17 O on the turbulence factor as shown in Figure 3 are opposite to the relative humidity conditions, i.e. the higher the turbulence factor the higher the change in Δ 17 O. The dependence of turbulence have been discussed in several previous publications (Merlivat, 1978;Horita et al., 2008). Specific studies investigated its influence on Δ 17 O of surface water in arid zones (Surma et al., 2018) or on lake water (Gibson et al., 2016). A turbulent value, n 1, corresponding to completely stagnant conditions, i.e. soil water or water in plant leaves, leads to fully developed diffusional conditions and therefore to maximal kinetic fractionation for the prevailing humidity value. In contrast, a turbulence value of zero corresponds to fully turbulent conditions with absent kinetic fractionation independent on prevailing humidity conditions. More information regarding the turbulence conditions for the Swiss precipitation sites is given below.
Regarding disentangling kinetic from equilibrium fractionation from measurements performed within the Swiss network of isotopes on precipitation (ISOT), we proved that the simple Rayleigh approach yields the correct range of water vapor fractions remaining at the site of condensation. Additionally, it also documents the seen anti-correlation between Δ 17 O and δ 17 O, δ 18 O, and δD. This means that there seems to be limited admixture influence of completely different water isotope signatures, e.g. re-evaporated water on land or from lakes. Furthermore from the log-log plots, we derived the exponents for the relationship between δ 17 O and δ 18 O for all sites. All these values are below 0.529, corresponding to the equilibrium fractionation exponent, which first tells us that there is most probably a combination of kinetic and equilibrium fractionation at action to form the precipitation. Yet, as nicely documented by Passey and Levin (Passey and Levin, 2021) a lower slope than 0.529 (equilibration fractionation) on meteoric waters does not necessarily mean that a combination of equilibrium and kinetic fractionation is needed due to the sequential rain out along the path.
The balance between the source and the site signature influence is crucial to understand what drives the isotopic precipitation signal. This balance of influence has been studied with our stations in Switzerland giving access to a north south transect as well as an altitude range. From the experiment, we see that for lower elevated sites the source signature is more important compared to the higher elevated sites. Minimal source influence has been found for intermediate altitudes, i.e. Guttannen and Grimsel stations ( Table 3). The altitude dependence for Δ 17 O shown in Figure 5 documents the highest values for these intermediate sites. A negative correlation is therefore obtained between Δ 17 O and the source-site balance values y. This points to an influence from both the source as well as from the site conditions on Δ 17 O. Relative humidity together with the turbulence parameter are the driving forces for the water stable isotope fractionation at both the formation location of the humidity as well as at the site of precipitation. Optimization of the turbulence factor and relative humidity at the source and site lead to moderate variations for the relative humidity at the source of 0.65-0.98 whereas it is strongly restricted to small range close to unity for the relative humidity at the site (0.94-0.98). This indicates that re-evaporation after precipitation formation according (Uechi and Uemura, 2019) is strongly limited and therefore the type of precipitation formation (snow or rain) might be of minor influence. Yet, further studies are necessary in particular due to the fact that the seasonal dependence of d ex values are not well matched at all.
In contrast, the turbulence factor values use the complete range (0.1-1) for the source of water vapor whereas it is limited to higher values (0.63-1) for the precipitation site. The latter is favoring kinetic fractionation whereas the high relative humidity at the site hinders it. Since relative humidity and the turbulence factor have opposite influences on Δ 17 O (Mixing of Kinetic and Equilibrium Fractionation), it might be an artifact of the optimizing routine that leads to the full range in n for the water vapor source location or due to the simple model approach as presented here. In literature a value of 0.6 for n is often used (Merlivat and Jouzel, 1979) to account for the influence of turbulent conditions prevailing at an evaporation site. Yet, there are contrasting studies regarding the value of n. A recent publication indicates that a value near 1 should be favored (Bonne et al., 2019) based on many direct observation. Another recent publication (Gonfiantini et al., 2020) documents laboratory experiments resulting in a strong wind dependence with values similar as generally assumed with 0.5 for windless conditions and lower values with increasing wind velocities. Yet, the authors reported that the values for δD and δ 18 O are deviating the higher the applied wind speed (up to 2.5 m/s) gets. In summary, the whole range seems to be plausible.
The optimization lead to an improvement of the balance value y which reaches values within its expected range between 0 and 1. This is also the case of the humidity values which are close to unity for the site of precipitation and lower for source locations of three precipitation sites, namely Guttannen, Grimsel and Jungfraujoch. These stations also have rather unrealistically low turbulence factors. This might indicate that our assumption of the same conditions for the humidity source for all seven sites is not valid. As we know from other studies, sites north of the Alps are mainly governed by North Atlantic sea sources whereas those south of the Alps from the Mediterranean sea. Further studies are required to account for this Alpine barrier for air circulation and its influence on precipitation isotopes. From Figures 7-13, it can be learnt that the agreement for the primary stable isotope ratios δ 17 O, δ 18 O and δD is very good. Regarding the secondary parameters, Δ 17 O is in much better agreement than d ex . Since Δ 17 O is significantly less temperature dependent than d ex it might point to a temperature influence that has not been taken into account yet. Here, one can think of temperature differences between condensation and site temperatures. This has to do with the mean cloud height above lowlands and the Alpine area. Most certainly the temperature difference between condensation and site temperatures is higher for lower altitude sites. Additional studies are required to further shed light on this issue.

CONCLUSION
We demonstrated the non-linear dependence of Δ 17 O in contrast to the linear dependencies of δ 18 O and δ 17 O when water mixing is applied. We proved this experimentally by measuring three laboratory internal water standards with significantly different Δ 17 O and δ 18 O values. It is important that the primary isotope ratios scales linearly in contrast to Δ 17 O which show a higher order dependence as expected from its definition. A simple Rayleigh model approach yields to a rather good agreement for four out of the five isotope parameters for each of the seven stations with the exception of d ex . It documents a clear interplay of kinetic with equilibrium fractionation. It has been noticed that the contribution of source and site on the corresponding fractionation is important to match the measurements. The source signal contribution is more important for lower than higher elevated sites and least important for intermediate heights.
The turbulence factor is difficult to judge since the complete range from zero to unity has been obtained when matching the data with minimal deviations. In contrast, humidity could be rather well determined. This somewhat inconclusive results might be due to our assumption of similar conditions for all seven stations since they are situation on a north-south transect through the Alps exhibiting different source locations, i.e. North Atlantic and the Mediterranean sea and are situated at different altitudes.

DATA AVAILABILITY STATEMENT
The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.