1. Introduction

Precipitation over land is composed of a mixture of oceanic, as well as terrestrial, sources of moisture. The relative contribution of each source exhibits large variability in space and time. There is a growing need to improve the understanding of the sources and sinks of moisture that contribute to precipitation events, especially over land (Trenberth et al. 2003). Atmospheric moisture within a region can be advected from external sources, or can come from local evapotranspiration within the region. The terrestrial sources of water vapor are of particular interest for the hydrologic community. We refer to the precipitation originating from local sources as “recycled precipitation.” Evapotranspiration from land supplies moisture for the overlying atmosphere, so changes in land surface hydrology, caused by anthropogenic or natural forces, will impact precipitation. The present study focuses on the contribution of evapotranspiration to precipitation events and the role that atmospheric moisture storage plays in the recycling process.

The study of precipitation recycling has been approached in a variety of ways, such as physical analysis using isotope data, numerical tracer experiments, and analytical theories. Analysis of isotopic data in rainfall provides information about the origin of the water molecules. Isotopic studies in the Amazon region are numerous and date back to the 1970s with the work of Salati et al. (1979) and Salati and Vose (1984). These studies find that evapotranspiration within the Amazon basin contributes to about 50% of precipitation. Recently, isotopic data across Russia have also been used to study recycling processes (Kurita et al. 2004). While isotopes provide valuable information about the sources of precipitable water, there are large uncertainties in the calculations because of the variability in the isotopic content of the moisture that contributes to the observed precipitation (Kurita et al. 2004).

The work of Joussaume et al. (1984) and Koster et al. (1986) introduced the idea of numerical water vapor tracers in order to “tag” water from its evaporative source. Numaguti (1999) used water vapor tracers to study the water cycle over the Eurasian continent, finding that oceanic sources dominated moisture supply in the winter, but land sources were important during the summer. In a similar study, Bosilovich and Schubert (2002) used the Goddard Earth Observing System (GEOS-3) GCM, implementing three-dimensional tracers to study the regional hydrologic cycle for North America and India. This study found that 50% of precipitation in the midwestern United States came from continental sources while oceanic sources dominate the supply in India. Dirmeyer and Brubaker (1999) implemented a back-trajectory algorithm to study the moisture sources over the United States during the 1988 drought and 1993 flood, finding that oceanic sources increased during the 1993 flood, while the recycling ratio reached a maximum in 1988. Brubaker et al. (2001) later used the same methodology to construct a 36-yr climatology of evaporative sources of warm season precipitation over the Mississippi River Basin. Recently, Stohl and James (2004) have proposed a Lagrangian technique that uses meteorological analysis data and a particle dispersion model to trace the evaporative sources of water that falls as precipitation. Numerical models have the advantage of providing detailed information about source–receptor relationships. The drawback of numerical models is their inherent dependence on the physical parameterizations of the model.

Analytical models provide a simple and computationally efficient way of estimating the relative contribution of recycled to advected precipitation within a region. The simplicity of analytical models implies that they should not be used for detailed quantitative calculations, but rather as a qualitative tool to diagnose precipitation recycling within a region, and compare across different regions (Trenberth 1999; Bosilovich and Schubert 2001). In the 1950s a widely accepted theory to describe moisture transport was formulated by Budyko and Drozdov (1953), and the one-dimensional model was later described by Budyko (1974). Brubaker et al. (1993) later extended Budyko's model in two dimensions. This model has been used in several studies including a global recycling analysis by Trenberth (1999) and a regional analysis over Europe and the North Atlantic (Schär et al. 1999). Savenije (1995) studied precipitation recycling over the Sahel and proposed a one-dimensional model in a Lagrangian framework. This model, which includes assumptions specific for dry regions, predicts higher recycling over the Sahel than Brubaker et al. (1993). The author attributes this bias to the assumption of constant evaporation and precipitation in the model proposed by Brubaker et al. (1993). Relaxing the assumption of spatial invariance, Eltahir and Bras (1994) formulated a two-dimensional recycling model using a grid-based approach that takes spatial heterogeneities into account. Bosilovich and Schubert (2001) have used this model to study recycling over the central United States. Burde and Zangvil (2001b) propose a two-dimensional recycling model that incorporates the effects of inhomogeneity of the fluxes and nonparallel flow effects. This work addresses a major shortcoming of the Brubaker et al. (1993) model, which is only valid for a parallel flow field. The authors analyzed several primary flow fields and found that the deviations from parallel flow significantly affect the recycling ratio.

