Geoscience Reference
In-Depth Information
then we have from
(15.46)
+∞
e
iκ
1
r
1
φ
11
(
κ
)d
κ
R
11
(r
1
,
0
,
0
)
=
−∞
+∞
e
iκ
1
r
1
+∞
−∞
φ
11
(
κ
)dκ
2
dκ
3
dκ
1
.
=
(15.48)
−∞
We define the one-dimensional spectrum
F
11
as
+∞
F
11
(κ
1
)
=
φ
11
(
κ
)dκ
2
dκ
3
.
(15.49)
−∞
From
Eq. (15.48)
it is the one-dimensional Fourier transform of
R
11
(r
1
,
0
,
0
)
:
+∞
e
iκ
1
r
1
F
11
(κ
1
)dκ
1
.
R
11
(r
1
,
0
,
0
)
=
(15.50)
−∞
As with the one-dimensional scalar spectrum, the superscript 1 in
F
11
indicates the
direction of the separation vector
r
. Similar one-dimensional spectra can be defined
for other combinations of velocity;
F
13
, for example, goes with
R
13
(r
1
,
0
,
0
)
.
If the two lower indices are the same, then the one-dimensional spectrum is a
real, even function. If they differ, then the one-dimensional spectrum will not be
real, so it can be split into co- and quadrature spectra. The co- and quadrature one-
dimensional spectra can be related to the full co- and quadrature spectra in a way
similar to
(15.49) (Problem 15.13)
.
The
three-dimensional spectrum
is also defined with the indices contracted,
φ
ii
(
κ
)
2
E(κ)
=
dσ,
(15.51)
κ
i
κ
i
=
κ
2
the factor of two being included so it integrates to the turbulence kinetic energy per
unit mass (TKE):
∞
u
i
u
i
2
=
E(κ)dκ.
(
2
.
63
)
0
This three-dimensional spectrum represents the contribution of Fourier modes of
wavenumber magnitude
κ
to the TKE, regardless of the orientation of the Fourier
modes. Spherical averages of other functions (for example,
Co
ij
,
Section 15.4.3
)
are also used.
15.4.2 Isotropy
We introduced the implications of isotropy for scalar spectra in
Section 15.3.3
.Here
we'll extend the discussion to turbulent velocity fields.
