Geoscience Reference
In-Depth Information
15.3.3 Isotropy
As we discussed in
Chapter 3
, isotropy is unlikely to appear in the energy-containing
range of “natural” (as opposed to computational) turbulence. But what
Batchelor
(
1960
) calls
axisymmetry
, invariance to rotations and reflections about an axis, is
possible - as in turbulent free convection, for example, which has axisymmetry
about the vertical axis. Axisymmetry is isotropy in the plane normal to the axis of
symmetry.
An isotropic scalar field has simple relationships among the functions
φ
,
F
1
and
E
c
discussed in the last section. Under isotropy
φ(
κ
)
=
φ(κ)
and
Eq. (15.37)
becomes
4
πκ
2
φ(κ).
E
c
(κ)
=
κ
2
φ(κ)dσ
=
(15.39)
κ
i
κ
i
=
The one-dimensional spectrum
F
1
(κ
1
)
then is, from
(15.34)
,
∞
∞
E
c
(κ)
F
1
(κ
1
)
=
φ(κ)dκ
2
dκ
3
=
4
πκ
2
dκ
2
dκ
3
.
(15.40)
−∞
−∞
As indicated by the wavenumber-space diagram in
Figure 15.2
,
we can integrate
over circular rings and write
(15.40)
as
∞
∞
E
c
(κ)
4
πκ
2
2
πκ dκ
E
c
(κ)
2
κ
F
1
(κ
1
)
=
=
dκ,
(15.41)
0
κ
1
from which we find by differentiation with respect to
κ
1
∂F
1
(κ
1
)
∂κ
1
E
c
(κ
1
)
2
κ
1
=−
.
(15.42)
This equation allows the three-dimensional spectrum
E
c
to be found frommeasure-
ments of the one-dimensional spectrum
F
1
if the field is isotropic. Furthermore,
Figure 15.2 A wavenumber-space diagram of the integral in
Eq. (15.40)
.
