The model estimates the joint effect of equilibrium and kinetic isotopic fractionation during the phase transition from liquid water to vapor. When resistance to transport in the liquid phase is neglected, the isotopic composition of the water vapor flux can be expressed as follows: 1δE=δL-ε+/α+-h⋅δA-εk1-h+10-3⋅εk, where δL and δA indicate the isotopic compositions of the evaporating surface and the atmosphere, h is the relative humidity of the atmosphere, α+ and ε+ are equilibrium fractionation factors, and εk is a kinetic fractionation factor. Here (and elsewhere in this paper), δ values and fractionation factors may refer to either hydrogen or oxygen isotopes unless otherwise noted. The δ notation expresses water isotope ratios as deviations, in parts per thousand, from Vienna Standard Mean Ocean Water .

The equilibrium fractionation factor α+ (–) describes differences between the isotopic compositions of liquid and vapor phases at isotopic equilibrium, and is expressed here as the super-ratio of liquid to vapor isotope ratios. Its value is slightly larger than one, reflecting the fact that lighter molecules break their bonds more readily and thus are more abundant in the vapor phase. The values of α+ can be computed as a function of temperature T (K) using the well-established experimental results by : 103ln⁡α+2H=1158.8T3/109-1620.1T2/1062+794.84T/103-161.04+2.9992109/T3 103ln⁡α+18O=-7.685+6.7123103/T3-1.6664106/T2+0.3504109/T3. The equilibrium isotopic separation between liquid and vapor is then computed as ε+ = (α+ - 1) 103 (‰).

The kinetic fractionation factor εk quantifies isotopic effects during net evaporation associated with the higher diffusivities of isotopically lighter molecules. Variations in εk are generally dominated by the relative humidity (h) of the air overlying the evaporating surface. Several expressions have been derived specifically for εk in soils see. Here we use a simplified expression given as 4εk=θn(1-h)1-Di/D103(‰). The weighting term θ (–) accounts for the possible influence of the evaporation flux on the ambient moisture, and is usually assumed to equal one for small water bodies . The term Di/D is the ratio between the diffusivities of the heavy and light isotopes. Commonly accepted values are provided by : Di/D(2H) = 0.9755 and Di/D(18O) = 0.9723. The term n (–) accounts for the aerodynamic regime above the evaporating liquid–vapor interface. It ranges from n = 0.5 (fully turbulent transport that reduces kinetic fractionation, appropriate for lakes or saturated soil conditions) to n = 1 (fully diffusive transport, appropriate for very dry soil conditions). According to Eq. (), in a dry atmosphere (h = 0), the kinetic fractionation factor is roughly 12.2–24.5 ‰ for εk(2H) and 13.8–27.7 ‰ for εk(18O).

We now consider the case of an isolated volume of water with initial isotopic composition δ0 that evaporates into the atmosphere. As evaporation is the only flux, the remaining liquid volume decreases over time (a case sometimes referred to as a “desiccating” water body). We use x (–) to represent the fraction of the initial volume that has evaporated. The fraction remaining as liquid thus equals 1 - x. Assuming that the fractionation factors do not change during the evaporation process, the equation describing the isotopic composition of the residual liquid δL is 5δL=δ0-δ*(1-x)m+δ*, where δ* [(‰) represents the limiting isotopic composition (i.e., the composition that the desiccating water volume would approach upon drying up) and the term m (–) is referred to as “temporal enrichment slope” . These two terms can be computed as follows: 6δ*=hδA+εk+ε+/α+/h-10-3⋅εk+ε+/α+ and 7m=h-10-3⋅εk+ε+/α+/1-h+10-3⋅εk. Equation () can represent an isolated volume of precipitation with initial isotopic composition δP that progressively evaporates into an atmosphere with an isotopic composition of δA. If the isotopic composition of the atmospheric vapor is unknown, it is common to assume that it is in equilibrium with precipitation : 8δA=δP-ε+/α+.

As an introductory example, we modeled the isotopic evolution of an individual water volume by implementing Eqs. ()–() with parameters T = 20 ∘(C), h = 0.75 and n = 1. Figure  illustrates the resulting increase in the hydrogen and oxygen δL of the residual water during the evaporation process. The dual-isotope plot (Fig. c) shows the simultaneous behavior of the hydrogen and oxygen isotope ratios. As more of the water evaporates, the composition of the residual liquid gradually departs from the LMWL following a nearly linear trajectory. This trajectory is termed the evaporation line. Depending on the atmospheric parameters used in Eqs. ()–(), the slopes of evaporation lines will typically range from 2.5 to 5, markedly shallower than typical meteoric water lines, which usually have slopes of roughly 8 .

