VOL. 14, NO. 5 WATER RESOURCES RESEARCH

Climate, Soil, and Vegetation 6. Dynamics of the Annual Water Balance

Deparrmenr of Civil Engineer~ng. Massachurerrs lnsrlrure of Technology Cambridge. Massachurerrs 02139

Mass conservation is employed to express the natural water balance of climate-soil-vegetation systems In terms of the average annual values of precipitation, evapotransplratlon. surface runoff, and groundwa- ter runoff as derived from the probability distributions of storm properties and from the physics of the appropriate storm and Interstorm soil molsture fluxes. The resulting conservation equatlon is used to define the dimensronless parameters governing the dynamic s~mllarity of the annual water balance. 4 n asymptotic~analys~s of this water balance equation yields a set of rational criteria for the classification of climate-soil-vegetation systems. Sensitivity wlth respect to the primary climate, soil, and vegetal parame- ters demonstrates that qualitative changes In water balance behavior are primarily dependent upon the exfiltration effectiveness of the so~l . A natural selection hypothesis is presented wh~ch specihes the stable vegetation density and the plant coefficient for a giv-:n climate-soil system In which water and not nutrition or light IS limiting.

Analytical formulation of the annual water balance f rom mass conservation principles began early in the twentieth cen- tury when hydrologists, notably Gunner [I9081 and Mqver [1915], recognized that annual runoff was a residual deter- mined by the difference between annual prec ip~ta t ion and an- nual evapotranspiration.

Thornfhwaire [I9441 appears to have been the first to realize that evapotranspiration must have two aspects, actual a n d potential, and that both play a role in the water balance depending upon the 'deficiency' o r 'sufficiency' of soil mois- ture. He developed an empirical relationship for potential evapotranspiration as a function only of the atmospheric tem- perature and a 'temperature efficiency' index [Thornrhwaire. 19311.

In a later classic paper. Thornrhwaire [I9481 used the con- cept of potential evapotranspiration as a factor along with precipitation in what he termed a 'rational classification' of climate. H e correctly recognized the key climatic role of the precipitation-potential evapotranspiration ratio but took n o account of the role of the soil properties as he performed a month by month moisture accounting to arrive at a soil mois- ture index of climate.

T h e moisture-accounting method became increasingly pop- ular as a means of relating s torm precipitation and runoff beginning perhaps with the work of Snyder [1939], who used temperature and time as an index to soil moisture. As the role of evaporation began t o be understood, better, Limley and Ackermann [I9421 used cumulative pan evaporation a s an index to soil moisture in their accounting scheme. Kohler and Limley [I9511 followed this with a more elaborate moisture- accounting technique, which they applied on a daily basis.

Features of these' various methods are combined in t he moisture-accounting method recommended today by the Soil Conservation Service [Mockus. 19641 for calculation o f basin water yield on a monthly basis. Here although the so-called 'water-holding' capacity of the soil is introduced, t he soil's capacity to infiltrate water and t o deliver soil moisture t o the surface by capillarity plays n o role.

T h e advent of the computer made it practical to perform moisture accounting o n even shorter (i.e., s torm and sub-

Copyright @ 1978 by the American Geophysical Union.

Paper number 8W0207. 0043- 1397/78/058W-0207$03.00

OCTOBER 1978

s torm) time scales and to account in a more physically realistic manner for soil moisture variations. A pioneering paper in this field is the well-known work of Crawford and Linsley [1966].

In summary. we see that the trend of hydrologic water balance research has been to use the ~ncreased understanding of evapotranspiration and the power of the digital computer to elaborate schemes for evaluating the loss term in the basic conservation relation applied over small t ime intervals at spe- cific locations. In so d o ~ n g , however, the dynamlcs o f the soil moisture movement processes have been represented, if a t all. by gross indices which d o not lend themselves t o general- ization. There appear to have been few attempts to use o u r current high level of physical understanding of all the natural

processes involved to develop a generalized model ot' the an- nual water balance. Such a model might produce valuable insights into the interactive role of soil moisture in the determi- nation of climate and would provide a tractable basis for der iv~ng generalized probability distributions of such impor- tant water balance components a s annual basin yield.

Of all previous work on this subject, that coming closest t o these goals appears to be the 'evaporation climatonomy' for- mulation of Lerrau [I9691 and Lerrau and Baradas [1973]. They write a dimensionless water balance equation in which the evaporation term is replaced by the latent heat term of a n energy balance relation. In this approach, however, there is n o explicit consideration of the effect o f t he soil and vegetal properties which will control the evaporation under all but the most humid conditions. He accomplishes this necessary shift from climate control of evaporation t o soil control of evapora- tion (as aridity increases) in an implicit manner through the introduction of an empirical 'interpolation function' which lacks the physical basis being sought here.

T h e annual conservation of water mass for a unit watershed area is usually written

where

PA annual (seasonal) precipitation, centimeters; R,, annual (seasonal) surface runoff, centimeters;

R,, annual (seasonal) groundwater runoff, centimeters;

ETA annual (seasonal) total evapotranspiration, centime- ters;

AS," annual (seasonal) change in surface storage, centime- ters;

as,, annual (seasonal) change in soil moisture and groundwater storage, centimeters.

Recognizing the presence of a storm surface retention E, which is evaporated between storms [Eagleson, 197861, we may define two new annual quantities. annual rainfall excess R,,', where

R,,' = REA + E,, (2)

and annual evapotranspiration from soil moisture ETAm, where

E T ~ ' = ErA - ErA (3)

In both of these, E," equals annual surface retention. These quantities allow ( I ) to be rewritten to account only for that moisture I , which infiltrates the soil annually. That is,

I ~ = P , - R , ~ * - A s , , = E ~ ~ * + R , ~ + A s , , (4)

Taking the time average of (4) term by term and assuming the system to be stationary in the long term allow us to discard the troublesome storage terms and to write the average annual water balance

E[IAI = mpA - E[R,"'] = E[ET;] + E[R,,l ( 5 )

in which E[ ] equals the expected value of [ 1, and E[P,] = mp,.

Using the observed probability distributions of storm prop- erties and simplified infiltration dynamics. Eagleson [I97861 derived the average annual infiltration (for climates without significant snowfall):

when

Otherwise,

EIIA1 = rnpA - EIErAl ( 6 ~ )

In the above.

