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Climate, interseasonal storage of soil water, and the annual water balance
Advances in Water Resources 17 (1994) 19-24
© 1994 Elsevier Science Limited Printed in Great Britain. All fights reserved 0309-1708,,'94/$07.00
ELSEVIER
Climate, interseasonal storage of soil water, and the annual water balance P. C. D. MiHy US Geological Survey and Geophysical Fluid Dynamics Laboratory/NOAA, Princeton, New Jersey, USA The effects of annual totals and seasonal variations of precipitation and potential evaporation on the annual water balance are explored. It is assumed that the only other factor of significance to annual water balance is a simple process of water storage, and that the relevant storage capacity is the plant-available water-holding capacity of the soil. Under the assumption that precipitation and potential evaporation vary sinusoidally through the year, it is possible to derive an analytic solution of the storage problem, and this yields an expression for the fraction of precipitation that evaporates (and the fraction that runs off) as a function of three dimensionless numbers: the ratio of annual potential evaporation to annual precipitation (index of dryness); an index of the seasonality of the difference between precipitation and potential evaporation; and the ratio of plant-available water-holding capacity to annual precipitation. The solution is applied to the area of the United States east of 105°W, using published information on precipitation, potential evaporation, and plant-available water-holding capacity as inputs, and using an independent analysis of observed river runoff for model evaluation. The model generates an areal mean annual runoff of only 187mm, which is about 30% less than the observed runoff (263 nun). The discrepancy is suggestive of the importance of runoff-generating mechanisms neglected in the model. These include intraseasonal variability (storminess) of precipitation, spatial variability of storage capacity, and finite infiltration capacity of land.
INTRODUCTION
soils are penetrated by plant roots, which extract stored water for transport through the plant to the leaves; inside the leaves, available energy is consumed in the phase change of liquid water to vapor. The plantavailable water-holding capacity (or, herein, storage capacity) of most soils is around 15% of the bulk soil volume of the root zone. The volume of soil whose water is accessible to plant roots is relatively uncertain, but is usually assumed to include all soil above the maximum root depth, which is typically of the order of one meter. This implies an effective storage capacity of the order of 0-15m. In principle, the temporal mismatch of energy and water supplies can be modified not only by water storage, but also by energy storage. Energy may be stored in the soil during periods of strong insolation, thereby allowing evaporation to occur when precipitation is falling onto the heated soil. However, such timeshifting of energy supply does not play a major role in the dynamics of water balance, especially at the scale of the annual cycle, because the water-equivalent heat storage is relatively small in comparison with the water storage capacity of most soils. (It is possible, however, that the effect of energy storage could be
A fundamental problem in hydrology is the identification of the basic controls on the time-mean partitioning of precipitation into runoff and evaporation - the annual water balance. On the basis of numerous empirical data, it has been established 2 that the fraction of precipitation that eventually runs off is controlled, in great measure, by the ratio of annual potential evaporation to annual precipitation (index of dryness). Indeed, if precipitation and potential evaporation were constant through time, evaporation would simply equal the lesser of these two variables and runoff would equal any precipitation in excess of potential evaporation. (Throughout this paper the terms 'evaporation' and 'potential evaporation' refer to total vapor flux from the surface to the atmosphere, including transpiration by plants.) Because the necessary water (precipitation) and energy (potential evaporation) for evaporation do not arrive at the same time, the problem of water balance also requires the consideration of storage. Soils have the ability to retain water by capillary forces, which prevent or retard their drainage by gravity. The upper layers of 19
20
P. C. D. Milly
important at the shorter storm-interstorm time scale for bare surfaces of solid rock or for extremely coarse granular media, both of which are incapable of retaining water for evaporation after the precipitation event.) One may hypothesize that the annual water balance is determined mainly by the temporal distributions of water and energy supplies to the land surface and by the plant-available water-holding capacity of the soil. Such a hypothesis is implicit in a long tradition of water-balance modeling, 1,9 although there has been little attention given to precise articulation and testing of the hypothesis. Milly (1993) 7 used this hypothesis as the basis for a stochastic-storage model of water balance in the absence of seasonality, with allowance for the random nature of precipitation. He did not test the model extensively, but simply noted that the implied production of runoff appeared insufficient to explain the empirical data on annual water balance. He noted that the discrepancy did not necessarily invalidate the hypothesis, but that it might rather indicate the importance of seasonality or soil spatial variability. Empirical evidence for the dependence of the annual water balance on the seasonality of climate has been reported by Budyko and Zubenok (1961). 3 They noted that some of the variability of runoff across different basins having identical values of the index of dryness could be explained by seasonal characteristics of precipitation and potential evaporation. Where precipitation and potential evaporation are in phase, evaporation is enhanced and runoff is suppressed, relative to the average response over many basins. Where the water and energy supplies are out of phase, the opposite effect is observed. The objective of this paper is to develop and test a special case of the hypothesis that annual water balance is determined by the temporal interplay of water and energy supplies, through a simple water storage process. The assumption is made that the most important source of temporal variability in water and energy supplies is associated with the seasonal cycle of climate; the random intraseasonal fluctuations of precipitation considered by Milly (1993) 7 are ignored here. The resulting problem of water balance is formulated mathematically and solved analytically. The derived expression for partitioning of the annual water balance is evaluated using available climatic, hydrologic, soil, and vegetation data for a large section of the United States.
