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Bulk Recycling Models with Incomplete Vertical Mixing. Part I: Conceptual Framework and Models GEORGY I. BURDE Jacob Blaustein Institute for Desert Research, Ben-Gurion University of the Negev, Sede Boker, Israel (Manuscript received 25 June 2004, in final form 25 July 2005) ABSTRACT A conceptual framework, within which the bulk recycling models could incorporate the effects related to an incomplete vertical mixing of water in the tropospheric column, is developed. As an example, the bulk model with the modified well-mixed atmosphere condition, incorporating the “fast recycling” process, is constructed. The model contains a tunable, spatially varying parameter K that characterizes the level of the incomplete vertical mixing effects. The simplified analytical models based on the modified well-mixed atmosphere condition are developed. The analysis of the simplified model results shows that the regional recycling ratio may depend significantly on the average of K, but it depends only slightly on the rate of variation of K, which indicates that models with a constant K can be used instead of the variable K model.
1. Introduction Feedback processes between the soil and the atmosphere are of major importance for climate studies. One of the classical issues in this respect is precipitation recycling. Recycling, as defined, refers to how much evaporation in a continental region contributes to the precipitation in the same region. It is commonly characterized by the recycling ratio expressing the fraction of local-origin precipitation to the total precipitation in the region. The concept of precipitation recycling has been debated for many decades [for reviews see Brubaker et al. (1993) and Eltahir and Bras (1996)]. Estimates of precipitation recycling, even though they are based on observational data, inevitably represent indirect estimates since no possibility exists in observations to separate water molecules of local origin from those of external origin in precipitating water. Another approach is the use of stable isotopes of water (e.g., Salati et al. 1979; Henderson-Sellers et al. 2002; Kurita et al. 2004). There are, however, some difficulties in interpreting isotope analysis for water recycling studies because the isotope data reflect a number of
Corresponding author address: Dr. Georgy I. Burde, Department of Solar Energy and Environmental Physics, Jacob Blaustein Institute for Desert Research, Ben-Gurion University of the Negev, Sede Boker 84990, Israel. E-mail:
[email protected]
© 2006 American Meteorological Society
processes simultaneously. Thus, estimation of precipitation recycling necessarily involves some underlying assumptions and limitations—often referred to as a “recycling model” [see review of recycling models in Burde and Zangvil (2001a)]. The existing recycling models use the vertically integrated water vapor transport and assume that the local source of water is mixed well with all other sources of water in the whole tropospheric column—therefore they may be called “bulk” models. In the bulk models, time-averaged data is used—commonly monthly, and it is assumed that the change in storage of water vapor is small compared with the atmospheric water vapor fluxes. The first recycling model of such type is a highly simplified model developed by Budyko and Drozdov (1953) and described in Budyko (1974)—Budyko’s model. The estimates of regional precipitation recycling were mostly made using Budyko’s model either in its original or, in recent studies, Brubaker et al.’s (1993) form. The latter variant is, sometimes, called the Brubaker model, for example, in Trenberth (1999) and Bosilovich and Schubert (2002). It is commonly identified with a bulk model, and the studies of the continental-scale water cycle with numerical atmospheric models mostly compare the results of recycling calculations with the estimates by Budyko’s model. Several generalizations of Budyko’s model have been proposed among which we mention the Drozdov and
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Grigor’eva (1965) generalization and the Integral Moisture Balance (IMB) model—see Burde and Zangvil (2001a) for details. The particular cases of the Drozdov and Grigor’eva model have been used for recycling calculations in Shiklomanov (1989), Brubaker et al. (1993), and Savenije (1995). The IMB model has been used under different names, such as the “one grid-cell method” in Eltahir and Bras (1994), “Budyko’s model” in Schär et al. (1999), and the “tank model” in Zangvil et al. (2004). Eltahir and Bras (1994) were the first to develop a numerical method for precipitation recycling calculations on the basis of a two-dimensional bulk model. The relations expressing conservation of water vapor mass of different origins, together with the condition of a well-mixed atmosphere, are applied to a cell of a grid covering the region, and an iterative procedure is used to find the recycling ratio values satisfying those relations for all cells. Eltahir and Bras (1994) applied their method to studying precipitation recycling in the Amazon basin. They also considered a possibility of applying the method to all of the region as one grid cell: in such a form it becomes exactly equivalent to the IMB model. The Eltahir and Bras numerical model was applied for quantifying precipitation recycling over the central United States in Bosilovich and Schubert (2001) and for North American regions in Bosilovich and Schubert (2002). It is not clear what form of the Eltahir and Bras method, the grid iterative calculations or the one cell approach, was used since the issue of the grid spatial resolution does not arise in Bosilovich and Schubert (2002). In Szeto (2002), the Eltahir and Bras (1994) numerical method was applied for investigating moisture recycling over the Mackenzie basin. In Burde and Zangvil (2001a,b), the most general bulk recycling model, based only on the vertically integrated two-dimensional water vapor transport equations for the total moisture content and its fraction of local origin and the condition of a well-mixed atmosphere, is developed. The condition of a well-mixed atmosphere is directly used to eliminate the quantities related to the moisture of local origin, which results in a system consisting of two partial differential equations. This general bulk recycling model has not been used yet to obtain estimations of precipitation recycling for specific land regions on the globe. In Burde and Zangvil (2001b), it is used to analytically study an influence of the nonparallel flow and other effects omitted in the simplified bulk models; this approach to the problem was initiated in Burde et al. (1996). A direct way to estimate precipitation recycling with numerical models is to incorporate tagged water experiments into GCM simulations to trace the origin and