6 reciprocal of average ralnstorm duration, equal to mt,-', s-';

c soil pore disconnectedness index; m soil pore size distribution index.

Eagleson [19786, c] also gives, to the first approximation,

where

m, mean number of storms per rainy season; h, storm surface retention capacity, centimeters; k, plant coefficient (equal to potential rate of transpiration

divided by potential (soil surface) rate of evaporation ip,/ep and approximately equal to the effective transpir- ing leaf surface per unit of vegetated land surface):

M vegetated fraction of surface: K parameter of gamma distribution of storm depth: X = r i m H , cm-'.

from which he finds the average annual surface runoff to be

when

Otherwise,

Using the time average rate of potential evapotranspiration, the probability distribution of interstorm periods, and sim- plified exfiltration dynamics, Eagleson [1978c] derived the an- nual (rainy seasonal) average total evapotranspiration E[ET,] as

L

I E[ET*] = EIEpAIJ(E. M, ku, ho) ( ( 1 )

a = [ 5nqZK( 1 )*( I)( 1 - so)z$r ' I 3

6r6m (8) where J(E, M , k,, ho) is the evapotranspiration function. E[EpA] is the weighted average rainy season potential eva-

where potranspiration and is given by [Eagleson, 1978~1

a . reciprocal of average rainstorm intensity, equal to E[EpA] = rn,mt,Tp' = m,mtB[l - M(l - &,)IFp (12) m,- I, seconds per centimeter;

K(1) saturated hydraulic ,-onductivity of soil, ten- We Can find the average from soil timeters per second; moisture ,!?[ETAw] from (3) as

so space and time average soil moisture concentration in surface boundary layer;

E[ET*'] = E[ET*] - E[E,*] (13)

w apparent velocity of capillary rise from water T O within 2% or 3% error, we can satisfy (13) by setting the centimeters per second; storm surface retention capacity h, equal to zero in the ex-

n effective porosity of soil; pression for J(E, M , k,, ho), while replacing the potential total q reciprocal of average rainstorm depth, equal to ~ H - ' T evapotranspiration EpA by the potential soil moisture evapot-

cm- I; ranspiration EpA'. That is, *(I) saturated soil matrix potential, centimeters (suction);

4, dimensionless infiltration sorptivity; E[ET*'I .r E[EPAw]J(E, M, ku) (14)

EAGLESON: CLIMATE, SOIL, A N D VEGETATION 75 1

in which, accounting for the reduced opportunity for soil moisture evapotranspiration.

E[Ep,'l = ElEp,,] - E[E,,,l (15)

and 7 ..

.{[I + Mk, + (2B)IJz E]e-BE

- [Mk, t (2C)lJZ E]e-CE

- (2E)11z[~(i, CE) - y(3, BE)]} (16)

where

I - M W k , + (1 - M)wfZp B = +

1 + Mk, - wfFp 2(1 + Mk, - w,EP)' (17)

where w is the apparent velocity of capillary rise from the water table and

where @, the reciprocal of the average time between storms, equals rntb-' in s- ' , 6, is the dimensionless desorption diffusiv- ity, and d is the diffusivity index of the soil.

The average annual groundwater runoff is given by the difference between the average wet season percolation to the water table and the average annual capillary rise from the water table. By using the components derived by Eagleson [I9784 this difference becomes

where m, is the average length of the rainy season in seconds and T is 1 year in seconds; and where the velocity of capillary rise w is given by [Eagleson, 19784

7 - - -

Z being the depth to the water table in centimeters. Substituting (6), (14), and (20) in (5) gives the average

annual water balance equation for soil moisture,

When the surface runoff is nonzero. (The term to the left of the equal sign is infiltration, the first term to the right is evapot- ranspiration from soil moisture, and the last two terms are groundwater runoff (the first is groundwater recharge and the last is groundwater loss).) Otherwise,

mp, = B[Ep,] J(E, M, k,, h,) + m,K(l)soc - Tw (22b)

SENSITIVITY

T o understand more fully the physical significance of (22). it is helpful to examine the sensitivity of the water balance ele- ments to changes in the soil, climate, and vegetal parameters. The highlights of such an examination are presented in this section on the basis of ( 22~ ) . the more widely applicable of the two relations. A mure detailed analysis is given elsewhere [Eagleson, 1978fl.

We will begin by examining, in Figure 1. the effect of the primary soil properties k ( l ) and c on the average annual water balance components in two very different climatic regimes under the simplifying restrictions of no vegetation ( M = 0) and no surface retention (h, = 0). It will be seen later that the bare soil (i.e., M = 0 ) behavior characterizes the system evapotranspiration quite well (qualitatively); thus this sim- plification will not distort the sensitivity analysis. Neglect of the surface retention will produce significant distortion only in the most arid climates.

The climatic parameters for the two locations studied, Clin- ton. Massachusetts, and Santa Paula, California, are given in Table 1. Because the effective porosity has a very limited range (in comparison with k ( l ) and c), we will keep it constant throughout this study at the value n = 0.35.

On Figure 1 , we should note the following. 1. Even though we are discussing the behavior of mean

annual values, we have, for convenience, dropped the 'mean' and 'expectation' notations.

2. The number written following the label on the depen- dent variable axis is the maximum plotted value of that vari- able.

3. The indicated classification of the systems as subhumid or arid will be explained later.

T o understand the physical significance of these figures, we need to remember that as log k ( l ) increases, k ( l ) increases. and the soil becomes intrinsically more permeable. The param- eter c measures the disconnectedness of the soil pores; thus smaller values will indicate higher permeability (for constant k ( l ) and so).

Soil moislure. Look first at the so-k(l) plane in Figure la , which represents the subhumid climate of Clinton, Massachu- setts. Keeping c constant at its origin value of 4 and increasing the intrinsic permeability k(l), we find (although it is not shown) that for very small k ( l ) the soil moisture increases with k( l ) due to the increasing ability of the soil to accept moisture. As k ( l ) continues to increase, beyond the origin value of lo-" cm2, the soil becomes unable to hold the infiltrated water against gravitational percolation, and the soil moisture falls off. This is accompanied by a rapid rise in the percolation to groundwater as can be seen at the bottom of this figure. Holding c constant at the large value, c = 11, we find a relative insensitivity of so over the full range of k ( l ) because of the extremely small permeabilities produced there by the factor soC.

