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with:
z
z
1
−
z
L
z
L
u
r
=
ln
Ψ
+
Ψ
u
0
am
m
m
κ
u
0
*
(3.44)
z
z
1
−
z
L
z
L
r
=
ln
θ
Ψ
+
Ψ
θ
0h
ah
h
h
κ
u
0h
*
Note that Eqs. (
3.43
) and (
3.44
) are not new, but rather are special cases of (
3.24
) and
(
3.31
), respectively. Again, an iteration procedure is needed to determine the luxes,
since the resistances depend on stability, which depends on
u
*
and
θ
*
. The latter are
the quantities we want to solve for.
The difference between the roughness lengths for momentum and scalar (e.g.,
heat) can also be interpreted in the framework of resistances (see
Figure 3.20
). The
aerodynamic resistance for momentum transport
r
am
is used between the upper level
z
θ
and
z
0
. But at
z
0
the temperature proile has not yet reached its surface value, that
is,
θ θ
0
≠
. The additional step in temperature between
z
0
and
z
0h
is related to an
additional resistance. This excess resistance (or boundary-layer resistance,
r
bh
) is
due to the molecular exchange of heat directly at the surface. If we consider the
total temperature difference between the surface and the observation level
z
θ
, the
resistance to obtain the correct sensible heat lux is the sum of the aerodynamic
resistance for momentum and the boundary-layer resistance (
rr r
ah
()
z
s
=+).
am
bh
Question 3.27:
Describe the iteration procedure needed to compute the sensible heat
lux from single level observations of wind speed and temperature, in combination with
assumed values for the roughness lengths for momentum and heat (similar to the itera-
tion procedure given in
Section 3.6.1
).
In atmospheric modelling the concept of drag coeficients is frequently used to deter-
mine luxes from vertical differences of, for example, wind speed or temperature.
Fluxes are then determined using the following expressions:
2
τ
≡
ρ
u
2
=
ρ
C
u z
()
,
H
=−
ρ
c
C
u z
() ()
θ θ
θ
z
−
(3.45)
*
dm
u
p
dh
u
s
If we compare Eqs. (
3.45
) and (
3.43
), we see that the friction velocity implicitly con-
tained in the resistance used in Eq. (
3.43
) is now replaced by an explicit occurrence of the
mean velocity in the drag laws of Eq. (
3.45
). Using the expressions for the velocity and
temperature proiles in Eq. (
3.42
), the expressions for the drag coeficients become:
−
2
z
z
−
z
L
z
L
2
u
C
=
κ
ln
Ψ
u
+
Ψ
0
dm
m
m
0
1
(3.46)
−
1
−
z
z
−
z
z
−
z
L
z
L
z
L
z
L
=
2
θ
+
u
C
κ
ln
Ψ
θ
Ψ
0h
ln
Ψ
u
+
Ψ
0
dh
h
h
m
m
0h
0
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