Geoscience Reference
In-Depth Information
7.2.1 Penman Derivation
Penman (1948) derived a well-known expression for the evaporation from wet sur-
faces. Here we follow a formal derivation, using the knowledge gathered in
Chap-
ters 2
and
3
. The starting point is the energy balance equation, and the resistance
expressions for sensible and latent heat luxes:
QGHLE
v
*−=+
(7.5)
ρ
θ
θ
p
()
z
r
−
H
=−
c
s
(7.6)
ah
qz
()
−
q
LE
=−
ρ
L
s
(7.7)
v
v
r
ae
where
θ
s
and
q
s
are the potential temperature and speciic humidity at the surface, and
r
ah
and
r
ae
are the aerodynamic resistances for heat and moisture transport. Now, we
make a number of small modiications to Eqs. (
7.5
) to (
7.7
):
•
Because the upper observation level will be relatively close to the ground (order of
metres), the difference between potential temperature and regular temperature are small,
and
θ
is replaced by
T
.
The speciic humidity will be replaced by the vapour pressure,
•
1
which is more commonly
c
L
R
R
e
p
p
v
γ
where
γ
is the psychrometric constant (Pa K
-1
),
R
d
,
and
R
v
are the gas constants for dry air and water vapour (J kg
-1
K
-1
), respectively (see
Appendix B
).
The indic
a
tion
of
the observation height is dropped, and the values related to air are indi-
used in this context:
q
≈
d
=
e
v
•
cated by
T
a
and
e
a
, and the surface values by
T
s
and
e
s
.
This gives the following new versions for Eqs. (
7.6
) and (
7.7
):
T
T
−
a
s
H
=−
ρ
p
c
(7.8)
r
ah
c
e
−
e
p
a
LE
=−
ρ
γ
s
(7.9)
v
r
ae
Let us assume that we have observations for the available energy
Q*
-
G
, the aerody-
namic resistances
r
ah
and
r
ae
(where the dependence of the resistances on stability, hence
1
Note, that the derivation of the Penman equation is equally well possible when using speciic humidity as the
humidity variable (in that case the slope of the saturated vapour pressure curve
s
, used later on, will indicate
dd
sat
qT
/
rather than dd
sat
eT
/
; see, e.g., van Heerwaarden et al. (
2009
)).
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