VOL.
14, NO. 5
WATER
e
RESOURCES
RESEARCH
OCTOBER
1978
Climate, Soil, and Vegetation
The ExpectedValueof Annual Evapotranspiration
PETER S. EAGLESON
Department
of Cit)ilEngineering,
Massachusetts
Instituteof Technology
Cambridge,Massachusetts02139
The depthof interstorm
evapotranspiration
fromnaturalsurfaces
is composed
(by proportionto
vegetalcanopydensity)of evaporation
frombaresoilandtranspiration
fromvegetation.
The formeris
obtained
in termsofrandomvariables
describing
initialsoilmoisture,
timebetween
storms,
andpotential
rateof evapotranspiration
froman exfiltration
analogyto the Philipinfiltrationequationmodifiedto
incorporate
moistureextraction
by plantroots.The latteris assumed
to occurat the potentialratefor
natural vegetalsystems.In a zeroth-orderapproximationthe initial soil moistureis fixedat its climatic
spaceand time averagewherebyusingan exponentialdistributionof time betweenstormsand a constant
potentialrateof evapotranspiration
theexpected
valueof interstorm
evapotranspiration
is derived.This
meanvalueis usedto obtainthe annualaveragepointevapotranspiration
asa fractionof thepotential
valueandasa functionof dimensionless
parameters
definingtheclimate-soil-vegetation
system.
INTRODUCTION
influencethe soil moistureavailablefor actualevapotranspiration and hence for vegetalgrowth, it will affect the albedo of
the compositesoil-vegetationsurface,the aerodynamicproperties of the vegetation(density and height), and the vapor
content of the ambient near-surfaceatmosphere.This feedback and its climatic significancehave been pointed out by
are the alternateintervalsof infiltrationandevapotranspira-many investigators,including Bouchet [1963], Morton [1965,
tion,theratesofrainfallandpotential
evapotranspiration,
and 1968], and more recently Charney [1975].
In this work we will decouplethe atmosphere-landsurface
therate-controlling
soilmoistureconcentration.
Process
physfeedback
at this point and treat e•,as an independentvariable,
icscan beintroducedinto the parametersof a statisticaldistribution of evapotranspiration
by derivingthis distribution remembering,however, that in application we must compute
systembeing
from the knowndistributions
of theindependent
climaticvari- its value for the particular climate-soil-vegetation
considered.
To
facilitate
this
computation,
which
at best can
ables by means of an analytical relation betweenthe interonly
be
approximate,
we
have
formulated
the
problem
using
stormperiod,the potentialrate of evapotranspiration,
and the
volumesof exfiltratedand transpiredsoil moisture.We will separate potential rates for the bare soil surface and for the
vegetated surface.The fact that these may have different nufind only the first moment of this distribution.
This approach follows one taken earlier by the author merical values results primarily from the different effective
[Eagleson,1972] to incorporatecatchment-stream
dynamics elevationsfor the evaporation and transpiration, the effective
transpiration area per unit of land surface,the configuration
into a derivation of flood frequency.
(with respect to insolation and wind) of the effective transPOTENTIAL EVAPOTRANSPIRATION
piration surfaces,and the shieldingof the bare soil component
The exchangeof water massbetweensoil and atmosphere by the vegetal component.
acrossthe land surfaceis driven by the solar energywhich is
PROBLEM FORMULATION
available at the surface for evaporation. This atmospheric
evaporativecapacityis measuredby the potentialrate of evapWe will consider permeable natural surfaceswhich are a
oration e,. It is most commonlydefinedin terms of the residhomogeneous mixture of vegetated and bare soil fractions.
ual of an energybudgetcomputationin which, for practical
Allowing only vertical moisture fluxes, we will subdividethe
applications,advectionand storageare usuallyneglected[Penevapotranspirationprocessinto three separateelements.
man, 1948].