In Figure Ib, which corresponds to the Santa Paula, Califor- nia, climate, we see much less sensitivity of so to either soil property and a shift of the maximum soil moisture to the region of low k(1) and high c, a region of low permeability. T o explain the qualitative difference between Figures la and Ib, we need to examine the fundamental difference between the two climates as demonstrated by the variability of their annual evapotranspiration. This is shown in the second row of Figure - I .

Evapotranspiration. In Clinton, the rate of potential evaporation is about one half that for Santa Paula, while B for Clinton is about 3 times that of Santa Paula. These two factors combine to make the exfiltration effectiveness 2E(1) (twice (19) with so = I ) large for Clinton and small for Santa Paula. This means that evapotranspiration in Clinton is limited by the climatically imposed potential value, while that in Santa Paula will be controlled by the soil.

Therefore in Clinton, where there is inadequate atmospheric moisture capacity, we expect the actual evapotranspiration

lop [kl l ) ] log [kl I)]

a SUB -HUMID CLIMATE (CLINTON. MA.) b. ARID CLIMATE [SANTA PAULA. CA.)

Fig. 1 . Sensitivity of annual water balance to changes in soil parameters (M = 0, h, = 0 and w / Q << I ) .

EAGLESON: CLIMATE. SOIL. A S D V E G E T A T I O ~ 753

TABLE I . Independent So11 and Climate Paramelers for Two Climate-So11 Systems

Locat~on

Parameter Clinton, Mass. Sanla Paula. Calif. -

n k(l). cm2

C

mpA, cm e,. cm/d mt,, d

m,,. d m,. d Z, m K

To, "C h,, cm k c

rate to be insensitive to the soil's ability to deliver moisture to the surface and hence insensitive to changes in the soil proper- ties. Looking at Figure la , we see that in Clinton. for low to moderate c, the evapotranspiration ETA/PA is insensitive to k ( l ) and to c, being limited by and essentially equal to the potential value. As the soil becomes very impermeable (high c and low k(1)). E( I ) will decrease. ultimately putting theevapo- transpiration under soil .control and causing a decline in ETA/ PA.

In Santa Paula, where there is excess atmospheric moisture capacity. we expect the evapotranspiration to follow the soil's ability to deliver moisture to the surface and hence to be sensitive to changes in soil properties. This is borne out in Figure Ib.

Thus where the evapotranspiration is controlled by the climate's evaporation potential rather than by the mois- ture supply to the surface, the soil moisture will be highest in the region where the permeability readily admits water and holds it effectively against gravity. This is the low c moderate k( l) region for a humid climate.

Where the evapotranspiration is controlled by the moisture supply to the surface. either by the annual precipitation itself or by the soil properties, the lowest soil moistures will occur where the soil properties most effectively permit soil moisture movement. This will occur in the region of largest E(I), which is small c and large k( l) . Conversely, because of difficulty of moisture movement the largest soil moisture will occur where k ( l ) is small and c is large.

Surface m o f f The behavior of surface runoff is qualita- tively the same in both climates. The surface runoff component Ra4/P, increases with increasing c due to decreasing per- meability and decreases with increasing k( l ) due to increasing permeability.

Groundwater runoff The effect of soil characteristics on groundwater runoff is shown in the last row of Figure 1. For a given c, R, , /P , increases monotonically with k(l), the rate being governed by the particular value of c and by the soil moisture so. Since the Santa Paula soil moisture is smaller than that for Clinton, the rates of increase are larger in the latter case. For large values of k( l ) we note a saddle in R , , / P A as c increases, particularly for Santa Paula. This results from the behavior of the factor soc, where so is less than one and in- creases with c.

Climate parameters. A similar study of sensitivity to changes in the climate parameters [Eagleson, 197813 merely

reinforces the above observation that qualitative changes in behavior of climate-soil systems are linked directly to the degree of humidity or aridity as defined by the primary param- eter E. .Any soil or climate change which significantly alters E will cause a qualitative alteration in the water budget. In approximate order of importance in their effect uvon E are (1 ) the rate of potential evaporation P,. (2 ) the annual precip- itation P, (:hrough so), (3) the season length r . and (41 the average time between storms m,..

This dominance of E is illustrated most strikingly by an- other sensitivity illustration. In Figure 2 we compare the water balance comvonents of the Santa Paula climate-soil svstem (Figure Ib) with those of the Clinton climate-soil system, in which the rate of potential evaporation has been raised to the Santa Paula value. In both cases we have again used the M = 0 and h, = 0 simplification.

Notice in Figure 2 that changing only the rate of potential evaporation causes the Clinton water balance surfaces to take on soil-controlled shapes identical to. but slightly displaced from. the corresponding Santa Paula surfaces. The displace- ment resuits primarily from the different season length which causes higher EpA/P, in Clinton and hence smaller runoff components there.

Vegeror~on. T c understand the system sensitivity to vegetal characteristics. we must first recall and restate the principal assumptions utilized in modeling the interaction between vege- tation and soil moisture [Eagleson, 1978dl.

We speak of an 'equilibrium' state of our climate-soil-vege- tation system. B y this. we do not mean a state in which all evolution has ceased but rather one in which the natural evolutionary change over 'engineer~ng time' is small enough io be negligible at the desired level of accuracy.

More importantly perhaps, we are considering only those vegetal systems wh~ch are water limited as opposed to those that are limited by nutrient supply, by available light, or by other ecosystem factors.

We have hypothesized that under natural equilibrium condi- tions, each vegetal species will operate, on the average. in the unstressed state. This is proposition I of our natural selection hypothesis for equilibrium of natural vegetal systems. This hypothesis permitted us to approximate the average annual rate of transpiration per unit of vegetated surface by the poten- tial value k,Z,.

We next assumed that the canopy density and root system have evolved so that at natural equilibrium they draw soil moisture uniformly from the entire volume above the maxi- mum root depth, i.e., from the soil beneath the unvegetated surface fraction as well as from that beneath the vegetated portion. This is proposition 2 of our natural selection hypothe- sis. That is, natural vegetal systems exist at an equilibrium density and species mix which makes full use of the available soil moisture in the root zone.

