A Lagrangian Perspective on Stable Water Isotopes During the West African Monsoon

We present a Lagrangian framework for identifying mechanisms that control the isotopic composition of mid‐tropospheric water vapor in the Sahel region during the West African Monsoon 2016. In this region mixing between contrasting air masses, strong convective activity, as well as surface and rain evaporation lead to high variability in the distribution of stable water isotopologues. Using backward trajectories based on high‐resolution isotope‐enabled model data, we obtain information not only about the source regions of Sahelian air masses, but also about the evolution of H2 O and its isotopologue HDO (expressed as δ D) along the pathways of individual air parcels. We sort the full trajectory ensemble into groups with similar transport pathways and hydro‐meteorological properties, such as precipitation and relative humidity, and investigate the evolution of the corresponding paired { H2 O, δD } distributions. The use of idealized process curves in the { H2 O, δD } phase space allows us to attribute isotopic changes to contributions from (a) air mass mixing, (b) Rayleigh condensation during convection, and (c) microphysical processes depleting the vapor beyond the Rayleigh prediction, i.e., partial rain evaporation in unsaturated and isotopic equilibration in saturated conditions. Different combinations of these processes along the trajectory ensembles are found to determine the final isotopic composition in the Sahelian troposphere during the monsoon. The presented Lagrangian framework is a powerful tool for interpreting tropospheric water vapor distributions. In the future, it will be applied to satellite observations of { H2 O, δD } over Africa and other regions in order to better quantify characteristics of the hydrological cycle.

The values for  , eq c E vary with temperature as also the saturation vapor pressure does. They were determined for liquid and ice condensation during various laboratory studies (e.g., Horita & Wesolowski, 1994;Majoube, 1971;Merlivat & Nief, 1967). Non-equilibrium fractionation is assumed to occur in addition to equilibrium fractionation for processes that enforce a fast isotope flux between vapor and liquid, for instance when ventilated or unsaturated conditions prevail.
A simple framework for the isotopic fractionation in a precipitating air parcel is the Rayleigh distillation process (Dansgaard, 1964;Rayleigh, 1902). In this model, a moist adiabatic ascent is assumed with immediate removal of the condensate (Johnson et al., 2001). As soon as the dew point temperature is reached, condensation begins and condensate forms from ambient vapor under equilibrium conditions. While this process enriches the condensate with heavy isotopes, the ambient vapor gets depleted according to The conditions at the starting point of the ascent are defined by 0 v E q and the isotopic ratio 0 v E R . For condensation above the frost point (263 K, according to Noone, 2012 andCiais &Jouzel, 1994) the liquid fractionation factor (3) is used and at colder temperatures, the factor over ice is applied. A typical Rayleigh line for convective condensation over West Africa is shown in green in Figure 1b (see Sections 4.2 and 4.4.2).
If a liquid hydrometeor falls into unsaturated air, evaporation takes place and acts as a reversed distillation process . If the evaporated fraction of the rain drop is large, then this process has an enriching effect on the surrounding water vapor (e.g., Risi et al., 2021;Tremoy et al., 2014). In contrast, if the evaporated fraction is small (i.e., partial evaporation), this leads to a depletion in  D v E , as lighter isotopes evaporate preferentially Lee & Fung, 2008;Noone, 2012). While enriching the rain water, the ambient vapor content increases due to the input of relatively more depleted evaporated rain water. If both partial rain evaporation and Rayleigh condensation occur at the same time, this leads to a drop to below the Rayleigh curve in the } space (Dansgaard, 1964;Rozanski et al., 1992) and creates a so-called Super-Rayleigh signal, representing a Rayleigh process with an increased fractionation factor (Noone, 2012): Starting at different positions on the Rayleigh curve, this creates the signals indicated by magenta lines in Figure 1b. According to Noone (2012), the Super-Rayleigh hypothesis excludes cases, where either the H E O within its whole hydrological cycle (Pfahl et al., 2012). Fractionation is assumed whenever phase changes occur that involve the vapor phase. A one-moment microphysical scheme is used and the isotopic composition is calculated for water vapor, liquid and ice clouds as well as rain and snow. For this purpose, it includes the fractionation schemes of Stewart (1975) for rain evaporation and Jouzel and Merlivat (1984) for snow formation. Further, it uses the isotope-enabled multi-layer soil moisture scheme iso TERRA E for fractionating soil evaporation and non-fractionating plant transpiration . Fractionation during ocean evaporation is represented by the Craig-Gordon-model (Craig & Gordon, 1965).