THEORY
A single water store (w) is assumed to be replenished or depleted at a rate equal to the difference between rates of precipitation (P) and potential evaporation (Ep), unless such storage change is prevented by the finite capacity
(w0) of the store, 7
dw = dt
0 0 (e - Ep)
for (P > Ep) and (w = w0) for (P < Ep) and (w = 0) otherwise
(1)
Conceptually, the capacity w0 is the plant-available water-holding capacity of the soil, and w (the plantavailable water) is the amount of water stored, in the root zone, in excess of the water content at the wilting point of the vegetation. Any excess of P over Ep when the store is saturated is assumed to produce runoff, Y, by rapid gravity drainage from the root zone. The actual routing of this runoff through the subsurface media and the surface channel network is not considered here, but in the long run, the mean value of Y corresponds conceptually to the sum of fiver and ground-water discharge from a given area. The rate of evaporation E is assumed to equal its potential rate whenever w is greater than zero or P is greater than Ep. When w is zero and Ep exceeds P, E is taken to be equal to P. Both P(t) and Ep(t) are assumed to be periodic. The mechanism that drives the seasonality of climate is the tilt of the earth's axis relative to its orbit around the sun and the resulting seasonality of solar irradiance normal to the top of the atmosphere. At extratropical locations, this produces a strong sinusoidal signal with period of one year in most climatic variables. At the equator, the fundamental period is one-half year. For simplicity of analysis, and without great loss of generality, the temporal variations in water and energy supplies are assumed to be sinusoidal and to have their extrema at the same times of year:
P(t) = b + APsinwt
(2)
Ep(/) =/~p + AEp sinw/
(3)
where w is the angular frequency (2~r/r, where "r is the period), and an overbar denotes an annual average value. (In the application presented later, the time origin is chosen to be at the end of the month of April, so that extrema of P(t) and Ep(t) will be at the ends of January and July, consistent with typical midlatitude climatological data.) Note that A p and AEp may have either sign, and that P and Ep may be either in phase or one-half cycle out of phase. Integration of (1), subject to (2) and (3), yields the annual cycle of w(t). From this, the annual cycle of E(t) and its mean value /~ can be deduced; annual-average runoff can be found as the difference between t5 and/~. The derived evaporation ratio is
[
E / P = roan 1, R, w + ~
+
arcsin
(4a)
Interseasonal storage of soil water when l
and
E / P = min[1, R]
(4b)
when
in which R is defined as an index of dryness,
R = Ep/P
(5)
S is a non-negative, dimensionless index of the seasonality of the difference P - Ep, S = l A P - AEpI/P
(6)
and W is a dimensionless water storage capacity,
21
the origin represents energy-limited evaporation, and the horizontal segment represents precipitation-limited evaporation. Curves falling below these two asymptotes represent situations where the limiting factor switches between water and energy during the year, and storage capacity is insufficient to balance their temporal mismatch; such a situation may be considered to be storage-limited. High values of S imply temporal mismatch between supply rates of water and energy and lead to potentially high runoff (low evaporation) for a given index of dryness. The vertical position of the storage-controlled curves is determined by the value of W; high storage capacity implies high evaporation. For comparison, the empirical curve of Budyko2 is also shown in Fig. 1, and can be seen to lie among the curves implied by eqn (4). A more critical comparison can be made by constraining the relation between R and S. In middle latitudes, the seasonal variability of potential evaporation is typically much greater than that of precipitation. Taking AEp /~p and AP = 0, we have S = R. The solution to eqn (4) with S = R (and W = 0.2) is plotted as the dotted curve in Fig. 1. Although there is a qualitative similarity to the curve of Budyko,2 the theory prescribes significantly less runoff. This difference is discussed further in the 'Summary and Conclusion'. In the development that has been presented, snowfall and snowpack have been ignored. Nevertheless, the derived solution for annual water balance might be expected to apply reasonably well to areas whose hydrology is affected by snow. For the given assumptions, it can be shown that the water balance would be unchanged by the presence of snow, as long as snow is present only during the season when precipitation exceeds potential evaporation. On the other hand, frozen, near-surface soil water could potentially provide a barrier for infiltration near the end of the cold, wet season (when P - Ep is positive) and prevent use of the full storage capacity, thereby increasing annual runoff. =
w0
W = --Pr
(7)
Equation (4b) applies to the trivial case where P(t) and Ep(t) never cross; in that situation, annual evaporation is simply the minimum of annual water and energy supplies. Equation (4a) applies to the case where the quantity P - E p changes sign twice per period. In the limit of negligible storage capacity, the annual evaporation would equal the time integral of the minimum of P and Ep throughout the cycle. For non-zero storage capacity, the evaporation would be increased by an amount equal to the storage capacity, except that it could not exceed the annual water or energy supply. The water-balance solution (eqn (4)) is plotted in Fig. 1, which shows the annual evaporation ratio as a function of the index of dryness for various values of the seasonality index S, with W given a typical value of 0"2 (consistent, for example, with w0 -- 0.15 m, and P'r = 0.75 m). The unity-slope line segment starting at '
'
t
APPLICATION
// 0
0
Budyko(1974). . . . . 1
J 2
R=Ep/P Fig. 1. Fraction of precipitation consumed by evaporation, as a function of index of dryness R, for various values of the seasonality index S, with W = 0-2. Dotted curve is for S = R, with W = 0.2, dashed curve is from Budyko.2
It is useful to test eqn (4) with actual hydrologic and climatic data. For this purpose, the area of the United States east of the Rocky Mountains (more precisely, east of 105°W) was chosen; this area contains a wide range of climatic conditions, and data availability is generally good. The independent variables (R, S, and W) were estimated a priori on the basis of information from published studies. These variables were used with eqn (4) to calculate the evaporation ratios, which were combined with observed precipitation to obtain estimates of annual runoff. These estimates were compared with independent estimates of runoff obtained from analysis of river discharge.
