Journal of Hydrology, 111 (1989) 1 7
1
Elsevier Science Publishers B.V., Amsterdam -- Printed in The Netherlands
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P O T E N T I A L E V A P O R A T I O N A N D ~'THE C O M P L E M E N T A R Y
RELATIONSHIP"
J.E. NASH
Department of Engineering Hydrology, University College, Galway (Ireland)
(Received and accepted for publication February 27, 1989)
ABSTRACT
Nash, J.E., 1989. Potential evaporation and "The complementary relationship". J. Hydrol., 111:
17.
It is shown that Morton's concept of a complementary relationship between actual and
potential evaporation is in no way incompatible with Penman's concept of potential evaporation.
A difference in terminology, with some lack of precision in the difinition of what was being kept
constant, has led to an apparent, not a real, conflict. The conclusion is drawn that Morton's work
provides a valuable extension of Penman's, leading to a method of calculating actual evaporation
under the condition of a less than abundant water supply.
INTRODUCTION
It is p r o b a b l e t h a t t h e w o r k of P e n m a n (1948) on e v a p o r a t i o n c o n s t i t u t e s one
of t h e m o s t i m p o r t a n t c o n t r i b u t i o n s to m o d e r n a p p l i e d h y d r o l o g y . It is p o s s i b l e ,
t h a t s u i t a b l y e x p o i t e d , M o r t o n ' s (1983) c o n t r i b u t i o n to t h e s a m e s u b j e c t will
attract equal recognition.
On a c u r s o r y r e a d i n g , t h e s e t w o c o n t r i b u t i o n s seem to be d i a m e t r i c a l l y
o p p o s e d b u t on a c l o s e r a n a l y s i s , it is seen t h a t b o t h i n v o k e t h e s a m e c o n c e p t
of p o t e n t i a l e v a p o r a t i o n . M o r t o n , h o w e v e r , r e j e c t s P e n m a n ' s s u g g e s t i o n t h a t
w h e n t h e w a t e r s u p p l y is n o t a b u n d a n t , a c t u a l e v a p o r a t i o n w o u l d be prop o r t i o n a l to p o t e n t i a l e v a p o r a t i o n a n d a f u n c t i o n of t h e a v a i l a b i l i t y . I n s t e a d of
u s i n g P e n m a n ' s p o t e n t i a l e v a p o r a t i o n as a f o r c i n g f u n c t i o n , to w h i c h t h e
a c t u a l e v a p o r a t i o n is a r e s p o n s e c o n d i t i o n e d by t h e a v a i l a b i l i t y of w a t e r ,
M o r t o n , in t h e f u r t h e r d e v e l o p m e n t of a c o n c e p t first p r o p o s e d by B o u c h e t
(1963), uses t h e s a m e m e s a u r e of p o t e n t i a l e v a p o r a t i o n as a n e g a t i v e i n d i c a t i o n
of t h e e v a p o r a t i o n w h i c h is a c t u a l l y o c c u r r i n g . He v i e w s p o t e n t i a l e v a p o r a t i o n
in a t w o f o l d way, first as t h e e n e r g y a v a i l a b l e for e v a p o r a t i o n u n d e r c o n d i t i o n s
of a n a b u n d a n t s u p p l y of w a t e r , a n d s e c o n d as a n e g a t i v e i n d e x of a c t u a l
e v a p o r a t i o n w h e n t h e s u p p l y is l i m i t i n g . T h e l a t t e r is t h e e s s e n t i a l M o r t o n
c o n t r i b u t i o n w h i c h e n a b l e s one to b y p a s s P e n m a n ' s a s s u m p t i o n of a r e l a t i o n s h i p b e t w e e n a c t u a l a n d p o t e n t i a l e v a p o r a t i o n d e p e n d i n g on t h e a v a i l a b i l i t y of
0022-1694/89/$03.50
i~ 1989 Elsevier Science Publishers B.V.
water. Morton's analysis, however, contains an empirical factor which will
require evaluation by further observations in the field.
The two views of potential evaporation, on the one hand as the cause, and
on the other as the effect of actual evaporation, are not incompatible. They
arise when different sets of factors are considered constant. In terms of the
constancy of external factors which are themselves unaffected by actual
evaporation, potential evaporation is indeed a (negative) index of actual
evaporation, as stated by Morton, while in Penman's interpretation it remains
a positive index of evaporation, under the assumption of constant factors of
radiation, windspeed, humidity, air temperature and velocity.
