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e sat, s
s
e sat, a
e sat
T a
T s
T
Figure 7.1 Linearzation of the saturated vapour pressure curve, as used in the Penman
derivation. Grey points are actual combinations of e sat and T , whereas the white point
is the combination of e sat and T at the surface, as implied by the linearization.
H , is temporarily neglected) and the temperature and humidity at a given height. Then
the system of equations (Eqs. ( 7.5 ), ( 7.8 ) and ( 7.9 )) cannot yet be solved because there
are four unknowns ( H, L v E, T s and e s ) and only three equations. However, given the fact
that we consider a wet surface, the vapour pressure at the surface can be assumed to be
equal to the saturated value at the surface temperature (see Appendix B ). Thus:
eeT
s
= ()
(7.10)
sat
s
Now there are as many equations as unknowns and the system can be solved. But, this
system has a nonlinear component, since the saturated vapour pressure curve is a non-
linear function of temperature, and thus the system cannot be solved explicitly, but
only iteratively. The main trick of the Penman derivation is that we can eliminate the
surface temperature altogether. To this end, the surface vapour pressure is estimated
from the saturated vapour pressure at the observation level z :
eT e
()
() ()
T
+
s
T
T
T
(7.11)
a
a
a
sat
s
sat
s
where s ( T ) is the slope of the saturated vapour pressure curve, d
d
e
T
sat (see Appendix B ).
As can be seen in Figure 7.1 , this linearization is only an approximation, but for situ-
ations in which the difference between air temperature and surface temperature is not
too large (i.e., small sensible heat lux) it is suficient. Combining Eqs. ( 7.11 ) and
( 7.9 ) results in the following expression for the latent heat lux:
ρ
γ
ρ
γ
c
ee
r
()
T
T
T
p
LE
=−
sat
a
+
s
()
T
a
s
v
a
r
ae
ae
(7.12)
c
ee
(
T
)
s
()
T
r
r
p
=
a
sat
a
+ γ
a
ah
H
r
ae
ae
 
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