Journal of Hydrology, 111 (1939) 31 38
E1-.evier Science Pubh,,hers B V. Amsterdam
31
Printed in The Netherlands
[41
A COMPLEMENTARY RELATIONSHIP APPROACH FOR
EVAPORATION FROM NONSATURATED SURFACES
RJ, GRANGER
Division of Hydrology, College of Engineering, University of Saskatchewan, Saskatoon, Sash,
(Canada)
(Received November 14, 1988; accepted after revision February 6. 1989)
ABSTRACT
Granger, Rd., 1989. A complementary relationship approach for evaporation from nonsaturated
surfaces, J. Hydrol., 111: 31 38.
The complementary relationship between actual and potential evaporation* is evaluated,
Definitions for the potential and wet-surface evaporation rates are chosen which allow both these
parameters to be derived from the energy balance and mass transfer equations, and to be expressed
in terms of appropriate vapor pressure gradients. Using a development similar to that of Bouchet,
is then derived. The resulting equation,
the general form of the relationship between F, E and
unlike Bouchet’s, shows that the changes in actual and potential evaporation are not equal, but
A.
are described by iE iE
EfE) the modified complementary
By introducing the concept of relative evaporation (G
relationship reduces to a general equation describing evaporation from nonsaturated surfaces**.
Thus both the combination approach and the complementary relationship yield the same result,
INTRODUCTION
Bouchet (1963), utilizing an analysis based on the energy balance, demon
strated that as a surface dried from initially moist conditions the potential
evaporation increased while the actual evaporation was decreasing; he thus
corrected the misconception that a larger potential evaporation necessarily
signified a larger actual evaporation. The relation which he derived, which has
come to be known as the complementary relationship between actual and
potential evaporation, states that as the surface dries the decrease in actual
evaporation is accompanied by an equal, but opposite, change in the potential
evaporation; the potential evaporation thus ranges from its value at saturation
to twice this value Bouchet cautioned that this relationship wa an approxi
mate one Nonetheleas, the complementary relationship has formed the basis
Introduced b Bouchet (1963)
Identa al to that derived b Granger and Grai (th-, volumc) ucir g a developmcnt similar to that
of Penman (1948)
*
0022 1694 89 303.50
1989 Elsevier Science Publ shcrs B V
for the development of some evaporation models. (Morton. 1983: Brutsaert and
Stricker. 1979). Its behavior for different scales of space and time has been
analyzed (Seguin. 1975: Fortin and Seguin. 1975).
The complementary relationship makes use of two potential evaporation
parameters, the potential evaporation. E. and a second parameter, E. which
Bouchet defined as the value of the potential evaporation when the actual
regional evaporation rate is equal to the potential rate. The major benefit
accruing from the use of two potential evaporation parameters, rather than a
single one, is that the resulting relationship appears to be universally
applicable, without the need for locally-optimized coefficients. On the other
hand, it is now dependent on the proper selection, interpretation and appli
cation of two parameters. rather than one. This should not normally represent
a drawback, however the concept of potential evaporation has been the source
of some ambiguity: a variety of definitions can be found in the literature, of
which none has been universally accepted. The choice of definitions used to
date in the development of complementary relationship evaporation models
(Morton, 1983: Brutsaert and Stricker. 1979) has not permitted these authors to
examine or derive the exact form of the relationship, and has thus forced them
to assume that it is truly complementary, or that dE/dE
1. This paper
analyses the Bouchet relationship, using definitions for potential evaporation
and wet-surface evaporation which allow for the derivation of the general form
of the relationship.
—
POTENTIAL EVAPORATION PARAMETERS
Both Morton (1983) and Brutsaert and Stricker (1979). in the development of
their complementary relationship evaporation models, utilized the Penman
combination equation. or an adjusted version of this equation, to describe
potential evaporation. The Penman equation represents the evaporation from
a “wet” surface: its reliability for providing an accurate estimate of the
potential evaporation when the surface and atmospheric conditions are signifi
cantly differently from those of a wet environment is questionable.