The degree of evaporative fractionation will vary seasonally, reflecting seasonal changes in temperature and relative humidity. The isotopic composition of precipitation will also vary seasonally, reflecting seasonal shifts in moisture sources, air mass trajectories, and cloud processes . With this in mind, we explore how these two seasonal patterns jointly shape the isotopic composition of the residual liquid remaining after rainfall partly evaporates from a soil.

We consider a 12 month period and for each month we use the mean isotopic composition of precipitation as source water for the model outlined above. Each month's precipitation then undergoes a seasonally varying amount of evaporation, and the isotopic composition of the residual water is determined separately for each month using Eqs. ()–(), along with that month's average temperature and relative humidity. In this approach, the isotopic composition of monthly residual water depends only on precipitation and evaporation during the same month. This simplified approach does not explicitly account for in-soil mixing processes, whose effects are discussed in Sect. .

We apply this approach to real-world weather and precipitation data from the Vienna Hohe Warte station, Austria. The full isotopic dataset is freely available, along with temperature and vapor pressure data, from the Global Network of Isotopes in Precipitation (GNIP Database), provided by IAEA/WMO and accessible at: https://nucleus.iaea.org/wiser. From GNIP we obtained long-term mean monthly values of precipitation δ18O at Hohe Warte, along with mean monthly air temperatures and vapor pressures, from which we calculated mean monthly relative humidities. Rather than using the precipitation δ2H values from GNIP, we instead computed them from δ18O using the equation for the LMWL at Hohe Warte: δ2H = 2.12 + 7.45δ18O. This ensured that each precipitation sample plotted exactly on the LMWL, thus aiding visualization.

The long-term mean monthly time series exhibit pronounced seasonality (Fig. ). The seasonal temperature excursion is about 20 ∘C, and monthly average δ18O ranges from -13 ‰ in winter to -6 ‰ in summer. The relative humidity ranges from roughly 0.85 in winter to 0.65 in spring and summer.

To investigate the effect of seasonality in evaporation rates on residual liquid composition, we represented the evaporation-to-precipitation fraction (the variable x) by sinusoidal cycles with different amplitudes and timing. We did not consider transpiration fluxes, since the isotopic effects of transpiration are generally considered to be negligible. Moreover, to keep the example simple, we did not consider the seasonality of precipitation flux, although this could be easily included. The parameter n in Eq. () was fixed at 0.75 throughout the year. A numerical code to implement Eqs. ()–() and apply them to the case of seasonal sources and climatic conditions is freely available at https://github.com/pbenettin/evaporation-lines and provided in the Supplement.

The isotopic compositions of different source waters (mean monthly values of precipitation from the Vienna Hohe Warte station) and of the residual liquid water after evaporation (computed through Eqs. –) are shown in dual-isotope space in Fig. . For this figure, we generated two hypothetical evaporation cycles, both peaking in July and having the same mean value x‾ of 0.10 (–) but with different degrees of seasonality. The weakly seasonal cycle had a peak-to-peak amplitude of just 0.02, and the more strongly seasonal cycle had an amplitude of 0.16. These x values are modest, representing conditions of limited evaporation that may be found in many temperate regions.

The source waters (shown as yellow stars) vary along the LMWL reflecting the seasonal variability of atmospheric moisture sources and conditions, with isotopically lighter precipitation during colder months. The simulated residual water samples (shown as green dots) plot below the LMWL, with summer samples plotting farther from the LMWL than winter samples, reflecting their greater evaporative enrichment. The evaporation lines connecting individual source waters and residual waters are longer and shallower in summer than in winter, reflecting seasonal differences in temperature, relative humidity, and evaporated fraction x. As a result, the summer residual water samples plot farther away from the LMWL than the winter samples do, by an amount that reflects the seasonality in the evaporation process. The residual water samples follow a nearly linear trend (shown as a dashed line), which is markedly steeper than the evaporation lines for the individual source waters (shown as grey lines). The slopes of the evaporation lines range from 3.1 to 3.4; by contrast, the trend lines for the residual waters have slopes of 6.1 (Fig. a) and 7.1 (Fig. b), close to the assumed LMWL slope of 7.45. Note that whenever the residual water trend line has a slope that is close to that of the LMWL, the location of the intersection between these two lines will be highly uncertain.