Here we use data from a iso COSMO E simulation with a focus on West Africa during the WAM season 2016. The simulation period is chosen to match the DACCIWA campaign (01 June-31 July 2016, Knippertz et al., 2017) and the model output frequency was set to 1 hr. Data provided by the global isotope-enabled model wiso ECHAM5 E (Werner et al., 2011) are used as initial and boundary data as well as for a spectral nudging of the horizontal wind fields above 850 hPa. This serves to keep the meteorology close to reality, as the wiso ECHAM5 E simulation was nudged to ERA-interim reanalyses provided by the European Center for Medium Range Weather Forecasts (ECMWF). The model domain of the iso COSMO E simulation is chosen such that it covers the dominant moisture source regions of the WAM (see Figure 2). The model configuration has 40 vertical hybrid levels between the surface and 22.7 km and a horizontal grid spacing of 14 km (similar to the horizontal pixel size of Metop/IASI data, Diekmann et al., 2021). Vergara-Temprado et al. (2020) stated that for a horizontal grid spacing below 25 km switching off the convection parameterization, such that convection is explicitly simulated by the dynamical core of the model and the convective precipitation is determined by the microphysical scheme, leads to overall better results than increasing the horizontal resolution. Specifically when simulating the WAM, various studies reported significant improvements when using explicit convection (Berthou et al., 2019;Crook et al., 2019;Marsham et al., 2013;Martínez & Chaboureau, 2018;Maurer et al., 2017;Pante & Knippertz, 2019). Additionally, the shallow convection parameterization in iso COSMO E is a non-precipitating scheme (similar to most shallow convection schemes) and does currently not include the water isotope physics. Therefore we decided not to use a parameterization for deep and shallow convection despite the relatively coarse grid spacing of 14 km, with the aim to have as few as possible non-resolved processes influencing the isotope and water budget.

The Trajectory Tool LAGRANTO
LAGRANTO is a Lagrangian analysis tool that allows calculating backward and forward air trajectories based on 3D wind fields and tracing physical variables along individual trajectories by interpolating model fields onto the trajectory path (Sprenger & Wernli, 2015;Wernli & Davies, 1997). Using the iso COSMO E data from Section 3.1 as input, we calculate backward trajectories starting from the Sahelian mid-troposphere during the WAM season 2016. In accordance with the typical residence time of atmospheric water the trajectory length is set to 7 days (Sodemann, 2020). This implies that each trajectory contains 169 time steps, that is, one initialization step plus 24 time integration steps per day. Trajectories are started daily at 09 and 21 UTC from 08 June to 30 July 2016, at 575 and 625 hPa (corresponding to around 4.8 and 4.1 km) and for approximately every Figure 2). If in the following not explicitly mentioned, all trajectories from both starting altitudes and both starting times are analyzed. In total, this results in 12,720 trajectories. In addition to various meteorological variables, we trace the specific contents x E q of 2 H E O and HDO in vapor, sedimenting (rain and snow) and non-sedimenting condensates (liquid and ice clouds). As the trajectory setup is chosen with respect to the characteristics of the remotely } data from Metop/IASI (peak of vertical sensitivity smoothed around 600 hPa, local overpass times around 09.30 and 21.30, results given as volume mixing ratios; see Diekmann et al., 2021), we convert v E q for 2 H E O and HDO into volume mixing ratios (ppmv) and calculate  D v E along each trajectory. We will refer to the starting point (first calculation step, day 0) as the target time and to the last calculation step (day −7) as the trajectory origin.

