Forcing, intermittency, and land surface hydrologic ... - CiteSeerX

Published in: WATER RESOURCES RESEARCH, VOL. 33, NO. 1, PAGES 167–175, JANUARY 1997

Forcing, intermittency, and land surface hydrologic partitioning Marco Marani,1,2 Giovanna Grossi,3 Francesco Napolitano,4 Michael Wallace,5 and Dara Entekhabi1 Abstract.¬ Because of sampling problems, the precipitation estimates from satellite remote sensing are aggregated over time (typically monthly) and over space. Since land surface hydrologic processes have threshold and nonlinear dependencies on precipitation, coarse-resolution precipitation observations may not be directly used in hydrologic models. Differences in the character of intermittency of precipitation, the averaged values remaining the same, can in fact yield large differences in the hydrologic partitioning and therefore in the resulting climate. In this paper an equilibrium hydrologic model is used to study the influence of intermittency on the way precipitation is partitioned into different hydrological quantities. The parameters defining intensity and duration of storms are varied (keeping total precipitation volume constant), and the resulting effects on the partitioning into runoff, evaporation, recharge, and soil moisture storage are determined. It is found that the character of intermittency in storm arrivals has a large impact on the hydrologic partitioning. Furthermore, investigations on the sensitivity of hydrologic partitioning on soil type and water table depth show that rainfall intermittency plays a major role irrespective of these other factors. Runoff generation, evaporation, and groundwater recharge are the three main components of the water balance. Depending on the soil type and climate, the main competition is between different combinations of these three loss mechanisms. It is concluded that the temporal structure of storms has a strong influence on the long-term equilibrium state of the hydrological system.

Introduction A major focus in applications of surface hydrologic science is the partitioning of atmospheric forcing at the land surface. Incoming radiation and precipitation form the principal forcing of the surface energy and water balances, respectively. For example, precipitation is partitioned between runoff and infiltration depending on the surface characteristics. The runoff is generally lost from the local system, and the infiltration is either returned to the atmosphere as evaporation during the interstorm period or lost to net recharge of the regional groundwater system. The key challenge for surface hydrology is to determine the relative partitioning rates given the specification of atmospheric forcing. Surface gauge or remote sensing (satellite and radar) may be used to specify the atmospheric forcing. The surface control on the partitioning is generally modeled by accounting for changes in a state variable (soil moisture and temperature) and relating the partitioning as functions of the state variables. There is a wide spectrum of parameterizations, ranging from the complicated to the empirical, commonly used in relating partitioning factors to the state variables. 1 Ralph M. Parsons Laboratory, Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge. 2 Permanently at Istituto di Idraulica “G. Poleni”, Universita ` di Padova, Padouva, Italy. 3 Dipartimento di Idraulica del Politecnico di Milano, Milan, Italy. 4 Dipartimento di Idraulica, Trasporti e Strade, Universita ` degli Studi di Roma “La Sapienza,” Rome. 5 RE/SPEC Inc., Albuquerque, New Mexico.

There are two major problems associated with developing parameterizations for hydrologic partitioning. First, spatial heterogeneities in soil composition, vegetation, roots, and terrain physiography are all first-order factors. Thus developing parameterizations involving the reduction of complexity in order to define a simple lumped (or per unit area) model is inherently limited. Second, the flux of water into and out of the porous medium at the surface separating soil and atmosphere is temporally a complex phenomenon. The partial differential equations (conservation of mass and moisture flux under energy gradients) governing soil moisture dynamics have surface boundary conditions that switch between concentration (Dirichlet) and flux (Neumann) conditions. This process occurs during both storms and interstorm periods. When the sorption or desorption capacities of the porous medium exceed the atmospheric forcing in magnitude, then the governing boundary condition is determined by the atmospheric forcings. This is generally referred to as atmosphere-control or climatelimited regime for land surface partitioning. As the infiltration (during storms) or evaporation (during interstorms) proceeds and reduces the sorption/desorption rates, the boundary condition switches to that of a concentration condition. In this soil-limited regime the fluxes are independent of the atmospheric forcing. There are numerous theoretical [e.g., Eagleson, 1978c; O’Kane, 1991; Kuhnel et al., 1991; Philip, 1957] and experimental investigations [e.g., Idso et al., 1974; Jackson et al., 1976] concerning switches between atmospheric- and soillimited regimes. The time compression approximation is used to approach this problem [Reeves and Miller, 1975; Eagleson, 1978c; Milly, 1986; Salvucci and Entekhabi, 1994a]. Runoff losses also occur by saturation-excess and subsurface flow, which depend on the vadose zone saturation level. Now that the infiltration and exfiltration sequences during storm and interstorm periods are functions of the atmospheric