Bouchet referred to the second potential evaporation parameter (E) as the
value of the potential evaporation when the actual regional evaporation is
equal to the potential. Morton (1983) calls this parameter the wet-environment
evaporation and Brutsaert and Stricker (1979) refer to it as the equilibrium
evaporation: both take it to represent the evaporation rate from a large,
advection-free, moist surface, and both use a regression against the radiation
term of the energy balance to provide an estimate of its i alue. Fortin and
Seguin (1975. an the other hand, suggest that thc Penman equation provide
a better enmate nf the term described by Bouchet.
in a companion paper IGranger, this olum a vstematic approach wa
used to develop definitian fo a erics of potential e aporation parameters. Of
these, the following are relevant to this discussion: EP’2
th evaporation
rate which would occur if the surface were brought to saturation and the
energy supply (radiation and soil heat flux) to the surface were held constant:
EP3
the evaporation rate which would occur if the surface were brought to
saturation and the atmospheric parameters and the energy supply to the
surface were held constant: and EP5
the evaporation rate which would
occur if the surface were brought to saturation and the atmospheric
parameters and the surface temperature were held constant.
The parameter EP2 cannot be calculated directly since there are too many
variables and too few equations describing the system. It has been empirically
related to the radiant energy supply cPriestley and Taylor. 1972) and is
equivalent to the parameter referred to as wet-environment evaporation by
Morton (1983) and “equilibrium” evaporation by Brutsaert and Sticker (1979).
EP3 can be calculated directly using the energy balance and mass transfer
equations: it represents the situation described by the Penman combination
equation. EP5 can he caculated from the mass transfer equation if the surface
temperature is known.
These potential evaporation parameters are related in the following manner:
EP5EP3EP2E
(1)
where E is the actual evaporation rate. They behave in a complementary-like
manner in that the difference between the respective values is greatest for
initially dry conditions and they are of course identical for initially wet
conditions.
THE RELATIONSHIP BETWEEN E, E, AND
Bouchet (1963) postulated that as a wet surface dried the decrease in actual
evaporation was matched by an equivalent increase in potential evaporation.
ie, iE
This resulted in the following relationship:
—
E
E,
2E
(2)
in which
is defined as the evaporation which would occur under conditions
where E
E. In deriving his relationship (2) Bouchet did not use known or
derived expressions for E, E or
he simply based his analysis on the
assumption that as E was reduced the energy liberated must necessarily go to
increasing E. The inequality (1) presented above provides a framework which
supports the concept introduced by Bouchet: however, it does not necessarily
support the assumption that E
cEL.
Those complementary relationship models developed to date have utilized
the parameters EP3 and EP2 to represent potential and wet-environment cv
aporatio ,repectivelv Becaus. LP2 cannot bc dcrivcd directly from cnergy
I aim e and mass t ansfer tquations. the form of t e iciat mship ctwcen F
EP and FF2 was rot dens d. hut wa m,sumed t he truly complementarr, The
parameter EP’2 ha been obtained through regression. Brutsaert and Stricker
1979) utilize a regresion similar to that presented by Priestley and Taylor
1972. Morton 193) establishes a regression in such a way as to make it equal
34
to one half of the potential which he adopts for a ‘dry” environment; by doing
so he is in fact redefining the parameter and ensuring that the relationship will
truly be a complementary one.
However, since all three potential evaporatinn parameters (EP2, J’P3, and
FF53 behave in a complementary -like manner, it should be possible to develop
a “complementary relationship model” using any two of these, In addition, if
the two parameters selected can be derived from energy balance or mass
transfer equations and expressed in terms of appropriate vapor pressure
gradients, it should also be possible to derive the general form of the relation
ship between the actual evaporation and the two potential evaporation
parameters. FP3 and FF5 can both be calculated directly; FF3 is represented
by the Penman combination equation, and FF5 is derived directly from the
mass transfer equation. There is some justification for utilizing the parameters
FF5 and FF3 to represent the potential evaporation (Fr) and wet-surface
evaporation (F), respectively; FPS is the largest of the potential evaporation
parameters, and Van Bavel (1966) adopts this definition for potential evapora
tion: Fortin and Seguin (1975) suggests that the Penman equation, which is
independent of the surface parameters, represents the wet-surface evaporation
rather than the potential evaporation.
Figure 1 is a schematic representation of an evaporating surface depicting
the vapor pressure gradients governing the evaporation rates F, FF3, and FF5.
