Geoscience Reference
In-Depth Information
13.11 Use the expression for the convergence of a time average to the ensemble
average,
Chapter 2
, to show why dispersion experiments are much more
readily done in the convection tank than in the CBL.
13.12 Defend the argument that laboratory convection-tank studies of dispersion
should be phased out in favor of numerical computations through turbulence
simulation.
13.13 Refute the argument that laboratory convection-tank studies of dispersion
should be phased out in favor of numerical computations through turbulence
simulation.
13.14 Discuss the key step, concept, or definition on which Lundgren's derivation
of the evolution equation for the pdf of velocity in a turbulent flow is based.
Problems
13.1 Determine the probability density of a sine wave experimentally by sampling
it a large number of times and putting each sample into the appropriate
amplitude bin. Why must you make sure that your sampling period is not an
integral or fractional multiple of the period? Discuss the scaling required to
convert the bin counts into the probability density.
13.2 Determine the probability density of the sum of samples taken in
Problem
13.1
. Let the number of samples in the sum become large. What form does
the probability density take?
13.3 Showhow the probability density of
u(t)
, say, is related to the joint density for
u(t)
and
v(t)
if the processes are stationary. What simplification in the joint
density ensues if the processes are statistically independent? Demonstrate
the latter with the joint Gaussian probability density.
13.4 Estimate the uncertainty in a probability density determined by binning and
counting, as in
Problem 13.1
.If
β
m
(u)
is the measured probability density
(determined through binning and counting a total of
N
samples, say) and
β(u)
is the true density, how does
2
vary with
N
? (You might want
to adapt the averaging-time formula
(Chapter 2)
for this purpose.) Discuss
the benefits of sampling slowly enough that neighboring samples are statis-
tically independent. For what values of
u
is the probability density least well
determined? What are the implications for higher moments of the variable?
13.5 Discuss the proof of the Central Limit Theorem.
13.6 Using
Eq. (13.9)
, showhow the value of
u
at the peak of the integrand defining
the
n
-th moment of a Gaussian variable
u
depends on
n
for even values of
n
.
13.7 Prove
Eq. (13.39)
.
13.8 Discuss, using physical examples, when youwould expect turbulence to have
nonzero third moments.
β
m
(u)
[
−
β(u)
]