Trajectory Sorting
For a meaningful interpretation of the full ensemble of 12,720 trajectories, the trajectories will be sorted according to geographical and meteorological criteria along their atmospheric pathways. By considering the geographical position and altitude of the trajectory origins, we aim to build clusters that represent the dominant transport patterns of the WAM (Niang et al., 2020). As transport is an important control factor for atmospheric moisture, such a dynamical clustering will give a useful first overview of the characteristic moisture evolution of the defined clusters (Nieto et al., 2006;Salih et al., 2015;Sy et al., 2018 per time step on average), the trajectory is assumed to be non-precipitating. An individual trajectory data point is classified as precipitating, if the specific content ( r E q for rain, s E q for snow) is at least 5 10 E g O decreases,  D v E remains mostly constant, leading to a mixing signature that follows the dry mixing curve (see orange arrow from 2 to 3). Figure 4) represents a trajectory with both strong mixing and precipitating effects. It originates in the lower troposphere (  tra E z 3 km) over the Gulf of Guinea and exhibits moistening and enrichment, while subsiding below 1 km and taking course toward the Guinea Coast. This moistening is associated with surface evaporation, while the trajectory penetrates into the boundary layer ( Figure 4h). This leads to an enrichment following the moist mixing line (see blue arrow from marker 1 to 2, in Figure 4e) and results in higher moisture contents than for T1. Over the Sahel, a local convection event (see precipitation patterns in Figure 4b) lifts the trajectory abruptly from  E 1-6 km altitude (see marker "x" in Figure 4b). As a consequence, precipitation forms and depletes the trajectory of its heavy isotopes in the vapor phase following a clear Rayleigh signature (green arrow from 2 to 3 in Figure 4e). Thereafter, the air parcel appears to leave the convective cell and weak mixing with drier surrounding air occurs (orange arrow, from 3 to 4),

T2 (middle column in
in the Sahelian mid-troposphere (600 hPa, see dashed box in Figure 2) during 08 June-30 July 2016. The two-dimensional contours indicate the data distributions during the different monsoon stages as described by Knippertz et al. (2017). For each stage, the contours summarize 50 and 95% of the respective data points with the respective total numbers given in the legend. Additionally, this plot includes the idealized process curves that are marked with arrows in Figure 1. above into the considered air parcel. While 2 H E O decreases slightly, but still remains high during this event, shows a sharp drop by more than 50 ‰( Figure 4f, magenta arrow from marker 1 to 2). This depletion is stronger than would be predicted using the Rayleigh model and thus penetrates into the Super-Rayleigh regime. The hypothesis is that these depleted and slightly drier trajectory data are a result of an isotopically processed cold pool associated with the squall line marked by "x" in Figure 4c. Afterward, along its northeastward path over the Sahel, the trajectory enriches, likely due to surface evapotranspiration (blue arrow from 2 to 3), until it finally interacts with a second squall line and exhibits once again an isotopic pull toward the Super-Rayleigh regime (magenta arrow from 3 to 4). However, at this time the air parcel is lifted to 4 km and changes its flow direction by  180 E , consistent with the easterly wind component of the AEJ and the southwestward propagation of the observed squall line. A subsequent enrichment (blue arrow from 4 to 5) defines the isotopic composition for the injection into a further convective updraft, which is part of the southwestward propagated squall line system observed at the trajectory points 3 and 4 and where the occurrence of snow particles (Figure 4i) is accompanied by Super-Rayleigh signals (magenta arrow from 5 to 6).
In summary, the analysis of the selected trajectories reveals that, by using the theoretical process curves from Figure 1 } pairs along air parcels can be divided into moist and dry mixing, drying and depletion due to Rayleigh condensation, and processes that deplete the vapor beyond the prediction by the Rayleigh model. Only by considering the whole isotopic history of an air parcel, it is possible to fully explain its target position in the

Identification of Dominant Transport Patterns
In the next step, the aim is to test the usefulness of the idealized process curves for interpreting larger trajectory ensembles during the monsoon period 2016. Therefore, as discussed in Section 3.4, we first sort the full ensemble of 12,720 trajectories into meteorologically meaningful clusters of trajectories that experience a similar moisture history. Taking into account the characteristic regions of the trajectory origins as well as the relative position of origin altitude against target altitude, we roughly distinguish between rising (R1 to R3) and subsiding (S1 and S2) transport clusters (see Section 3.3). Their main averaged properties (see (see Figures 6c and 6d). It advances on an anticyclonic path toward the Sahel, where it is lifted into the middle troposphere due to moist convection (see Figure 6a; Marsham et al., 2013). This ascent into colder and dryer regions is associated with intense precipitation (Figures 6e and 6f), leading to a strong depletion in  D v E (Figure 6d). R2. The orange trajectories in Figure 5 indicate the subtropical Atlantic low-level inflow with the trade winds that get deflected eastward toward the Sahel as a response to the Saharan heat low (Lavaysse et al., 2009;Nieto et al., 2006). Its initial moisture is lower than for R1, but increases during the transport over the Atlantic (Figures 6c and 6d). Similar to R1, it experiences a convective lifting over the Sahel (Figure 6a), but ends up with more enriched  D v E . R3. The trajectories in yellow ( Figure 5) originate in the lower troposphere over the Mediterranean Sea and follow the Etesian winds toward the African continent (Tyrlis & Lelieveld, 2013). Over North Africa, this cluster moves along the eastern side of the Atlas mountains and then feeds the relatively dry Harmattan (Hall & Peyrillé, 2006). As the surface evaporation over North Africa is small, there is hardly any change in moisture (Figure 6c) as well as no significant contribution to the Sahelian precipitation (Figures 6e and 6f). At the target location, it shows  D v E values similar to R2 (Figure 6d). S1. The African Easterly Jet inflow is represented by the dark blue trajectories ( Figure 5). It is characterized by a low-latitude easterly flow that transports dry air masses from the upper troposphere (Figure 6a) from East Africa down to the Sahelian mid-troposphere (Cook, 1999;Sy et al., 2018). Through deep tropical convection, frozen precipitation falls into the AEJ (Figure 6f). During its subsiding path into moister tropospheric regions, 2 H E O and  D v E increase and converge toward the values of R1 (Figures 6c  and 6d). S2. The cyan trajectory ensemble in Figure 5 describes extratropical mid-level dry intrusions, which feed into the anticyclonic circulation above the Saharan surface heat low (Cook, 1999;Lavaysse et al., 2009).