1. Surface retention loss Er is the depth of free-standing
It is importantto rememberthroughoutthis work that e•, water left on all surfacesat the conclusionof precipitation and
representsthe rate of evaporation which is expectedfrom a surface runoff. This retention will be assumed to be removed
In seekinga physicallybasedanalyticaldescriptionof the
averageannual water balance[Eagleson,1978a]we must deal
with the random variability of the variousclimaticvariables
involvedin the physicalprocesses
definingthe separatewater
balancecomponents.Primary amongtheserandom variations
particularsurfaceunderconditionsof unlimitedmoisturesupby evaporationat the wet surfacepotential rate e•, from the
ply at that surface.This meansthat e•, will be a function not
bare
soilsurfaceandat thevegetalpotentialratee•,vfromthe
only of the climaticenergyinput (insolation,cloudcover,etc.)
vegetation.
but also of the surfaceitself through its albedo, through its
effective evaporation area per unit of land surface, and
through the parametersgoverningthe vapor transport in the
atmosphericboundary layer (i.e., area size,surfaceroughness,
wind speed, and ambient vapor concentration). We need to
know all thesefactorsbeforee•,can be specified.Sincee•,will
Copyright¸
1978 by the AmericanGeophysicalUnion.
Paper number 8W0187.
0043-1397/78/058W-0187501.00
731
2. Bare soil evaporation Es is the depth of soil moisture
evaporated from the bare soil fraction of the surface. This
volume is broughtto the surfaceagainstgravity by capillarity
in the soil moisture movementprocesscalled exfiltration. It
takesplace at rate fe.
3. Transpiration Eo is the depth of soil moisture evaporated metabolically by plants from the vegetatedfraction of
the surface.This processtakes place at rate eo.
732
EAGLESON:CLIMATE, SOIL, AND VEGETATION
that the randomlyvariablee•,maybe
'Thematerial,
including
vegetation,
andslope
ofanysurface indicateour assumption
will determine
the maximumdepthof waterh0,equalto sur- replacedby its long-termaveragevalue;that is,
faceretention
capacity,
which
canbeheldtherebysurface
e•,(t) -• e-•
forcesagainsttheforcesof capillarity
andgravity.Anyrain-
Extendingthe infiltrationequationof Philip[1969],Eagle-
stormmustexceedthisdepthbeforeinfiltrationand/or surface
runoff can occur. At the conclusion of each storm the surface
(5)
son[1978c]hasrepresented
the exfiltrationrate by
retentionwill be evaporatedat the potentialrate, beingcom-
o. et- 1/9.-- Meo + w
f e -- -zS
pletelyexhausted
provided
theinterstorm
periodissufficiently
long.Withthisconcept
andtherainfalleventseries
modeled
in where
Figure 1, we will now look at the physicsof thesethree
M
components.
(6)
vegetatedfractionof surface;
eo vegetaltranspirationrate(a functionof the spatialaverage soil moistureconcentration
in the root zone),cen-
BARE SOIL EVAPORATION
timetersper second;
At the beginningof eachinterstormperiodtherewill be a
thin surfaceretentionfilm of depthErson all soilsurfaces.
This
film will be evaporatedat the soil surfacepotentialrate ep
w apparentvelocityof capillaryrisefrom watertable,centimeters per second.
and must be exhaustedbefore the soil can begin delivering The exfiltration'desorptivity'Seis definedfor a dry surfaceby
moistureto the surfacefor evaporation.We can expressthis
Se2Sol+a/•'
[nK(
l)xIt(
l)c)e(d)
]1/•'
surface retention loss as
=
h < ho
Ers - h
to > h/ev
,
= 0
•m
(7)
where
h > ho
Er, = h0
Er• = epto
to> ho/ev
(1)
n effectiveporosity of soil;
otherwise
K(1) saturatedeffectivehydraulicconductivityof soil,cen-
whereh equalsstormdepth,whichhas the probability
densityfunction[Eagleson,
1978b]
X(Xh)K-le
-xh
fH(h)
=
I'(K)
timetersper second;
•(1)
h> 0
(2)
½e(d) dimensionless
desorptiondiffusivityof soil;
d diffusivity index of soil;
rn pore sizedistributionindex of soil;
So initial soil moisture which is assumed constant
throughoutthe surfaceboundarylayer.
And to is the time betweenstorms,whichhasthe probability
densityfunction[Eagleson,1978b]
fr(to) = t•e-l•t•
saturatedmatrix potential of soil, centimeters(suction);
to> 0
(3)
Using(7) in (6) givesthe exfiltrationcapacity
where
fe*
=Sol+a/ø'
[nK(1)xlt(1)qbe(d)
]
i•-1 = mtø
,
- Meo + w
•r rn t
(8)
is the averagetime betweenstorms.