THE CONTROLOF EVAPORATION
While there may be a set of external circumstances which determine evaporation, without themselves being affected by it, this does not preclude the
existence of a valid relationship between evaporation and another set of
factors which would be so affected. Such a relationship might be described as
"a feedback" relationship.
The difficulty arises in the use, not in the concept of such a relationship. If
the evaporation were expressed in terms of external factors only, which were
themselves unaffected by evaporation, the effect of a change in one of these
factors might readily be assessed. However, when the relationship is of a
feedback kind, such that a change in one of the "independent" variables affects
others, in a manner which is not fully accountable, the effect on the evaporation cannot easily be calculated. Irrigation of the Sahara desert would
produce complex changes in the atmosphere which would themselves affect the
potential evaporation to an extent which would be difficult to assess, if we
possessed only a relationship of a feedback kind. To neglect the feedback effect
would lead to the obviously incorrect conclusion that the actual evaporation
from an irrigated Sahara would equal the present potential rate.
POTENTIAL EVAPORATION
In defining potential evaporation (Ep), Penman envisaged a surface sufficiently extended to obviate any significant advection of energy from outside,
with an abundant water supply and subject to radiation at a constant rate.
Assuming the constancy of the net radiation and all other external factors
which might influence evaporation (for example through affecting the
windspeed) the steady-state evaporation rate (and indeed the sensible heat flux,
the air temperature, humidity, etc.,) would depend on the net radiation and
such other external factors. A complete analysis of the relationships would, in
principle, be possible and its results would be expressible by a set of equations,
one for each atmospheric variable including evaporation, in terms of the
external independent variables (principally the net radiation).
Such a complete analysis has not yet been undertaken, but a partial solution
is inherent in two equations expressing the energy used in evaporation and
sensible heat flux, separately, in terms of more proximate causes, factors which
are indeed functions of the independent net radiation but which would
themselves react to changes in the moisture availability in a feedback way.
These are:
(latent heat flux)
E = (es
(sensible heat flux)
K
=
ea)f(u)
y(T s -
Ta)F(u)
(1)
(2)
where es is the vapour pressure near the surface, e~ the vapour pressure in the
air above, T~ and Ta the corresponding temperatures, f ( u ) is the relevant
function of velocity, and y is the psychrometer constant.
Also, it is obvious that under conditions of zero heat storage and advection,
the net radiation H is disposed of between the sensible and latent heat fluxes,
E and K:
H
=
E + K
(3)
For an abundant water supply, the saturation vapour pressure-temperature
relationship must apply near the evaporating surface:
e~ = f ( T s )
(4)
Thus, assuming that we know f ( u ) and have measured H, there are four
equations in seven unknowns:
u, es, ea, To, T ~ , E a n d K
Penman obtained an incomplete solution by assuming that T~, e~ and u were
measured on site, thus reducing the number of remaining unknowns to four E,
K, es, Ts. By using eqns. (1)-(4) simultaneously, he obtained a solution for E in
terms of T~, ea, u and H:
Ep -
HA ÷ E'y
h + y
(5)
where A is the slope of the saturation vap0ur p r e s s u r ~ t e m p e r a t u r e relationship, evaluated at the air temperature Ta, y is the psychrometer constant and
E' is the evaporation which would be obtained from eqn. (1) if the vapour
pressure near the surface equaled the saturation vapour pressure corresponding to the temperature of the air, Ta. In fact, difficulties in the algebraic
analysis rendered the solution for Ep approximate only.
Equation (5) is Penman's equation for the energy available for evaporation
under the conditions prescribed, viz., an extensive area, abundantly supplied
with water a n d in the a b s e n c e o f a n y t e m p o r a l v a r i a t i o n in the r e l e v a n t factors.
Equation (5) can readily be expressed in terms of evaporation depth rather than
energy.
Application of eqn. (5) to the measured ~'independent" factors (Ta, e~, u and
H) under any circumstances of water availability, produces an estimate of
Penman's "potential evaporation" for that set of values of these factors,
implying that if the single condition of water availability were changed to
provide an abundant supply and the other factors of eqn. (5) on which the
evaporation depends remained unchanged, the evaporation would assume the
potential value as given by eqn. (5).
There is nothing inconsistent in Penman's expression of potential evaporation in terms of Ta, ea, u and H, provided one does not fail to appreciate that
these factors are themselves subject to feedback and therefore would change
with a change in the supply of water for evaporation. Failure to appreciate this
point would lead to the erroneous conclusion mentioned above viz., that,
evaporation from an irrigated Sahara would occur at thepresent potential rate.