P. C. D. Milly
22
Estimates of precipitation are based on the work of Legates and Willmott. s They analyzed global fields of monthly mean precipitation, adjusted for gauge-induced biases, at a spatial resolution of 0.5 °. Their data were collected mostly during the period 1920-1980, with heavier weight on more recent years. The annual total of these fields was used to estimate/~. Fourier analysis of the monthly values was used at each gridpoint to estimate Ap, with the assumption that extrema occur at the ends of January and July. Estimated values of A p are positive (summer maximum of P) and exceed 40% of/~ in the western third of the study area and in the peninsula of Florida. Summer minima are found throughout the lower Mississippi basin and coastal New England; in these regions, IAPI does not exceed about 30% of P. Seasonal variations of P(t) are relatively small in the Great Lakes region and along the middle Atlantic coast. Gebert and coworkers 4 have mapped the distribution of runoff from the United States on the basis of streamflow observations for the period 1951-1980. Their map was digitized by subjective interpolation, to the nearest inch, at 0'5 ° resolution. Preliminary estimates of potential evaporation were computed at 0.5 ° resolution by direct application of Thornthwaite's method, using the monthly temperature fields of Legates and Willmott. 6 The time period of temperature data collection was similar to that for precipitation. Because Thornthwaite's method is based on a relatively small set of measurements, the preliminary estimates of potential evaporation were checked for consistency against observed evaporation, which was taken as the difference between observed precipitation and runoff. Figure 2 is a plot of observed evaporation against the preliminary potential evaporation. In principle, all data in Fig. 2 should lie below the 1-1 line. In ,
,
,/,
,
12011 -
)/
. ...7.~ .;,f.]..
-
]
..'-"
-:
_
400
1.2
..:
..,-~..5.: .:i::..-:" .:.
...,;:.p ...-;..
..-."
-i
..-"
..
•
"
- -
,S 0
I.
0
I I [ 400 800 1200 II-10RI~q'I-IWAITIr'SPOITNTIAI 1:3//ff~0l~n0N (ram)
Fig. 2. Annual observed evaporation (mm), inferred from precipitation minus runoff, plotted against annual potential evaporation (mm), estimated by direct application of the method of Thornthwaite. 9
fact, 55% of the points lie above the 1-1 line. Given the relatively high quality of the precipitation and streamflow data sets and the improbability of any significant continental-scale groundwater flow divergence, it is difficult to avoid the conclusion that the estimates of potential evaporation are biased downward significantly. For the present study, therefore, it was necessary to adjust the values of potential evaporation. This adjustment was made by applying a single scale factor to all monthly and annual values at all locations. In principle, the adjustment should be large enough to bring all points below the 1-1 line in Fig. 2. However, it must be kept in mind that there are undoubtedly random errors in addition to the apparent bias. A scale factor of 1.2 was selected, quite subjectively, by balancing the need for consistency among the input data sets against the preference to minimize the adjustment and the recognition that there are local errors in all data fields. (With a factor of 1'2, the inferred evaporation still exceeds the potential value at about 5% of the gridpoints. A factor of 1.1 left 19% of the gridpoints out of balance, and a factor of 1.35 would have been needed to bring the proportion down to 1%.) At each gridpoint, Fourier analysis of the monthly values yielded AEp. Estimated values of AEp range from 80% to 130% of/~p over most of the study area. (Negative winter values of Ep are implied where AEp exceeds /~p; this presents no problem for the theory, and was allowed, so that the seasonal integrals of P - E p implicit in eqn (4) would have magnitudes consistent with climatological data.) Values of w0 were assigned by using the 0"5° global estimates of 'soil total-available water-holding capacity' produced by Patterson. 8 She defined this quantity as the difference between moisture contents of soil at field capacity and at the permanent wilting point of the predominant vegetation, integrated over the root zone of the soil. The thickness of the root zone was taken as the maximum rooting depth for the local vegetation. The necessary soil and vegetation parameters were estimated by Patterson 8 from global data sets describing soil characteristics and vegetation type. Inferred values of the storage capacity in the study area are mostly in the range from 175 to 250 mm. The computed and observed distributions of annual runoff are shown in Fig. 3, and their difference is mapped in Fig. 4. There is a qualitative similarity in the runoff distributions, but the observed runoff exceeds the modeled runoff greatly over much of the area. In Florida and the Appalachians, the model underestimates runoff by more than 150mm; errors are smaller in the arid west, where the aridity causes most precipitation to evaporate. For the entire study area, the areal mean annual runoff modeled by eqn (4) is 187mm, and the average, from the observations, is 263 mm. It is clear that the model, and the underlying hypothesis, fails to explain the large-scale water balance of the study area.