These two assumptions lead us to one-dimensional mod- eling of the root extraction of soil moisture as a distributed sink of aggregate strength M k , i p , which acts in a way that reduces the bare soil exfiltration rate. In the presence of ade- quate nutrients and light this rate of water use should be proportional to the rate of biomass production by the vegetal cover.

With this background, we will now examine the role of vegetal canopy density in the average annual water balance. For a given set of soil and climate parameters and for a given k,, (22) does not define so uniquely but rather as a function of M . To explore the form of this function, we have used the f o u ~

log [k( I I] i.a[klll]

C

o SANTA PAULA C L I M A T E b. CLINTON CLIMATE WITH SANTA PAULA Z p

Fig. 2. Dominant importance to water balance of rate of potential evaporation (M = 0, h , = 0, and w / e , << I ) .

representative sets of soil parameters listed in Table 2. The so versus M relations, obtained from (22a) by using all four soils, are presented in Figure 3 for both the Clinton and Santa Paula climates under conditions of average annual precipitation, i.e., PA/mpA = 1 and k, = 1. Notice that there is a particular M = M , for each climate-soil (and k,) combination at which so is a maximum. This may be explained as follows. For small but

TABLE 2. Independent Parameters of Representative Soils

Parameter Clay Clay Loam Silty Loam Sandy Loam

nonzero M , where bare soil exfiltration is the dominant mode of soil moisture depletion, the composite evapotranspiration rate will be reduced over that for bare soil yielding higher equilibrium soil moisture. As M increases, the vegetated frac- tion, transpiring at the potential rate, becomes dominant and the composite evapotranspiration rate will begin to increase, thereby reducing the equilibrium soil moisture. This peak in the so versus M curve is apparent at Clinton only for the less pe. .neable clay and clay loam soils because it is only for these soils that the evapotranspiration is not controlled by the cli- mate (see Figure 1). The point of maximum so in Figure 3 corresponds to maximum surface and groundwater runoff, which means for fixed precipitation there is minimum evapo- transpiration from soil moisture. We thus expect a minimum in E[fiTA*] = E[EPA*Y(E, M , k,) at M = MD. For the common

I .o I I I I

CLINTON. M A .

1 0.6 k C L ~

SI-L - SO i u

/ I I

a4 r Mo

! SA-L I V

I

SANTA P A U L A , CA

Fig. 3. Sensit~v~ty of mean annual soil rnolsture ro vegetal canopy density for typical soils ( k c = I and w i i , << ! ).

case, k, = 1, all M sensitivity comes from J(E, M. k,), which will then have a minimum at M = M,. We see this in Figure 4, where the information of Figure 3 is used (with (16)) to display the corresponding J versus M relations. Note that in Santa Paula, the clay and clay loam soils cannot absorb enough water to produce canopy densities greater than 0.3 and 0.8. respectively, as long as k, = 1 .

For a given M, (22) does not define so uniquely but rather as a function of k,. In Figure 5 we see that for constant M and P,/mp, (0.5 and 1, respectively, in this case), increasing the plant coefficient k, causes a monotonic decline in so. This decline becomes precipitous at a value of k, which is very sensitive to climate for a given soil, being smaller for arid climates.

We thus see that there is an infinity of M-k, combinations which can satisfy (22) for a given climate and soil.

While M may be evaluated observationally and k , may be estimated from the literature once the predominant species are known, it is natural to wonder what determines the M-k, values in a given climate-soil system.

It seems reasonable to assume that in natural systems the plant coefficient changes on an evolutionary time scale while the canopy density may change in response to short-term variability in climate. We thus hypothesize separate equilib- rium states: ( I ) a short-term or growth equilibrium reached by the canopy density of a given species (i.e., of given k,) in response to short-term fluctuations of soil moisture and (2) a long-term or evolutionary equilibrium reached by natural se- lection of species to be optimally compatible with the given climate and soil. We seek first to propose principles that guide

the selection of the equilibrium canopy density of a given species.

We hypothesize that natural vegetal systems of given species will develop a canopy density which produces minimum stress under the local climatic conditions. A necessary condition for zero stress is a soil moisture which is sufficiently larger than the largest critical soil moisture in a given species mix, so that stress will not be produced during the periods of dry weather. A necessary condition for minimum stress is that the soil moisture take on the maximum value possible. This is proposi- tion 3 of our natural selection hypothesis. That is, natural vegetal systems of given species tend toward a growth equilib- rium in which soil moisture is maximized.

If we use this hypothesis, the given climate, soil, and plant coefficient determine the equilibrium canopy density M = Ma through the water balance equation, where the soil moisture is maximum or. equivalently, where the soil moisture evapo- transpiration is a minimum (see Figures 3 and 4). Differentia- ting, we have the new relationship

aEIEo,*I aJ(E, M. k,) aE's'*l = J ( E . M , k . ) a M

a M + EIEP~*l a M

(23)

in which M = ,Ma and J(E, M, k , ) are given by (16). Exactly fork, = I and within about 2% otherwise. (23) can be replaced by

Substitution of (24) in (16) and letting M = M, give the equilibrium soil moisture evapotranspiraton function J(E, M,, k,) for natural climate-soil-vegetation systems. This is plotted as the solid lines in Figure 6 for values of k, covering the

o b i 0'7 WON I CL , MA, 1 i

.-

S A N T A PAULA , C A .

Fig. 4. Sensitivity of evapotranspiration function to vegetal canopy density for typical soils (k, = I and w/P, << I ) .

EAGLESON: CLIMATE. SOIL, A N D VEGETATION

1 0 i ! ; i ! I / ! I I SANTA PAULA. CA A

Fig. 5. Sensitivity of mean annual soil moisture to plant coefficient for typical soils (M = 0.5 and w / e , << I).

reported range of this parameter. Every point on each of these curves has a separate value of M,, small for small E where the soil is dry and approaching M, = I a s the moisture supply becomes 'unlimited.' A few loci of M, = consr are shown as dashed lines on Figure 6. and the M, versus E relationship of (24) is plotted in Figure 7 for k , = I . It is important ro remember that these M, versus E relationships a r e valid only under equilibrium conditions, that is, for PA/mpA = 1. '4s P, varies from mpA, E and thus J(E, M,, k,) will change accord- ingly. while t o the first approximation, M will remain constant at M,.