The time t* at which the surface retention is completely We will assumeit to apply in the interval t* < t < to, even
thoughthe historyof theexfiltrationprocess
wasnotin accord
evaporatedis
with the conditions of its derivation.
t* = Er•/ev
(4)
But what are to and te? To estimate to, we will use the
[Linsleyet al., 1958,p. 179]that
As canbe seenin Figure2, for stormdurationtr > t* the customaryassumption
actualrate of evaporationfrom the soilsurfacedepends
upon
the relativemagnitudes
of thepotentialrateof evaporation
and the potentialrate of exfiltration(calledthe 'exfiltration
capacity')
re*.Thislatterquantityrepresents
therateatwhich
the soil candelivermoistureto a dry surfaceandis a function
of the internal soil moistureconcentrations, the rate of mois-
RAINFALL
INTENSlTY
tureextraction
by vegetation,
andtherateof moisture
supply
from the watertable[Eagleson,
1978c].Earlyin an interstorm
periodwhiletheinternalsoilmoisture
ishigh(seeFigure2),
thispotentialdeliveryrateis highandnormallyexceeds
the
ratee•,atwhichtheclimatecanremove
waterfromthesurface.
In response
to thisinequality
thesurface
soilmoisture
sltakes
on a valueof 0 < sl < 1 suchthat the actualdeliveryrate
exactlymatches
e•,.As timegoeson, moisture
is evaporated,
o
o
and the internal soil moisture falls. The surface soil moisture-
RAINFALL
mustalsofall to maintainfe = e•,.Thiscontinues
untiltimet*
+ to,whensl = 0 andfe = fe*. For t > t* + to,fe* < ev,and
INTENSITY
(o)
ACTUAL
evaporation
fromthesoilsurface
proceeds
at theratefe*. The
process
ceases
eitherwhent - te,at whichtimefe* m0, or at t
= to, whenthe next rainfallbegins,whicheveroccursfirst.
The relationship
amongtheseratesand timesis illustrated
for a typicalinterstormperiodin Figure2. In thisfigurewe
(b)
MODEL
Fig. 1. Modelof precipitation
eventseries.
EAGLESON:CLIMATE, SOIL, AND VEGETATION
733
i\
I\
I \
tb > t*+ te
II \
RATES
EVAPORATION
OF
I
I
._.SURFACE
RETENTION
EVAPORATION
I
\
OF SOIL
\
I
MOISTURE
I
I
I
'"---'
Er/•P
•e•I
to
I
t* + to
TIME, t
o
Fig. 2.
t*+ te
Interstorm evaporation from bare soil.
e-o= re*
(9a)
as given by (6) and (7) when
where
qT
•I,•
•I't
r•
transpiration rate per unit width, cm• s-•;
soil moisture potential, bars;
fe*(•'e) = fe*(•t•to)
(9b)
leaf moisture potential, bars;
resistanceto moisture flow in soil, s cm-•;
where V'e(t) is the cumulative volume of moisture extractions
at time t _• t*.
r, resistanceto moistureflow in plants, s cm-•;
To calculateV'e,we mustconsiderall processes
removing 'yw specificweight of soil water, dyn cm-•.
soil moisturethrough the soil-atmosphere
interface.This requiresthat we first examinetranspirationby vegetation.
TRANSPIRATION
In a steadystate approximation to plant moisture movement, vandenHonert [1948]drewa physiological
analogyto
Ohm's law to write the volumerate of transpirationper unit
width of plant communityas
qT=(l)ev=
[•8--•t
r8+ r, 1%ø-•
Under conditions of unlimited water supply to the plant
roots we expectthe transpirationrate evto equal the potential
rate of transpiration•,• for the particularvegetaltype. As the
soil moistureis progressively
reduced,the soil moisturepotential becomeslarger (negatively).That is [Eagleson,1978c],
% = ,I,•(s)= ,I,(•)s-""
If we assumethat the resistances
r• and r•,remainconstantand
remember that xI,t is negative also, eo will tend to be reduced
(10) and thus the plant will be put in a stateof stress.The plant
kv
ev
•p
S_>S c
.
•lc
(•)
•sc
XI/soil,BARS
Fig. 3. Steadystateinterstormtranspiration.
734
EAGLESON:
CLIMATE,SOIL,ANDVEGETATION
respondsby closingits stomatasufficientlyto make 'I•t larger
(negatively) and thus to maintain the difference•8 - •t. As
the soil drying progresses,the plant eventuallyreachesits limit
of stomatal control at which point ,I•t = 'I•tc, the critical leaf
moisture potential. For still lower soil moistures,% becomes
a larger negative number, while ,I•t remains at 'I•tc. To the
extent that the resistancesremain constant,we can thus expect
eo to decline linearly with ,I• as the soil is dried beyond this
point. Using (10), we can find this critical soil moisturepoten-
Under the assumptionthat koreflects the effectivearea of
transpiringleaf surfaceper unit of vegetatedland surface,we
will useit as an amplificationfactor to approximatethe surface
retention lossErofrom the vegetation.