Interseasonal storage of soil water 105
100
95
90
85
80
75
SUMMARY AND CONCLUSION
70
OHSERVEb~
40
i
i 30
30
....
.. -:;.
50
25
1 5
25
100
200
300
400
500
600
mm
Fig. 3. Modeled (top) and observed (bottom) annual runoff (mm). 105
I00
95
90
85
80
75
23
70
45 45
40 40
35 35
30 30
This paper has explored the hypothesis that the annual water balance is determined by the interaction of the seasonal cycles of precipitation and potential evaporation through a simple process of storage in the soil. An idealized representation of the seasonal variations of water and energy supplies, forcing a simple storage reservoir, was used to determine the partitioning of annual precipitation into annual evaporation and annual runoff. Under the assumptions noted, the annual water balance is determined by three dimensionless numbers: the index of dryness (ratio of annual potential evaporation to annual precipitation); a simple index of the seasonality of the difference between precipitation and potential evaporation; and the ratio of plant-available water-holding capacity to annual precipitation. According to the theory, seasonality tends to reduce annual evaporation below the absolute limits imposed by annual water and energy supplies, and storage tends to counteract this effect of seasonality. In application, the theory did not reproduce the observed distribution of runoff over the United States east of the Rocky Mountains. Milly 7 explored a hypothesis similar to that developed here. As in the present analysis, the water balance was assumed to be the result of interaction of precipitation and potential evaporation, with storage allowed in the soil. In contrast to the present study, however, seasonal variations were ignored, and only the storminess of precipitation was considered. Such intraseasonal variability of forcing was judged insufficient, on its own, to generate observed amounts of runoff. What is the cause of the fraction of the observed runoff left unexplained in the present study? It is possible that the simple water balance model used both here and by Milly 7 could explain observed water balance if both seasonal and intraseasonal variability of forcing were considered jointly. Spatial variability of plantavailable water-holding capacity could also explain some part of the runoff in this same framework. On the other hand, explanation of the observed water balance may require consideration of physical processes neglected altogether in these studies. There is a widespread assumption among hydrologic modelers, for example, that the infiltration capacity of soil is a major factor in the partitioning of precipitation between runoff and evaporation. Further tests of hydrologic models and their underlying hypotheses against observational data might be beneficial.
25
ACKNOWLEDGEMENTS lii!242=i~iiiiiiii:;i:.i~!!;;!i;Tii i;;;;!!;[ : : : : ::::: ::l -150 -100 50
I 0
50
mm
Fig. 4. Difference between modeled and observed annual runoff (mm).
Krista A. Dunne provided technical support. She and Gregg J. Wiche provided helpful reviews, as did two anonymous reviewers.
24
P. C. D. Mill),
REFERENCES 1. Budyko, M. I. Heat Balance of the Earth's Surface (in Russian). Gidrometeoizdat, 1956, 255 pp. 2. Budyko, M. I. Climate and Life. Academic Press, New York, 1974, pp. 321-30. 3. Budyko, M. I. & Zubenok, L. I. Determination of evaporation from the land surface (in Russian). Izv. Akad. Nauk SSSR, Set. Geogr., 1961, 3-17. 4. Gebert, W. A., Graczyk, D. J. & Krug, W. R. Average annual runoff in the United States, 1951-1980. Hydrologic Investigations Atlas HA-710 US Geological Survey, Reston, VA, 1987. 5. Legates, D. R. & Willmott, C. J. Mean seasonal and spatial
6. 7. 8. 9.
variability in gauge-corrected, global precipitation. Int. J. ClimatoL, 10 (1990) 111-27. Legates, D. R. & Willmott, C. J. Mean seasonal and spatial variability in global surface air temperature. Theor. Appl. Climatol., 41 (1990) 11-21. Milly, P. C. D. An analytic solution of the stochastic storage problem applicable to soil water. Water Resour. Res., 29 (1993) 3755-8. Patterson, K. A. Global distributions of total and totalavailable soil water-holding capacities. Masters thesis, Dept. of Geogr., Univ. Delaware, Delaware, USA, 1990. Thornthwaite, C. W. An approach toward a rational classification of climate. Geogr. Rev., 38 (1948) 55-94.
