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and thus the general expression for the turbulent diffusivities is
κ
uz
z
L
κ
uz
z
L
K
=
*
,
K
=
*
(3.27)
m
h
φ
φ
m
h
If we now introduce the actual functional forms of the lux-gradient relationships
(from Eq. (
3.21
)) we obtain:
14
/
z
L
Ku z
=
κ
116
−
m
*
z
L
fo
r
≤
0
12
/
z
L
Ku z
=
κ
116
−
(3.28)
h
*
κ
*
uz
z
L
z
L
KK
==
+
for
≥
0
m
h
15
From this we can see that for unstable conditions a decrease of
z/L
(more negative,
hence more unstable) causes
K
m
and
K
h
to increase. On the other hand, for stable con-
ditions an increase of
z/L
(more stable) causes
K
m
and
K
h
to decrease. Both tendencies
are in accordance with what was seen in
Figure 3.3
.
The results in Eq. (
3.28
) can be further analysed in terms of the turbulent Prandtl
number (
Pr
t
≡
K
m
/
K
h
), which is an indication of the extent to which momentum and
heat are transported in an equivalent way. According to Eq. (
3.28
)
Pr
t
= 1 for neutral
conditions, whereas for unstable conditions
Pr
t
< 1. This relatively eficient transport
of heat can be understood from the fact that under unstable conditions vertical wind
speed and temperature correlate well as buoyancy is an important cause of vertical
motion (
R
wT
roughly doubles from 0.25 to 0.5, going from neutral to very unstable
conditions). On the other hand, vertical wind speed and horizontal wind speed show
a decrease in correlation. As luctuations in horizontal wind speed become increas-
ingly dominated by boundary-layer scale motions, the correlation coeficient
R
uw
in
the ASL decreases from about 0.6 to 0.15 when going from neutral to unstable (based
on Wilson,
2008
).Under stable conditions the consequence of Eq. (
3.28
) is that
Pr
t
=
1 (owing to the fact that the used lux-gradient relationships for momentum and heat
are equal). However, there are indications that
Pr
t
is larger than 1 with increasing sta-
bility: transport of momentum is hampered less by stability than transport of heat (see
for example the lux-gradient relationships of Högström,
1988
; Kondo et al.,
1978
and Zilitinkevich et al.,
2013
). There is no full consensus, however, about this depen-
dence of Pr
t
on stability (Grachev et al.,
2007
).
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