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transport of water substances. In this way, water is “tagged” at its surface source and followed through atmospheric processes until precipitated from the atmosphere. Such a methodology was first used in Koster et al. (1986) and Joussaume et al. (1986), and later in a range of subsequent studies (e.g., Druyan and Koster 1989), but the amount of local water that actually contributes to precipitation was not typically quantified in those simulations. Recently, this methodology has been used for studying the regional sources of precipitation and precipitation recycling in Numaguti (1999) and Bosilovich and Schubert (2002). In Dirmeyer and Brubaker (1999) and Brubaker et al. (2001), a quasiisentropic back-trajectory method was used to trace and estimate the evaporative source of precipitation. The recycling calculations using the tagged water experiments with a GCM confirmed the view, received from the earlier GCM numerical experiments, that Budyko’s model [here and everywhere below, the model modified due to Brubaker et al. (1993) is meant] does not provide correct estimates of precipitation recycling. For some regions the numerical results showed a reasonable agreement between these two estimations, and in other cases considerable differences, up to hundreds of percents, were revealed. This is in accordance with the analysis of Burde and Zangvil (2001b) indicating that the nonparallel flow effects omitted in Budyko’s and other simplified bulk models may considerably affect the recycling ratio. Thus, the discrepancies between the recycling estimates by the simplified bulk models and the results of the GCM simulations can be attributed to inadequate assumptions made in the simplified models and should not compromise the bulk model approach, in general. The basic assumption of bulk models is the so-called condition of a well-mixed atmosphere. The well-mixed assumption is generally stated as follows: the ratio of locally originating water wm in the tropospheric column to the total water content w is equal to the ratio of locally originating precipitation Pm to total precipitation P: Pm wm ⫽ . P w
共1.1兲
The atmosphere above most land regions is well mixed vertically (see, e.g., discussion in Eltahir and Bras 1996). There are, however, indications that, for some land regions, the local water may not necessarily be vertically well mixed through the column, which implies that Eq. (1.1) may not be valid for those regions. The recent results on the sources of water for precipitation over the United States, obtained by a short general circulation model simulation with water vapor traces in
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Bosilovich (2003), show that the two ratios in (1.1) are similar for most regions but, within a fairly small sample of data points, those ratios can be substantially different. Generally, for most regions, the local fraction of precipitation is larger than the local fraction of the total precipitable water—even though they do not differ substantially. For the Amazon basin region, it was suggested in Lettau et al. (1979) that part of evaporation may be returned to the regional air/soil interface by “fast recycling,” which refers to local showers yielding rain before all cloud water is mixed with the total precipitable water in the average tropospheric column above the region. To allow for such a process, Lettau et al. (1979) assumed that the depleting process may be divided in two parts, one of which is related to the fast recycling and the other is proportional to the amount of precipitable water in the tropospheric column. In Lettau et al., one-dimensional calculations of precipitation recycling over the Amazon basin, based on this assumption, are made. However, in addition to the oversimplification due to the one-dimensional formulation, the recycling part of precipitation is assessed incorrectly in Lettau et al. (1979)—see more details in section 3a. Despite the fact that Lettau et al.’s (1979) work is mentioned every time when the issue of precipitation recycling (and, in particular, precipitation recycling in the Amazon basin) is discussed, Lettau et al.’s assumption was neither disproved nor confirmed. Even the fact that, with acceptance of this assumption, the well-mixed atmosphere condition (1.1) becomes invalid was not addressed. The estimates of precipitation recycling in the Amazon basin by Lettau et al. are usually considered to be on an equal footing with the estimates by the other models based on the common well-mixed condition (1.1). Correspondingly, no attempts were made to produce consistent precipitation recycling estimates based on Lettau et al.’s (1979) assumption. Some other effects related to incomplete vertical mixing of water in the tropospheric column could also influence the bulk model precipitation recycling estimates. As an example, we will mention vertical stratification of the fraction of local water in total precipitable water that was indicated in the recent GCM simulation with water vapor traces by Bosilovich (2003). It is reported in Bosilovich that vertical variations of local moisture in the tropospheric column are apparent and, for some regions, significant. The percent contribution of local water is greater in the lower troposphere and less in the middle and upper troposphere. The vertical stratification of local and remote sources of water may result in that the precipitation draws from each reservoir of moisture (of local and advective origin) in the
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tropospheric column not in proportion to the abundance of moisture due to each source. Although the assumption that the atmosphere is well mixed vertically is evidently satisfied with sufficient accuracy for most regions of the globe, a possibility of incomplete vertical mixing of water in the tropospheric column introduces an uncertainty in the bulk model estimates that has not been quantified before. The purpose of this work was threefold. First, we wished to develop a conceptual framework, within which the bulk recycling models, formulated in terms of the vertically integrated water vapor transport, could incorporate the effects related to an incomplete vertical mixing of water in the tropospheric column. The main idea is that it can be done via the well-mixed atmosphere condition modified in a proper way. To give an example of such bulk model formulation, we have constructed the bulk model with the modified well-mixed condition, which reflect effects like the fast return of the local moisture to the air–soil interface (fast recycling) in the spirit of Lettau et al.’s (1979) conjecture. This modification, on the one hand, uses the well-established fact that, in a fully mixed atmosphere, molecules of precipitable water of different origins have equal probabilities to fall as precipitation but, on the other hand, it incorporates the fast recycling process suggested in Lettau et al. (1979). As a result, in the modified model the ratio of locally originating water in the tropospheric column to total water content is not equal to the ratio of locally originating precipitation to total precipitation. The model formulation includes one additional interregional parameter, which controls the level of fast recycling, so that the field of this parameter is to be added to the model input in precipitation recycling calculations. A generalized formulation of the model, which is not straightforwardly related to the fast recycling process, is also possible. The analytical onedimensional bulk models incorporating the modified well-mixed condition, in particular, the modified Budyko’s model, are developed. The effects related to the new interregional parameter and its spatial variability are studied analytically in the one-dimensional formulation. We also show that other effects, in particular those related to the vertical stratification of local and remote sources of water in the tropospheric column, can be included into the bulk recycling model within this conceptual framework via a proper modification of the well-mixed condition. The second objective of this work was to develop an efficient method providing a numerical realization of the general bulk recycling model formulated in Burde and Zangvil (2001a,b) and put it into a form suitable for
