Geoscience Reference
In-Depth Information
Now, along the same lines as in Section 3.5.4 , we integrate the gradients in Eq.
( 3.20 ) vertically to obtain information on vertical differences:
z
z
∫ u
∂
∂
κ
z
u
2
u
2
z
L
∫
dz
=
φ
dz
⇔
m
zu
z
*
z
u
1
u
1
z
z
u
−
z
L
z
L
u
u
u
2
() (
z
−
z
1 )
=
*
ln
Ψ
+
Ψ
u
2
u
1
u
2
u
m
m
κ
u
1
(3.29)
z
z
∂
∂
θ
κ
z
z
L
θ
2
θ
2
∫
∫
dz
=
φ
dz
⇔
h
z
θ
*
z
z
θ
1
θ
1
z
z
θ
−
z
L
z
L
θ
2
θ
() ()
z
−
θ
z
=
*
ln
Ψ
+
Ψ
θ
2
θ
1
θ
2
θ
1
h
h
κ
θ
1
Here Ψ m and Ψ h (psi-functions) are the integrated lux-gradient relationships,
deined as:
− ()
zL
/
∫ 1
0
φ ζ
ζ
′
z
L
Ψ y
≡
y
d
ζ
′
with subscript y=m,h
,e, or x.
′
For each φ -function the integral yields a different Ψ-function. Using the expressions
for the Businger-Dyer lux-gradient relationships (given in Eq. ( 3.21 )), the following
expressions for the Ψ-function can be derived (Paulson, 1970 ) 17 :
14
/
2
arctan() π
z
L
1
+
x
1
+
x
−
z
L
Ψ m
=
2
ln
+
ln
2
x
+
, wit
h
x
= −
116
2
2
z
L
2
for
≤
0
14
/
z
L
2
z
L
1
+
x
Ψ
=
2
ln
,
with
x
= −
11
6
h
2
=−
z
L
z
L
z
L
z
L
Ψ
=
Ψ
5
for
≥ 0
m
h
(3.30)
These functions are depicted in Figure 3.16b . Now, Eq. ( 3.29 ) can be rewritten to
express the luxes in terms of vertical differences. The general expression is identical
to Eq. ( 3.24 ), but the expressions for the resistances are different:
z
z
−
1
z
L
z
L
u
2
r
=
ln
Ψ
u
2
+
Ψ
u
1
am
m
m
κ *
u
u
1
(3.31)
z
z
1
−
z
L
z
L
θ
2
r
=
ln
Ψ
+
Ψ
θ
2
θ
1
ah
h
h
κ
u
θ
1
*
17 Wilson ( 2001 ) provides alternative expressions for the Ψ -functions, both for the Businger-Dyer lux-gradient
relationships used here, as well as for alternative forms of the φ -functions.
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