Geoscience Reference
In-Depth Information
+∞
e
iκ
1
ξ
1
F
1
(κ
1
)dκ
1
,
=
(15.34)
−∞
+∞
F
1
(κ
1
)
=
φ(
κ
)dκ
2
dκ
3
.
−∞
F
1
is called a one-dimensional spectrum. Its superscript indicates the direction
of
the spatial separation vector
ξ
.
†
Equation (15.34)
indicates that
F
1
(κ
1
)
and
θ
2
ρ(ξ
1
,
0
,
0
)
are a Fourier transform pair:
+∞
e
iκ
1
ξ
1
F
1
(κ
1
)dκ
1
=
θ
2
ρ(ξ
1
,
0
,
0
),
−∞
(15.35)
+∞
1
2
π
F
1
(κ
1
)
e
−
iκ
1
ξ
1
θ
2
ρ(ξ
1
,
0
,
0
)dξ
1
.
=
−∞
Evaluating the second of
Eqs. (15.35
)at
κ
1
=
0gives
+∞
θ
2
2
π
θ
2
π
F
1
(
0
)
=
ρ(ξ
1
,
0
,
0
)dξ
1
=
1
,
(15.36)
−∞
where
1
is the integral scale of
θ
in the
x
1
direction.
Equation (15.36)
says that
the one-dimensional spectrum does not vanish at zero wavenumber. This happens
because a Fourier mode of wavenumber magnitude
κ
, oriented nearly normal to
the
x
1
direction, appears to have a very small wavenumber in that direction.
The
three-dimensional spectrum E
c
(the subscript
c
differentiates it from
E
,the
three-dimensional energy spectrum) is the integral of
φ(
κ
)
over a spherical shell of
radius
κ
:
E
c
(κ)
=
κ
2
φ(
κ
)dσ.
(15.37)
κ
i
κ
i
=
As indicated,
E
c
(κ)
contains contributions from all Fourier modes of wavenumber
magnitude
κ
regardless of their direction. From
(15.30)
,
κ
2
φ(
κ
)dσ
dκ
+∞
∞
∞
θ
2
=
φ(
κ
)d
κ
=
=
E
c
(κ) dκ.
(15.38)
−∞
0
κ
i
κ
i
=
0
=
Like
E
,
E
c
vanishes at the origin because there is no energy at
κ
0; we are
working with zero-mean variables. Thus, unlike the one-dimensional spectrum, its
shape does reflect the relative importance of the contributions of eddies of spatial
scale 1
/κ
to the variance.
E
c
is the scalar spectrum traditionally used in turbulence theory. The Obukhov-
Corrsin arguments
(Chapter 7)
for the behavior of the scalar spectrum in the inertial
subrange, for example, are made for
E
c
.
†
Some authors indicate this direction by writing
F(κ
1
)
, but since a function depends on the
value
of its argument,
not the
name
, this is inappropriate.
