The time it takes before evaporated water from land and oceans is returned to the land surface as precipitation is a fundamental characteristic of the Earth's hydrological cycle. This atmospheric residence time of moisture is not often discussed in the scientific research literature. The global average residence time of atmospheric moisture is mostly seen as non-controversial knowledge in textbooks e.g.,, general water literature e.g., and educational web pages e.g.,. All of these examples estimate the global average residence time of atmospheric moisture based on the size of the atmospheric reservoir divided by the incoming or outgoing flux and as such arrive at estimates in the range of 0.022–0.027 years or 8–10 days.

While the global average residence time is a simple estimate, spatial and temporal pictures of residence times are more difficult to provide. Local depletion times, given by W/P, and local restoration times, given by W/E (where W is water in the local atmospheric column, P is precipitation and E is evaporation), were computed near-globally by and . When horizontal moisture transport is small compared to precipitation and evaporation they provide a proxy for the residence time. However, it is safer to interpret them as local timescales of atmospheric moisture recycling . found a global average residence time of atmospheric moisture of 8.9 days based on evaporation and 9.1 days based on precipitation, and attributed this difference to the input data having a non-closure of the global water balance. On the other hand, global spatial average local depletion times and restoration times were found to be 8.1 and 8.5 days, respectively. The difference with the global moisture-weighted values was explained by the heterogeneous distribution of evaporation and precipitation over the Earth and their spatial correlation with atmospheric moisture.

Moisture tracking models also allow for the estimation of atmospheric moisture residence time. A semi-Lagrangian method of passive water vapor tracers in a general circulation model was used to perform a special experiment in which all atmospheric water was tagged, after which it was evaluated how quickly the tagged water rains out . To clarify, in this experiment the atmosphere was replenished by evaporation, but this water was not tagged. The outcome of this analysis was a global average residence time of 8.5 days in May and 9.2 days in an undefined time period , respectively. The same types of experiments were performed by with an offline one-layer Eulerian moisture tracking model using the global GAME reanalysis data. They found a global mean residence time of between 7.3 days (April) and 9.2 days (August).

The global experiments of and also suggest that the time it takes before atmospheric water rains out has a negative exponential distribution, which belongs to a Poisson process e.g.,. The median residence time can be estimated from the graphs provided by and as around 6 days. On a more local scale with a pure Lagrangian moisture tracking method applied in India, showed probability density functions of the time that evaporated water remains in the atmosphere before it precipitates again. They found that evaporated water has the highest probability of staying in the atmosphere for around 5 days, but that there is a long tail of the probability distribution as many water particles still reside in the atmosphere more than 25 days after evaporation, suggesting the mean to be much higher than the median.

In a near-global study extended Eulerian moisture tracking model WAM-2layers to calculate both the amount and age of tracked water. Forcing their model with precipitation, wind and humidity from ERA-Interim (ERA-I) , and evaporation from global hydrological model STEAM , they computed atmospheric residence times over land. As such, they found that the residence time of land precipitation is 9.7 days and that of precipitation recycled over land is 6.4 days. For evaporation they found a residence time of 8.7 days, with fast evaporation processes (interception, soil and open water evaporation) having a residence time of 8.1 days vs. a residence time of 9.1 days for transpiration. In addition to these averages, they also showed spatial maps of residence times, but these refer to the fraction that recycles only, which is relatively fast compared to the average residence times. South America popped up as the continent with the lowest atmospheric residence times, which were shown to be around 4 days for moisture recycling over land. As explained above, residence times of atmospheric moisture recycled above land are substantially lower than the mean residence times of total moisture. The method of age tracers was previously also used by , who found residence times of multiple months. However, started counting from sea evaporation in his experiments and continued the aging after water precipitated on land, infiltrated into the soil and re-evaporated. Therefore, his results cannot be interpreted as atmospheric residence times directly.