This film will be evaporatedat the vegetalpotentialrate k• v
and must be exhausted before transpiration can begin. We
expressthis surfaceretention lossas
Ero = h
tial ß •c to be
h < koho to > h/ko•5,
E•o = koho
•,c = •'d•vo(r, + rv) + •tc
(12)
In a manner similar to that suggestedby Cowan [1965] our
transpirationmodel is then written for a unit area of vegetated
surface,
h > koho to> koho/kot'v
(17)
otherwise
EVAPOTRANSPIRATION
Consideringonly plant communitiescomposedof homogeneousmixtures of vegetationand bare soil and assumingthe
plant growing seasonto be coincidentwith the rainy season,
we will proportion the total evapotranspirationEr from a unit
land surfaceaccording to
W(s)
= ev/•p
= xI•(1
)s-i/m- xI•lc
e-•(r8
+ rv),y•
,
•tc-•• •-•,c (13a)
and
Er = (1 - M)Es + MEv
W(s) = ko
,I•, > •c
(13b)
(18)
in which
where
Es = E, + E•,
ko = 7vo/•v
(19)
(14)
Ev = Eo +
This model is illustrated in Figure 3. Typical values of the
resistancesand critical potentialsare given in the plant physiological literature, including the works of Evans [1963], Cowan
[1965], Slatyer [1967], and Kramer [1969].
To use (13) in the calculation of soil moisture extraction
requiresspecificationof the species-dependent
critical leaf potential 'I•tcand transpirationefficiencyk,as well asthe speciesand time-dependent plant-soil resistancer8 + r,. Generalization of thesevegetalparametersis beyondthe scopeof this
work, and an expedientsimplificationwill be introduced.
We will neglectthe time variation of W(s) during the interstorm period, replacing W(s) by its value at the spaceand time
average soil moisture. That is,
W(s) = W(So)
(15)
And as was mentioned before M equals the canopy density,
which is the fraction of land area covered by leaves.
INTERSTORM
EVAPOTRANSPIRATION
We are now ready to calculate the interstorm volume of
evapotranspirationas indicatedby Figure 2. This first requires
estimation of the times to and te.
If we begin at t = t*, soil moisturewill be exhaustedby both
exfiltration and transpiration.Assumingboth processes
draw
from the samesoil moisturereservoir,we can use(6) and (18)
to write the volume Ve of extracted soil moisture as
•'e = {(1 -- M)[Se(t - t*) v2 + (w - Mkv•v)(t - t*)]}
+ Mk•v(t-
t*)
t-
t* < te
Confining our attention to natural vegetalsystems(i.e., no while from (6),
irrigation or artificial fertilization), we hypothesize that
through natural selection,all specieswill operate under avert - t* = {Se/[2(fe* -- W + Mko•v)]}•
age soil moisture conditions for which • >_•c, sinceunder
these conditions they are unstressed.It is suggestedthat the Substituting(21) in (20) and using (9), we obtain
stressedcondition is unstablein the long term due to increased
plant susceptibilityto drought and to disease.
Se2
t0 -Under this hypothesis,a unit area of vegetationwill tran2Y•?(1+ Mko - w/•,)
spire at the potential rate•,o and give
(16)
(20)
(21)
ß(l-M+M•ko
(1--M)w/•,
2(1 ++
Mko
w/•v) )
(22)
The numerical value of k,is a matter of some controversy.
interstormdurationmto
, weobtainthe
Slatyer [1967, p. 53] statesthat becausethe actual area of Dividingbythe average
evapotranspiration
effectiveness
evaporating surfacemay be much greater than the equivalent
land area, the total evapotranspirationfrom a plant community, per unit of land area, may exceedthat from a similar area
of bare wet soil. Linacre et al. [1970] found values of k,for
water plants which ranged from 0.6 to 2.5 dependingupon
species.Penman[1963, Table 18, p. 52], however,showspeak
water use to be independentof crop type in a given climate,
and Kramer [ 1969,p. 338] statesthat evaporationfrom a plant in which
community never exceedsthat from a similar area of wet soil
E = l•Se•'/2•v
•
(24)
with the same exposure.