Because the simulated residual water samples can be fitted easily with a simple trend line, it may seem logical to interpret this trend line as an evaporation line, and to infer an apparent source water end-member from its intersection with the LMWL. In the case of an isolated water parcel that is progressively evaporated (as in Fig. c), this approach could yield a reasonable estimate of the original source water. However, when residual water samples do not come from a single source, the trend line is not an evaporation line, and the intercept of this trend with the LMWL can lie far away from the average source water (Fig. a); the intercept can even lie far outside the range of all the source waters (Fig. b).

Figure  illustrates how different degrees of seasonality in evaporation patterns may yield different trend lines in residual water samples, with different intercepts with the LMWL. The individual source waters and evaporation lines are the same as in Fig. . The five trend lines in Fig.  are associated with different seasonal evaporation cycles, which feature similar low evaporation fractions in winter (roughly x = 0.04), but different evaporation fractions in summer (roughly x = 0.15 to x = 0.60). The evaporation cycles with higher summer peaks correspond to trend lines with shallower slopes and less negative intersections with the LMWL. All of the intersections lie far from the true mean source water; indeed none of them lie within the range of the individual monthly source waters.

If the seasonal cycle of evaporative fractionation is not in phase with the seasonal cycle in source water composition (that is, if the most strongly fractionated sample is not also the one with the heaviest initial isotopic signature), the residual water samples will trace out a hysteresis loop. In Fig. , the source waters are the same as those in Fig. , but the seasonal evaporation cycle has been shifted by two months. The width of the resulting hysteresis loop depends on the amplitude of the seasonal cycle in evaporation, and how far out of phase it is with the seasonal cycle in precipitation isotopes. Even where such hysteresis loops exist in nature, they may be difficult to detect due to measurement uncertainties and environmental noise.

In Figs. –, each residual water sample is derived from a discrete monthly precipitation source water sample. Real-world soil waters, by contrast, can be expected to contain mixtures of waters with different ages, and thus different source water signatures and evaporative fractionation trajectories. For simplicity, we simulated the soil as a well-mixed reservoir that integrates each month's residual waters. Mathematically this means that the composition of the soil pool is an exponentially weighted running average of the residual water samples shown in Fig. a. For purposes of illustration we used a time constant of six months, such that the same-month contribution to each sample is roughly 15 % and the contribution from the previous 12 months is roughly 86 % of the total. The results are shown in Fig. a.

The same procedure was used to create Fig. b, except we considered each of the approximately 600 individual monthly δ18O and δ2H values available at Hohe Warte (the cloud of grey dots) as meteoric source waters. These source waters were not constrained to lie along the LMWL, in contrast to the analyses presented above. These source waters were individually evaporated and fractionated, by amounts that depended on the individual monthly temperature and relative humidity (and the same seasonal cycle in the evaporated fraction x that was assumed in Figs. a, , and a). We then applied the same running weighted time averaging used in Fig. a, with the resulting cloud of residual water samples shown in Fig. b. The more that the residual water samples are time-averaged, the more their scatter will be compressed and the smaller the portion of the dual-isotope plot they will occupy, but their trend line will remain almost the same. The exponentially weighted averaging used here also introduces a time lag of roughly 3–4 months between the seasonal cycle in the source water and the seasonal cycle in the time-averaged soil water. For this reason, the isotopically heaviest soil water samples are found in October even though the isotopically heaviest precipitation falls in the summer. (Different time constants in the weighted averaging would yield different lag intervals.) Similar lag periods of several months are often found in experimental studies e.g.,.

Due to the scatter among the source water samples in Fig. b, the evaporatively fractionated residual water samples are less collinear than in Fig. a. Nonetheless, in both cases the trend lines intersect the LMWL far from the true mean source water. Because the intersection point lies within the range of the individual winter precipitation samples, however, there is a risk that one could incorrectly infer that it represented a winter-precipitation source water for the evaporated soil samples (when in fact the winter precipitation in these simulations has hardly been evaporated at all).