The established knowledge of the residence time of water in the atmosphere being 8–10 days was recently challenged by . They used backward trajectories computed by Lagrangian particle tracking method FLEXPART (forced with ERA-I) and concluded that the global mean residence time of water in the atmosphere is 4–5 days, i.e., half of the traditional estimates. More precisely, they calculated the residence time to be 3.9 ± 0.8 days (spatial variability indicated by 1 standard deviation) for 15-day backward trajectories and 4.4 days for 20-day backward trajectories, and showed spatial figures of the first estimate. An obvious candidate for the discrepancy with the prevailing knowledge is the length of their trajectories; however, they stated that almost 100 % of the original moisture can be attributed to evaporation after 20 days, and that further backtracking is unphysical and can never come close to the 8–10-day estimate. Here, we would like to point out another important assumption, which was not addressed by , namely that their methodology can accurately estimate evaporation. It should be known that FLEXPART is generally only used to look at specific humidity changes (evaporation - precipitation, or E - P). Attribution of evaporation is difficult as note that a number of moisture transport processes are neglected, which are moisture changes due to convection, turbulence, numerical diffusion, and rainwater evaporation. even note that specific humidity fluctuations along a trajectory may be entirely unphysical. , who evaluated the FLEXPART methodology, found that when FLEXPART is used to evaluate E and P separately, evaporation is highly overestimated. Specifically, they obtained a global average evaporation of E = 1380 mm yr-1, which corresponds to E = 704 × 103 km3 yr-1. Using the numbers obtained by Stohl and James (2004) and ERA-Interim atmospheric storage, a global average residence time of 12.4 × 103 km3 / 704 × 103 km3 yr-1 = 0.017 years = 6.4 days is obtained. Overestimation of evaporation will thus bias the estimates of residence times downward. How this assumption influences the results by LS16 was not evaluated in their paper. Whatever the methodological reason for the low global mean residence time estimates by , we will argue in this paper that these numbers are physically impossible.

The objective of this paper is to revisit the current knowledge and provide a state-of-the-art view in time and space of the residence time of water in the atmosphere using several different approaches. Table 1 provides a non-exhaustive overview of the global average residence time of water in the atmosphere found by the studies mentioned in this introduction. Section 2 explains the methods used in this study. In Sect. 3 we explain why the global average residence time estimates based on quantifications of the global hydrological cycle are valid and the counterarguments provided by are not. In Sect. 4 we provide near-global spatial pictures of atmospheric residence time above land and sea. Section 5 analyzes the probability density function of the residence time of atmospheric water particles, and, finally, in Sect. 6 we state the conclusions of this paper.

If one would like to know the average residence time τ in any reservoir, one simply divides its average mass M‾ (or volume when assuming constant density) by its average outgoing mass flux F‾ (which equals the ingoing flux when there is no change in mass): τ=M‾F‾. While this is a simple formula, computation of reliable residence times in, for example, a surface water lake may be difficult due to many uncertainties in a lake's volume, hydraulic flow, precipitation, evaporation and seepage e.g.,. Moreover, a lake may be permanently stratified (i.e., there is permanent dead storage), and one could argue that the actual volume participating in the water cycle of the lake does not equal the lake's total volume, meaning that the actual average residence time becomes lower. If one can, however, reliably estimate a lake's volume and inflow or outflow, it is not necessary for a lake to be well mixed for Eq. (2) to hold, nor is it necessary for F to be constant. The mere necessity is that the entire volume participates in the water cycle. Of course, one could still have significant local differences, but the average can reliably be calculated by Eq. (2).

When the Earth's entire atmosphere is considered to be the reservoir of study, its residence time can actually be calculated much more easily than that of a lake. In the global case the only inflow is evaporation and the only outflow is precipitation. Moreover, due to the turbulent nature of the atmosphere all water that resides in the atmosphere also participates in the atmospheric water cycle; i.e., there is no such thing as permanent dead water storage e.g.,. This is furthermore supported by the online passive moisture tracking experiments of . Note that this online tracking method does not suffer from the well-mixed assumption for precipitation. Even so, they found that by tagging all moisture in the atmosphere, after 30 days, there was only 3 % of the tagged passive moisture left; thus, at least 97 % of the total moisture in the atmosphere must have been participating in the hydrological cycle within 30 days.