Also shown as dashed lines on Figures 6 and 7 are the bare soil (M, = 0) evaporation function and its asymptotes. It is very important for a later a rgument t o note that the asymp- totes of J (E , M,, k,) are identical with those of J (E) . This can be proved from (1 6) and (24).

W e come now t o the plant coefficient k,. The primary ap- pearance of k, in all cases is through the rate of potential transpiration from vegetated surfaces MC,, = Mk,tp. W e have already pointed ou t tha t where nutrients o r light are

not limiting, this average rate of water use Mep, will be pro- portional t o the average rate o f production of vegetal biomass. I t has been hypothesized [Odum, 1971, p. 2521 that for a given energy supply, biological systems evolve toward a condition of maximum biomass production. This leads us t o 4 of our natural selection hypothesis. That is, equilibrium natu- ral vegetal systems which are water limited evolve toward maximum water utilization.

If these propositions hold. we expect natural vegetal systems which are a t growth equilibrium t o have the canopy density M = M,, which gives maximum average soil moisture for the given k,. If these systems are also water limited, we expect them to have an equilibrium k, = ku0 such that the index o f potential biomass MokuD is a maximum. Tha t is.

In the upper portions of Figure 8 we present the variation of so through its surrogate, 2.1 - E[ETAL], with changes in both k , and M for one soil at each of the two example climates, Clinton. Massachusetts, and Santa Paula, California. under the condition rnpA/P, = 1. T h e locus of s, maxima (i.e., as,/ aM = 0 ) is shown by the line connecting the plotted points in each surface. This defines M, for each k, in the manner of Figure 3.

In the lower portions of Figure 8 the set of values (M, and k,) defining the locus of so maxima in each case is plotted to show the variability of M,k, along the locus. Along M = I , so has only a local maximum. since for small k,. the derivative as,/'?M vanishes at the infeasible density M > I . T h e maxi- mum value Moku0 represents the hypothesized equilibrium state of a natural system limited only bv water supply. We assume that the relative rapidity of cover growth or death (i.e., change in M ) in ccinparlson with the evolutionary process producing species change (r.e., change in k,) will always'keep the evolving vegetal system in a M = M, state o f growth equilibrium. It should then move from one growth equilibrium to another as k, evolves toward the hypothesized maximiza- tion of M,k,.

The values of M, and kuO determined as they are above for each climate-soil combination are listed in Table 3. Notice that both M, and kUD get smaller for a given soil as the climate becomes more arid. Since many (perhaps most) natural sys- tems will be nutrient limited o r light limited and hence will operate at a lesser value o f M,k,, we call the peak value Moku0

I I

J(E.Y,LJ

0.5 0 .5

CLIMATE CONTROLLED

0.1 0.1 10- lo-' I 10 E

Fig. 6. Evapotranspiration function for natural systems (w /+ , << I ).

BARE SOIL ASYMPTOTES / \ i

Fig. 7 . Equilibrium vegetal'canopy density ( k , = 1 and w/P, << I ).

the index of potential biomass and will retain the suboptimal notation for k,.

Many questions remain concerning the validity of this hy- pothesis which must be answered through comparison with field observations. In a later paper [Eagleson. 1978el we will compare these hypothesized equilibrium states with those for which our derived annual yield frequency is in best agreement with the observed annual streamflow frequency a t Clinton and at Santa Paula.

Equation ( 2 2 ~ ) defines the soil moisture balance in terms of 73 variables and parameters, mpA, m,, e,, m,., mfb, mH, m,, m,, c, d. m. n, Wl ) , O,, Of, Z, so, k(1). u , h,,, Ma, k,, and the normal annual surface temperature T,,. W e have, in addition t o (220). nine supplementary relations: '

By definition

T o solve the water budget relation for the dependent vari- able so, we must therefore specify the values of 13 parameters (i.e., 23 variables minus 10 equations).

For the soil system we have the six independent parameters ha, k ( l ) . c, n. To, and Z.

For the vegetation, k , is the only independent parameter. For the climate we may choose six independent parameters

from among the set of seven. K , mp,. Pp, mH, mf,, m,, and m,. Let us now examine the qualitative behavior of (22a) by

allowing the individual mean annual water balance com- ponents to vary with the mean annual precipitation, while keeping all other independent variables constant.

Consider the simple case in which the water table is at Z =

a. Since we are varying mp,. (26) tells us that either mH o r m, (or both) must vary also. Fo r simplicity of argument we will hold m~ fixed as an independent variable over the full range of mpA and accomplish the mpA variation solely through variation in m,.

.4s mpA - 0.

By definition

m , = m , ( m , , + m , , ) m . 2 1 and

With the assumption of independence o f i and I ,

ml = r n ~ / m ~ , (28)

From Brooks and Corey, [I 9661

m = 2/(c - 3 )

By definition

(29) The separate terms of (220) will approach the asymptotes given by

F rom Eagleson [ I 9 7 8 4 (empirical) E[I,] -- mpA{l - e-G(0)-2do)I'(u(O) + I )a(O)-do)]

*(I ) = *(n. k ( l ) , Ta) (31) mpA - 0 (38)

From' Eagleson [ I 9 7 8 4

4' = &t(d. so)

From Eagleson [I 9 7 8 4

We saw earlier (Figure 6 ) that the asymptotes of J ( E , Mo, (32) k,) are identical with those for the bare soil evaporation func-

tion J(E). These have been shown to be [Eagleson. 1978~1

From natural selection hypothesis (24) and

Mo = M(so) (34) J (E, Ma , k,) - J ( E ) - [*E/2I1'* SO -- 0 (40)

TABLE 3. Equilibrium Properties of Vegetal Cover

Clay Clay Loam Silty Loam Sandy Loam

Mo kuO Mo kuo ML7 k uO Mo k , Clinton, Mass. 0.2 2.1 0.90 0.70 1.00 0.90 1 .OO 0.70 Santa Paula. Calif. 0. I 0.7 0.75 0.30 1.00 0.50 1 .OO 0.30

We thus have

E[ErA*] - E[Ep,*][*E/2]"2 m p A - o (41)

where [Eagleson, 1978~1

E[E *)[nE/2]l/2 - s o ' + ( C + l ~ / ' P A

Since c > 3 [Eagleson, 19786), c > I + (c + I)/J, and for so << 1,

and the groundwater recharge term will become negligible very rapidly as so -- 0. Equation (5) then gives

which implies that the mpl -- 0 asymptotes of these two terms are identical. Equating them, we have, as mpA - 0.