Equation (5) was however obtained under the condition of an abundant
water supply and in the absence of any advection of energy. Under these
circumstances, the factors on the right-hand side are themselves dependent on
the radiation and because of this they are also mutually related. For the
Penman scenario, including the abundant water supply, a steady state and
constant net radiation, these factors must adjust to mutually consistent values
on which the evaporation will depend in accordance with eqn. (5). At this stage,
it is not clear what would be the meaning of the "potential evaporation"
obtained by eqn. (5) for an inconsistent set of"independent" factors, arbitrarily
chosen, or observed in a different scenario, e.g., under relatively dry
conditions. In other words, we cannot easily interpret eqn. (5) for a set of
observations not consistent with the original scenario of an abundant water
supply, and a steady state.
Morton, insisting quite correctly, that Penman's definition of potential
evaporation did not correspond to any definable physical situation in which the
evaporation would persist unchanging, in the presence of unchanging
causative factors, re-defined potential evaporation as the evaporation which
would occur from an irrigated point in an extensive area not necessarily
abundantly supplied with water, Morton's definition obviously permits of a
stable relationship with the "independent" factors, even when arbitrarily
chosen, or observed under other than a "wet" scenario. There is however no
real conflict between Penman's and Morton's definitions-though Morton's is
clearer. Morton himself habitually used Penman's equation to calculate
"potential evaporation".
ACTUAL EVAPORATION (Ea)
In the absence of an abundant water supply, eqn. (4) would no longer apply,
and, therefore, eqn. (5) could not be obtained as an estimate of actual evaporation under conditions of water scarcity. Penman assumed that actual evaporation would be proportional to the potential rate multiplied by some function
¢ of the availability (e.g., soil moisture deficiency):
Ea = Ep¢(smd)
(6)
By first applying eqn. (5) on the assumption of an abundant supply, Ep is
obtained and using an empirical relationship for ~b in eqn. (6), Ea is obtained.
This is also the manner in which potential evaporation is commonly used in
current rainfall-discharge modelling and other practical applications in
hydrology.
MORTON'S CONTRIBUTION
Morton saw the weakness of eqn. (6). Why should the actual evaporation be
proportional to the potential rate, when the limitation is one of water supply
not of energy, or any inadequacy in the transportation mechanism? His more
important contribution, however, apart from clarifying the concept of
potential evaporation, was to see in the potential evaporation, as calculated by
Penman's or any similar formula, or as indicated with some distortion, by a pan
measurement, a hereto unused source of information. Morton sees Ep as a
reflection of the energy available for evaporation, but unused because of the
unavailability of water. That is, he sees Ep, in the steady state, as a negative
index of actual evaporation.
THE COMPLEMENTARYRELATIONSHIP
To make this concept quantitative, and to express it in the form of an
equation which could be used with eqns. (1), (2) and (3) to find the actual
evaporation (and, if required, the other three unknowns, es, T~ and K), Morton,
like Penman, considered an extensive area, thus avoiding advection of energy,
and assumed all the external factors including the water supply to be time
invariant. Under these conditions, he considered how the actual and potential
evaporation rates would vary under different conditions of water supply. He
assumed that for a greater water supply the actual evaporation rate would also
be greater and, as a consequence, the potential evaporation, as expressed by
Penman's formula for the current values of the variables, or as indicated by a
pan measurement, would be less. He suggested, as an hypothesis, that, for
constant external conditions, the changes in actual and potential evaporation
might be equal and opposite, thus implying a constant sum:
E~ + Ep = constant
(7a)
Morton intended, that after evaluation of the constant, eqn. (7a) would be
used with eqns. (1), (2) and (3), to solve for the actual evaporation Ea (and, if
required, the other three unknowns, es, T~ and K) in terms of net radiation and
the measured quantities Ta, ea and u. To evaulate the constant of eqn. (7a), he
considered what the equation would imply under conditions of zero and
abundant water supply (the dry and wet environment conditions) respectively.