The use of Eq. (2) to calculate the global mean residence time of atmospheric water has recently been criticized by . They argued that Eq. (2) is (a) not a reliable estimator for local residence times as it does not involve horizontal moisture transport, (b) should be corrected for the surface area of the Earth where most precipitation is observed, and (c) that the temporal characteristics of global precipitation cannot be measured by depletion time constants. However, we disagree with these arguments as (a*) horizontal moisture transport is irrelevant for the global average value, (b*) the surface area of the Earth is irrelevant as it is not in Eq. (2), but nonetheless all areas participate in the hydrological cycle as there is also transport even over the Sahara e.g.,, and (c*) the values in Eq. (2) correspond to the elemental physical concept that an average residence time can be calculated from dividing a stock by its average influx or outflux under the assumption that there is negligible net stock change over a longer period; whether these average fluxes are constant or not is irrelevant, and the temporal characteristics of precipitation and evaporation can only affect the probability density functions of the residence time (Sect. 5), but not the average. In the Supplement we dispute the counterexamples, objecting to Eq. (2) by Supplement Sect. 4, in more detail. Moreover, we argued above that the entire atmospheric volume participates in the hydrological cycle. Thus, Eq. (2) can, in our opinion, safely be used to calculate the global average residence time of atmospheric water.

Applying Eq. (2) to estimates of the global hydrological cycle (Fig. 1) yields a global mean residence time of atmospheric water of 8.9 ± 0.4 days (uncertainty indicated by 1 standard deviation). The calculation of the mean is as follows: τ=12.6×103(403.5+116.5)×103=0.024years=8.9days, and the standard deviation was calculated by general uncertainty propagation theory. The 1st and 99th percentiles of this estimate are 7.9 and 9.8 days, respectively. All previous global average estimates referred to in this paper roughly fall within this uncertainty range (Table 1), except for the estimate provided by , which is less than half. Based on the arguments provided in this section we believe that the latter estimate is incorrect (the probability of an atmospheric residence time lower than 3.9 days equals 1 × 10-30). Even if we make the assumption that water outside the troposphere, and thus in the stratosphere or mesosphere (where aerosol lifetimes on the order of 1 year have been found; ), does not participate in the hydrological cycle, the atmospheric storage is reduced by ∼ 1 % only. The corresponding global average residence time of water in the troposphere then is 8.8 ± 0.4 days, and thus nowhere near the estimates by . In the following example we show that their findings violate global mass balance: let us start from the fact that the average atmospheric water storage in ERA-I is 12.4 × 103 km3 and the average precipitation rate in ERA-I is 531 × 103 km3 yr-1 = 1.45 × 103 km3 day-1 = 2.85 mm day-1 (Fig. 1). For the sake of the example, let us neglect the 1 % moisture outside the troposphere. Then, the resulting active atmospheric storage is 12.3 × 103 km3. For that atmospheric water to have a residence time of 3.9 days, there must have been an evaporation rate of 12.3 × 103/3.9 = 3.15 × 103 km3 day-1 = 6.03 mm day-1, which is physically impossible as that means an enormous imbalance in P and E. Alternatively, it would require an enormous part of the atmospheric water to be dead storage (i.e., never participate in the hydrological cycle), namely 12.4 × 103 - 1.45 × 103 × 3.9 = 6.7 × 103 km3 or 54 % of all water in the atmosphere.

Figure 1 shows several estimates of global residence times. ERA-I fluxes fall well within the uncertainty ranges provided by , but they are generally on the high side. Thus, the global mean residence time estimate of 8.4–8.6 days, based on global ERA-I data, is slightly on the low side of the uncertainty spectrum. The estimates from our moisture tracking methods (WAM-2layers and 3D-T), which use ERA-I, match quite well for global mean atmospheric residence time. The moisture tracking estimates split out for land and ocean are, therefore, slightly lower than the most likely value of 8.9 days. However, it is clear that the turnover of water in the atmosphere is faster over the ocean than over land, having a difference of about 2 days. The logical explanation here is that the hydrological cycle over the ocean is not limited by dry conditions on land, and, as a consequence, is more intense.