MARANI ET AL.: LAND SURFACE HYDROLOGIC PARTITIONING

as modified by Salvucci and Entekhabi [1994b] (see Appendix A). Rainstorms are conceptualized as the Poisson arrival of rectangular pulses with random (single) storm duration and storm rain intensity. In this conceptual framework the characteristics of storms are defined by the values of mean period rainfall m P , mean storm intensity ¯ i , and mean storm duration ¯ t r . To study the effects of storm intermittency, three different climate types have been considered as characterized by a fixed value of the mean period rainfall, given by ¯ t rT i¯ mP 5¯ ¯ tr 1 tb

Figure 1.¬ Mean number of storms as a function of mean storm intensity and mean storm duration. forcing intensity, it is presumably possible that the timing and time aggregation of the atmospheric forcing may have significant influence on the hydrologic partitioning. Sensitivity of the hydrologic fluxes to the timing of the atmospheric forcing needs to be examined. In this paper, a statistical-dynamical model of soil water balance is used to examine how surface hydrologic partitioning of atmospheric forcing changes when the storm and interstorm temporal structure is varied. In these tests the cumulative period atmospheric forcing (mean climate) is held constant and only the temporal structure is modified. The Eagleson [1978a] statistical-dynamical model and its modifications by Salvucci and Entekhabi [1995] for shallow water tables are used. The Eagleson model and its modifications solve the water balance equation in terms of an equilibrium soil moisture value, which serves as the initial condition for infiltration/exfiltration dynamics during storm/interstorm periods. Salvucci and Entelhabi [1994a, b] consider this assumption in detail and test it across a wide range of soils and climates. Major sensitivities to temporal storm structure are quantitatively evaluated in this paper. Situations in which these sensitivities may pose serious problems in hydrologic investigations are when the atmospheric forcing observations are space-time averaged over some characteristic length scales and timescales at and above the storm structure space scales and timescales. For example remotely sensed satellite precipitation estimates (like those proposed to be used for Global water balance modeling in the Earth Observing System [EOS] era) are generally limited both by relatively long aggregation times (of the order of a month [Adler et al., 1994]) and relatively large spatial integration elements (of the order of 103 km2). While such measurements certainly constitute a useful resource for hydrology, the implications of the use of coarse spatial and temporal resolution data must be carefully considered in each application. It will be argued that the use of atmospheric forcing data, which have been temporally aggregated, as input into hydrologic partitioning models must be accompanied by relevant space-time disaggregation schemes for the forcings.

where T is the length of the period and ¯ t b is the mean interstorm time [Salvucci and Entekhabi, 1994b]. The model is solved for ranges of ¯ i and ¯ t r values around a reference value. The total period precipitation m P is held constant. It should also be noted that there is considerable within-storm rain rate variability. Such fine statistical structure and short-timescale intermittency, not considered here, may also have significant effects on hydrologic partitioning of precipitation into runoff. In this study both storm and interstorm effects due to storm arrival structure are analyzed. In this framework an important role in the interpretation of results is played by the mean number of storms, m v , T mv 5¯ ¯ tr 1 tb

The sensitivity of hydrologic partitioning on the character of rainfall is investigated using Eagleson’s [1978a, b, c, d, e] model

(2)

t b from (1) gives for m v the hyperbolic Substitution of ¯ tr 1 ¯ form represented in Figure 1. Changing storm characteristics ¯, ¯ (i t r ) while requiring that the total precipitation volume remain the same results in adjusting the mean number of storms per period. The intermittency character of the precipitation forcing of land surface hydrology is thus affected. The assigned ¯ i and ¯ t r are normalized by the nominal values in Table 1 for nondimensionalization and ease of interpretation. In the model solutions the three basic sets of values for the relevant climatic parameters defined by Salvucci and Entekhabi [1994b] are used as references and to obtain dimensionless parameters for ¯ i and ¯ t r (Table 1). These values include semiarid and humid climates. It is important to include such diverse climatic regimes because they have contrasting behaviors with respect to switches between soil-controlled and climatecontrolled storm and interstorm periods. The model application considers three soil types, following Salvucci and Entekhabi [1994b], to address the influence of soil type on the hydrologic partitioning. The values of soil properties are defined so that a wide range of soil hydraulic characteristics are tested. These values are difficult to determine experimentally, and they are susceptible to large uncertainties.

Table 1.¬ Definition of the Climate Types Considered in the Model Solutions

Climate

Effects of Intermittency on Hydrologic Partitioning

(1)

Humid¬ Semihumid¬ Arid¬

Mean Storm Intensity, cm d21 1.61¬ 5.07¬ 2.99¬

Mean Storm Duration, days 0.72¬ 0.25¬ 0.48¬

After Salvucci and Entekhabi [1994b].

Mean Interstorm Duration, days 3.77¬ 3.44¬ 6.46¬

Mean Annual Precipitation, cm 94.2 125.4 75.5


9

Table 2.¬ Definition of the soil textures and Brooks-Corey hydraulic parameters considered in the simulations

Soil Clay¬ Silt¬ Sand¬

Saturated Hydraulic Conductivity, cm s21 3.4 3 1025 3.4 3 1024 3.4 3 1023

Porosity 0.45¬ 0.35¬ 0.25¬

Pore Size Distribution Parameter 0.44 1.2 3.3

After Salvucci and Entekhabi [1994b].