The figure illustrates that for a given set of conditions (net available energy,
Q; drying power, Ea, defined by the vapor pressure deficit of the air; surface
temperature, T and surface vapor pressure, e
), the actual evaporation (F) is
8
governed by the vapor pressure gradient (e,
ea), and the potential evapora
tion (Fr) is defined by the gradient (e
ea), where e’is the saturation vapor
pressure at the surface temperature 7
’. If the conditions of Q and Ea were
w
I
N
F g 1 Vapor pressure grad er t above a C r saturated surfaCE (e*
e
the patE ntial evaporation e
e ) and the gradu nts defir ing
maintained while the surface was supplied with water, the evaporation rate
would increase and the surface temperature would decrease to some value T
would
with a saturation vapor pressure e. The wet-surface evaporation,
the
than
less
somewhat
be
would
and
3
(e
gradient
the
ea).
be defined by
approached
conditions
surface
(initial>
actual
the
As
E.
evaporation.
potential
Using a
saturation both E and E would approach the wet-surface value.
Dalton-type formulation, these rates can be written as follows:
f(u>(e
Wet-surface evaporation
Potential evaporation
(3b)
e)
f(u)(e
E
Drying power
(3a)
e)
f(u)(e,
E
Actual evaporation
E
-
f(u)(e’
(3c>
ea)
(3d)
e)
For a given set of conditions (Q and Ea) the wet-surface evaporation rate
(E) will be fixed: the potential evaporation rate (En) is then governed by the
surface temperature, and the actual evaporation (E by the actual vapor
pressure at the surface.
Referring to Fig. 1. and using eqns. (3a), (3c), and (3d) to describe E, E, and
E, respectively, we can parallel Bouchet’s development and write:
=
—
E
f(u)(e
ea)
f(u) (e
ea)
—
f(u)(e.
ea)
=
f(u)(e
—
e)
(4)
e)
(5)
and:
—
E
f(u) (e
ea)
—
f(u)
(
—
Equations (4) and (5) can be combined in the following manner:
F
=
(Es,
(6)
The slope of the saturation vapor pressure curve at the surface temperature,
T, could be estimated by:
—
(e
(T.
e)
T
)
3
For the situation presented in Fig. 1. where the available energy and the
atmospheric conditions are held constant, only the latent and sensible heat
The
fluxes are allowed to change as the evaporation rate changes from F to
integrity of the energy balance then requires that the change in latent heat flux
result in an equal but opposite change in the sensible heat flux. (Again it is
assumed that the soil heat flux is negligible in comparison with the other
fluxes, In fact a change in surface temperature will result in a change in soil
heat flux which is very small compared to the resultant change in turbulant
sensible heat flux), The Bowen ratio relating these changes in latent and
sensible heat fluxes can be expressed as:
/3
(T*.
1
T
e)
-_
(e
where ; is the psychrometric constant. Equation (6) can then he simplified to:
2
E
E
(E.
or:
E
E
E (1
(7)
-
which replaces Bouchet’s complementary relationship eqn. (2). Equation (7)
holds for a wet surface where F
and reduces to Bouchet’s rela
tionship only when A
;. a condition which. for normal atmospheric pressure.
occurs when the temperature is near 6°C. Equation (7) shows that for constant
available energy and atmospheric conditions (i.e. constant
i’E/?E
A, and that this “complementary” relationship is not normally symmetric as
Bouchet had assumed.
This result does not necessarily void all the other developments which have
been advanced relative to Bouchet’s relationship. The exact form of the com
plementary relationship will depend on the selection of the potential evapora
tion parameters. Morton (1983), for example, uses a variation of the Penman
method to calculate Er,. and actually redefines E in such a way as to ensure
that Bouchet’s relationship, eqn. (2), holds.
Equation (7) can be used to provide estimates of actual evaporation if
appropriate expressions are available for estimating
and E.
is given by
the Penman combination equation: however, E requires that the surface
temperature be known. Substituting the Penman equation into eqn. (7) and
introducing the concept of relative evaporation (G
E/E), the ratio of actual
to potential evaporation, results in the following general equation for evapora
tion:
—
F
—
AGQ
(AG
y)
-
3
-GE
(AG
(8)
-
This is identical to the general equation describing evaporation from nonsaturated surfaces which was developed by Granger and Gray (this volume)
using an approach similar to that of Penman (1948). Thus both the combination
approach (paralleling Penmans development) and the complementary
approach (Bouchet’s development) yield the same general evaporation
equation, this, however, should he expected since the definitions used for
potential and wet surface evaporation are consistent. Equation (S) is similar in
form to the Penman equation, hut differs through the inclusion of the relative
evaporation. G. which accounts for departures from saturated conditions,
The use of eqn. (8) for estimating actual evaporation requires that a suitable
expression for the relative evaporation. G. be found. Granger and Gray (this
,
volume) showed that there exists a unique relationship between G and a
E (Q + Es), the
parameter which they called the relative drying power, D
ratio of the drying power to the sum of the available energy and the drying
power. An exponential function fitted to 158 measurement data points re
presenting various surfaces gave the following expression for the G D relation
ship:
(9)
ls0.028e
with a standard error of the estimate of G of 0.051. Equations (8) and (9) form
the basis for a simple model for the calculation of evaporation from nonsaturated surfaces. This method is independent of surface parameters (tem
perature and vapor pressure) and does not require a prior estimate of potential
evaporation.
