Geoscience Reference
In-Depth Information
Figure 15.5 The geometry of a common spectral integral.
Integrals of the form of
(15.107)
appear frequently in these analyses. We can
generalize it to
†
π/
2
cos
α
θdθ.
I(α)
=
(15.108)
0
cos
2
θ
we can rewrite this as
If we define
v
=
1
1
2
v
α
−
1
1
2
dv.
v)
−
I(α)
=
(
1
−
(15.109)
2
0
The Beta function is defined by
1
v
m
−
1
(
1
v)
n
−
1
dv.
B(m, n)
=
−
(15.110)
0
When
m
and
n
are any positive real numbers,
(m)(n)
(m
+
n)
.
B(m, n)
=
B(n, m)
=
(15.111)
In our expression
(15.109)
for
I(α)
we have
n
=
1
/
2and
m
=
(α
+
1
)/
2, so that
α
+
2
√
π
2
(
2
)(
α
+
1
)
2
I(α)
=
2
)
=
1
2
,
(15.112)
α
α
2
(
1
+
+
from which we find
I(
1
3
)
0
.
61
.
Equation (15.107)
for the circularly averaged
vertical velocity spectrum in the plane then becomes
=
0
.
61
α
2
/
3
κ
−
5
/
3
E
(
2
)
=
,
(15.113)
w
h
†
I am indebted to Ricardo Munoz for this development.