As can be seen in Table 2, there is an order-of-magnitude increase in saturated hydraulic conductivity from clay to silt, and from silt to sand. By plotting the calculated partitioning variables (runoff, evaporation, and recharge) as functions of dimensionless¯ i and ¯ t r for a given climate and soil type it is possible to examine how a climatic variation with respect to a base state affects hydrologic partitioning. The complete set of nine combinations of different soil and climate types has been explored (though not presented here). The water table depth was initially considered infinite, in order to study the effects of climatic variations alone. Another set of analyses was then run, and will be described later, in which the influence of a shallow water table is considered. Among the possible combinations of soils and climates, the case for semihumid climate and silt-loam soil was observed to exhibit the relevant important features that will be discussed in detail in this paper. Nevertheless, the general conclusions that will be drawn hold for all the combinations of cases analyzed. The partition variables considered are mean bare soil evaporation, groundwater recharge, and runoff, expressed as fractions of mean total precipitation. The value of equilibrium relative soil saturation is also considered, since this is the basic state variable of the system. The results of the model solutions are plotted in Figures 2 through 5. Soil moisture as calculated exhibits only a weak sensitivity and small dynamic range in response to changes in mean storm intensity or mean storm duration (Figure 2). This does not, however, imply that the hydrologic fluxes are weakly sensitive

Figure 2.¬ Dependence of relative soil saturation on mean storm intensity ¯ i and mean storm duration ¯ t r for the semihumid climate and silt soil case. Both ¯ i and ¯ t r are normalized by the values indicated in Table 1 for the semi-humid climate.

Figure 3.¬ Dependence of runoff (normalized by m P ) on mean storm intensity ¯ i and mean storm duration ¯ t r for the semihumid climate and silt soil case. Both¯ i and ¯ t r are normalized by the values indicated in Table 1 for the semihumid climate.

to storm structure. Both evaporation and runoff, in fact, show rather strong sensitivity. It is possible that recharge infiltration and evapotranspiration events organize into different temporal patterns (and season totals) while yielding nearly identical long-term mean surface soil saturation. The relatively large variability of the hydrological partitioning of water fluxes (evaporation, recharge, and runoff) compared with the relatively low variability of soil moisture storage is not surprising. As an example, one can consider the relationship between one of the hydrologic fluxes, recharge q, and relative soil saturation s given by a unit gradient flow in a Brooks-Corey soil [Brooks and Corey, 1966]: q 5 k~1!s c,

(3)

where k(1) is the saturation hydraulic conductivity and c is a pore size distribution parameter. The sensitivity of the recharge to soil moisture is then given by

Figure 4.¬ Dependence of bare soil evaporation (normalized by m P ) on mean storm intensity ¯ i and mean storm duration ¯ tr for the semihumid climate and silt soil case. Both ¯ i and ¯ t r are normalized by the values indicated in Table 1 for the semihumid climate.


Figure 5.¬ Dependence of recharge (normalized by m P ) on mean storm intensity ¯ i and mean storm duration ¯ t r for the semihumid climate and silt soil case. Both¯ i and ¯ t r are normalized by the values indicated in Table 1 for the semihumid climate.

~dq/q!/~ds/s! 5 c¬

(4)

Hence a unit percent change in soil moisture corresponds to a percent change c times greater in recharge. According to Brooks and Corey [1966], c is given by c5

2 1 3m m

(5)

Using the values given in Table 2, one has c 5 7.5 for clay, c 5 4.7 for silt and c 5 3.6 for sand. Thus, in the case of Figures 5 and 2, a change in soil moisture of about 16% corresponds to a 75% change in recharge. Runoff and evaporation show a strong symmetry along the diagonal, correlating well with the mean number of storms (Figures 3 and 4). Infiltration excess runoff (Figure 3) is less than 5% of the period precipitation for the nominal climate and for most intermittency patterns. Only when storms are characterized by long duration and high intensities (conditions that lead to storm duration exceeding the time to ponding and hence significant infiltration-excess runoff) do runoff losses approach the high values as much as 25% of the period precipitation. Evaporation losses during the interstorm (Figure 4) show very strong sensitivity to the manner in which the period precipitation is delivered. Over the range of storm structure parameters considered, the interstorm evaporation loss ranges from 10% to 60% of the period precipitation. When storms are relatively infrequent and ¯ i is relatively high, the correlation between bare soil evaporation and m v is quite high, as is evident in the diagonal pattern in Figure 4. In fact, the mean time with no rain (from the end of one storm to the beginning of the next), ¯ t 0 , is related to m v by ¯ tb 2¯ tr 5 t0 5¯