© 1994 Elsevier Science Limited Printed in Great Britain. All fights reserved 0309-1708,,'94/$07.00
ELSEVIER
Climate, interseasonal storage of soil water, and the annual water balance P. C. D. MiHy US Geological Survey and Geophysical Fluid Dynamics Laboratory/NOAA, Princeton, New Jersey, USA The effects of annual totals and seasonal variations of precipitation and potential evaporation on the annual water balance are explored. It is assumed that the only other factor of significance to annual water balance is a simple process of water storage, and that the relevant storage capacity is the plant-available water-holding capacity of the soil. Under the assumption that precipitation and potential evaporation vary sinusoidally through the year, it is possible to derive an analytic solution of the storage problem, and this yields an expression for the fraction of precipitation that evaporates (and the fraction that runs off) as a function of three dimensionless numbers: the ratio of annual potential evaporation to annual precipitation (index of dryness); an index of the seasonality of the difference between precipitation and potential evaporation; and the ratio of plant-available water-holding capacity to annual precipitation. The solution is applied to the area of the United States east of 105°W, using published information on precipitation, potential evaporation, and plant-available water-holding capacity as inputs, and using an independent analysis of observed river runoff for model evaluation. The model generates an areal mean annual runoff of only 187mm, which is about 30% less than the observed runoff (263 nun). The discrepancy is suggestive of the importance of runoff-generating mechanisms neglected in the model. These include intraseasonal variability (storminess) of precipitation, spatial variability of storage capacity, and finite infiltration capacity of land.
INTRODUCTION
soils are penetrated by plant roots, which extract stored water for transport through the plant to the leaves; inside the leaves, available energy is consumed in the phase change of liquid water to vapor. The plantavailable water-holding capacity (or, herein, storage capacity) of most soils is around 15% of the bulk soil volume of the root zone. The volume of soil whose water is accessible to plant roots is relatively uncertain, but is usually assumed to include all soil above the maximum root depth, which is typically of the order of one meter. This implies an effective storage capacity of the order of 0-15m. In principle, the temporal mismatch of energy and water supplies can be modified not only by water storage, but also by energy storage. Energy may be stored in the soil during periods of strong insolation, thereby allowing evaporation to occur when precipitation is falling onto the heated soil. However, such timeshifting of energy supply does not play a major role in the dynamics of water balance, especially at the scale of the annual cycle, because the water-equivalent heat storage is relatively small in comparison with the water storage capacity of most soils. (It is possible, however, that the effect of energy storage could be
A fundamental problem in hydrology is the identification of the basic controls on the time-mean partitioning of precipitation into runoff and evaporation - the annual water balance. On the basis of numerous empirical data, it has been established 2 that the fraction of precipitation that eventually runs off is controlled, in great measure, by the ratio of annual potential evaporation to annual precipitation (index of dryness). Indeed, if precipitation and potential evaporation were constant through time, evaporation would simply equal the lesser of these two variables and runoff would equal any precipitation in excess of potential evaporation. (Throughout this paper the terms 'evaporation' and 'potential evaporation' refer to total vapor flux from the surface to the atmosphere, including transpiration by plants.) Because the necessary water (precipitation) and energy (potential evaporation) for evaporation do not arrive at the same time, the problem of water balance also requires the consideration of storage. Soils have the ability to retain water by capillary forces, which prevent or retard their drainage by gravity. The upper layers of 19
20
P. C. D. Milly
important at the shorter storm-interstorm time scale for bare surfaces of solid rock or for extremely coarse granular media, both of which are incapable of retaining water for evaporation after the precipitation event.) One may hypothesize that the annual water balance is determined mainly by the temporal distributions of water and energy supplies to the land surface and by the plant-available water-holding capacity of the soil. Such a hypothesis is implicit in a long tradition of water-balance modeling, 1,9 although there has been little attention given to precise articulation and testing of the hypothesis. Milly (1993) 7 used this hypothesis as the basis for a stochastic-storage model of water balance in the absence of seasonality, with allowance for the random nature of precipitation. He did not test the model extensively, but simply noted that the implied production of runoff appeared insufficient to explain the empirical data on annual water balance. He noted that the discrepancy did not necessarily invalidate the hypothesis, but that it might rather indicate the importance of seasonality or soil spatial variability. Empirical evidence for the dependence of the annual water balance on the seasonality of climate has been reported by Budyko and Zubenok (1961). 3 They noted that some of the variability of runoff across different basins having identical values of the index of dryness could be explained by seasonal characteristics of precipitation and potential evaporation. Where precipitation and potential evaporation are in phase, evaporation is enhanced and runoff is suppressed, relative to the average response over many basins. Where the water and energy supplies are out of phase, the opposite effect is observed. The objective of this paper is to develop and test a special case of the hypothesis that annual water balance is determined by the temporal interplay of water and energy supplies, through a simple water storage process. The assumption is made that the most important source of temporal variability in water and energy supplies is associated with the seasonal cycle of climate; the random intraseasonal fluctuations of precipitation considered by Milly (1993) 7 are ignored here. The resulting problem of water balance is formulated mathematically and solved analytically. The derived expression for partitioning of the annual water balance is evaluated using available climatic, hydrologic, soil, and vegetation data for a large section of the United States.