As this flow originates from the mid-latitude upper troposphere, it reveals very low moisture contents (Roca et al., 2005), even lower than for S1 (Figures 6c and 6d). During its subsiding transition toward the Sahel (Figure 6a), moistening and enrichment takes place. Several studies have documented similar abundances of relatively high moisture contents over Saharan regions Schneider et al., 2016), where moisture advected from adjacent sea basins (e.g., North Atlantic and Mediterranean Sea) converges at low levels (Dahinden et al., 2021). This moist air is then lifted by dry convection during the day in the SHL and thereby moistens S2 (Figure 6b). At its target position, the  D v E of S2 resembles the values of the rising extratropical clusters (R2 and R3). Figure 7a shows the relative contributions of each transport cluster to the target region as a function of time. The clusters represent together up to 90% of the air transported into the Sahelian mid-troposphere. The unclassified trajectories mainly originate above the West African continent with no characteristic large-scale transport. Even though the relative contribution of the monsoon inflow (R1) is comparably low in terms of number of trajectories (  E 10%), it is nonetheless the major driver of precipitation for the Sahel during the post-onset stage (Phase 2, e.g., compared to R2 in Figure 7b). The pre-onset Phase 1 shows marked fluctuations associated with synoptic-scale disturbances described in Knippertz et al. (2017), leading to single rainfall events during June (e.g., Maranan et al., 2019). As the monsoon has not fully developed yet, the fraction of trajectories from the subtropical Atlantic (R2) is higher than in the other Phases (Figure 7a). The actual monsoon onset is characterized by a breakdown and then re-establishment of the AEJ as indicated by the dark blue trajectories S1. The fraction of monsoon trajectories in Phase 2 clearly increases compared to Phase 1 and precipitation events are now more frequent (Figure 7b). Finally, the unusual flow situation during Phase 3 (and to a lesser extent Phase 4) is reflected in a clear shift of the fractions of transport clusters.   Extratropical intrusions almost disappear entirely with a surge in AEJ inflow. The monsoon inflow, which causes marked precipitation events, increases at the expense of the Harmattan inflow.
In summary, the trajectory clustering according to their source regions reflects well the major transport contributions for the Sahelian troposphere during the monsoon season 2016. The clusters separate the trajectories into rising and subsiding transport patterns that bring moist and dry air masses to West Africa from different regions.

Isotopic Process Attribution Along Transport Clusters
In this section, we investigate the importance of different processes along the transport clusters presented in Section 4.3. We address the question to which extent and for which meteorological conditions the mixing, Rayleigh, and Super-Rayleigh process curves from Figure 1 are useful to explain the isotopic signals along the transport clusters. As these clusters are most representative during the active monsoon (see Figure 7), we focus in the following on trajectories during the post-onset stage (Phase 2).