1-M
to/mto
= I + Mko
- w/•,
(1--M)w/[v
+ M•ko
2(1++
Mko
w/•v?)E (23)
EAGLESON:
CLIMATE,SOIL, AND VEGETATION
f(h,
735
Regime C
tb )
Esj
=h+•vto
+
f •* dt
•t o
(25d)
te > to > to + h/•p
h<ho
and
0
hø
•--•
h
Esj
=h+7pto
+
fe* dt
to
(25e)
to
to > te
h<ho
Here for t = t•, f•* = 0. From (6) this gives
tb/
t• = IS•/[2•p(mkv-
Fig. 4. Integrationregionsfor calculationof expectedvalueof interstorm evapotranspiration.
w/•,)]} 2
Againdividingby into,
te/rnt•= E/2(Mkv- w/-go)
•
is the bare soil (M = 0) evaporationeffectiveness.
We can also find to (and thus E) by solving the linearized
diffusion equation under a constantflux boundary condition
Because
E • to/mto,E represents
the relativeimportance
of
soil properties in the dynamics of exfiltration. Referring to
Figure 2, we see that for large to relative to storm duration,
evapotranspirationwill occurpredominatelyat the potential
rate, and the climate-soil-vegetation
systemis largely under
climate control insofar as evapotranspirationis concerned.
When to is small relativeto to, the evapotranspirationis con-
(27)
The expectedvalueof interstormevaporationisgivenby the
integral under the joint probability densityfunction
E[Es•]
=ffEs,
f(h,
to)
dh
dto
to obtain the time at which s• = 0. The form of this solution is
the same as that given in (24) except that (by using the same
effective diffusivity) the constant coefficient is •r2/16 rather
than • [Carslawand Jaeger, 1959, p. 75].
(26)
(28)
If we assumeh and to are independentand use(2) and (3),
the joint distribution is written
(29)
f(h,
to)=f•(h)fr(to)
=[•X(Xh)'•-le-Xn-tttø
r(•)
Equations(25a)-(25e) divide the integration region for bare
soil into the three regimesseparatedby the solidlinesin Figure
trolledby the soilproperties.
The ratio E • to/mt•measures 4. Equation (28) can now be written as follows:
the averagestate of this relationshipfor the givenclimate-soilIn regime A
vegetationsystem.
From (23) we find that t* = Er/-•, and to are of the same
I*o
order of magnitude. The surface retention will thus have a
significanteffect on the exfiltration dynamics,and we must
incorporate it in our further calculations.
EA[Es•]
= f(h)
dh tø •,tof(to)
dto
+
To calculate
thevolumeEr• of evapotranspiration
during
f(h) dh
•,tof(to) dto
(30)
•0
thejth interstormperiod, we will treat the bare soil evapora-
tionEs•andthetranspiraton
Ev•separately.
Lookfirstat the In regime B
bare soil evaporation. We can use Figure 2 to identify three
regimes of behavior:
Ea[Es•]
=f••f(h)dh
Oto•ho/70
chø•oto
Regime A
+
Es•= •,to
0 _<to< t* + to
+
Regime B
Esj= ho+ 7pto+
fe* dt f(to) dto
(25a)
fe* dt
f to-ho/•
+
(250)
•t o
te > to > to+ ho/•,
h>ho
•
fe* m f(to)dto
In regime C
and
Es•= ho+ 7pto+
f te-ho/•
pf** dt
(25c)
to
+
to > te
h > ho
fe* m f(to) dto
(31)
736
EAGLESON:CLIMATE,SOIL, AND VEGETATION
wher•
e h/'•p
q-
my-- E[v]
re* dt f(to) dto
(32)
to
(41)
Following our assumptionof a constantpotentialevapotranspirationrate the weightedmeanannual(seasonal)
potential evapotranspirationis
The averageinterstormevaporationfrom a unit area of bare
soil is then givenby addingthe abovethreecomponentsto get
•
E[EvA
] = tn,,tnt,?.*= rn•.*/l•
(42)
(33) where
•p*, theweighted
average
potential:evapotranspiration
Using (6) and (29) to integrate (30), (31), and (32) and rat•, is givenby
E[Esj]= EA[Esj]
+ EB[Es•]
+ Ec[Es,]
adding accordingto (33) givesfor bare soil evaporation,
e_Br+
{1_ •'-(Z•{1- e-•r-t•ho/•,
I'(K)
-- 1q-l•ho/•p
Xh0 •[K,
I'(K)
•fl-•E[Es,]
='•[K,
Xh0]
Xho
+l•ho/['p]
ß[1 q-Mko + (2B)•/•'E- W/7p]+ e-Cr-t•hor•.[Mko
+ (2C)•/•'E- W/7p]+ (2E)•/•'e-t•no;•.