THEORY
A single water store (w) is assumed to be replenished or depleted at a rate equal to the difference between rates of precipitation (P) and potential evaporation (Ep), unless such storage change is prevented by the finite capacity
(w0) of the store, 7
dw = dt
0 0 (e - Ep)
for (P > Ep) and (w = w0) for (P < Ep) and (w = 0) otherwise
(1)
Conceptually, the capacity w0 is the plant-available water-holding capacity of the soil, and w (the plantavailable water) is the amount of water stored, in the root zone, in excess of the water content at the wilting point of the vegetation. Any excess of P over Ep when the store is saturated is assumed to produce runoff, Y, by rapid gravity drainage from the root zone. The actual routing of this runoff through the subsurface media and the surface channel network is not considered here, but in the long run, the mean value of Y corresponds conceptually to the sum of fiver and ground-water discharge from a given area. The rate of evaporation E is assumed to equal its potential rate whenever w is greater than zero or P is greater than Ep. When w is zero and Ep exceeds P, E is taken to be equal to P. Both P(t) and Ep(t) are assumed to be periodic. The mechanism that drives the seasonality of climate is the tilt of the earth's axis relative to its orbit around the sun and the resulting seasonality of solar irradiance normal to the top of the atmosphere. At extratropical locations, this produces a strong sinusoidal signal with period of one year in most climatic variables. At the equator, the fundamental period is one-half year. For simplicity of analysis, and without great loss of generality, the temporal variations in water and energy supplies are assumed to be sinusoidal and to have their extrema at the same times of year:
P(t) = b + APsinwt
(2)
Ep(/) =/~p + AEp sinw/
(3)
where w is the angular frequency (2~r/r, where "r is the period), and an overbar denotes an annual average value. (In the application presented later, the time origin is chosen to be at the end of the month of April, so that extrema of P(t) and Ep(t) will be at the ends of January and July, consistent with typical midlatitude climatological data.) Note that A p and AEp may have either sign, and that P and Ep may be either in phase or one-half cycle out of phase. Integration of (1), subject to (2) and (3), yields the annual cycle of w(t). From this, the annual cycle of E(t) and its mean value /~ can be deduced; annual-average runoff can be found as the difference between t5 and/~. The derived evaporation ratio is
[
E / P = roan 1, R, w + ~
+
arcsin
(4a)
Interseasonal storage of soil water when l
and
E / P = min[1, R]
(4b)
when
in which R is defined as an index of dryness,
R = Ep/P
(5)
S is a non-negative, dimensionless index of the seasonality of the difference P - Ep, S = l A P - AEpI/P
(6)
and W is a dimensionless water storage capacity,
21
the origin represents energy-limited evaporation, and the horizontal segment represents precipitation-limited evaporation. Curves falling below these two asymptotes represent situations where the limiting factor switches between water and energy during the year, and storage capacity is insufficient to balance their temporal mismatch; such a situation may be considered to be storage-limited. High values of S imply temporal mismatch between supply rates of water and energy and lead to potentially high runoff (low evaporation) for a given index of dryness. The vertical position of the storage-controlled curves is determined by the value of W; high storage capacity implies high evaporation. For comparison, the empirical curve of Budyko2 is also shown in Fig. 1, and can be seen to lie among the curves implied by eqn (4). A more critical comparison can be made by constraining the relation between R and S. In middle latitudes, the seasonal variability of potential evaporation is typically much greater than that of precipitation. Taking AEp /~p and AP = 0, we have S = R. The solution to eqn (4) with S = R (and W = 0.2) is plotted as the dotted curve in Fig. 1. Although there is a qualitative similarity to the curve of Budyko,2 the theory prescribes significantly less runoff. This difference is discussed further in the 'Summary and Conclusion'. In the development that has been presented, snowfall and snowpack have been ignored. Nevertheless, the derived solution for annual water balance might be expected to apply reasonably well to areas whose hydrology is affected by snow. For the given assumptions, it can be shown that the water balance would be unchanged by the presence of snow, as long as snow is present only during the season when precipitation exceeds potential evaporation. On the other hand, frozen, near-surface soil water could potentially provide a barrier for infiltration near the end of the cold, wet season (when P - Ep is positive) and prevent use of the full storage capacity, thereby increasing annual runoff. =
w0
W = --Pr
(7)
Equation (4b) applies to the trivial case where P(t) and Ep(t) never cross; in that situation, annual evaporation is simply the minimum of annual water and energy supplies. Equation (4a) applies to the case where the quantity P - E p changes sign twice per period. In the limit of negligible storage capacity, the annual evaporation would equal the time integral of the minimum of P and Ep throughout the cycle. For non-zero storage capacity, the evaporation would be increased by an amount equal to the storage capacity, except that it could not exceed the annual water or energy supply. The water-balance solution (eqn (4)) is plotted in Fig. 1, which shows the annual evaporation ratio as a function of the index of dryness for various values of the seasonality index S, with W given a typical value of 0"2 (consistent, for example, with w0 -- 0.15 m, and P'r = 0.75 m). The unity-slope line segment starting at '
'
t
APPLICATION
// 0
0
Budyko(1974). . . . . 1
J 2