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both the original bulk model and the modified model, incorporating effects related to an incomplete vertical mixing of moisture in the tropospheric column. The third objective of our work was to apply the numerical procedure developed for the general bulk recycling model, both in its original and modified forms, to the Amazon basin region. Choosing this region, we were motivated, first of all, by the fact that Lettau et al. (1979) suggested a possibility of fast recycling just for the Amazon basin, but a permanent scientific interest to the interactions of land surface hydrology and regional climate of the Amazon basin [see, e.g., Avissar et al. (2002), for an overview of a recent effort], as well as the discrepancies existing between different previous estimates for this region, served as an additional motivation. We have also included some results related to another case study over the central United States for which information on the degree of vertical mixing is available from the results of the water vapor tracer calculations with a GCM (Bosilovich 2003). The results related to the last two objectives are reported in Burde et al. (2006, hereafter Part II). The present paper (Part I) is organized as follows. In section 2, following the introduction, the basic definitions concerning the precipitation recycling characteristics are given, the assumptions common to all recycling models are discussed, and the system of equations for the general bulk recycling model is formulated. In section 3, the general bulk recycling model is modified to incorporate the fast recycling process. A generalized formulation of the model, which is not straightforwardly related to the fast recycling process, is also pre-
u⫽
F 共x兲 , w
⫽
F 共y兲 1 ; F 共x兲 ⫽ w Lg
冕
共2.3兲
and correspondingly the precipitation P is composed of the parts Pa and Pm of advective and local (evaporative) origins: P ⫽ Pa ⫹ Pm.
2. Basic definitions, assumptions, and equations a. Basic definitions Consider an atmospheric control volume above the land region of interest into which water vapor is brought by air currents through the sides of the volume. The water vapor content w in the air, moving across the region with a horizontal velocity V, varies within the region, decreasing due to precipitation with a vertical flux P and increasing due to evaporation with a vertical flux E. The water vapor content w is the vertically integrated water vapor depth (precipitable water) represented by the vertical integral of specific humidity q( p) from the surface to an elevation where pressure p vanishes as follows: w⫽
共2.4兲
The problem consists in determining the relative contributions of advective moisture and local evaporation to precipitation for a given domain. This relation may be characterized by the precipita-
1
Lg
冕
0
q共p兲 dp,
共2.1兲
p0
where L is the liquid water density and g is the acceleration due to gravity. The velocity V ⫽ (u, ) can be defined through the vertically integrated water vapor flux F ⫽ (F(x), F(y)), as follows:
q共p兲uˆ 共p兲 dp, F 共y兲 ⫽
where uˆ (p) and ˆ (p) are the wind components in the x and y directions, respectively. The water vapor content is composed of an advective portion wa and an evaporative portion wm: w ⫽ wa ⫹ wm,
sented. In section 4, the analytical one-dimensional bulk models based on the modified well-mixed condition are developed, and an influence of a new interregional parameter, appearing due to the model modification, is studied in the one-dimensional formulation. In section 5, a bulk model incorporating effects related to the vertical stratification of local and remote sources of water is outlined. In section 6, some comments on the results are made.
0
p0
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1
Lg
冕
0
q共p兲ˆ 共p兲 dp,
共2.2兲
p0
tion recycling ratio representing the fraction of precipitation due to local evaporative origin. The precipitation recycling ratio for the total land region r is defined as the ratio of total precipitation derived from evaporation to the total area precipitation:
r⫽
冕 冕
Pm共x, y兲 dA
A
,
共2.5兲
P共x, y兲 dA
A
where A is the area of the region and the areal integrals are taken to find the total area precipitation. The same may be expressed as the ratio of areal averages:
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r⫽
Pm P
共2.6兲
.
r⫽
Here and everywhere overbars denote horizontal averaging over the region, defined as
⫽
1 A
冕
共x, y兲 dA,
共2.7兲
A
where (x, y) is any variable. The regional recycling ratio defined in such a way depends on the size of the area considered: it ranges from 0 to 1, being small for small spatial scales and increasing for larger scales. The limiting values 0 and 1 correspond to the extreme cases of the area reduced to a point (the evaporation contribution is zero) and the whole globe (all water evaporated from the earth’s surface precipitates back to the surface). For a given region of an intermediate scale, the recycling ratio r depends on the processes involved in the atmospheric branch of the regional hydrological cycle. It is useful to define an auxiliary function (x, y) for a specific point (x, y) of the region under consideration as
共x, y兲 ⫽
共2.8兲
which is a result of the passage to the limit
共x, y兲 ⫽ lim共⌬A→0兲
⌬A
Pm共x, y兲 dA
⌬A
共x, y兲P共x, y兲 dA
A
冕
共2.10兲
, P共x, y兲 dA
A
which is obtained by introducing Pm from (2.8) into (2.5). The averaged data (monthly) is commonly used so that all quantities taking part in the above definitions represent monthly averages. To calculate (x, y) and r for a longer period (e.g., annual) the following formulas are to be used: 12
兺
共x, y兲PM共x, y兲
M
ann共x, y兲 ⫽
M⫽1
,
Pann共x, y兲 12
P
共x, y兲 ⫽
ann
兺P
共x, y兲
M
M⫽1
rann ⫽
冕
ann共x, y兲Pann共x, y兲 dA
A
冕
,
共2.11兲
共x, y兲 dA
ann
P
A
Pm共x, y兲 , P共x, y兲
冕 冕
冕
,
共2.9兲
where the quantities with the superscript M (M ⫽ 1, . . . , 12) denote monthly averages. Since summation over months and integration over the region (in the numerical procedure it represents summation over the grid points) can be interchanged, the last formula can be represented in the form
P共x, y兲 dA
where ⌬A is a small area reduced to the point (x, y) in the limit. It should be emphasized that in this definition Pm(x, y) is the contribution of evaporation from the total area of the domain to precipitation at this specific point. In other words, Pm(x, y) is the water molecules that appeared in the atmosphere because of an evaporation event from any point within the region, stayed in the atmosphere for some time, and precipitated from the atmosphere on the area ⌬A including the point (x, y). Thus, the quantity defined by (2.8) represents the regional contribution to local precipitation. This definition should not be confused with the contribution of evaporation within the small area ⌬A to precipitation in that same area, which evidently tends to zero when the area is reduced to a point. [In Burde and Zangvil (2001a,b), the quantity was named “local recycling ratio” but afterward we found that this term might be misleading.] Having determined the distribution (x, y), the regional recycling ratio r may be found from the relation
12
兺r
M
rann ⫽
PM
M⫽1
12
兺P
,
rM ⫽
M
M⫽1
冕
A
M共x, y兲PM共x, y兲 dA
冕
, PM共x, y兲 dA
A
共2.12兲 which enables one to calculate the annual value of the regional recycling ratio using only its monthly values.