All of the analytical models mentioned above assume that the change in storage of atmospheric water vapor is negligible when compared to the atmospheric water vapor fluxes. To justify this assumption, the models are applicable only at monthly or longer time scales. Zangvil et al. (2004) overcome this assumption and propose a model that, although not formally derived from the equation of conservation of mass, can be used at a daily level. Their results show that the monthly averaging masks key relationships between total precipitation and recycled precipitation that become evident at a daily time scale. They also illustrate the importance of relaxing the assumption of negligible atmospheric storage and analyzing shorter time scales. Our study has greatly benefited by the previous work done by Burde and Zangvil (2001a) in which the authors review several recycling models developed in the last decades and present them in a unified framework.

In the present work we use National Centers for Environmental Prediction–Department of Energy (NCEP–DOE) Reanalysis-II (R-II) data (Kanamitsu et al. 2002) to reevaluate the assumption of negligible change in water vapor storage at a monthly time scale. We then present a new analytical model of precipitation recycling derived from the equation of conservation of atmospheric water vapor. The storage term is not neglected, so the model is applicable at a range of time scales, from daily to monthly and longer. This new dynamic recycling model is expressed in a simple and computationally efficient form. The model is then used to explore the spatial and temporal variability of the recycling ratio over the continental United States. Empirical orthogonal function (EOF) analysis is performed on the monthly recycling ratio anomalies over the entire region. The two dominant EOFs, which capture a large percentage of the variability in the data, are discussed. The recycling ratios predicted by the new model are then compared to traditional models of precipitation recycling that do not include the change in moisture storage.

This paper is organized as follows: Section 2 presents a brief overview of the underlying principles of precipitation recycling models and their basic assumptions. A simple magnitude analysis is performed to evaluate the validity of the assumption of negligible change in atmospheric storage. Section 3 presents the new dynamic recycling model and its formal derivation. Section 4 illustrates the recycling ratio as a function of spatial scale. A detailed spatial and temporal analysis of the recycling ratio over the conterminous United States is presented in section 5 and is then compared to the results from traditional recycling models in section 6. The final summary and discussion of the major results of this study are presented in section 7.

2. Recycling models and the assumption of negligible change in moisture storage

a. Basic equations

The first widely accepted theory to quantify the relative contributions of advected moisture and local evaporation to precipitation within a domain was formulated by Budyko (1974). The underlying principle of this, and the numerous models that stemmed from Budyko's Model, is the conservation of atmospheric water vapor mass. Following the notation of Burde and Zangvil (2001a), we can express the vertically integrated balance equation for water vapor as

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where

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where w is the amount of water vapor contained in a unit area column of air; E is evapotranspiration; P is precipitation; and q, q′, , û′, , and υ̂′ are the time mean and transient eddy values of specific humidity and zonal and meridional wind components, respectively. Notice that û are υ̂ are the wind velocities that do not necessarily correspond to u and υ.

Most recycling models make two basic assumptions. The first is that the atmospheric column is well mixed. This means that the ratio of advected to evaporated water vapor in the atmospheric column is equal to the ratio of advected precipitation to recycled precipitation, that is,

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where the subscripts a and m refer to the advected and recycled components, respectively.

The second assumption is that at monthly or longer time scales, the change in atmospheric moisture storage is small compared to the other terms in the mass balance equation [Eq. (1)] and can be neglected. We will briefly discuss two of the most commonly used two dimensional precipitation recycling models that use these two assumptions, and in section 6 we will compare their predictions to that of the proposed dynamic model.

Brubaker et al. (1993) extended the traditional Budyko model to a two-dimensional land region. In their work, the regional recycling ratio is expressed as

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where F⁺ is the inflow of atmospheric moisture over all the boundary of the region, E is evapotranspiration averaged over the region, and A is the area of the region. It has been argued that this method underestimates the recycling ratio because it uses area-averaged values of precipitation and evapotranspiration (Savenije 1995). The second model, developed by Eltahir and Bras (1994), overcomes this drawback by including the spatial variability of the data in a grid-cell-based approach. The region is divided into a rectangular grid, and the recycling ratio within each cell is expressed as

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where I_m and I_a denote the moisture inflow to the grid box from recycled and advective origin, respectively, and E is evapotranspiration. Notice the difference between F⁺, which corresponds to advected inflow into the entire region, and the inflow to a grid box I, which has both advective and recycled moisture.