R=Ep/P Fig. 1. Fraction of precipitation consumed by evaporation, as a function of index of dryness R, for various values of the seasonality index S, with W = 0-2. Dotted curve is for S = R, with W = 0.2, dashed curve is from Budyko.2
It is useful to test eqn (4) with actual hydrologic and climatic data. For this purpose, the area of the United States east of the Rocky Mountains (more precisely, east of 105°W) was chosen; this area contains a wide range of climatic conditions, and data availability is generally good. The independent variables (R, S, and W) were estimated a priori on the basis of information from published studies. These variables were used with eqn (4) to calculate the evaporation ratios, which were combined with observed precipitation to obtain estimates of annual runoff. These estimates were compared with independent estimates of runoff obtained from analysis of river discharge.
P. C. D. Milly
22
Estimates of precipitation are based on the work of Legates and Willmott. s They analyzed global fields of monthly mean precipitation, adjusted for gauge-induced biases, at a spatial resolution of 0.5 °. Their data were collected mostly during the period 1920-1980, with heavier weight on more recent years. The annual total of these fields was used to estimate/~. Fourier analysis of the monthly values was used at each gridpoint to estimate Ap, with the assumption that extrema occur at the ends of January and July. Estimated values of A p are positive (summer maximum of P) and exceed 40% of/~ in the western third of the study area and in the peninsula of Florida. Summer minima are found throughout the lower Mississippi basin and coastal New England; in these regions, IAPI does not exceed about 30% of P. Seasonal variations of P(t) are relatively small in the Great Lakes region and along the middle Atlantic coast. Gebert and coworkers 4 have mapped the distribution of runoff from the United States on the basis of streamflow observations for the period 1951-1980. Their map was digitized by subjective interpolation, to the nearest inch, at 0'5 ° resolution. Preliminary estimates of potential evaporation were computed at 0.5 ° resolution by direct application of Thornthwaite's method, using the monthly temperature fields of Legates and Willmott. 6 The time period of temperature data collection was similar to that for precipitation. Because Thornthwaite's method is based on a relatively small set of measurements, the preliminary estimates of potential evaporation were checked for consistency against observed evaporation, which was taken as the difference between observed precipitation and runoff. Figure 2 is a plot of observed evaporation against the preliminary potential evaporation. In principle, all data in Fig. 2 should lie below the 1-1 line. In ,
,
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Fig. 2. Annual observed evaporation (mm), inferred from precipitation minus runoff, plotted against annual potential evaporation (mm), estimated by direct application of the method of Thornthwaite. 9
fact, 55% of the points lie above the 1-1 line. Given the relatively high quality of the precipitation and streamflow data sets and the improbability of any significant continental-scale groundwater flow divergence, it is difficult to avoid the conclusion that the estimates of potential evaporation are biased downward significantly. For the present study, therefore, it was necessary to adjust the values of potential evaporation. This adjustment was made by applying a single scale factor to all monthly and annual values at all locations. In principle, the adjustment should be large enough to bring all points below the 1-1 line in Fig. 2. However, it must be kept in mind that there are undoubtedly random errors in addition to the apparent bias. A scale factor of 1.2 was selected, quite subjectively, by balancing the need for consistency among the input data sets against the preference to minimize the adjustment and the recognition that there are local errors in all data fields. (With a factor of 1'2, the inferred evaporation still exceeds the potential value at about 5% of the gridpoints. A factor of 1.1 left 19% of the gridpoints out of balance, and a factor of 1.35 would have been needed to bring the proportion down to 1%.) At each gridpoint, Fourier analysis of the monthly values yielded AEp. Estimated values of AEp range from 80% to 130% of/~p over most of the study area. (Negative winter values of Ep are implied where AEp exceeds /~p; this presents no problem for the theory, and was allowed, so that the seasonal integrals of P - E p implicit in eqn (4) would have magnitudes consistent with climatological data.) Values of w0 were assigned by using the 0"5° global estimates of 'soil total-available water-holding capacity' produced by Patterson. 8 She defined this quantity as the difference between moisture contents of soil at field capacity and at the permanent wilting point of the predominant vegetation, integrated over the root zone of the soil. The thickness of the root zone was taken as the maximum rooting depth for the local vegetation. The necessary soil and vegetation parameters were estimated by Patterson 8 from global data sets describing soil characteristics and vegetation type. Inferred values of the storage capacity in the study area are mostly in the range from 175 to 250 mm. The computed and observed distributions of annual runoff are shown in Fig. 3, and their difference is mapped in Fig. 4. There is a qualitative similarity in the runoff distributions, but the observed runoff exceeds the modeled runoff greatly over much of the area. In Florida and the Appalachians, the model underestimates runoff by more than 150mm; errors are smaller in the arid west, where the aridity causes most precipitation to evaporate. For the entire study area, the areal mean annual runoff modeled by eqn (4) is 187mm, and the average, from the observations, is 263 mm. It is clear that the model, and the underlying hypothesis, fails to explain the large-scale water balance of the study area.