S

D

T 1 2¯ tr 2¯ tr 5 mv mv

S

mP T22 ¯ i

D

time to drying, the exfiltration process can proceed at the energy-limited potential evaporation rate, while large values of ¯ t 0 tend to bring the system to a soil-controlled state more often. In this regime the evaporative loss of soil moisture is significantly reduced. On the other hand, when ¯ i is low, ¯ t 0 is ¯ sensitive both to changes in m v and i , resulting in a more complex interplay between time to drying, ¯ t 0 , and mean intensity, as is shown by the nonhyperbolic behavior of isolines in the upper left portion of Figure 4. When both¯ i and m v are low, the storm periods must last longer (shorter interstorms), and the evaporation regime is more dominantly energy-limited. It is important to note that the space over which runoff showed sensitivity in Figure 3 corresponds to the region in Figure 4 where evaporation has least sensitivity. It is evident that runoff and evaporation are not competing for this given soil texture. There is a third hydrologic flux, recharge to the regional groundwater system, that must be considered in analyzing the partitioning of the same period precipitation volume in all cases. Recharge shows a monotonical increase as a function of mean storm duration, while reaching a peak at an intermediate value of mean storm intensity (Figure 5). In fact, given a specified value of ¯ i , increasing ¯ t r decreases the time available for evaporation, thus favoring recharge. On the other hand, for a specified value of¯ t r , increasing¯ i favors recharge with respect to evaporation until the rain rate is large enough to trigger runoff. Further increase of ¯ i results in increased runoff at the expense of both evaporation and recharge. Recharge competes with both runoff and evaporation in their respective conditions of strong sensitivity to storm parameters. When soil can absorb (or yield) moisture at the rate that the overlying atmosphere provides (or demands), the system is said to be climate-controlled. This condition occurs at the onset of a rainfall or evaporation period and applies until either the maximum infiltration capacity is exceeded by the storm precipitation rate or capillary rise is unable to satisfy the atmospheric evaporative demand. If and when it does occur, the system is called soil-controlled or profile-controlled, and a switch in boundary conditions occurs (from a flux condition to a concentration condition). Soil-controlled periods are responsible for infiltration-excess Hortonian surface runoff production and for reducing evaporation below its potential value. The occurrence of switching in boundary conditions depends

(6)

Here¯ t b is the time from storm origin to storm origin. It is seen from (6) that when ¯ i is high there tends to be a one-to-one relationship between m v and ¯ t 0 , whose value controls the way in which evaporation occurs. When¯ t 0 is smaller than the mean

¯b as a function of¯ Figure 6.¬ Values of E > ¯ t d /t i and¯ t r for the semihumid climate and silt soil case.


1

Figure 7.¬ Fraction of total rainfall that goes into recharge as a function of ¯ i and ¯ t r for the arid climate and sandy soil case.

Figure 9.¬ Fraction of total rainfall that is evaporated as a function of ¯ i and ¯ t r for the arid climate and clay soil case.

on the sequence and nature of the forcing, the equilibrium value of soil moisture, and the soil characteristics. For the case of a semihumid climate and a silt soil, the effects of intermittency on such changes in boundary conditions are examined. Figure 6 is a plot of the ratio E between mean time to drying, ¯ t d , and ¯ t b [Eagleson, 1978d]. When E is greater than 1, climate-controlled states prevail, and changes in boundary conditions are rare. When E is smaller than 1, the system tends to be soil-controlled, and switching between the two types of boundary conditions occurs more frequently. Figure 6 shows that, when ¯ i is large and storms are infrequent, E correlates well with m v , as may be seen by comparison with Figure 1. The values of E lower than unity in this area indicate that switching of boundary conditions occurs often. When ¯ i is small, on the other hand, the system tends to be in an atmosphere-controlled regime almost regardless of the value of ¯ tb, and the correlation with E breaks down, in agreement with the observations made on Figure 4.

where the recharge component is represented for the two extreme climates. The fraction of total rainfall that goes into recharge of the ground water system is almost insensitive to changes in ¯ i, ¯ t r , and climate type in sands. These results and those based on other combinations of soils and climates suggest that in the case of soils with very high hydraulic conductivity, the effects and characteristics of the regional groundwater system play a dominant role in the partitioning of hydrologic quantities. The influence of climate on the way soil-atmosphere interactions yield different partitioning of hydrologic quantities are given in Figures 9 and 10, which show the dependence of runoff and evaporation fractions on¯ i and¯ t r for an arid climate. It may again be noted that for the lowest values of ¯ i and ¯ t r it is increasingly evident that short interstorm periods do not allow frequent switches of exfiltration to soil control. As a result, evaporation generally proceeds near its potential (energylimited) rate and returns much of the period precipitation to the atmosphere. On the other hand, when ¯ i and ¯ t r reach their maximum values, evaporation periods are reduced, while the mean storm intensity is high enough to produce runoff (which therefore dominates over recharge). The variables may be seen to correlate better with m v than in the semihumid case. This is

Influence of Soil and Climate The influence of soil type on hydrologic partitioning is studied for several different climate-soil combinations. Two significant results for the case of sand are shown in Figures 7 and 8,

Figure 8.¬ Fraction of total rainfall that goes into recharge as a function of¯ i and¯ t r for the humid climate and sandy soil case.