b. Common assumptions The following assumptions are commonly made in precipitation recycling studies. 1) Time-averaged data are used (e.g., monthly). The time-averaged moisture fluxes represent the result of averaging the flux values obtained from daily or twice-daily (sometimes four times daily) measurements. It is important to note that u and , as defined by (2.2), represent the normalized vertically integrated moisture fluxes, and therefore their time averages include contributions from both the mean
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flow and the high-frequency transient eddies. Thus, the transient eddy effect is included. 2) The next assumption states that at sufficiently long time scales the change in storage of atmospheric water vapor is small compared with the atmospheric water vapor fluxes [see Eltahir and Bras (1994) for the data supporting this assumption]. 3) The atmosphere is assumed to be well mixed, which means that water molecules of external (advective) origin and those of internal (evaporative) origin have equal probabilities to be precipitated. Such a formulation of the assumption of a well-mixed atmosphere allows us to include the models with incomplete vertical mixing into the framework. If all evaporated moisture is mixed with the total precipitable water in the tropospheric column, as it is assumed in the existing bulk recycling models, then assumption 3 implies that the ratio of locally evaporated and advected water molecules in the precipitation is the same as that in the vertically integrated atmospheric moisture wa Pa ⫽ . Pm wm
共2.13兲
This, with allowance for (2.3) and (2.4), may be expressed as (1.1).
c. Equations The underlying principle for all recycling models is the conservation of atmospheric water vapor mass, which is expressed by the equations of conservation of the total water vapor content (with sources and sinks due to evaporation and precipitation): ⭸共wu兲 ⭸共w兲 ⫹ ⫽E⫺P ⭸x ⭸y
共2.14兲
共2.15兲
The last equation can be represented with the use of (2.8) and (1.1) as an equation for the function (x, y) in the form ⭸共wu兲 ⭸共w兲 ⫹ E ⫺ P. ⭸x ⭸y
vertically integrated water vapor flux components F (x) and F (y) as follows: ⭸F 共x兲 ⭸F 共y兲 ⫹ ⫽ E ⫺ P, ⭸x ⭸y
共2.16兲
Equations (2.14) and (2.16), which are the basic equations of our model, may be expressed in terms of the
共2.17兲
⭸共F 共x兲兲 ⭸共F 共y兲兲 ⫹ ⫽ E ⫺ P. ⭸x ⭸y
共2.18兲
It is useful for the following discussion to remember that, according to the definition (2.8) of and the relation (1.1), the quantities F (x) and F (y) represent the (y) components F (x) m and F m of the flux of water vapor of local (evaporative) origin, so Eq. (2.18) can be written (y) in terms of F (x) m and F m as ⭸F 共mx兲 ⭸F 共my兲 ⫹ ⫽ E ⫺ P. ⭸x ⭸y
共2.19兲
d. Formulas for the precipitation recycling estimates provided by simplified one-dimensional models In Budyko’s model and the IMB model (see Burde and Zangvil 2001a,b for details), the moisture flux into the region is used as a parameter of a recycling model. We define the influx F⫹ as the integral of the atmospheric moisture inflow over all of the boundary of the region: F⫹ ⫽ ⫺
冕
⌫in
w␥V␥ · n␥ d␥ .
共2.20兲
In the boundary integral (2.20), w␥ and V␥ ⫽ (u␥, ␥) are respectively the moisture content and the flow velocity vector on the boundary, n␥ is the outward unit normal vector, and ⌫in denotes a part or parts of the boundary across which the atmospheric motion is inward. Then, an estimation of the regional recycling ratio by Budyko’s model rB and by the IMB model rIMB is rB ⫽
and its fraction of local (evaporative) origin ⭸共wmu兲 ⭸共wm兲 ⫹ ⫽ E ⫺ Pm. ⭸x ⭸y
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rIMB ⫽
EA ⫹
EA ⫹ 2F EA
⫹
EA ⫹ F
⫽
⫽
1 1 ⫹ 2F⫹ⲐEA 1
1 ⫹ F⫹ⲐEA
,
, 共2.21兲
where A is an area of the region (A ⫽ LH for a rectangular region: 0 ⱕ x ⱕ L, 0 ⱕ y ⱕ ⌯ ) and E is the average evaporation flux. The dimensionless regional parameters ϒ and ⌳ may be introduced as ϒ⫽
F⫹ E , , ⌳⫽ P PA
共2.22兲
where P is the average precipitation flux. The formulas for the regional recycling ratio produced by the Budyko
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FIG. 1. Simplified schematics illustrating precipitation recycling processes over continental regions for two cases: (a) complete mixing—all evaporated moisture is well mixed with the total precipitable water above the region—and (b) incomplete mixing—a part of evaporation is returned to the regional air/soil interface by local showers yielding rain before all cloud water is mixed with the total precipitable water. Solid arrows refer to the fluxes of moisture of external (advective) origin and dashed arrows refer to the fluxes of moisture of local (evaporative) origin; Fin and Fout are the moisture influx and outflux, P is precipitation, E is evaporation, and pE is the interregional parameter serving to express (by pEE ) the part of evaporation returned to the regional air/soil interface by fast recycling.