Interseasonal storage of soil water 105
100
95
90
85
80
75
SUMMARY AND CONCLUSION
70
OHSERVEb~
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100
200
300
400
500
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Fig. 3. Modeled (top) and observed (bottom) annual runoff (mm). 105
I00
95
90
85
80
75
23
70
45 45
40 40
35 35
30 30
This paper has explored the hypothesis that the annual water balance is determined by the interaction of the seasonal cycles of precipitation and potential evaporation through a simple process of storage in the soil. An idealized representation of the seasonal variations of water and energy supplies, forcing a simple storage reservoir, was used to determine the partitioning of annual precipitation into annual evaporation and annual runoff. Under the assumptions noted, the annual water balance is determined by three dimensionless numbers: the index of dryness (ratio of annual potential evaporation to annual precipitation); a simple index of the seasonality of the difference between precipitation and potential evaporation; and the ratio of plant-available water-holding capacity to annual precipitation. According to the theory, seasonality tends to reduce annual evaporation below the absolute limits imposed by annual water and energy supplies, and storage tends to counteract this effect of seasonality. In application, the theory did not reproduce the observed distribution of runoff over the United States east of the Rocky Mountains. Milly 7 explored a hypothesis similar to that developed here. As in the present analysis, the water balance was assumed to be the result of interaction of precipitation and potential evaporation, with storage allowed in the soil. In contrast to the present study, however, seasonal variations were ignored, and only the storminess of precipitation was considered. Such intraseasonal variability of forcing was judged insufficient, on its own, to generate observed amounts of runoff. What is the cause of the fraction of the observed runoff left unexplained in the present study? It is possible that the simple water balance model used both here and by Milly 7 could explain observed water balance if both seasonal and intraseasonal variability of forcing were considered jointly. Spatial variability of plantavailable water-holding capacity could also explain some part of the runoff in this same framework. On the other hand, explanation of the observed water balance may require consideration of physical processes neglected altogether in these studies. There is a widespread assumption among hydrologic modelers, for example, that the infiltration capacity of soil is a major factor in the partitioning of precipitation between runoff and evaporation. Further tests of hydrologic models and their underlying hypotheses against observational data might be beneficial.
25
ACKNOWLEDGEMENTS lii!242=i~iiiiiiii:;i:.i~!!;;!i;Tii i;;;;!!;[ : : : : ::::: ::l -150 -100 50
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Fig. 4. Difference between modeled and observed annual runoff (mm).
Krista A. Dunne provided technical support. She and Gregg J. Wiche provided helpful reviews, as did two anonymous reviewers.
24
P. C. D. Mill),
REFERENCES 1. Budyko, M. I. Heat Balance of the Earth's Surface (in Russian). Gidrometeoizdat, 1956, 255 pp. 2. Budyko, M. I. Climate and Life. Academic Press, New York, 1974, pp. 321-30. 3. Budyko, M. I. & Zubenok, L. I. Determination of evaporation from the land surface (in Russian). Izv. Akad. Nauk SSSR, Set. Geogr., 1961, 3-17. 4. Gebert, W. A., Graczyk, D. J. & Krug, W. R. Average annual runoff in the United States, 1951-1980. Hydrologic Investigations Atlas HA-710 US Geological Survey, Reston, VA, 1987. 5. Legates, D. R. & Willmott, C. J. Mean seasonal and spatial
6. 7. 8. 9.
variability in gauge-corrected, global precipitation. Int. J. ClimatoL, 10 (1990) 111-27. Legates, D. R. & Willmott, C. J. Mean seasonal and spatial variability in global surface air temperature. Theor. Appl. Climatol., 41 (1990) 11-21. Milly, P. C. D. An analytic solution of the stochastic storage problem applicable to soil water. Water Resour. Res., 29 (1993) 3755-8. Patterson, K. A. Global distributions of total and totalavailable soil water-holding capacities. Masters thesis, Dept. of Geogr., Univ. Delaware, Delaware, USA, 1990. Thornthwaite, C. W. An approach toward a rational classification of climate. Geogr. Rev., 38 (1948) 55-94.