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Figure 3.16 Similarity relationships for momentum and heat. ( a ) Flux-gradient rela-
tionships according to Eq. ( 3.21 ). ( b ) Integrated lux-gradient relationships accord-
ing to Eq. ( 3.30 ).
lux-gradient relationships between different experiments may be due to low distor-
tion by sensors and comes up with slightly different expressions: φ h =−
(
112
zL
/
)
12
/
,
/) /
14
for unstable conditions and, φ h =17.
zL , φ m =14.
zL for
φ m =−
(
1193
.
zL
stable conditions.
Question 3.15: Various expressions exist for the lux-gradient relationships. One
could wonder how much they differ. Determine the relative difference between the
Businger-Dyer lux-gradient relationships and the expressions proposed by Högström
( 1988 ) (see earlier). Take the Businger-Dyer relationships as a reference. Do this for
both heat and momentum, and for the following values of z /L : -2, -1, 0, 0.5 and 1.
3.5.4 Gradients and Proiles Under Neutral Conditions
Before dealing with the general case where both shear and buoyancy play a role
( Section 3.5.5 ), we irst analyse the consequences of similarity theory for neutral con-
ditions. Because the lux-gradient relationships are identical (and equal to 1) for all
quantities, we take only the gradients of horizontal wind speed and potential temper-
ature as examples. Because we are interested in the lux as a function of the gradient,
we rewrite the similarity relationship Eq. ( 3.20 ) in such a way that the luxes occur at
the left-hand side. To that end we use the fact that φ h =1 (neutral conditions) and that
τ = u 2 and Hc
=− ρ p** . Then we obtain:
u
u
z
θ
τρ
=
κ
uz
,
H
=−
ρ κ
cuz
(3.22)
*
p
*
z
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