and the IMB models expressed with the parameters ϒ and ⌳ are rB ⫹
1 , 1 ⫹ 2⌳Ⲑϒ
1 rIMB ⫽ 1 . 1 ⫹ ⌳Ⲑϒ
共2.23兲
3. Bulk model incorporating fast recycling a. Lettau et al.’s model Lettau et al. (1979) developed a model in which the assumption is made that the depleting process is divided in two parts, P ⫽ pEE ⫹ rpw,
共3.1兲
where pE and rp (here we adhere to Lettau et al.’s notation) are interregional parameters that may vary within the region. The dimensionless parameter pE serves to express (by pEE ) the part of evaporation returned to the regional air/soil interface by fast recycling, which refers to local showers yielding rain before all cloud water is mixed with the total precipitable water in the average tropospheric column above the region (see Fig. 1). The dimensional rp (yr⫺1) is the flushing frequency of atmospheric moisture. In Lettau et al. (1979), the one-dimensional calculations (one airflow component u is present and all quantities depend only on x) using Eq. (2.14) jointly with (3.1) are made to find the distributions of precipitable water w and precipitation P along the flow. In their estimations of precipitation recycling, Lettau et al. consider pEE as the recycling part of the regional precipitation and rpw as precipitation derived from advected precipitable water and correspondingly treat the ratio pEE/P as the recycling ratio [they calculate Budyko’s recycling coefficient P/Pa as the ratio P(rpw)⫺1, which is equivalent]. Evidently, it
is not correct: Eq. (2.14) and its solution incorporate replenishment of precipitable water w by the evaporative flux and thus the quantity rpw includes also part of the precipitation of evaporative origin (in addition to pEE ). Thus, besides the oversimplification due to the one-dimensional formulation, the use of the assumption (3.1) in Lettau et al.’s (1979) recycling calculations is incorrect. Therefore, their estimations of precipitation recycling in the Amazon basin cannot be considered to be valid even though one accepts their conjecture expressed by (3.1).
b. Basic equations for the modified model Below we will show that a correct use of the assumption (3.1) results in Eq. (1.1) expressing the condition that a well-mixed atmosphere changes its form. If the atmosphere is fully mixed, which means that water molecules of different origins have equal probabilities to be precipitated, then the flushing frequency of atmospheric moisture rp is to be the same for the moisture of different origins. Thus one may write, in addition to (3.1), a similar relation for the moisture of local (evaporative) origin; namely, Pm ⫽ pEE ⫹ rpwm.
共3.2兲
Eliminating rp from (3.1) and (3.2) yields Pm ⫺ pEE wm ⫽ , P ⫺ pEE w
共3.3兲
which is the relation replacing (1.1) in the new model. With the use of (3.3) and (2.8), wm may be expressed through w by wm ⫽ w
⫺K , 1⫺K
共3.4兲
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where
c. Generalized formulation K⫽
pEE P
共3.5兲
is a new (dimensionless) variable of the model. Introducing Eq. (3.4) into Eq. (2.15) yields
冉
冊 冉
冊
⭸ 共 ⫺ K兲w ⭸ 共 ⫺ K兲wu ⫹ ⫽ E ⫺ P, ⭸x 1⫺K ⭸y 1⫺K
共3.6兲
which is a counterpart of Eq. (2.16), and correspondingly a counterpart of Eq. (2.18) will be
冉
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冊 冉
冊
⭸ 共 ⫺ K兲F 共y兲 ⭸ 共 ⫺ K兲F 共x兲 ⫹ ⫽ E ⫺ P. ⭸x 1⫺K ⭸y 1⫺K
共3.7兲
Note that K is a function of coordinates. Conditions for at the boundary segments with inflow through the boundary in the modified model also differ from those of a model not involving the assumption (3.1). Now, despite the fact that there is no moisture of local origin in the air entering the region (wm ⫽ 0), the precipitation recycling takes place in the near boundary region through the fast recycling process reflected by the term pEE in Eq. (3.2). Thus, it should be taken as
Here we show that the modified model can be formulated in terms of the variables Pm/P and wm/w without referring to the fast recycling process and correspondingly without use of the parameter pE of Eqs. (3.1) and (3.2). Having determined the two percentages Pm/P and wm/w (e.g., from the tagged water experiments with a GCM), one can estimate the parameter K of the modified model. as the amount of precipitation Let us introduce P(add) m of local origin that is additional to that drawn from the reservoir of moisture in the tropospheric column in proportion to the abundance of moisture due to local origin. This amount should not be obligatory, parameterized by pEE as in equations originating from the Lettau et al. (1979) assumption. Then Eqs. (3.1) and (3.2) take the form P ⫽ P共madd兲 ⫹ rpw,
Pm ⫽ P共madd兲 ⫹ rpwm,
共3.10兲
which leads again to Eq. (3.4) but with the parameter K defined in a more general way than in (3.5); namely, K⫽
P共madd兲 . P
共3.11兲
Rewriting Eq. (3.4) in the form Pm pEE ⫽ ⫽K ⫽ P P
wm Pm ⲐP ⫺ K ⫽ w 1⫺K
for the inflow boundaries. In applying a numerical method to the problem it is useful to introduce the new variables
L ⫽ ⫺ K, EL ⫽ E ⫺ KP, F 共Lx兲 ⫽
F 共x兲 F 共y兲 , F 共Ly兲 ⫽ , 1⫺K 1⫺K
共3.8兲
which permits Eq. (3.7) to be represented in a form identical to that of Eq. (2.18); namely, ⭸共LF 共Lx兲兲 ⭸共LF 共Ly兲兲 ⫹ ⫽ EL ⫺ LP. ⭸x ⭸y
共3.9兲
Based on (3.8) and (3.9), one can apply a numerical method designed for the unmodified model to find the spatial distribution of the variable L without any alterations since the equation for L has the same form as Eq. (2.18) for , and the condition L ⫽ 0 at the boundary segments with inflow through the boundary is also identical with that for in the unmodified model. The spatial distribution of is found as ⫽ L ⫹ K afterward. An input for the numerical procedure must include the spatial distribution of the regional parameter pE (or K ) in addition to those of F (x), F (y), and E.