Importance of Mixing Processes
As discussed in Section 4.2, air mass mixing plays a crucial role for the isotopic evolution along a trajectory, in particular if no rain processes occur. Therefore, to extract pure mixing effects in the } phase space, we select all non-precipitating trajectories (see Section 3.3). } pair data along the non-precipitating trajectories for each transport cluster. Even though the rising clusters R1 and R2 show on average strong occurrences of precipitation along their pathways (see Figure 6e), still non-precipitating trajectories appear for both (7% and 56%). The non-precipitating trajectories of R1, R2, and R3 show clear isotopic signals toward the moist mixing line. Moisture uptake from ocean evaporation and continental evapotranspiration represents a very moist mixing member and is opposed to the relatively dry conditions in the free troposphere. For instance, the non-precipitating trajectories in the monsoon cluster R1 (only 7%) start with very moist and enriched values above the Gulf of Guinea and subsequently mix with the drier and more depleted mid-tropospheric air masses while they } pair distributions for the non-precipitating trajectories of each transport cluster. The relative fractions tra E f of corresponding trajectories in each cluster are given in the respective plots. The solid, colored contours comprise 95% of the data for the last 24 hr of the trajectory before reaching the target region (day 0), the dashed, dark gray contours the data of 2 days before arrival and the light gray contours the data of 5 days before arrival. The underlain gray process curves are the same as in Figure 1. Note the much larger axis ranges shown in the bottom two panels.
advance over West Africa. Similar mixing structures are apparent for the Atlantic inflow (R2) and the Harmattan (R3), with substantially larger numbers of non-precipitating trajectories (56% and 84%, respectively). As their initial moisture is much more variable than for R1, both moistening and drying occurs along the non-precipitating trajectories of the R2 and R3, closely following the moist mixing curve.
For the subsiding clusters S1 (AEJ) and S2 (extratropical intrusions), the non-precipitating trajectories are predominant (  E 90%). As they typically originate in the upper troposphere, their starting points constitute very dry and depleted end members, while in this case the mid-tropospheric air masses act as moister end members. Thus, during the subsidence of S1 and S2 strong signals along the dry mixing curve develop, until the moisture approaches values similar to the target moisture of the rising trajectories.
In summary, even though the non-precipitating trajectories of the rising and subsiding transport clusters start with significantly different isotopic signals, mixing homogenizes to first order their } pairs when arriving over the Sahel. Dehydration and moistening along the respective trajectories is well described by the theoretical moist and dry mixing curves.

Importance of Rayleigh Processes
To identify Rayleigh processes along the transport clusters, we now focus on the precipitating trajectories (see Section 3.3). Here, we consider only the transport clusters R1 and R2, because only these two clusters include trajectories that exhibit a significant rain amount and therefore fulfill the rain criterion (see Figure 7b).
In addition to signatures along the moist mixing curve, a clear Rayleigh signal is evident for both clusters (see Figure 9). In particular during the last 24 hr before arrival, when the convection peaks, the } pairs are distributed along the theoretical Rayleigh curve and indicate a depletion that cannot be explained with the mixing curves alone. Additionally, also values appear below the Rayleigh curve. Either this is due to further Rayleigh processes with initial conditions that lead to Rayleigh curves that are shifted toward low- values (which is rather unlikely, since the plotted Rayleigh curve is already chosen for relatively high surface temperature and relative humidity, see Figure 1), or there are processes that lead to an enhanced depletion and create signals in the Super-Rayleigh area, as documented for trajectory T3 in Section 4.2.
rain drops with the ambient vapor. In case of sub-saturation, rain evaporation can take place. Both effects have the potential to further deplete the water vapor (see Section 2) and may thus explain the depleted Super-Rayleigh signatures inside the melting zone.
During the sedimentation of the liquid drops through a convective system, the Super-Rayleigh signatures are less pronounced for the saturated trajectory points, but still remarkable for unsaturated and cloud-free conditions (Figures 11c and 11g). This depletion results, for instance, from rain evaporation in the unsaturated area below the stratiform cloud shield of a squall line. Figures 11d and 11h show the trajectory points, where rain drops occur near the surface and below the convective cloud base. Here, the air parcels are mostly unsaturated and indicate sharp tendencies toward the Super-Rayleigh area. In agreement with , this hints toward effects of subcloud rain evaporation in unsaturated downdrafts.
for the corresponding classes of precipitation and disequilibrium. while equilibration and rain evaporation proceed and reduce the grade of disequilibrium (see Section 2). Eventually, in the sub-cloud zone the imbalance in  , D v deq E changes sign (Figure 12f), featuring equilibrium vapor from precipitation with a higher  E D than the sub-cloud vapor. As here unsaturated conditions prevail (Figures 11h), rain evaporation is strongly enhanced, leading to an enrichment of heavy isotopes in the rain drops and as a consequence to negative  , D v deq E .
Figures 11 and 12 reveal another interesting feature with more enriched values toward the mixing curves (at around −150‰). In particular for the mid-level liquid precipitation, a clear mixing signal stands out that correlates with sub-saturation (Figures 11g) and negative disequilibrium (Figure 12b). We suspect that this feature is a result of synoptic-scale intrusions that transport dry and depleted air masses as rear-to-front flow into a convective system (Kurita, 2013).