Figure 10.¬ Fraction of total rainfall that goes into surface runoff as a function of¯ i and¯ t r for the arid climate and clay soil case.


Figure 11.¬ Values of runoff, normalized by mean storm depth, characterized by a return period equal to 100 years.

due to the fact that when runoff production is limited, evaporation is high enough to balance rainfall input, thus inhibiting recharge. Our results also showed that unlike the case for the humid climate, the values of the parameter E are below unity for the whole range of parameters, indicating that the average time to drying is shorter than the mean interstorm time and that the system thus very frequently switches from an atmospherecontrolled to a soil-controlled regime.

Extreme Values Analysis The Eagleson [1978e] model also allows us to examine the influence of intermittency on extreme events, such as the 100year surface runoff event. The probability of dimensionless runoff Z (here defined as storm runoff divided by mean storm depth m r 5 ¯ i¯ t r ) exceeding a value z may be found to be P~Z $ z! 5 2 Îz exp ~2G 2 2 s ! K 1~2 Îz!

G~ s 1 1! , ss

(7)

where G is a parameter accounting for gravitational infiltration rate, s is a capillary infiltration parameter, and K 1 ( x) and G( x) are the Bessel function of first order and the gamma function, respectively (see Appendix B). Setting the recurrence interval to 100 years allowed us to back calculate z for values of ¯ i and ¯ t r in the selected case of a semi-humid climate and a silt soil. Examination of Figure 11 shows that the characteristics of this extreme-value parameter do not strictly follow the same patterns as those exhibited by the mean-value parameters (Figure 3). The 100-year runoff in fact, unlike mean runoff, appears to be almost insensitive to changes in ¯ t r in the area where ¯ i is small. Figure 11 shows that storm intensity and, to a lesser extent, storm duration control extreme events. High-intensity storms lead to short time to ponding. Generally, any storm duration will contain a ponding and runoff generation event. The sensitivity of hydrologic extremes to changes in storm intensity have important implications in assessments of climate change impacts on water resources.

Role of the Water Table The analyses previously described ignore the influence of the regional groundwater system and, in particular, the influence

of a shallow water table. A modified version of Eagleson’s [1978a, b, c, d, e] model has been developed by Salvucci and Entekhabi [1995] that takes this factor into account. Using the modified model, solutions are found considering a semihumid climate and a clay soil. Clay soil is chosen because its capillary flow properties lead to the strongest saturated-unsaturated zone coupling. The results show the clear influence of the water table, especially as far as recharge is concerned. In the finite water table depth case, if soil moisture cannot satisfy the atmospheric evaporative demand, water is provided by the regional groundwater system to the unsaturated zone through capillary rise. Therefore in contrast to the semi-infinite table depth case, recharge can take on negative values, which are bounded from below by the climate-controlled potential evaporation rate. The lower limit is reached for a certain critical depth of the water table that is unique for each system, i.e., depends on the climate and soil types that characterize it. The critical value of the water table depth for the semihumid climate clay soil combination was calculated by trial and error to be approximately 178 cm. Thus, the variability of rainfall partitioning was investigated for four different values of the water table depth exploring the behavior of the system in the transition zone between semi-infinite and finite water table depth regimes. The depths considered were (1) Z w 5 2200 cm (Figures 12 and 13), (2) Z w 5 2250 cm (Figure 14); (3) Z w 5 2500 cm, and (4) Z w 5 2750 cm. Results obtained for cases 3 and 4 were compared with the semi-infinite case. As expected, the semi-infinite table depth assumption was an acceptable approximation for deep water table conditions. Cases 1 and 2 are characterized by very similar distributions of hydrologic quantities as functions of ¯ i and ¯ t r . Both cases differed substantially from the semi-infinite water table depth case, as can be seen in Figures 12 and 13. In Figure 12 it may be seen that the maximum value of runoff (80%) was found to be higher than the maximum value (60%) obtained with the semi-infinite assumption. Here both infiltration excess and saturation from below runoff generation mechanisms can occur, while in the previous case only the former could take place. Moreover, runoff values are insensitive to mean storm duration, especially when the mean intensity is very low. In fact, for low rainfall intensities, runoff can be generated

Figure 12.¬ Total runoff (normalized by period precipitation) as a function of dimensionless ¯ i and ¯ t r . Water table depth is 200 cm, the climate is semihumid, and the soil texture is clay.


only through the saturation excess mechanism because infiltration capacity is rarely overwhelmed by storm intensity, no matter what the duration of the storm. As was demonstrated by Salvucci and Entekhabi [1994b], this is true for most soil textures as well. Figure 13 shows that values obtained for bare soil evaporation are also generally higher, when compared to the output referring to the semi-infinite depth assumption, but the distribution is quite similar. Higher runoff and higher evaporation mean a larger water demand, which is provided by capillary rise and the regional groundwater system. The recharge to the groundwater from the unsaturated zone shows an interesting behavior in Figure 14: the recharge rate changes sign depending on the storm arrival structure. Generally, when storms have low duration and interstorm periods are long, the water table becomes a source of moisture for evaporation. Low-intensity and long-duration storm characteristics promote greater recharge of groundwater for the surface water balance.