Figure 2 provides a near-global spatial view of the annual average hydrological cycle, atmospheric residence times and age. Globally averaged, precipitation residence time, evaporation residence time and age of atmospheric water should be the same, but may differ due to imbalances in the data of the atmospheric hydrological cycle (see Fig. 1). More importantly, these three metrics have a different physical meaning, and thus, a different spatial pattern (Fig. 2c–e). Let us consider a particular location in the world, in this case Portugal, as an example. As can be seen from Fig. 2d, moisture which evaporates from Portugal stays in the atmosphere on average about 14–15 days before it rains out again. In other words: the atmospheric residence time of evaporation is 14–15 days. The local recycling of atmospheric water is only a few percent e.g.,, and much of the evaporated atmospheric moisture is, in fact, transported towards relatively dry regions in the Mediterranean and Africa e.g.,, hence the relatively long atmospheric residence time of evaporation. On the other hand, the precipitation in Portugal comes for a large part from oceanic sources relatively nearby e.g.,, and we estimate that is has resided in the atmosphere for about 7–8 days (Fig. 2c) before it fell as precipitation in Portugal. In other words: the atmospheric residence time of precipitation is 7–8 days. The spatial image of the age of atmospheric water (Fig. 2e) is very similar to the precipitation residence time (Fig. 2c). For our Portugal example, the average age of atmospheric water is about about 8–10 days. Precipitation draws its water from the atmospheric reservoir with a certain age, but apparently, the atmospheric moisture in the drier months has a higher age. Hence, for Portugal, the time-averaged age of atmospheric moisture can be somewhat higher than the precipitation-weighted atmospheric residence time of precipitation.

With Fig. 2 we would like to stress that there are multiple ways of looking at the residence of atmospheric moisture. Whether you look at residence time from a precipitation perspective (Fig. 2c) or an evaporation perspective (Fig. 2d) gives an entirely different picture. In the precipitation perspective, the time from the previous evaporation is stressed, while in the evaporation perspective, the time to the next precipitation event is stressed. The definition of an atmospheric residence time – for both precipitation and evaporation – is analogous to the definition of other metrics for the atmospheric branch of the hydrological cycle, which are also defined for both the precipitation and evaporation perspectives e.g.,. Moreover, you can also look at the actual age of atmospheric water as it resides in the atmosphere (Fig. 2e). Figure 2f provides the latitudinal averages of Fig. 2c–e, as well as of the estimates from WAM-2layers and 3D-T separately (Fig. S1), for which a discussion is attached in the Supplement.

Figure 2c is directly comparable to a recent estimate Fig. 2a. Their spatial patterns are very similar, however, we observe that they underestimate the residence time everywhere with a factor 2–3 compared to our results. As pointed out in the introduction, the method of relies on the unevaluated assumption that they can accurately attribute evaporation, and on a rather short length of their trajectories (15 or 20 days). In contrast, our methods use longer trajectories (WAM-2layers: continuous; 3D-T: 30 days), and use the fields of ERA-I evaporation directly. Moreover, our global average values fit with the atmospheric water balance (Eq. 2 and Fig. 1). For an extensive discussion of the counterarguments of against the use of Eq. (2), and our rebuttal, we would like to refer the reader to the Supplement attached to this paper.

We make the following observations based on Fig. 2.

The places of low precipitation residence times (Fig. 2c) coincide mostly with areas of low precipitation (Fig. 2a). This indicates that if there is precipitation, its water content has recently evaporated and is most likely of local origin. Note that the reverse statement, low precipitation (Fig. 2a) coinciding with low precipitation residence times (Fig. 2c), is not necessarily true.

Figure 3 shows the age and residence times of atmospheric water in January and July. In January (Fig. 3a) the age of atmospheric water is relatively low in the Northern Hemisphere, and relatively high in the Southern Hemisphere. In July (Fig. 3c) the pattern is reversed. In much of the Southern Hemisphere the atmospheric moisture storage (see Fig. S3) is lower in July (Fig. S3b) compared to January (Fig. S3a), precipitates rates are higher in July (Fig. S3d) compared to January (Fig. S3c), and evaporation rates are also higher in July (Fig. S3f) compared to January (Fig. S3e). The higher evaporation rates in July in the ERA-I data may seem counterintuitive, but correspond to previous studies e.g.,. It is, therefore, quite logical that lower storage and higher fluxes lead to lower moisture ages in the Southern Hemisphere in July (Fig. 3c) compared to January (Fig. 3a). Note that Animation 1 in the Supplement provides a view of moisture age throughout the year. The seasonal patterns we observe for precipitation residence times (Fig. S2a and c) are quite similar to Fig. S3, albeit that their figures are for DJF and JJA, while ours are for January and July, and, as for the yearly average figures, their absolute values are much lower.