In summary, to account for the Super-Rayleigh signals in water vapor in the presence of precipitation, it is not sufficient to think of an isolated process but rather to consider the full interaction of microphysical processes that occur within and around a convective cell. The depletion due to Rayleigh condensation during the convective updraft is superposed by additional depleting contributions of evaporation and equilibration of the falling rain drops. However, the two Super-Rayleigh lines marked in the } phase space constitute rough bounds for the Super-Rayleigh area, as they frame the altitude range, where interactions between vapor and liquid precipitation can occur (from the melting zone to the surface).
pair distributions of R1 and R2 toward the Rayleigh line. Additionally, the increased convection enhances effects such as diffusive equilibration and partial rain evaporation. Since we here consider data in the free troposphere, that is, in the melting zone of falling snow particles, strong isotopic signals develop for R1 and R2 toward the lower Super-Rayleigh line. Because of the strong relation of monsoon precipitation with the air masses transported by the AEJ (Niang et al., 2020;Sy et al., 2018), the isotopic composition of cluster S1 merges with the signals of R1. By contrast, the northwesterly subtropical clusters R3 (Harmattan) and S2 (extratropical mid-level dry intrusions) remain around the mixing curves with only slight tendencies toward the Rayleigh curve. This emphasizes the existence of a subtropical mixing barrier that hinders the isotopic exchange between subtropical and tropical transport clusters as discussed in Yang and Pierrehumbert (1994) and Niang et al. (2020). The resulting contrast between the effects of mixing and microphysical processes are well represented in the contours of the full ensemble.
Finally, in the unusually wet Phases 3 and 4 (right column in Figure 13) the Sahelian free troposphere remains overall moist, but also becomes even more isotopically depleted. The monsoon inflow (R1) and the AEJ inflow (S1) further drop to lower  D v E , as convective processes increase and foster Rayleigh and Super-Rayleigh signatures. During this period convection is so widespread that also the sub-tropical clusters R3 and S2 show indications of reduced mixing and increased Rayleigh signals. As already shown in Figure 7, the low-level Atlantic inflow cluster R2 is not present during these phases (Knippertz et al., 2017). The isotopic composition of all trajectories clearly reflects the shift from the mixing to the Rayleigh line with a marked extension toward the Super-Rayleigh area.
To summarize, the comparison of the } pairs of the transport clusters from Figure 13 against the iso COSMO E grid point values from Figure 3 reveals that the identified process curves along different transport } pair distributions in the target region (Sahelian free troposphere) for each transport cluster (colors, see Figure 5) and monsoon phase in 2016 (columns, see introductory text of Section 4). The shown contours mark 50% and 95% of all data points for (a-c) clusters R1, R2, and R3, (d-f) clusters S1 and S2, and (g-i) all trajectories.
Following the definitions from Section 2, the ratio between the specific water contents of HDO and x eq x x x eq x x eq q q R R q q (A1) with D x E q and x E q referring to the initial, isotopically non-equilibrated state and , D x eq E q and , x eq E q denoting the water contents after isotopic equilibration. Analogous to Equation 3, the equilibrium fractionation factor for the equilibrated states is In saturated conditions there is no net exchange of 2 H E O between the rain drop and the ambient water vapor, that is,  , x x eq E q q . However, as saturation does not imply automatically that also HDO is in equilibrium, an isotopic flux may be enforced between the rain drop and the vapor that fulfills following criterion: and after further rewriting this results in the following expression: Equation A6 relates the ratio of the isotopic composition in the vapor between the equilibrated ( , v eq E R ) and non-equilibrated ( v E R ) state to the ratio of the initial, non-equilibrated isotopic compositions of the falling rain drop ( r E R ) and the vapor ( v E R ). As the rain drops form typically further aloft from more depleted vapor, r E R is in this case lower than , r eq E R . Therefore, we can assume that the ratio of r E R and v E R is lower than  eq E , such that Equation A6 results in That is, the vapor in equilibrium with the more depleted rain drop is more depleted in  D v E than the non-equilibrated vapor. This shows that isotopic equilibration can account for an enhanced depletion in

Data Availability Statement
Simulations were conducted at the Swiss National Supercomputing Centre (CSCS). The trajectory data can be accessed via the DOI: 10.35097/469 (Diekmann & de Vries, 2021).