Conclusions The results presented suggest that a very high sensitivity in the hydrologic partitioning is to be expected when storm structure is changed even for a given fixed basic climate (i.e., total mean period rainfall). While soil moisture may have moderate variations, the hydrological fluxes show in fact a strong dependence on the storms’ arrival process, duration, and intensity. These results are particularly relevant for proper use of hydrological data with limited time resolution. They show in fact that long-term-averaged values of rainfall are not sufficient to uniquely determine the other hydrological variables. On the contrary, because of the nonlinear character of the system, the nature of intermittency in the detailed distribution of precipitation must be taken into account. Since during storms and interstorms infiltration and exfiltration switch between climatecontrolled (storm precipitation rate and potential evaporation) and soil-controlled regimes, the timing and sequences of storms (even when they deliver the same total volume) are important factors in the determination of surface hydrologic fluxes. The results furthermore confirm the understanding that soil type has an important influence on the way the precipitation input is partitioned into the different components of the

Figure 14.¬ Recharge as a function of dimensionless ¯ i and ¯ tr. Water table depth is 250 cm.

hydrologic system. The characteristics of the regional groundwater system must be considered where a shallow water table is possible because of variations in topography. In the latter case a two-way water exchange is possible between the atmosphere and the saturated zone, giving rise to a nonlinear interaction that depends on the memory of the system and further complicates the way in which partitioning is achieved across seasons and years.

Appendix A: Equilibrium Water Balance Equations Following are the equations needed to evaluate the water table dependent equilibrium water balance as derived by Salvucci [1994] and Salvucci and Entekhabi [1995] as a modification to the work of Eagleson [1978a, b, c, d, e]. The reader is referred to Salvucci [1994] and Salvucci and Entekhabi [1995] for a complete and detailed derivation relating to the inclusion of shallow water table position as the lower boundary condition of an unsaturated soil column. The mean period bare soil evaporation Ê S s &, the expected exfiltration from the bounded soil column under a mixture of climate-controlled and soil-controlled periods within interstorms with durations that are exponentially distributed, is dependent only on the number of storms, the interstorm duration, the value of potential evaporation, and the soil type. It is given by Ê sS& 5

ep mv $1 2 ~1 1 b

1 ~~2V! 21/ 2 1 1

Î2LE 1 ~2V! 21/ 2!e 2LE

Î2VE!e 2VE

Î2E@ g ~ 23 , VE! 2 g ~ 23 , LE!#%

Ê sS& 5

ep mv b 1

Figure 13.¬ Evaporation (normalized by period precipitation) as a function of dimensionless ¯ i and ¯ t r . Water table depth is 200 cm, the climate is semihumid and the soil texture is clay.

3

H S 12

11

w,

Î2LE 1

Î2E@G~ 23 ! 2 g ~ 23 , LE!#

J

K~1!s *c 2

(A1a)

D

K~1!s *c w 2LE 2 e 2e p ep (A1b)

K~1!s *c K~1!s *c ,w, 1 ep 2 2


Ê sS& 5

ep mv b

w.

K~1!s *e 1 ep 2

l;

(A1c)

where w K~1!s *c 2 L511 4e p 2e p

S

K~1!s *c w 11 2 2e p ep

S

K~1!s *c 2w V52 2 ep ep

s* ;

S

(A2) Z9 ;

22

(A3)

~11~c11!/4!

*

h 5¯ i 21

(A21)

F

S

n eK~1!uC su f e mp

11

^q& K~1!

2~1/ 2!@h2dSi2#1/3

G

Appendix B: Runoff Volume Distribution 1/ 2

(A7)

(A8)

DS D D Zw Cs

2mc

1/c

(A9) (A10)

~1 2 e 2d t s~1 1 d t *s!!

S 2i ;e Si 1 2 A0 2 A0 4A 20

*

(A11)

D

Eagleson [1978e] derives the probability density function of point storm rainfall excess by integration of the Philip infiltration equation. Such integration is applied to a rainstorm of uniform intensity yielding the depth of point surface runoff in terms of random variables defining the initial soil moisture, the rainfall intensity, and the storm duration. The initial soil moisture is, for convenience, fixed at its climatic space and time average, while exponential distributions are assumed for storm intensity and duration. The derived probability density function is

SÎD

R ¯21¯ f~R! 5 2i t 21 exp ~2G 2 2 s ! K 0 2 ¯¯ r i tr

G~ s 1 1! , ss (B1)

where G is a parameter accounting for gravitational infiltration rate, s is a capillary infiltration parameter, K 0 (¬ ) is the Bessel function of order zero, and G(¬ ) is the gamma function. Since i and t r are assumed to be independent, the product ¯ i¯ t r can be replaced by the mean storm depth m r . The cumulative distribution function obtained from (B1) is P

S

R .r mr

D

5 2 Îr exp ~2G 2 2 s ! K 1~2 Îr!

