Geoscience Reference
In-Depth Information
∂β(
1
)
∂t
∂β(
1
)
∂x
(
1
)
v
(
1
)
+
i
i
⎡
β(
1
,
2
)d
x
(
2
)
d
v
(
2
)
⎤
∂
∂x
(
1
)
v
(
2
)
2
∂
∂v
(
1
)
1
4
π
1
∂
∂x
(
2
)
⎣
−
⎦
+
j
x
(
1
)
x
(
2
)
|
−
|
i
i
j
β(
1
,
2
)d
v
(
2
)
v
(
2
)
∂
∂v
(
1
)
∂
∂x
(
2
)
∂
∂x
(
2
)
+
lim
x
(
2
)
→
x
(
1
)
ν
=
0
.
(13.40)
i
i
j
j
The conservation equation for
β(
1
)
involves also the two-point (joint) probability
density
β(
1
,
2
)
. Lundgren also derived the equation for
β(
1
,
2
)
, which involves
β(
1
,
2
,
3
).
Thus a closure approximation is needed in order to use the pdf conser-
vation equation. This approach forms the basis of a type of turbulence modeling
called
pdf modeling
.
Questions on key concepts
13.1 The terms probability
density
and probability
distribution
are frequently
misused. Explain how the two differ.
13.2 Interp
ret
Eq
. (13.9)
physically.
13.3 Write
f(u)
,where
u
is a random variable, as an integral of the probability
density of
u
. Explain the integral physically.
13.4 What is the Central Limit Theorem? Why is it important in applications?
13.5 Given a joint pdf of two variables, show formally how one obtains a pdf of
one of the variables.
13.6 Explainwhy the broadening of the tails of the pdfs in
Figure 13.3
is consistent
with the notion of dissipation intermittency discussed in
Chapter 7
.
13.7 Explain physically, using the concepts of scalar fine structure discussed in
Chapter 7
, why the kurtosis of velocity derivatives in turbulent flows tends
to be large.
13.8 Using the properties of the pdf, explain why a variable with a pdf having a
sharp peak at the origin has a larger kurtosis than one with a broad maximum
there.
13.9 Interpret the pdf of vertical velocity in a CBL,
Figure 13.4
,
as being consistent
with a zero-mean field with narrow, strong updrafts separated by broader
regions of weak downdrafts.
13.10 Explain why the behavior of effluent plumes in a CBL with
R
t
10
8
can
evidently be effectively modeled in a convection tank having 4-5 orders of
magnitude smaller
R
t
.
∼