[.•(•, CE) - •(•, BE)]}
+[l+l•ho/•.l-K
Xho
+l•ho/•p]
'•h0 'y[K.
r(K)
i(2E)•/•'['•(•].
CE)- •(•.BE)]
+ e-C•[Mko
+ (2C)
•/•'
E- w/•.]
- e-•r[Mk.
+ (2B)•/•'E-
W/•p]}
Here
•p* = (1 -M)•p+
1- M
(34)
M•k. + (1 - M)w/•p
B-- 1q-Mk.-W/•p
+ 2(1q-Mk.-W/•p)
•' (35)
M•.p. = [1 -M(1
-ko)]•p
(43)
Dividing (40) by (42) gives
and
E[Ev,]
C = l(Mkv - W/•p)•'
J(E,
M,ko,
ho)
- E[Ev.]
(36J
For a vegetalsurfacewe are assuming
that transpira-
(1 - M)l•E[Es,]/•p
+ Mi•E[Ev,]/•-v
tion is alwaysat the potentialrateko•p,thuswe havesimply
(l•/•.) E[Ev•]= k.
(1 - M + Mko)
(44)
(37} wherel•E[Es•]/•.andi•E[Ev•]/•.aregivenby(34)and(37),
Finally,we obtainthe expected
interstormevapotranspira- respectively.
Equation(44) isevaluatedasa functionof E for representa•, ko,Xh0,l•ho/g.,andM andis
and(41) according
tothecanopy
density.
Byfollowing
(18)we tivevaluesof theparameters
tionfrom
ahomogeneous
composite
surface
byweighting
(34)
presentedin Figures5 and 6 underthe usualconditionthat w/
get
E[Ev,]= (1 - M)E[Es,]+ ME[Ev,]
(38)
Baresoil. BylettingM = 0, W/•p << 1,andh0= 0 in (44),
In (34) we seetwo new dimension!ess
parametersXh0and the bare soil evaporationfunctionis
•ho/•,. The first of these,Xh0,appearswhen we account for
thosestormsWhichdo not fill the surfaceretentioncapacity. E[Er.]/E[EpA]= J(E)= 1
The evapotranspirationwill thus vary inverselywith this parameter.The secondparameter,•ho/•,, can be rewritten
- [1 + 2•/•'E]e-• + (2E)x/•'I' [•1,E]
(45)
where
•ho/•. = (ho/•.)/mt,
FromFigure
2, withE• = h0,weseethatthisparameter
r[a, x] =
e-tt a-1 dt
(46)
measures
the,potentialefffictof surfaceretentionon the dynamicsof the exfiltration process.
ANNUALEVAPOTRANSPIRATION
and E is given by (24) and (7) as
•
E = [2/•nK(1)•(1)/•rm•.•']½.s0•+•'
(47)
To expand(38) to the desiredmeanannual(seasonal)eva- Equation(45) is plottedin Figure5 alongwith its asymptotes.
potranspirationrequiresthat we recognizethat for v, the num- Let us look at these asymptotesto see what (46) tells us
berof interstorm
periodsin a year(season),
theannualevapot- physically.
As theprecipitationeventsoccurmorefrequently,asthe soil
ranspirationis
sorptivityincreases,
and/or as the potential?ateof evapora-
Ev,•
= • Erj
j=l
the expectationof whichis givenby
E[Ev.]= m,,E[Ev,]
tion decreases
(suchas in a cold moistclimate), the parameter
(39)E,
as givenby (47), increases.Theseconditionsall indicatean
increasein relativeevaporation,and hencefor a constantvalue
of the climatic parameter •v we expectthe actual annual
(40) evaporationto approachthe potentialin the limit. Indeed,if
EAGLESON:
CLIMATE,,SoIL,
ANDVEGETATION
E[E.PA]
737
'
E[ETA
] =1
d+2
-E=2Bn• mK(I)•(I)
•eSo
_
•
/' /
I/2
2
--
SOIL
CONTROLLED
•
CLIMATE
CONTROLLED
o.i
0-1
Fig.5. Baresoilevaporation
function
(w/•o << 1).
weletE -• •oin(45)andrecognize
thatF[{,•o]= 0,wehave
At the otherextreme,
whenprecipitation
eventsare few,
withlongtimesbetween,
whenthesoilsorptivity
islow,and/
(48) or whenthepotentialrateof evaporation
is high(suchasin
lim E[E•.A]/E[EpA
]= 1
hot dryclimates),theparameterE decreases.
