Geoscience Reference
In-Depth Information
from
(15.34)
we see that under isotropy the one-dimensional spectra are all equal:
F
1
=
F
2
=
F
3
.
15.3.4 The inertial subrange
Conserved scalar fluctuations in the inertial range, i.e., for scales much less than
and much greater than
η
, are traditionally argued
(Chapter 7)
to depend only on
r
,
,and
χ
c
. If so, then in this range
E
c
depends only on
κ
,
,and
χ
c
. Dimensionally,
this implies the inertial-range form (
Obukhov
,
1949
;
Corrsin
,
1951
)
βχ
c
−
1
/
3
κ
−
5
/
3
,
E
c
(κ)
=
(
7
.
9
)
with
β
a universal constant. Using this inertial-range form for
E
c
in
Eq. (15.42)
gives
3
10
βχ
c
−
1
/
3
κ
−
5
/
3
β
1
χ
c
−
1
/
3
κ
−
5
/
3
F
1
(κ
1
)
=
=
,
(15.43)
1
1
with
β
1
called the one-dimensional spectral constant.
15.3.5 Conventions in the literature
The one-dimensional spectrum is naturally defined as in
Eq. (15.34)
- as a function
of a variable that ranges from
. But since it is an even function it can be
convenient to use the “one-sided” form that integrates over 0 to
−∞
to
∞
∞
to the variance.
This doubles the value of
β
1
.
There are two conventions for the three-dimensional scalar spectrum
E
c
.
It inte-
grates either to the variance, as in
Eq. (15.38)
,or(asin
Tennekes and Lumley
,
1972
)
the half-variance.
15.4 Vector functions of space and time
In order to deal with turbulent velocity fields we shall now extend everything we
have done to vectors. The requirements for homogeneity and stationarity given
at the beginning of
Section 15.2
are the same. We will initially assume that the
turbulent velocity field is homogeneous in all three spatial directions, consider only
correlations at the same time, and not indicate explicitly the dependence on time
or other parameters. Although the results are more complicated notationally, they
do have direct and important implications for turbulence measurements.