For zero supply, E~ becomes zero and the constant becomes the value of the
potential evaporation which would occur under dry environment conditions Ed
(i.e., the evaporation which would be given by Penman's formula or, with some
distortion, by a pan, under the conditions which would develop in response to
6
constant external factors and no water supply). Therefore eqn. (7a) becomes:
Ea + Ep = Ed
(7b)
To relate the constant to the wet environment conditions, Morton
considered the equation with an abundant supply, for which obviously, Ea
equals Ep. The constant is therefore twice the potential evaporation under the
wet environment conditions Ew (the evaporation which would develop in
response to constant external factors and an abundant water supply). Equation
(7a) becomes:
E~ + Ep = 2Ew
(7c)
Figure 1, which is a copy of Morton's fig. 5 explains his hypothesis. Under
completely arid conditions (zero supply) the actual evaporation would be zero
and the potential evaporation would be high. For a less than abundant supply
the actual evaporation would equal the supply and for an abundant supply,
actual and potential evaporation would be equal. Morton's hypothesis is that
for all intermediate supply rates the slope of the two curves would be equal in
magnitude and opposite in sign thus implying a constant sum of potential and
actual evaporation.
In both eqns. (7b) and (7c), Ep must be interpreted as the value obtained by
applying Penman's, or similar, formula to the radiation, windspeed, humidity
and temperature actually occurring under the prevailing conditions. Ep is the
rate at which water would evaporate given an adquate supply and the current
values of the terms on the right of eqn. (5). Ew and ED, the wet and dry
environment potential evaporation rates are the values of the potential
evaporation which would develop under constant external conditions, with an
abundant supply and with no supply, respectively.
E (d)
ntial evaporation
E (w:
/ ~ A c t u a l
evaporation equal to
~the supply
0
Water supply
Fig. 1. The complementary relationship (from Morton's eqn. 5).
U n d e r c o n s t a n t e x t e r n a l c o n d i t i o n s (mainly net radiation), it is clear t h a t
the r i g h t - h a n d side of eqns, (7a), (7b) or (7c) must be r e l a t e d to the net r a d i a n t
energy. M o r t o n suggests t h a t it is a p p r o x i m a t e l y equal to this q u a n t i t y , but
w a r n s t h a t this i m p o r t a n t r e l a t i o n s h i p r e m a i n s to be confirmed or modified.
RELEVANCE TO CATCHMENT MODELLING
It would c e r t a i n l y be w r o n g in principle, as pointed out "by Morton, to t r e a t
p o t e n t i a l e v a p o r a t i o n as an i n v a r i a n t w h e n p r e d i c t i n g the effect of a c h a n g e in
w a t e r supply. Common p r a c t i c e in r a i n f a l l - d i s c h a r g e modelling, w h e r e
P e n m a n ' s f o r m u l a or the r e c o r d of an e v a p o r a t i o n pan is often so used, t h o u g h
usually for r e l a t i v e l y small c h a n g e s in the supply, is open to this criticism and
c o n s i d e r a t i o n must be given to c h a n g i n g this practice. M o r t o n ' s h y p o t h e s i s
expressed by eqn. (7), with the c o n s t a n t e v a l u a t e d in solne empirical way,
would be m u c h less open to this objection.
CONCLUSIONS
Despite initial a p p e a r a n c e s , the only conflict b e t w e e n M o r t o n ' s views and
those of P e n m a n , arises t h r o u g h M o r t o n ' s r e j e c t i o n of the a s s u m p t i o n t h a t
w h e n the w a t e r supply is limiting the a c t u a l e v a p o r a t i o n is p r o p o r t i o n a l to the
p o t e n t i a l e v a p o r a t i o n and some f u n c t i o n of the w a t e r supply (6). T h e r e is no
conflict b e t w e e n P e n m a n ' s and M o r t o n ' s definitions of p o t e n t i a l e v a p o r a t i o n ,
t h o u g h the l a t t e r is c l e a r e r and less open to m i s i n t e r p r e t a t i o n . M o r t o n ' s use of
the c o m p l e m e n t a r y r e l a t i o n s h i p is an extension, not a c o n t r a d i c t i o n of
P e n m a n ' s w o r k w h i c h enables one to estimate a c t u a l r a t h e r t h a n p o t e n t i a l
e v a p o r a t i o n , t h u s avoiding the necessity of assuming an empirical r e l a t i o n s h i p
b e t w e e n a c t u a l and p o t e n t i a l e v a p o r a t i o n as a f u n c t i o n of soil m o i s t u r e
deficiency.
REFERENCES
Morton, F.I., 1983. Operational estimates of areal evapotranspiration and their significance to the
science and practice of hydrology. J. Hydrol., 66:1 76.
Bouchet, R.J., 1963. Evapotranspiration Reelle et Potentielle, Signification Climatique. Int. Assoc.
Sci. Hydrol., Proc. Berkeley, Calif., Symp., Publ. 62: 134~142.
Penman, H.L. 1948.Natural evaporation from open water bare soil and grass. Proc. R. Soc. London,
Ser. A., 193: 12(~145.