Theseconditions
Theactualevaporation
is thuscontrolled
primarily
by the all indicatea decrease
in relativeevaporation.
Takingthe
climate
forlargeE through
thepotential
rateofevaporation. limit of (45) as E -• 0, we have
Xh
o=/•ho/•p
=0.1
.L_ f Xho= O.OI
I //l ,Oho/•p
=O.I
O.I-
•••
•.•"'"'•-/
Iho/gp = 0 .01; Xho = 0 01to 0
(a)
.....•
•
0
M=ho
=0ASYMPTOTE
0.1
I0-z
iO-I
E
i0o
Fig.6. Evapotranspiration
function
(K= 0.5,k, = 1,andw/•, << 1).
738
EAGLESON:CLIMATE,SOIL, AND VEGETATION
lim
E•0 E[ErA]/E[E•'A]
= [•rE/2]x/•'
(49)
Under these arid conditionswe see that the soil sorptivity
becomesthe limitingfactorin controllingthe actualevaPoration. For constantsoil propertiesthe sorptivity is directly
If we neglectcarryoverof unevaporatedretention,the bare soil
componentis
l•E[Ers,]/•o
= 1- e-Ono/ep
r[•,xh0]
F(K)
•ho/•o
]-K
7[K,
(Xh0
+flh0/•)]
(52)
related to the soil moisture So and hence to the avail-
ability of precipitation.
The intersectionof thesetwo asymptotesis given by setting (49) equal to 1, from which we obtain
Ec = 2/•r
(50)
which may be used as a rational criterion for the classification of climate-baresoil systemsas either soil controlled
(E < Ec), or climate controlled(E > E•), insofar as relative
evaporationis concerned.
We must remember that this development assumeses
>> w, which, although it is the 'normal' condition, will
break down in a cold wet climate with a high water table.
It is interestingto note that (45) gives a physicallyrational analytical basis for the form of the natural surface
evapotranspirationfunction developedempiricallyby earlier
investigatorssuch as Budyko [1956] and Pike [1964].
Bare soil with surface retention. Using empirical expressionsfor interception [Wigham, 1973, p. 4.6] to estimate
surfaceretention capacity,we find that h0 = 0(1) mm. Observationsof climateproperties[Eagleson,1978b]give• = 0(10-1
- 10-•') d-1, }x= 0(10-1 - 10-•') mm-1, and •, = 0(1) mm/d.
ThereforeXh0= 0(10-1 - 10-ø')and•h0/• = 0(10-1 - 10-9').
Equation (44) is evaluatedover this range of surfaceretention in Figure6a underbare soil conditions,M = 0 and w/•
<< 1. As couldbe expected,surfaceretentionmakesan appreciable differencein the annual evaporation only in the very
arid situations
where E is small.
there to changesin flh0/•.
Mixed vegetationand bare soil. In Figures6b and 6c, (44)
is plotted for variousvaluesof M from bare soil (M = 0) to
completevegetalcover(M = 1) both without surfaceretention
and with representativevalues of these parameters. In all
cases,w/• << 1. With k0 = 1, vegetalcoverapparentlyhas a
strongeffecton the annualevapotranspiration,
particularlyfor
smallE. However,in interpretingthesefigureswe mustkeepin
that
M
and E
are related
under
natural
conditions
through the soil moistureSo.For large E the systemhas excess
moisture and can support much vegetation,thus M will tend
to unity. For small E the reverseis true, and M will tend to
zero. Specificationof the anticipated functional relationship
betweenE and M will transform J(E, M, kv, ho) into J(E, kv,
h0). This important task is accomplishedin a later paper
[Eagleson,1978d]through introduction of a natural selection
hypothesis.Its existencewill make J(E, M)-• J(E), which
further emphasizesthe fundamental importanceof the bare
soil evaporationfunctionJ(E) in controllingthe evapotranspiration from vegetatedsurfaces.