共3.12兲
and solving it for K yields K⫽
冉
Pm wm ⫺ P w
冊冒冉
1⫺
冊
wm . w
共3.13兲
Thus, if the percentages Pm/P and wm/w are known from the GCM tracer transport modeling, the value of K can be estimated by (3.13). The results of Bosilovich (2003) indicate that, generally, the percentage of precipitation from a local source in the total precipitation is higher than that of water of local origin in the total precipitable water (although the differences may be insignificant), which implies that K is positive.
4. Modified one-dimensional analytical models a. One-dimensional equations In this section, we will consider the one-dimensional recycling models based on the modified form (3.3) of the well-mixed condition. In the one-dimensional models, the airflow is assumed to be parallel and all quantities vary only along streamlines. If the x axis is chosen to be parallel to the streamlines, the one-dimensional counterparts of the two-dimensional equations of con-
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servation of water vapor (2.14), (2.15), and (3.6) [replacing Eq. (2.16) in the present model] are d共wu兲 ⫽ E ⫺ P, dx
共4.1兲 r⫽
d共wmu兲 ⫽ E ⫺ Pm, dx
共4.2兲
d 共 ⫺ K兲wu ⫽ E ⫺ P. dx 1⫺K
共4.3兲
冉
冊
We will consider again a rectangular region, L ⫻ H, so that the flow is traversing the region from x ⫽ 0 to x ⫽ L parallel to the sides y ⫽ 0 and y ⫽ H. The boundary conditions for these equations are wu ⫽ w⫹u ⫽ F⫹ 1 ,
wmu ⫽ 0, ⫽ K
at x ⫽ 0. 共4.4兲
The last condition differs from that in the unmodified models (see discussion in the previous subsection). In (4.4), w⫹ is the moisture content at the entrance of the region and F⫹ 1 is the moisture influx per unit length related to the total moisture flux F⫹ by ⫹ F⫹ 1 ⫽ F ⲐH.
共4.5兲
b. Modified Budyko’s model In Budyko’s model, all of the vertical flux quantities P, Pm, and E are replaced by their average values. Since Pm is treated as a constant, the condition (3.3) for a well-mixed atmosphere can be imposed only on averages Pm ⫺ pEE P ⫺ pEE
⫽
wmu , wu
共4.6兲
and the regional recycling ratio r is then calculated by (2.6). To calculate the quantities wu and wmu one needs to solve Eqs. (4.1) and (4.2) subject to the conditions (4.4) and find the averages by (2.7), which for the onedimensional case takes the form
⫽
1 L
冕
L
共x兲 dx.
共4.7兲
0
Solving Eqs. (4.1) and (4.2) and averaging the results yields wu ⫽ F⫹ 1 ⫹
共E ⫺ P兲L , 2
wmu ⫽
共E ⫺ Pm兲L . 2
共4.8兲 Substituting (4.8) into (4.6) with the use of (4.5) yields r⫺K 1⫺K
⫽
共E ⫺ rP兲LH
2F⫹ ⫹ 共E ⫺ P兲LH
where K ⫽ (pEE)/P is defined as in (3.5) but with all variables replaced by their averages. Equation (4.9) can be readily solved for r to give
,
共4.9兲
2K⌳ ⫹ ϒ ⫺ K 2⌳ ⫹ ϒ ⫺ K
共4.10兲
,
where ⌳ and ϒ are regional parameters introduced in (2.22). It is seen that (4.10) turns into the first equation of Eq. (2.23) when K ⫽ 0. It is worth noting here that the modified Budyko’s equation (4.10) cannot be directly obtained from its unmodified counterpart (2.23) by applying the transformation (3.8) since the derivation of (4.10) involves operations (in particular, averaging) that are not invariant under the transformation (3.8).
c. General one-dimensional model (modified Drozdov and Grigor’eva model) The Drozdov and Grigor’eva (1965) model represents simply the one-dimensional counterpart of a general recycling model formulated in section 2. Correspondingly, the model developed below is the onedimensional counterpart of the modified general model. As distinct from Budyko’s model, the condition of constant Pm is not imposed and the condition of a well-mixed atmosphere is used in its local form (3.3) so that Eq. (4.3) obtained with the use of (3.3) may be applied. Then, the system of equations (4.1) and (4.3) may be solved to find (x) for given distributions E(x) and P(x), and the recycling ratio r for the total land region may be found by (2.10). The important specific case is that of E and P independent of x, which implies that the evaporation and total precipitation fluxes are taken to be equal to their average values (but the flux Pm is not). Drozdov and Grigor’eva studied extensively the results for this case, and the model considered in Brubaker et al. (1993), as Drozdov and Grigor’eva’s model, is just this case. We will also restrict ourselves to considering the results for the case of constant E and P. In the modified model, one more variable K is present. We will consider two cases: the case of constant K and the case when K varies linearly with distance downstream. We will start from the former case of constant K. Solving Eqs. (4.1) and (4.3) with the boundary conditions (4.4) for this case yields wu ⫽ F⫹ 1 ⫹ 共E ⫺ P 兲x
冋 冉 冊册
⫽ 1 ⫹ 共K ⫺ 1兲 1 ⫹
E⫺P F⫹ 1
⫹
⫽ 1 ⫹ 共K ⫺ 1兲e共K⫺1兲PxⲐF1
共E⫺KP兲Ⲑ共P⫺E兲
x
共E ⫽ P兲.