For large mpA the evapotranspiration term approaches the asymptote

E[ErA*1 -- E[Ep4*j mpA - (44 1

The intersection of the evapotranspiration asymptotes (41) and (44)) occurs a t E = 2/n, which separates soil-controlled from climate-controlled evapotranspiration [Eagleson, 1978~1. Putting E = 2/n in (43) gives

mpl/EIEp,*] = [I - e-c(o)-2#(0)r(o-(O) + I)u(O)-@)I-' (45

and is larger for rainfall excess producing soils such as clays than it is for sandy soils which have a high infiltration capac- ity. For very sandy soils the rainfall excess vanishes asymptot- ically. Using (22b) now gives

V)

W C - I- Z a 3 0

a a Z Z a Z a W I

0

Fig. 9.

SUB-MUMI" ,(MUMID 1 I R A I N F A L L

I I I I E X C E S S I

GROUNDWATER R U N O F F

M E A N A N N U A L PRECIPITATION, mpA

Mean annual water balance (unlimited lateral transmissivity)

which corresponds to the (common) so -- 0 asymptote of E[I,] and of E[ErAe], which has its steepest possible slope, 45". To justify the above approximation, we combine (IZ), (IS), and (27) to obtain

Remember that we are keeping mH constant, which means that as mpA -- 0, m, -- 0 and m,, -- m7 and gives

At this end of the climate spectrum we may thus consider E[Ep,*] to be independent of mpA and approximated by

At the other extreme we recognize that mpA cannot approach infinity solely through increase of m, while we maintain con- stant mT, m, , and m, because the product m n , , cannot exceed m,. Therefore in what follows, we tacitly assume the limiting soil moisture to be reached at a large mpA before this m, limit becomes effective.

As mpA - m.

u 4 u ( I ) = 0

and

The mean annual potential soil moisture evapotranspiration is still governed in this limit by (47). which now says since m,, -- 0,

It should be recognized, of course, that in this climatic ex- treme, cloud cover and low temperature would also operate to decrease E[EPA8] through their important effect upon the rate of potential evapotranspiration &. Since the right-hand side of (22a) approaches a constant in this limit, the left-hand side will approach the same value. This horizontal asymptote is given, for Z = m (and hence w = 0), by

Equation (48) implies that as mp, -- m, the mean annual rainfall excess and groundwater runoff approach the respective asymptotes

EIR.A*I = mp, - EIIA] = mP, - m,K(] ) (49)

and

El&,] = m,K( I ) (50)

With these asymptotic behaviors in mind, we present the mean annual water balance components as a function of mean

EAGLESON: CLIMATE, SOIL. A N D VEGETATION

r 1 - 1 A"'":" T" """MID $&lo' I SWAMP

I I

MEAN ANNUAL PRECIPITATION, mpA

Fig. 10. Mean annual water balance (limited lateral transmissivity).

annual precipitation, as is shown in Figure 9. The climate classification shown on this drawing is explained in the next section of this paper.

In the above analysis we have assumed no 'downstream control' on the lateral movement of the groundwater com- ponent of runoff. Rather, we have prescribed a fixed water table elevation independent of the groundwater discharge and of the aquifer transmissivity. This has allowed the soil to approach saturation asymptotically with increasing mpA and has kept the water table from rising to the surface.

A more practical water balance situation arises from the presence of limitations on the lateral transmissivity of the soil. This situation is sketched in Figure 10. We see that whenever the saturated percolation to groundwater exceeds the limiting lateral groundwater flow R ,A_, , i.e., for

the surface will be saturated continuously and give

and the infiltration becomes constant at the maximum value

The mean annual precipitation at which (52) and (53) apply defines the condition for the existence of a swamp. From (22) evaluated a t saturation, we have

This region is indicated in Figure 10 beginning at the point 4. Another extreme condition can be recognized for very low

values of mpA. There is a minimum value of the soil moisture, for each vegetal species, below which that species is unable to extract moisture from the soil. Slatyer [1967, p. 761 sets this value at an average soil matrix potential of 15 bars (suction) for a variety of plants. This gives the limiting soil moisture as [Eagleson. 197W

which will be very small. Equation (55) is used along with the soil moisture balance equation ((22)) to define the value of mpA below which the system may be classified as a vegetal desert.

The intersections with E[E,,*] of mp, and of the dry asymp- tote of E [ h A * ] and the intersection of the two asymptotes of EIIA] separate differing regimes of hydrologic behavior due to the dominance of different terms of the soil moisture con-

servation equation. and they can therefore be used to define a rational classification for hydrologic climate (during the wet season) by using the classical terminology of Thornrhwaire [1948]. These intersections are indicated by vertical dashed lines I, 2, and 3 in Figures 9, 10, and l I , respectively.

The most obvious of these climatic regimes is that defined by the ratlo mP,/E[Ep,*], which measures the potential humidity of a climate. Potentially humld climates will have

The climate is termed 'potentially' humid because realization of the humidity depends upon the ability of the soil to infiltrate moisture and then return it to the surface for evaporation.

The soil system on the other hand is free to behave humidly whenever both interfacial moisture exchange processes, EIIA] and E[ETA*], are controlled not by the soil properties but by the climate. This condition is indicated whenever the functions EIIA] and E[ETA*] are dominated by thelr mpA -- asymp- totes.

A necessary condition for humid behavior is thus obtained from the intersection of the E[ETA*] asymptotes as provided in (45) and as indicated by the points marked 2 in Figures 9, 10, and I I . This condition is

A second necessary condition for humid behavior is ob- tained from the intersection of the EIIA] asymptotes. Because E[ETA*] and EllA] share a common dry (i.e.. soil-controlled) asymptote, we can obtain this intersection from (45) by linear proportion. Normally, m.J(I) >> E[Ep,*]: thus regardless of lateral transmissivity, we may approximate this intersection by

which establishes [he point marked 3 in Figures9, 10, and 1 I. In this model, soil moisture evapotranspiration is an ab-

straction from infiltration; thus

which means that mpA as given by (58) will always exceed mpA as given by (45), and thus the necessary and sufficient condi- tion for fully humid behavior is

m PA

Fig. 11. Classification of climate.