3. Dynamic recycling model

a. Model derivation

If we consider a grid cell i of area ΔA_i within a region B (Fig. 2), we define the local recycling ratio ρ_i as the amount of precipitation that falls in that cell from evapotranspiration within region B to the total precipitation in that cell:

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where ρ is the recycling ratio, P_m is the precipitation from recycled origin, P is the total precipitation, and the subscript i denotes the grid cell. Notice that because of the assumption of a well-mixed atmospheric column [Eq. (5)], we can also write

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where w is the precipitable water in the atmospheric column. Following the grid-based approach of Eltahir and Bras (1994), we can then calculate the regional recycling ratio r within the region consisting of n cells, as a function of the local recycling ratio:

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For a given grid cell within the region, the water vapor conservation equation can be separated into its advected (a) and recycled (m) components. In the discussion that follows we will drop the subscript i for the grid cell, but it is understood that this applies only to one of the cells within the region. Recognizing that w = w_a + w_m and P = P_a + P_m, Eq. (1) can be decomposed to represent the advected and recycled components as

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Using the assumption of a well mixed atmosphere [Eq. (5)], the definition of recycling ratio ρ [Eqs. (8) and (9)], and the original mass conservation [Eq. (1)], we can rewrite Eq. (12) as

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or in terms of (1 − ρ):

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To solve this equation for ρ, we introduce a new coordinate system:

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With this new coordinate system we now represent evapotranspiration E(x, y, t), recycling ratio ρ(x, y, t), and precipitable water w(x, y, t) as ε(χ, ξ, τ), R(χ, ξ, τ), and ω(χ, ξ, τ), respectively. Burde and Zangvil (2001b) present a similar idea of coordinate transformation. However, the difference is that they do not include the temporal term, and in their derivation, only the y coordinate is transformed. Including the temporal term, and transforming the x and y coordinates, actually simplifies the calculation as shown below.

Through this transformation we are effectively following the paths defined by the u and υ velocities [Eq. (4)] and simplifying the mass conservation equation for the recycled component. Using the chain rule, we can transform Eq. (14) into the new coordinate system.

Note that

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and similarly,

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Substituting Eqs. (16)–(18) into Eq. (14), the partial differential equation is now a simple ordinary differential equation:

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Solving for R, we obtain the expression for the recycling ratio:

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The value of R can be transformed into ρ in the original coordinates by substitution of χ(x, y, t) and ξ(x, y, t) into the expression of R. This new model provides a simple tool to perform spatiotemporal analyses of precipitation recycling. Unlike most previous analytical models, it incorporates the moisture storage term. This is a significant improvement because the previous assumption of negligible storage restricted the applicability of the models to monthly or longer temporal scales, whereas the new model can be used to study all time scales longer than the time of boundary layer mixing. It also improves upon other models that do include this term (Zangvil et al. 2004), because it stems directly from the basic equation of conservation of mass. The model is also simple, and computationally efficient.

b. Numerical implementation

To ensure a homogeneous coverage of the target region, the numerical scheme was implemented to traverse back in time. Starting at time step n, with information at every grid point of the selected region, we trace the advected moisture until its location at time step n − 1 using a simple iterative technique (see Merril et al. 1986), that is,

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where the time step Δt is negative. For each time step, the value of ε/ω is calculated and used in Eq. (20). The integration is performed from the current time τ until time 0, when the x and y values have reached the border of the region. It is important to keep in mind that these are the trajectories defined by the velocities u and υ as defined by Eq. (4), and not by the actual zonal and meridional wind velocities. The moisture-weighted velocities u and υ are smaller than the mean zonal and meridional winds, because more weight is given to the velocities at lower altitudes (due to increased atmospheric moisture at low elevations) where velocities are reduced as a result of the interaction with the earth's surface. Figure 3 shows an example of the “paths” of the advected moisture for May 1986.

c. Budget calculation errors in Reanalysis-II data

In the sections that follow, we perform recycling estimates using 6-hourly data from the R-II project to calculate the different terms in the moisture budget including monthly precipitation, evapotranspiration, precipitable water, moisture flux, and u and υ values, which include the eddy contribution [Eqs. (1)–(4)]. When using assimilated data to perform moisture budget calculations, Eq. (1) must be modified to take into account the residual term α that arises from closure problems:

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This is an artificial residual forcing that appears in the four-dimensional data assimilation and is not part of the natural physics. Figure 4 shows the spatial variability of the average residual term at daily and monthly time scales. The importance of the residual term can be seen in Table 1, where the average value of each of the terms in Eq. (22) is presented, and α is of the same order of magnitude as the other terms in the equation. Unfortunately, this term is not random but a systematic correction due to different sources of error including space and time truncation errors and fundamental errors in the model physical parameterizations (Kanamitsu and Sha 1996; Roads et al. 2002).