共4.11兲 共E ⫽ P兲,
共4.12兲
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Using Eqs. (4.12) in (2.10) yields
冋 冉
r⫽1⫺⌳ 1⫺ 1⫹ 共K⫺1兲Ⲑ⌳
r ⫽ 1 ⫹ ⌳共e
ϒ⫺1 ⌳
⫺ 1兲
冊
共K⫺1兲Ⲑ共ϒ⫺1兲
册
共E ⫽ P兲,
Proceeding with the variable K case, we assume the following law of variation:
⫽1⫺e
冋 冉
共E ⫽ P兲,
K⫽k 1⫺b X⫺
共4.13兲
where the relation (4.5) has been used. Note that the values of the parameters ⌳ and ϒ must satisfy the condition ⌳ ⫹ ϒ ⫺ 1 ⱖ 0, which expresses a physically required condition that the moisture outflux should be positive.
bkXⲐ共1⫺ϒ兲
再 冋 冉 冊 册冎 冋 冉 冊 册 冉 冊 1 1⫹k b X⫺ ⫺1 2
ϒ⫺1 1⫹ X ⌳
Integrating Eqs. (4.15) and (4.16)with respect to X from 0 to 1 yields
冋
r ⫽ 1 ⫹ ⌳ ⫺1 ⫹ ebkⲐ共1⫺ϒ兲
冉
冊
bk⌳⫹共ϒ⫺1兲关k共1⫹bⲐ2兲⫺1兴 共ϒ⫺1兲2
⫺1
册
冊册
,
X⫽
x , L
共4.14兲
bk⌳⫹共ϒ⫺1兲关k共1⫹bⲐ2兲⫺1兴 共ϒ⫺1兲2
共E ⫽ P兲.
⫺1
共E ⫽ P兲,
共4.15兲 共4.16兲
Dependence of the regional recycling ratio on b, the rate of variation of K, is shown in Fig. 3. It is seen from Figs. 2 and 3 that, while the regional recycling ratio r may vary significantly with k ⫽ K, it depends only slightly on b, which indicates that the simpler models with a constant K can be used instead of the variable K model.
共E ⫽ P 兲
共4.17兲 r ⫽ 1 ⫹ ⌳共e共k⫺1兲Ⲑ⌳ ⫺ 1兲 共E ⫽ P兲.
1 2
where k is an average of K(x), b is the variation rate, and X is a dimensionless coordinate. Solving Eqs. (4.1) and (4.3) with the boundary conditions (4.4), where the value K(0) ⫽ k(1 ⫹ b/2) is taken for K at the boundary, yields
1 关k共2 ⫹ b ⫺ bX兲 ⫺ 2兴X ⫽ 1 ⫹ 关k共2 ⫹ b ⫺ 2bX兲 ⫺ 2兴exp 2 2⌳
ϒ⫺1 ⫻ 1⫹ ⌳
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5. Bulk model incorporating vertical stratification of local moisture
共4.18兲
It is seen that, in the case of E ⫽ P, the regional recycling ratio does not depend on the rate b of variation of the parameter K so that the expression (4.18), with k replaced by K, coincides with the second equation of (4.13).
d. Effects of the parameter K and its variability in the one-dimensional models Dependence of the regional recycling ratio r on the parameter K for both the modified Budyko model [Eq. (4.10)] and the modified Drozdov and Grigor’eva model [the constant K case, Eq. (4.13)], is illustrated in Fig. 2. For most values of the regional parameters ⌳ and ϒ, the values of r produced by those two models do not almost differ (the graphs are practically undistinguishable)—only for rather small values of ⌳ and ϒ (short-dashed lines) does a noticeable difference exist. Note that the same is valid for the original, unmodified, models (see Brubaker et al. 1993; Burde and Zangvil 2001b)—it is also seen from the positions of the K ⫽ 0 points in Fig. 2.
Some other effects related to incomplete vertical mixing of water in the tropospheric column could be also incorporated into the bulk model formulation via the modified well-mixed condition. As an example of such an effect, we will consider vertical stratification of the fraction of local water in total precipitable water that was indicated in the recent GCM simulation with water vapor traces by Bosilovich (2003). It is reported in Bosilovich (2003) that vertical variations of local moisture in the tropospheric column are apparent and, for some regions, significant. The percent contribution of local water is greater in the lower troposphere and less in the middle and upper troposphere. The vertical stratification of local and remote sources of water might result in the precipitation drawing from two reservoirs of moisture (of local and advective source) in the tropospheric column not in proportion to the abundance of moisture due to each source. This effect can be incorporated into a bulk model via modification of the well-mixed condition based on the assumption that the flushing frequency of atmospheric moisture rp (which relates the precipitation to the vertically integrated
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sphere that is different both from its common form (1.1) and from the relation (3.4) based on the Lettau et al. (1979) assumption. It is evident that Eq. (5.3) is equivalent to Eq. (1.1) if ⫽ 1 [or (rp)a ⫽ (rp)m].