Over the continents, especially Eurasia, the pattern of atmospheric water age is complex, and we observe an interesting ocean-to-land contrast (Fig. 3). In January, looking at the Northern Hemisphere, the ocean is relatively warm compared to the land, and the age of atmospheric water increases going inland, as there is little replenishment from land evaporation. In July the opposite situation occurs: the ocean is relatively cool compared to the land and the age of atmospheric water decreases going inland. This corresponds to high evaporation rates and corresponding high continental moisture recycling ratios e.g.,. As a result, the atmospheric water age over a large part of Asia is actually lower in July compared to January. Over the Southern Hemisphere we see the same ocean-to-land contrast, but it is less pronounced as there is relatively little land present. The latitudinally averaged residence times of precipitation and evaporation (Fig. 3b and d) also show the summer vs. winter reversal.

Figure 4 shows probability density functions (PDFs) for residence times. Theoretically, the global PDFs of evaporation and precipitation residence time should be identical. However, they slightly differ, which can be attributed to inconsistencies in the ERA-I forcing data as well as the assumptions made in the tracking model. Regarding the forcing data, it seems from Fig. 1 that the lifetime of atmospheric moisture is slightly too short, a common issue in all models , indicating that our results may be slightly skewed towards lower residence times. According to , precipitation also falls too early in the day in all models, thus the amplitude of residence times over land could also be affected, but it is unclear to what extent. Regarding the modeling assumptions, in 3D-T we assume a humidity-weighted well-mixed atmosphere during precipitation and the starting location of the trajectories is randomized over the grid cell. These assumption may lead to an underestimation of the number of water particles that undergo a very fast cycle, and may, thus, slightly skew our results towards higher residence times. By definition of mass balance, however, the actual mean of the distribution should not change. When more water particles undergo a faster(slower) cycle, as a logical consequence, also more water particles undergo a slower(faster) cycle. Adding age tracers to online tracking methods , but then applied to new global methods e.g.,, would allow to check the validity and consequences of these assumptions in more detail, however, would still depend on the model world.

Looking at Fig. 4 we see that short residence times have the highest probability, but there is a long tail with low probabilities and high residence times. We find the median residence time of precipitation and evaporation to be 5.7 and 4.6 days, respectively. Thus, the median is about 3 to 4 days less than the mean (see Fig. 1), indicating that the long tails skew the mean significantly. About 5 % of the moisture has residence times of more than 30 days, which we assumed to have a residence time of 30 days when we calculated the mean (Fig. 1). As a consequence, the estimates for the mean from 3D-T may be slightly lower than the “true” values. It is unclear how have dealt with the unattributed moisture after the end of their trajectories, but they already attributed 97 % of the initial precipitation to evaporation after just 15 days in their Lagrangian model. In Sect. 3, however, we argued that this is physically impossible from a global mass balance perspective.

We furthermore observe an interesting daily cycle in the residence time PDFs over land (Fig. 4), while the general shape of ocean and land PDFs is not very different when looking at timescales of multiple days. We suggest that the daily cooling and warming, resulting in a daily cycle of land evaporation, and a higher likelihood of precipitation occurring towards the end of the day, cause the daily cycle in the residence time PDFs. This phenomenon is only visible over land, as here surface cooling and warming occur with much greater amplitude than over the ocean. These cycles are still visible after multiple days (Fig. 4).

In this paper we studied the residence time of water in the atmosphere. We revisited the state-of-the-art knowledge and studied its properties in time and space on a global scale. As discussed in previous sections, our results are naturally limited by the validity of the input data and the assumptions in our tracking models. However, we trust our results enough to draw the following main conclusions.

Given the state-of-the-art estimates of the hydrological cycle, the global mean residence time of atmospheric water is 8.9 ± 0.4 days (uncertainty indicated by 1 standard deviation). The 1st and 99th percentiles of this estimate are 7.9 and 9.8 days, respectively.

The underlying ERA-Interim data are supplied by ECMWF and can be accessed at http://apps.ecmwf.int/datasets/data/interim-full-daily/. The model code to reproduce the results is accessible in the following way: the basic code for WAM-2layers is available for download from Github (van der Ent, 2016), and the code for 3D-T is available upon request from Obbe Tuinenburg (o.a.tuinenburg@uu.nl).