Geoscience Reference
In-Depth Information
15.4.1 The covariance, spectral density pair
With the Fourier-Stieltjes formalism we'll write our stationary, homogeneous
turbulent velocity field as
+∞
e
i
κ
·
x
dZ
i
(
κ
),
u
i
(
x
)
=
(15.44)
−∞
where the
dZ
i
(
κ
)
are randomvector functions. Aswith scalars, they have orthogonal
increments:
dZ
i
(
κ
)dZ
j
(
κ
)
κ
=
κ
;
dZ
i
(
κ
)dZ
j
(
κ
)
κ
=
κ
.
(15.45)
=
0
,
=
φ
ij
(
κ
)d
κ
,
The transform relationships are
+∞
e
i
κ
·
r
φ
ij
(
κ
)d
κ
,
u
i
(
x
)u
j
(
x
+
r
)
=
R
ij
(
r
)
=
−∞
(
2
π)
3
+∞
(15.46)
1
e
−
i
κ
·
r
R
ij
(
r
)d
r
.
φ
ij
(
κ
)
=
−∞
R
ij
is a
covariance
when
i
=
j
and a
cross covariance
when
i
=
j
. It is called the
correlation tensor
.
φ
ij
is the
spectral density tensor
.
By definition a homogeneous field is statistically unchanged under a translation
of its coordinate axes. From
Eq. (15.46)
this implies that
R
αα
is an even function;
R
αα
(
r
)
neither even nor odd, and
φ
12
is neither real nor imaginary.
Cospectra and quadrature spectra are defined as for scalars. We first write
R
ij
=
R
αα
(
−
O
ij
,
the sum of even and odd parts. The Fourier transform of
R
ij
is, from
(15.46)
,
=
E
ij
+
(
2
π)
3
+∞
e
−
i
κ
·
r
E
ij
(
r
)
+
O
ij
(
r
)
d
r
1
φ
ij
(
κ
)
=
−∞
(
2
π)
3
+∞
(
2
π)
3
+∞
1
i
=
cos
(
κ
·
−
sin
(
κ
·
r
)E
ij
(
r
)d
r
r
)O
ij
(
r
)d
r
−∞
−∞
=
Co
ij
(
κ
)
−
iQ
ij
(
κ
),
(15.47)
the sum of real and imaginary parts.
As for scalars, we can obtain the so-called
one-dimensional spectra
that can be
determined from measurements. If we measure
u
1
along the
x
1
axis, for example,