EVAPORATION
FROM SURFACE RETENTION
In a manner completelyanalogousto the above derivation
we may isolatethat portion Er of the total evapotranspiration
which comes from surface retention.
E[Era
]
E[E•A
]=
Also to the first approximation,the vegetalcomponentis
F[•,
Xk•h0]
OE[E,o,]/•,
=ko1- e-0n0/%
F(,)
- •l+Ohø/•"
-•T["'(xkohø+ohø
(53)
Xk•ho
This is
(1 - M)fiE[E•,j]/•,+
(1- M+ Mko)
(51)
F(•)
This component is useful in calculating surface runoff
[Eagleson,1978e].
SUMMARY AND CONCLUSIONS
Bare soil evaporation and vegetaltranspirationare calculatedfor a typicalinterstormperiodasfunctionsof properties
of the storm sequence,the surface,the soil, and the average
rate of potential evapotranspiration.The expectedvalue of
annualinterstormevapotranspiration
is thenderivedby using
observed distributions
of the random
climatic
variables.
The
normalized averageannual evapotranspirationis defined in
termsof the independentvariableE, which is a functionof the
spaceand time averagesoil moisturewithin the surfaceboundary layer. The evapotranspirationfunction for vegetatedsurfacesappearsto conform closelyto that definedby bare soil
conditions.
It can also be seen in this
figure that J(E) is very insensitive,even at small E, to variations in the relativesurfaceretentionXh0but is quite sensitive
mind
- 1+ Xh0
NOTATION
E
evaporationeffectiveness.
EvA averageannualpotentialevapotranspiration,
millimeters.
E• surface retention loss, millimeters.
Er8 surface retention loss from bare soil fraction, millimeters.
Err surfaceretention lossfrom vegetatedfraction, millimeters.
E8 soil moistureevaporationfrom bare soil fraction of
surface, millimeters.
Es total evaporationfrom bare soil fraction of surface,
millimeters.
Er
total evapotranspiration,millimeters.
ErA annualevapotranspiration,
millimeters.
Es• interstorm
baresoilevaporation,
millimeters.
Err interstorm
evapotranspiration,
millimeters.
Eo transpirationfrom vegetatedfraction of surface,millimeters.
Ev
total evapotranspirationfrom vegetatedfraction of
surface, millimeters.
Ev• interstorm
evapotranspiration
fromvegetated
fraction of surface, millimeters.
es potential(soil surface)evaporationrate, millimeters
per day.
•v long-term time averagerate of potential (soil surface) evaporation,millimetersper day.
•vo long-term time average potential rate of transpiration, millimetersper day.
es* potential evapotranspirationrate, millimeters per
day.
EAGLESON:CLIMATE,SOIL, AND VEGETATION
eo
fe
re*
h
h0
j
ko
rate of transpiration, millimetersper day.
exfiltrationrate, millimetersper day.
exfiltrationcapacity,millimetersper day.
storm depth, millimeters.
surfaceretention capacity, millimeters.
counting variable.
ratio of potential rates of transpiration and soil
surfaceevaporation.
M vegetalcanopy density.
rn pore size distribution index.
rnH mean storm depth, millimeters.
mto meantime betweenstorms,days.
rnv mean number of storms per year.
qv transpirationrate per unit width, squaremillimeters
per second.
rs resistanceto moisture flow in plant, secondsper
centimeter.
rs resistanceto moisture flow in soil, secondsper centimeter.
S• exfiltration sorptivity,cm/sx/•'.
s degreeof medium saturation(i.e., soil moistureconcentration), which equals volume of water divided
by volume of voids.
sx degreeof saturationat surfaceof medium.
,
t
to
t*
to
tr
V•
time, seconds.
duration of exfiltration at the rate 7 o, days.
time at which surface retention is exhausted,days.
time betweenstorms, days.
storm duration, days.
cumulative volume of interstorm exfiltration, millimeters.
739
Acknowledgments. This work was performed in part while the
author was a visitingassociatein the EnvironmentalQuality Laboratory of the California Instituteof Technologyon sabbaticalleavefrom
MIT. Publicationhas been made possibleby a grant from the Sloan
Basic Research
Fund of MIT.
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xI,sc
xI,(l)
xI,•
E[ ]
f( )
J( )
O( )
F( )
•,[a, x]
soil moisturepotential, centimeters(suction).
critical soil moisturepotential,centimeters(suction).
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(Received August 19, 1977;
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