Geoscience Reference
In-Depth Information
Both equation sets
(3.2)
,
(3.7)
,
(3.13)
and
(3.36)
,
(3.42)
,and
(3.43)
have more
unknowns than equations, giving what is called a
closure problem
. We shall explore
it in the following chapters.
Questions on key concepts
3.1 What is a flux? What three types of fluxes can emerge in an ensemble-mean
equation? Illustrate with the momentum or conserved scalar equations.
3.2 What is a Reynolds stress? A Reynolds flux? Explain G. I. Taylor's term
“virtual mean flux.”
3.3
Explain how symmetry arguments can be used to evaluate Reynolds fluxes.
3.4
Explain the statement “turbulent mixing need not be mixing at all.”
3.5
Explain the interplay of turbulent and molecular diffusion.
3.6
Give an example of an ensemble-mean model
in boundary-layer
meteorology.
3.7 Explain why it is unlikely that an ensemble-mean profile - for example, the
mean-wind profile in a boundary layer - would ever appear in a realization
of a turbulent flow.
3.8 If an advected constituent in a turbulent flow is conserved, how can it
ever mix?
3.9 Explain why a spatially filtered turbulent flow model can be more reliable
than an ensemble-averaged model.
3.10 How could a spatially filtered model be made to have the randomness
property - i.e., to be able to generate an ensemble of predictions?
3.11 How could a spatially filtered model of the type described in
Question 3.10
be used to predict both an ensemble mean and variances about that mean?
3.12 What is the
closure problem
? How does it originate?
3.13 With an eddy-viscosity closure the ensemble-mean equation of motion
(3.7)
can be written in the same form as the Navier-Stokes equation. Then why
does it not have turbulent solutions?
3.14 Why must we exclude mean pressure from homogeneity considerations?
What other mean variables fall into this category?
Problems
3.1
Starting from
Eq. (3.7)
, form the equation governing the evolution of the
kinetic energy of a steady, ensemble-averaged flow.
(a) Is the kinetic energy of the ensemble-averaged flow the same as the
ensemble-averaged kinetic energy? Discuss.
