Geoscience Reference
In-Depth Information
2.5.5 The restriction to homogeneous fields
You have now met some of the classical concepts in turbulence:
• the power spectral density, or spectrum;
• Kolmogorov's inertial subrange;
• the velocity scale
u(r)
of an eddy of size
r
.
We quantified these concepts through a lightly mathematical model of homo-
geneous turbulence, but engineering and geophysical flows are seldom homoge-
neous. The atmospheric boundary layer, for example, is inhomogeneous in the
vertical. So perhaps you wonder: do these classical concepts apply to real flows?
The answer has two parts. First, these concepts apply only in homogeneous
directions
. In the atmospheric boundary layer, for example, one applies spectral
analysis only in the homogeneous horizontal plane (with data from scanning radar
or lidar, or from numerical simulation) or along a homogeneous horizontal line
(with aircraft data). Second, at wavenumbers such that
κL
1, with
L
the scale
of the inhomogeneity (i.e., at turbulence scales small compared to the scale of the
inhomogeneity), it appears that spectra can be interpreted in the fully homogeneous
context
(Part III)
.
2.6 Turbulent vorticity
As with velocit
y, we
can define a characteristic amplitude
ω
of vorticity fluctuations
through
ω
2
=
ω
i
ω
i
.Weshowedin
Chapter 1
that at large
R
t
this vorticity is con-
tained in the smallest eddies, whose velocity and length scales are the Kolmogorov
scales
υ
and
η
. Thus we can express the characteristic vorticity
ω
as
υ
η
.
∼
ω
(2.67)
The vorticity characteristic of eddies of size
r
,
ω(r)
,isoforder
u(r)/r
,which
from
Eq. (2.66)
and
u
3
/
can be expressed as
∼
1
/
3
r
2
/
3
.
u(r)
r
ω(r)
∼
∼
(2.68)
Thus, as eddy size
r
decreases its characteristic vorticity
ω(r)
increases; at the
smallest scales it is
u(η)
η
υ
η
∼
ω(η)
∼
∼
ω,
(2.69)
as stated in
Eq. (2.67)
.
The contrast between the characteristic velocity and vorticity fluctuations
u
and
ω
at large
R
t
is striking:
u
∼
u()
, since the velocity fluctuations are dominated by