TABLE 4. Illustrative Classification o f Two Climate-Soil-Vegetal Systems

Value of m,, at Equation Classification Boundary* Defining

Classification Clinton, Santa Paula, Classificauon Boundary M a s s t Calif.$ -

.Arid (46) 47 57

Semiarid (45) 47 57

Subhum~d (58) 631 1830

Humid

'Values are given in centimeters. t For Clinton. Massachusetts, mpA = I I I m. $For Santa Paula, California, mp, = 54 cm.

There can be no climate control of soil moisture processes for mpA 2 E[E,,*]; thus the necessary and sufficient condition for fully arid behavior is

The point marked I in Figures 9, 10, and 1 I is located at the boundary of this arid zone where mpA = E[E,,*].

Equation (45) locates the point marked 2 in Figures 9, 10, and 1 I and serves to divide the partially arid, partially humid region into two additional climatic types.

For mp, larger than E[E,,*] but smaller than (45), the climate is potentially humid, but the infiltration and evapot- ranspiration are under soil control. We will call this region 'semiarid.'

For mpA larger than that in (45) but smaller than that In (58) the climate is potentially humid and the evapotranspiration is under climate control, but the infiltration is under soil control. We call this region 'subhumid.'

The sketch in Figure 1 1 illustrates this classification mecha- nism graphically. We will illustrate it numerically through its application to two climate-soil systems, that at Clinton and that at Santa Paula.

Values of the independent climate parameter for these areas. as determined elsewhere [Eagleson. 1978f1 by analysis of mete- orological data, are given in Table 1. N o data on soil or vegetal

o r on water table location were available, thus necessitating the following expedients.

1. The h, was chosen as 1 mm. 2. The k, was chosen as unity, thus making E[E,,*] inde-

pendent of M. 3. The water table was assumed to be deep enough to have

no influence. 4. The Clinton and Santa Paula soils were chosen as silty

loam. Among the four representative soils of Table 2 this gives the best fit when derived yield frequency is compared with observed streamflow frequency [Eagleson, 1978el.

In Table 4 are given the corresponding values of mpA satis- fying classification limits 1, 2, and 3 in Figure 11. From the magnitude of the observed mpA in relation to these we see that the Clinton system may be classified as subhumid, while the Santa Paula system is arid. The coincidence of the two asymp- totes in both cases tells us that for such a porous soil, surface runoff will tend to vanish in very dry years.

Using (24) in (222) and dividing through by mpA give the dimensionless average annual water balance equation for soil moisture,

which defines the dependent (dimensionless) variable so in terms of a set of independent (dimensionless) variables having physical significance and being the similarity parameters for the annual water balance. (In (62) the term to the left of the equal sign is infiltration, the first term to the right is evapo- transpiration from soil moisture, and the last two terms are groundwater runoff (the first is groundwater recharge and the last is groundwater loss).)

We identify these parameters as follows: I . Potential humidity ( a climate-soil-vegetation parame-

ter). Potential humidity is

As we have shown earlier, a necessary and sufficient condi- tion for a climate-soil-vegetation system to be fully arid is

while a necessary (but not sufficient) condition for fully humid behavior is

2 . Pore disconnectedness index c ( a soil parameter). As we decrease the interconnectedness of the soil interstitial passages, we expect the soil to be less permeable to fluid flow. In soils with highly interconnected pores. i.e.. small c, we expect the permeability to rise rapidly with increasing saturation as addi- tional flow passages are readily brought into play.

3. Gravitational infillration polential ( a climate-soil param- eter]. Equation (7) can be rewritten

Remembering [Eagleson, 1978dl that for so = 1 , w = 0, we can write (64) at saturation as

which equals the maximum gravitational infiltration rate di- vided by the average rainfall rate. When

the gravitational infiltration is potentially limited climatically, and when

G(l) I I

it is potentially limited by the soil. Whether these are true limits, of course, depends upon the actual value of G as modi- fied by the factor containing so, which depends in turn upon the full set of soil, climate, and vegetation parameters.

The, factor G is of major importance in dividing the annual precipitation into its infiltration and rainfall excess com- ponents. In general, when G(l) > 1, gravitational effects will dominate the infiltration process and produce large infiltration with little rainfall excess. When G(1) < I, the rainfall excess component becomes important.

We will thus define the gravitational infiltration potential as

4. fndex of water table influence ( a climate-soil parame- ter). The second bracketed term of (64) represents the rela- tive dynamic importance of the water table presence, since

w/K(1) is the characteristic capillary rise velocity divided by the characteristic gravitational percolation velocity. We will let

be the index of water table influence. For

the water table is of negligible importance to the dynamics of the soil moisture exchange process. For

the water table is very important, and we must question the superposition approximation used to incorporate its effects in this analysis. 5. Capillary infiltration effectiveness la climate-soil parame-

ter). From its derivation [Eagleson, 197861 we see that ( 2 ~ ) ~ ' ~ is the characteristic capillary infiltration depth divided by the average storm depth. Specializing a as defined in (8) for the condition so = 0, at which capillary infiltration will be a maximum, we obtain

in which

From the above definitions we see that when

the maximum characteristic capillary infiltration exceeds the storm depth. We thus define 2 4 0 ) as the capillary infiltration effectiveness.

Along with G(O), 2 4 0 ) governs the slope of the arid asymp- tote of I, and ETAo, as defined in (45). For Zu(0) > 1 the infiltration is high and rainfall excess becomes unimportant. For 2 4 0 ) < 1 the rainfall excess becomes important.

6. Exfiltration effectiveness (a climate-soil parameter). From its derivation [Eagleson, 1978~1, we see that (2E)1'2 is the characteristic interstorm exfiltration depth divided by the characteristic potential interstorm evaporatlon depth. Specializing E as defined in (19) for the condition so = 1, at which exfiltration will have its maximum value, we can write

where

From the above definitions we see that when

the maximum characteristic interstorm exfiltration exceeds the characteristic potential interstorm evaporation. We thus define =(I) as the exfiltration effectiveness.

7. Potential transpiration efficiency (a vegetal parame- ter). T h e ratio of potential rates of transpiration and soil surface evaporation defines the vegetal potential transpiration efficiency k,. That is, k, equals the potential rate of trans- piration divided by the potential (soil surface) rate of evapo- ration. This is often called the 'plant coefficient' by agricultur- alists.