There is no way to systematically account for the residual term in our analyses, however, we can evaluate which terms in the moisture budget equation could be significantly affected by this residual. The zonal and meridional winds, used to calculate the moisture flux terms, are type A variables [following the classification of Kalnay et al. (1996)], which are reliable data, strongly forced by observations. Precipitable water is derived from specific humidity, which is a type B variable, influenced by both the model and observations. When compared to the National Aeronautics and Space Administration (NASA) Water Vapor Project NVAP data (Randel et al. 1996), the precipitable water and change in moisture storage show excellent agreement (see Fig. 5). This gives us confidence about the quality of the moisture storage data and the flux terms. Precipitation is a type C variable, completely determined by model physics. Our results could be greatly influenced by this variable, but as we will show in section 5e, the recycling estimates are not significantly changed by the use of a different precipitation dataset.

We are left with evapotranspiration, also a type C variable. Figure 6 shows the average monthly evapotranspiration (1979–2000) used in this work. It is difficult to estimate the validity of this data as there are no alternate long-term datasets that could be used for comparison. As seen in Table 1, the residual term is around 11% of the evapotranspiration term at a monthly level. The negative sign of the residual could indicate an overestimation of the evaporation term, as other studies have indicated (Trenberth and Guillemot 1998). Furthermore, as seen in Fig. 4, the largest negative residuals at a monthly level are located in the Gulf Coast, a region of significant evapotranspiration. Keeping this strong limitation in mind, we believe that the R-II provides the best available data for the analysis presented in this work.

4. Spatial-scale analysis

The amount of recycled precipitation within a region is a function of spatial scale. Intuitively we know that as the size of the region increases, there is a larger possibility for evaporative sources to provide moisture for precipitation in the same region, so the recycling ratio increases. Furthermore, because of the spatial heterogeneity within the region, there is not a linear increase of recycling ratio with scale.

To get a quantitative estimate of the recycling ratio at different spatial scales, 22 yr (1979–2000) of R-II data was used to run the model for 10 nested regions of different sizes. As shown in Fig. 7, the smallest region is approximately 2.5 × 10⁵ km² centered over southern Illinois, and then gradually progresses to a 4 × 10⁶ km² region. The analysis is limited to the conterminous United States. Figure 8 shows the average monthly and average summer [June–August (JJA)] recycling ratio obtained from the 22 years of analysis. As expected from the previous discussion, the recycling ratio increases with scale, but the slope of the curve decreases, presenting a logarithmic relationship between spatial scale and recycling ratio. At larger scales, the spatial variability within the region will dampen the increase in recycling ratios. We can also see that the summer season shows higher recycling ratios, although the increase is not uniform, with larger regions presenting a higher percent increase than smaller ones. These results are similar to the findings of Brubaker et al. (2001), who analyze the spatial dependence of the recycling ratio over the Mississippi region using a back-trajectory algorithm.

5. Recycling analysis over the conterminous United States

To analyze the spatiotemporal variability of the recycling ratio over the conterminous United States we use the same data as in section 4 and divide the entire domain into 20 subregions (Fig. 9). As discussed before, the recycling ratio depends on the size of the region. For this reason, all the subdivisions have the same area (approximately 1 × 10⁶ km²), with the exception of those that extended into the oceans that were reduced.

Equation (20) is then used to calculate the value of R, which is equivalent to the recycling ratio ρ. For every grid point within the region we obtain a value of ρ, and the regional recycling ratio r corresponding to each of the 20 subregions shown in Fig. 9 is then obtained using Eq. (10). The analysis is done at a monthly time scale in order to compare the results with those obtained using traditional models. Only the summer season will be discussed, as it is the season of highest contribution from evapotranspiration.

6. Comparison to models without atmospheric moisture storage

In this section we compare the results obtained using the dynamic recycling model to more traditional analytical models that do not include the storage term. We will use the precipitation recycling models of Brubaker et al. (1993) and Eltahir and Bras (1994), which were described in section 2. Both analytical models were used to calculate the recycling ratio for the same 20 subregions of the United States (Fig. 9), using exactly the same data as in section 5.