G~ s 1 1! , ss (B2)

where K 1 (¬ ) is the Bessel function of order 1. 2

A0 . 0

(A12)

A50

Î2 x K~1!

(A13)

~~2 1 G!/3! K~1!

G $ 22

0

G , 22

2n e~1 2 s * !C s 1 ~1 1 2 ~c 2 3!~1 2 s * !sˆ~c11!/ 2! K~1! 2f C s/~1 2 s * ! 1 2

~1 1 ~c 2 3!~1 2 s * !sˆ~c11!/ 2! 1

(A22)

In (A22), K 2 (¬ ) is the Bessel function of the second type.

2m 2p 3~1 1 3m!~1 1 4m!

Si 5

G;

d 5¯ t 21 r

(A20)

(A6)

~ ; e/S i! 2

x;

~1 2 s * ! 2 l ~Z w 2 C s! l2f

2 1 3m m

where

H

(A19)

^R seS& 5 2m v ; eK 2~2 Î ; ehd !

^R ieS& 5 ^m P& exp ~2h A 0 2 @ h 2d S 2i # 1/3!G~1 1 21 @ h 2d S 2i # 1/3!

A0 5

2mc21

(A5)

In (A1), g( ) is the incomplete gamma function. For given soil parameters (K(1), C S , m, c, and n e ), climate parameters (e p , b , and m v ), and depth to water table (Z w ), the mean seasonal bare soil evaporation will depend, through equations (A1) through (A8), on the equivalent steady surface moisture state s * by way of the mean recharge or discharge ^q& (in (A9)). The mean annual infiltration excess runoff production during storms (^R ieS &) with exponentially distributed duration and intensities is given by

t *s 5

Zw Cs

E 5 b S 2e / 2e 2p

b 5¯ t 21 b

5S Î

S D

The mean seasonal storage excess runoff generation for the same storms following column saturation is given by

2^q& 1 K~1!

z ~ 21 @ h 2d S 2i # 1/3!

m~1 1 ^q&/K~1!!~s *12c! Cs

(A18)

(A4)

c5

fe 5

22

K~1!~Z w/C s! 2mc 1 2 ~Z w/C s! 2mc

w5

S e 5 2s

D

D

f;

m~1 1 ^q&/K~1!! Cs

(A14)

(A15)

(A16)

1

; e 5 2n e~1 2 s * ! Z9 1 2 n ef Z9 2 1 2 n el ~Z w 2 Z9 2 C s! (A17)

Notation A 0 second term of Philip-type infiltration equation [L/T]. c pore disconnectedness index of Brooks-Corey soil hydraulic model. E¬ dimensionless parameter group appearing in the expression of mean seasonal bare soil evaporation, an index of the relative ability of soil to evaporate at the climate-limiting rate. E s S seasonal bare soil evaporation [L]. e p seasonal average bare soil potential evaporation rate [L/T]. G parameter accounting for gravitational infiltration rate. H¬ soil depth of modeled hillslope domains [L]. ¯ i mean storm intensity [L/T]. k(1)¬ saturated hydraulic conductivity [L/T].


K 0 (¬ )¬ Bessel function of order zero. K 1 (¬ )¬ Bessel function of first type. K 2 (¬ )¬ Bessel function of second type. L¬ hillslope length [L]. m pore size distribution index of Brooks-Corey soil hydraulic model. m r mean storm depth [L]. m v mean number of storms per season. n e effective porosity, fraction of soil available to flow. m P seasonal precipitation [L]. ^q& seasonal recharge rate [L/T]. R ie S seasonal infiltration excess surface runoff for water table bounded soils [L]. R seS seasonal storage excess surface runoff for water table bounded soils [L]. s relative soil saturation. S e exfiltration desorptivity [L/T 1/ 2 ]. S i sorptivity [L/T 1/ 2 ]. s * value of equivalent steady soil saturation at the ground surface. T¬ length of the period [T]. ¯ t b mean interstorm duration [T]. ¯ t d mean time to drying [T]. ¯ t 0 mean time with no rain [T]. ¯ t r mean storm duration [T]. t *s approximated time to column saturation assuming soil-controlled infiltration history [T]. w potential rate of capillary rise from water table to dry ground surface [L/T]. Z w vertical Cartesian location of the water table relative to the ground surface [L]. z normalized runoff. b inverse of mean time between storms [T 21 ]. x time constant appearing in the derivation of the infinite series expansion of infiltration capacity [T]. V dimensionless parameter group appearing in the expression of mean seasonal bare soil evaporation. L dimensionless parameter group appearing in the expression of mean seasonal bare soil evaporation. d inverse of mean storm duration [T 21 ]. f e dimensionless desorption diffusivity. f i dimensionless sorption diffusivity. G dimensionless parameter group appearing in the derivation of the infinite series expansion of infiltration capacity. h inverse of mean storm intensity [T/L]. u slope of bedrock of modeled hillslope domains. C s bubbling head of Brooks-Corey soil hydraulic model, the value of the capillary tension head (negative) at which the soil first desaturates [L]. s capillary infiltration parameter. @ e storage capacity of the unsaturated zone under conditions of steady state equilibrium with a water table [L]. G(¬ )¬ gamma function. g ( z , z ) incomplete gamma function. Acknowledgments.¬ This paper reports on the student group project associated with the June 12–23, 1995, Summer School on Environmental Dynamics sponsored by the Istituto Veneto di Scienze Lettere ed Arti, Venice, Italy. D.E. was the instructor for this course. Completion of this study was supported by NASA subcontract NAS5-31721.