SUMMARY
Bouchet (1963) demonstrated a relationship between actual evaporation. E.
potential evaporation, E, and a second potential evaporation parameter,
which he defined as the value of the potential evaporation when the actual
regional evaporation is equal to the potential. The relationship, known as the
complementary relationship, is based on the assumption that as a surface dries.
the changes in the actual and potential evaporation rates are equal but
opposite in sign. This relationship has formed the basis of evaporation models
developed by Morton (1983) and Brutsaert and Stricker (1979). These models
use the Penman equation to describe the potential evaporation, and estimate
E using a regression with the radiation term of the energy balance.
In this study, the Penman equation is used to represent the wet-surface
evaporation, and potential evaporation is defined in a manner similar to that
presented by Van Bavel (1966). This choice of definitions allows both
parameters (E and E) to be derived from energy balance and mass transfer
equations and to be expressed in terms of appropriate vapor pressure gradients.
Using a development which parallels Bouchet’s (1963), the general form of the
is then derived. The resulting equation
realtionship between E, E and
which replaces Bouchet’s “complementary” relationship is not symmetrical:
the changes in actual and potential evaporation are given by (E(E
1.
5
whereas Bouchet assumed E ?E
Substituting the Penman equation into this new complementary relation
EE) results in a general
ship and introducing the relative evaporation (G
equation for evaporation from nonsaturated surfaces: this equation e idnticai
to that deriv’d by Granger and Gray thi olum iho used a development
sir lar to that of Penmar (1948) Thu both thr ‘eombination’ approach and
the complementar relationship approah yield the same result. Using th s
general evaporation equation in conjunction with an expression relating the
relative evaporation to a parameter called the relative drying power represents
2
WzW
Wj
TW W
38
a simple evaporation model which is independent of the surface parameters and
does not require a prior estimate of the potential evaporation.
REFERENCES
Bouchet. R.J. 1963. Evapotranspiration réelle et potentielle. signification elimatique. Tnt. Assoc.
Sci. Hydrol,, Proc. Berkeley, Calif. Symp.. Pubi. 62: 134 142.
Brutsaert, W and Stricker, H.. 1979. An advection aridity approach to estimate actual regional
ovapotranspiration. Water Resour, Ret.. 15(2): 443—450.
Fortin. JR and Seguin, B.. 1975. Estimation de l’ETR régionale S partir de IETP locale: utilization
de Ia relation de Bouchet S différentes Schelles de tempt. Ann. Agron., 26(5): 537 554.
Granger, R.J., 1989. An examination of the concept of potential evaporation. J. Hydrol.. 111:9 19
(this volume).
Granger, R.J. and Gray, D.M., 1989. Evaporation from natural nonsaturated surfaces. J. Hydrol.,
lii: 21 29 (this volume>.
Morton. FT.. 1983. Operational estimates of areal evapotranspiration anti their significance to the
science and practice of Hydrolorv J. Hvdrol.. 66: 1 76.
Penman, H.L., 1945. Natural evaporation from open water, bare soil and grass. Proc. R. Soc.
London. Ser. A., 193: 120 145.
Priestley, C.H.B. and Taylor. R.J.. 1972. On the assessment of surface heat flux and evaporation
using large-scale parameters. Mon. Weather Rev.. 100)21: 81 92.
Seguin. B.. 1975. Influence de l’évapotranspiration régionale sur Ia mesure locale d’évapotranspira.
tion potentielle. Agric. Meteorol. 15: 355 370.
Van Bavel. C.H.M,. 1966. Potential evaporation: the combination concept and its experimental
verification. Water Resour. Ret., 2(3): 455 467.