Figure 16 shows the smoothed regional recycling ratio using the recycling models with no storage. As expected, both models show similar spatial variability, but the recycling model proposed by Eltahir and Bras (1994), which includes spatial heterogeneity, predicts higher recycling ratios. Comparing Figs. 16 and 10, we see that the three models produce the same spatial pattern (with the exception of the southern California high recycling, which is lost in the Brubaker et al. model), but the new model predicts higher recycling values throughout the United States. This can also be seen in Fig. 17, which shows the time series of the predicted summer recycling ratio for a few selected regions using the three models. We can see that the temporal variability is almost identical, but the model that includes storage consistently predicts recycling ratios about 33% larger than the Budyko-type model proposed by Brubaker et al. (1993) and 12% larger than the Eltahir and Bras (1994) model. We can see that although the change in moisture storage is small when compared to the advection term at a monthly time scale (see section 2b), it plays an important role in the precipitation recycling process. By relaxing the assumption of negligible moisture storage in the atmosphere, significantly different recycling ratios are predicted, even at a monthly time scale. The precipitation of recycled origin shown in Fig. 17 closely follows the recycling ratio, with years of high (low) recycling ratio presenting high (low) recycled precipitation. The other regions (not shown) present the same type of behavior.

7. Conclusions

This work investigates the issue of precipitation recycling. It questions the assumption of negligible change in storage of atmospheric moisture, and shows how this term can be included in the formulation of analytical recycling models. We first perform a scaling analysis of the terms in the equation of conservation of atmospheric water vapor over the continental United States at a daily and monthly time scale. The magnitude of the storage term over the advection term is small at a monthly time scale, reaching a maximum of 2% over the arid south-central region of the United States. However, at a daily scale the storage is around 50% of the advection term over the eastern part of the country and southern Arizona. If precipitation recycling is to be modeled at daily time scales, this term must be included in the moisture budget equation.

To account for the change in moisture storage, a new recycling model is proposed that is derived formally from the equation of conservation of atmospheric water vapor. This is a significant improvement, as it enables the analysis of precipitation recycling at a range of scales, from daily to monthly and longer. The model is expressed in a simple and computationally efficient form, and a numerical algorithm is suggested. The model is then implemented at a monthly scale, for a 22-yr period over the conterminous United States using NCEP/DOE Reanalysis II data. On average, the southeastern part of the United States exhibits the largest recycling ratios as a result of the high temperatures and water availability. The northeast presents the lowest recycling ratios. The western United States also presents a strong gradient with low recycling ratios in the north and high recycling in the south. The Colorado region presents high recycling ratios because of the low precipitable water content in the mountainous regions.

The spatiotemporal variability of the recycling ratio over the conterminous United States is presented using EOF analysis. The first mode, accounting for 29% of the variability in the data captures strong recycling events over the western part of the country in the summers of 1986, 1992, and 1998 and low recycling values in 1994. The second mode, representing 21% of the variability in the data, captures recycling ratio anomalies in the central part of the country with low recycling ratios during 1980 and 1993 and high ratios during 1988 and 1989. The analysis of recycling ratios over a period of several years shows that the years of highest recycling ratios do not necessarily correspond to extreme drought or flood events. Our study shows that during the summers of high recycling, the moisture in the soil and vegetation contribute significantly to precipitation in these regions, but the triggering mechanisms were not linked to extreme drought or flood conditions over the region.

The results of the EOF analysis show an important aspect of analytical models of precipitation recycling. The simplicity of the models allows for extended spatiotemporal analysis, which serves as a diagnostic tool. Regions and time periods of anomalous recycling ratios can be selected, and then more complex models can later be used to study the physical processes in greater detail.

To evaluate the performance of the new model, we performed a recycling analysis using the same dataset but with two different traditional models of precipitation recycling that do not include the change in atmospheric moisture storage. When compared to recycling models of Eltahir and Bras (1994) and Brubaker et al. (1993), the new recycling model exhibits almost identical spatial and temporal variability. However, the predictions are consistently larger than the traditional models, with about a 12% and 33% increase, respectively. This indicates that despite its small contribution in the moisture budget equation at a monthly time scale, the change in moisture storage plays an important role in the precipitation recycling process.

The results presented in this study are dependent upon the quality of the data. We have used R-II data, which includes an artificial residual forcing in the moisture budget due to the assimilation process. We believe that this residual affects our results primarily through uncertainties in the evapotranspiration estimates. The results presented here must be compared with analysis using evapotranspiration data of better quality. These issues are under investigation.