Geoscience Reference
In-Depth Information
Thevarianceisgivenby
c
∗
(
κ
,α)e
i(
κ
−
κ
)
·
x
c
2
cc
∗=
=
c(
κ
,α)
ˆ
ˆ
κ
κ
φ(
κ
)(κ)
3
.
c
∗
(
κ
,α)
=
c(
κ
,α)
ˆ
ˆ
=
(2.55)
κ
κ
In the limit as both
L
and
N
approach infinity
Eq. (2.55)
becomes an integral:
∞
c
2
=
φ(
κ
)dκ
1
dκ
2
dκ
3
.
(2.56)
−∞
(κ
1
+
The integral in
Eq. (2.56)
can be done first over a sphere of radius
κ
=
κ
2
+
κ
3
)
1
/
2
andthenover
κ
.The
three-dimensional spectrum E
c
(κ)
†
is defined as
the integral of
φ
over the sphere,
E
c
(κ)
=
κ
2
φ(κ
1
,κ
2
,κ
3
)dσ,
(2.57)
κ
i
κ
i
=
so the variance is
∞
c
2
=
E
c
(κ) dκ.
(2.58)
0
2.5.3 Application to a homogeneous velocity field
For a random, zero-mean, homogeneous velocity field
u
i
(
x
;
α)
defined in a cube
of side
L
the Fourier coefficients are vectors:
α)e
i(
κ
·
x
)
.
u
i
(
x
;
α)
=
u
i
(
κ
;
ˆ
(2.59)
κ
The covariance is
u
i
u
j
u
j
(
κ
)
u
i
u
j
=
=
u
i
(
κ
)
ˆ
ˆ
κ
(2.60)
φ
ij
(
κ
)(κ)
3
.
=
κ
In the limit
Eq. (2.60)
becomes an integral,
∞
u
i
u
j
=
φ
ij
(κ
1
,κ
2
,κ
3
)dκ
1
dκ
2
dκ
3
.
(2.61)
−∞
†
This is so-named because it is a function of the magnitude of a three-dimensional wavenumber.
