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densities that are more “peaked” near the origin and have broader “tails” than the
Gaussian. Squaring such a variable makes it positive definite while maintaining
its intermittent character, and presumably the locally averaged dissipation rate
r
behaves this way. A useful model of such a variable is the “log normal,” a random
variable whose logarithm has a Gaussian or “normal” probability density. Thus
Obukhov
(
1962
)wrote
0
e
v
,
r
=
ln
r
=
ln
0
+
v.
(7.38)
Here
0
is the
geometrical mean
dissipation rate and
v
is a Gaussian random
variable with zero mean and variance
σ
2
. For this simple intermittency model
the mean value of the locally averaged dissipation rate raised to a power
p
,
say, is
e
p
2
σ
2
r
0
0
e
pv
=
=
2
.
(7.39)
Equation (7.39)
then implies
r
=
0
e
σ
2
,
e
9
σ
2
3
/
2
=
3
/
2
=
0
e
2
σ
2
.
,
r
(7.40)
8
r
0
The corresponding moments of
are
0
e
σ
2
,
e
3
σ
2
3
/
2
0
e
σ
2
.
(7.41)
3
/
2
(
r
)
3
/
2
2
(
r
)
2
=
r
=
=
=
,
=
=
4
0
α
These demonstrate that
r
1. Thus, with Obukhov's log-normal
model for the locally averaged dissipation rate the predictions
(7.37)
of the revised
hypothesis become
=
if
α
=
e
9
σ
2
e
2
σ
2
e
σ
2
e
3
σ
2
8
e
3
σ
2
e
σ
2
,
S
∂u/∂x
∼
∼
,
∂u/∂x
∼
∼
(7.42)
8
4
so that
(F
∂u/∂x
)
3
/
8
,
S
∂u/∂x
∼
(7.43)
which agrees well with the observations (
Figure 7.5
)
.
Kolmogorov
(
1962
) suggested that
σ
2
,
the variance of ln
r
,
Eq. (7.38)
,
behaves as
μ
1
ln
σ
2
=
A(
x
,t)
+
r
,
(7.44)
where
A(
x
,t)
depends on the large-scale structure of the flow and
μ
1
is a universal
constant. This predicts that the intensity of the dissipation fluctuations increases
