Geoscience Reference
In-Depth Information
7.16 Let
u
be a ra
ndom vari
able with
standard
deviation
σ
and let
u
=
u/σ
.Use
the identities
(u
2
−
u)
2
≥
0and
(u
2
+
u)
2
≥
0 to show that
|
S
|≤
(F
+
1
)/
2
,
u
.
7.17 Show how the inertial-range spectral forms,
Eqs. (7.5)
and
(7.9)
, follow from
their underlying assumptions on dimensional grounds.
7.18 Derive the fluctuating vorticity equation. Use it to derive a vorticity-
variance budget. Following the development of the TKE budget in
Chapter 5
, show that the leading terms in the budget are the rate of produc-
tion through vortex stretching and the rate of destruction through viscous
dissipation.
7.19 Show from
Eq. (7.65)
that the nature of
I
c
is the same in two- and
three-dimensional turbulence, but that
I
differs. What is the nature of the
differences in
I
?
7.20 The vector wavenumbers of the Fourier components contributing to a third
mome
nt add to zer
o (
Chapter 6
,
Section 6.5
). Demonstrate that this constraint
allows
(∂θ/∂x
1
)
3
to have contributions from the energy-containing range.
Then use the scaling implied by the observation that the skewness of
∂θ/∂x
1
is
with
F
and
S
the flatness factor and skewness of
u
and
˜
∼
1 to show that its principal contributions have all three wavenumbers in
the dissipative range, however.
7.21 We showed in
Eq. (5.35)
that
s/s
d
∼
R
1
/
t
,where
s
and
s
d
are the scalar inten-
sity scales in the variance-containing and dissipative ranges, respectively.We
assumed there that
γ
∼
ν
. Show that in the general case this generalizes to
Co
1
/
t
.
7.22 Explain physically the predicted faster-than-linear growth of a puff in the
inertial range of scales,
Eq. (7.21)
.
7.23 Explain why the
Kolmogorov
(
1941
) hypotheses are applicable to puff dis-
persion but not to the “Taylor problem,” diffusion from a continuous point
source (
Chapter 4
).
s/s
d
∼
References
Batchelor, G. K., 1950: The application of the similarity theory of turbulence to
atmospheric diffusion.
Quart. J. Roy. Meteor. Soc.
,
76
, 133-146.
Batchelor, G. K., 1959: Small-scale variation of convected quantities like temperature in
turbulent fluid.
Part I
. General discussion and the case of small conductivity.
J. Fluid
Mech.
,
5
, 113-133.
Batchelor, G. K., 1960:
The Theory of Homogeneous Turbulence.
Cambridge University
Press.
Batchelor, G. K., 1969: Computation of the energy spectrum in homogeneous
two-dimensional turbulence.
Phys. Fluids Suppl. II
,
12
, 233-239.
Batchelor, G. K., 1996:
The Life and Legacy of G. I. Taylor.
Cambridge University Press.