8. Groundwater recharge potential (a climate-soil parame- ter). The groundwater runoff terms of (62) can be written

The factor m,K(l)/mpA represents the potential unit wet sea- son groundwater recharge, since it is the value of the recharge term in (62), when so = I . It is called the groundwater recharge potential

When

the groundwater recharge potential is climate limited. When the ratio is less than one, the potential is soil limited. It must be remembered, however, that this is only a potential, and its realization depends upon the value of the modifying factor soc. The value of so is dependent upon all the other soil, climate, and vegetation parameters and processes.

9. Groundwater loss index ia ciimate-soii parameter}. The second bracketed term of (72) represents the potential fraction of maximum groundwater recharge which is lost to evapo- transpiration through capillary rise from the water table. That is, Tw/m,K(I) is equal to the annual volume of capillary rise from the water table divided by the maximum wet season re- charge of groundwater. It measures the volumetric importance of water table presence in the annual water balance. We will define

as the groundwater loss index. T o the extent that

the water table will have a negligible effect on groundwater losses.

We have defined the annual water budget in terms of nine independent dimensionless parameters. One is a soil parame- ter, one is a vegetal parameter, one is a climate-soil-vegetation parameter, and six are climate-soil parameters.

For the special case of negligible water table influence the parameter set is reduced to seven, one soil, one vegetal, one climate-soil-vegetation, and four climate-soil.

The average annual components of soil moisture movement (infiltration. evapotranspiration, and percolation to groundwater) expressed in terms of physical properties and parameters of the climate, soil, and vegetation are combined in a statement of mass conservation to yield a water balance equation.

A hypothesis of natural selection is proposed that consists of the following propositions.

I . Proposition 1 states that natural vegetal systems operate in an equilibrium state which is unstressed on the average.

2. Proposition 2 states that natural vegetal systems exist at an equilibrium density and root configuration which fully exploits the soil moisture above the maximum root depth.

3. Proposition 3 states that natural vegetal systems de- velop a growth equilibrium state in which average soil mois- ture is maximized.

4. Proposition 4 states that equilibrium natural vegetal systems which are water limited evolve toward maximum wa- ter utilization.

This hypothesis leads to specification of the equilibrium vegetal canopy density and of the plant coefficient k , for water-limited natural systems In terms of the system ex- filtration effectiveness E.

'4 sensitivity analysis of the water balance equation demon- ,[rates the critical importance of the exfiltration effectiveness in characterizing the behavior of climate-soil-vegetation sys- :ems.

An asymptotic analysis of this equation yields rational cri- teria for classifying climate-soil-vegetation systems during the rainy season.

The dimensionless form of the equation yields the set of nine physically significant dimensionless parameters which define the conditions for water balance similarity among regions of the world.

C

d E

EP"

EPA'

E, Er. ETA

ETA'

eP

P,'

G G(O)

ho 1.4

K(1)

k"

4 1 ) M

Mo m

mH m1

mpA

mb

mc, m. m7 n

P A R@A Re.,

Re" '

AS,"

pore disconnectedness index. diffusivlty index. exfiltration parameter. average annual potential evapotranspiration, cen- timeters. average annual potential soil moisture evapotranspi- ration, centimeters. storm surface retention. centimeters. annual surface retention, centimeters. annual evapotranspiration. centimeters. annual evapotranspiration from soil moisture, cen- timeters. long-term time average rate of potential (soil sur- face) evaporation. centimeters per second. long-term time average rate of potential evapotrans- piration from soil moisture. centimeters per second. gravitational infiltration parameter. dry soil gravitational infiltration parameter. saturated soil gravitational infiltration parameter. which equals gravitational infiltration potential. surface retention capacity. centimeters. annual infiltration. centimeters. saturated hydraulic conductivity, centimeters per second. potential transpiration efficiency, which equals plant coefficient. saturated intrinsic permeability, square centimeters. vegetal canopy density. equilibrium vegetal canopy density. pore size distribution index. mean storm depth, centimeters. mean storm intensity, seconds per centimeter. average annual precipitation, centimeters. mean time between storms, days. mean s torm duration, days. mean number of storms per year. mean length'of rainy season, days. medium effective porosity, which equals volume o f active voids divided by total volume. annual precipitation, centimeters. annual groundwater runoff, centimeters. annual surface runoff, centimeters. annual rainfall excess, centimeters. annual change in surface storage, centimeters. annual change in soil moisture and groundwater storage, centimeters.

so time and spatial average soil moisture concentration in surface boundary layer.

T one year. seconds. w upward apparent fluid velocity representing capil-

lary rise from the water table. centimeters per sec- ond .

Z depth to water table, centimeters. a reciprocal of average rainstorm intensity, equal t o

m , - ' , seconds per centimeter. 3 reciprocai of average time between storms, equal t o

rn,@-', days -'. 6 reciprocal of average storm duration equal to m,, - I ,

days- ' . 7 reciprocal of mean storm depth, equal to mH- ' , cm-I . K parameter of gamma distribution of storm depth. ,i groundwazer loss index X parameter of gamma distribution of storm depths,

equal t o K,/rnH. cm- ' . Z potential humidity. o capillary infiltration parameter.

a(0) dry soil capillary infiltration parameter r length of rainy season, days. T index of water table influence.

$, dimensionless exfiltration diffusivity. $, dimensionless infiltration diffusivity

*(I) saturated soil matrix potential, centimeters (suction) R groundwater recharge potential

E [ ] expected value of [ 1. J( ) evapotranspiration function. r( ) gamma function.

y [a, x ] incomplete gamma function.

Acknowledgmenrr. This work was performed in part during the 1975-1976 academic year while the author was a visiting associate in the Environmental Quality Laboratory at the California Institute of Technology on sabbatical leave from MIT. The sensitivity analysis was performed with the assistance of Pedro Restrepo, research assist- ant in the Department of Civil Engineering at MIT under NSF grant ENG 76-1 1236. Publication was made possible by a grant from the Sloan Basic Research Fund of MIT. The author is indebted to Franco~s M. M. Morel, associate professor and to Harold Hemond, assistant professor, both of the Civil Engineering Department of MIT, for their helpful discussions of the biological aspects of this work.

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(Received August 23, 1977: revised January 6. 1978;

accepted February 16. 1978.)