5

References Adler, R. F., G. J. Huffman, and P. R. Keehn, Global tropical rain estimates from microwave adjusted geosynchronous IR data, Remote Sens. Env., 11, 125–152, 1994. Brooks, R. H., and A. T. Corey, Properties of porous media affecting fluid flow, J. Irrig. Drain. Div. Am. Soc. Civ. Eng., 92(IR2), 61– 88, 1966. Eagleson, P. S., Climate, soil, and vegetation, 1, Introduction to water balance dynamics, Water Resour. Res., 14(5), 705–712, 1978a. Eagleson, P. S., Climate, soil, and vegetation, 2, The distribution of annual precipitation derived from observed storm sequences, Water Resour. Res., 14(5), 713–721, 1978b. Eagleson, P. S., Climate, soil, and vegetation, 3, A simplified model of soil moisture movement in the liquid phase, Water Resour. Res., 14(5), 722–730, 1978c. Eagleson, P. S., Climate, soil, and vegetation, 4, The expected value of annual evapotranspiration, Water Resour. Res., 14(5), 731–739, 1978d. Eagleson, P. S., Climate, soil, and vegetation, 5, A derived distribution of storm surface runoff, Water Resour. Res., 14(5), 741–748, 1978e. Idso, S. B., R. J. Reginato, R. D. Jackson, B. A. Kimball, and F. S. Nakayama, The three stages of drying of a field soil, Soil Sci. Soc. Am. Proc., 38, 831– 836, 1974. Jackson, R. D., S. B. Idso, and R. J. Reginato, Calculation of evaporation rates during the transition from energy-limiting to soil limiting phases using albedo data, Water Resour. Res., 12(1), 23–26, 1976. Kuhnel, V., J. C. I. Dooge, J. P. J. O’Kane, and R. J. Romanowicz, Partial analysis applied to scale problems in surface moisture fluxes, in Land Surface-Atmosphere Interaction: Observations, Models and Analysis, edited by E. Wood, pp. 221–247, Kluwer Acad., Norwell, Mass., 1991. Milly, P. C. D., An event-based simulation model of moisture and energy fluxes at a bare soil surface, Water Resour. Res., 22(12), 1680 –1692, 1986. O’Kane, J. P., Implications for remote sensing of natural switching from atmosphere-controlled to soil-controlled evaporation or infiltration, in Land Surface Evaporation, Measurements and Parametrization, edited by T. J. Schmugge and J.-C. Andre´, pp. 371–381, Springer-Verlag, New York, 1991. Philip, J. R., The theory of infiltration, 4, Sorptivity and algebraic infiltration equations, Soil Sci., 84, 257–264, 1957. Reeves, M., and E. E. Miller, Estimating infiltration for erratic rainfall, Water Resour. Res., 11(1), 102–110, 1975. Salvucci, G. D., Hillslope and climatic controls on hydrologic fluxes, Ph.D. thesis, Dep. of Civ. and Environ. Eng., Mass. Inst. of Technol., Cambridge, 1994. Salvucci, G. D., and D. Entekhabi, Equivalent steady soil moisture profile and the time compression approximation in water balance modeling, Water Resour. Res., 30(10), 2737–2749, 1994a. Salvucci, G. D., and D. Entekhabi, Comparison of the Eagleson statistical-dynamical water balance model with numerical simulations, Water Resour. Res., 30(10), 2751–2758, 1994b. Salvucci, G. D., and D. Entekhabi, Hillslope and climatic controls on hydrologic fluxes, Water Resour. Res., 31(7), 1725–1740, 1995. D. Entekhabi, Ralph M. Parsons Laboratory, 48-331, Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139. (e-mail: dara_entekhabi@ mit.edu) G. Grossi, Dipartimento di Idraulica Ambientale e del Rilevamento, Politecnico di Milano, Piazza Leonardo da Vinci, 32, I-20133, Milano, Italy. M. Marani, Istituto di Idraulica “G. Poleni,” Universita` di Padova, via Loredan 20, I-35121 Padova, Italy. F. Napolitano, Dipartimento di Idraulica, Trasporti e Strade, Universita` degli Studi di Roma “La Sapienza,” Via Eudossiana, 18, Roma, Italy. M. Wallace, Department 6849, Sandia National Laboratories, P.O. Box 5800, Mail Stop 1328, Albuquerque, NM 87185-1328.