6. Concluding remarks
FIG. 2. Dependence r(K ) from the one-dimensional models with a constant K [modified Budyko’s model—Eq. (4.10); modified Drozdov and Grigor’eva’s model—Eq. (4.13)] for different values of the regional parameters: ⌳ ⫽ 0.6 and ϒ ⫽ 0.6 for the short-dashed line, ⌳ ⫽ 2 and ϒ ⫽ 2 for the long-dashed line, ⌳ ⫽ 2 and ϒ ⫽ 0.6 for the thin solid line, and ⌳ ⫽ 0.6 and ϒ ⫽ 2 for the thick solid line. The results from the two models are practically undistinguishable except for the case of ⌳ ⫽ 0.6 and ϒ ⫽ 0.6.
moisture content) is different for the moisture of local and advective origin, as follows: Pm ⫽ 共rp兲mwm,
Pa ⫽ 共rp兲awa.
共5.1兲
This, with allowance for (2.3), (2.4), and (2.8), leads to
⫽
Pm 共rp兲mwm ⫽ , P 共rp兲mwm ⫹ 共rp兲a共w ⫺ wm兲
共5.2兲
which can be solved for wm to give wm ⫽ w
共rp兲a , ⫽ , 1 ⫺ 共1 ⫺ 兲 共rp兲m
共5.3兲
where is a new interregional parameter. It is another modification of the condition of a well-mixed atmo-
FIG. 3. Dependence r on the rate b of variation of K for k ⫽ 0.5 and for different values of the regional parameters: ⌳ ⫽ 0.6 and ϒ ⫽ 0.6 for the short-dashed line, ⌳ ⫽ 2 and ϒ ⫽ 2 for the long-dashed line, ⌳ ⫽ 2 and ϒ ⫽ 0.6 for the thin solid line, and ⌳ ⫽ 0.6 and ϒ ⫽ 2 for the thick solid line.
The problem of incorporating the effects, related to an incomplete vertical mixing of water in the tropospheric column, into a framework of bulk recycling models is not trivial due to the fact that those models are formulated in terms of the vertically integrated moisture transport. The main idea of the approach, which we developed, is that it can be done via modifications of the condition of a well-mixed atmosphere. As an example of such a modification, we have developed the bulk model incorporating the process of a fast return of a part of locally evaporated moisture to the air–soil interface before its complete mixing with the total precipitable water in the tropospheric column occurs. An existence of such a process (referred to as a “fast recycling”) was suggested by Lettau et al. (1979) for the Amazon basin region. They also proposed the relation quantifying the process together with possible values of an interregional parameter pE that controls the fast recycling level. Our approach allows us to remain within the bulk model framework, while accepting that relation, and it results in the modified well-mixed atmosphere condition, which includes an additional (variable) parameter K ⫽ pEE/P. A generalized formulation of the model developed in section 3c, which is not straightforwardly related to the fast recycling and its parameter pE, provides a more general view on the nondimensional parameter K. Another model, incorporating the effects related to an incomplete vertical mixing of water from different sources in the tropospheric column, outlined in section 5, also involves a new interregional parameter . The new interregional parameters, appearing in the modified models, allow us to quantify the degree to which the bulk model estimates may be affected by the incomplete vertical mixing effects. From a more general point of view, such parameters provide a possibility to partially allow for the atmospheric processes in the bulk model integral approach. In Part II of this paper we develop the numerical method for quantifying precipitation recycling, based on the general bulk model formulation, which is suitable for both the original and the modified models. We apply the method to the Amazon basin region and study the alterations in the level and distribution of precipitation recycling caused by the modification. Re-
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sults of application of the model to some other regions of the globe are briefly described. Acknowledgments. I am grateful to two anonymous reviewers for useful comments and constructive suggestions. REFERENCES Avissar, R., P. L. Silva Dias, and M. A. F. Silva Dias, 2002: The Large-Scale Biosphere–Atmosphere Experiment in Amazonia (LBA): Insights and future research needs. J. Geophys. Res., 107, 8086, doi:10.1029/2002JD002704. Bosilovich, M. G., 2003: On the vertical distribution of local and remote sources of water for precipitation. Meteor. Atmos. Phys., 80, 31–41. ——, and S. D. Schubert, 2001: Precipitation recycling over the central United States diagnosed from the GEOS-1 data assimilation system. J. Hydrometeor., 2, 26–35. ——, and ——, 2002: Water vapor tracers as diagnostics of the regional hydrologic cycle. J. Hydrometeor., 3, 149–165. Brubaker, K. L., D. Entekhabi, and P. S. Eagleson, 1993: Estimation of continental precipitation recycling. J. Climate, 6, 1077– 1089. ——, P. A. Dirmeyer, A. Sudradjat, B. S. Levy, and F. Bernal, 2001: A 36-yr climatological description of the evaporative sources of warm-season precipitation in the Missisipi River basin. J. Hydrometeor., 2, 537–557. Budyko, M. I., 1974: Climate and Life. Academic Press, 508 pp. ——, and O. A. Drozdov, 1953: Zakonomernosti vlagooborota v atmosfere (Regularities of the hydrologic cycle in the atmosphere). Izv. AN SSSR, Ser. Geogr., 4, 5–14. Burde, G. I., and A. Zangvil, 2001a: The estimation of regional precipitation recycling. Part I: Review of recycling models. J. Climate, 14, 2497–2508. ——, and ——, 2001b: The estimation of regional precipitation recycling. Part II: A new recycling model. J. Climate, 14, 2509–2527. ——, ——, and P. J. Lamb, 1996: Estimating the role of local evaporation in precipitation for a two-dimensional region. J. Climate, 9, 1328–1338. ——, C. Gandush, and Y. Bayarjargal, 2006: Bulk recycling models with incomplete vertical mixing. Part II: Precipitation recycling in the Amazon basin. J. Climate, 19, 1473–1489. Dirmeyer, P. A., and K. L. Brubaker, 1999: Contrasting evaporative moisture sources during the drought of 1988 and the flood of 1993. J. Geophys. Res., 104, 19 383–19 397. Drozdov, O. A., and A. S. Grigor’eva, 1965: The Hydrologic Cycle
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