Geoscience Reference
In-Depth Information
2
u
j
c
∂C
=−
(
mean-gradient production
)
∂x
j
∂c
2
u
j
∂x
j
−
(
turbulent transport
)
∂c
∂x
j
∂c
∂x
j
−
2
γ
(
molecular destruction
).
(8.72)
In each of these second-moment budgets we have used the large Reynolds
number, locally isotropic form of the molecular destruction term (
Part III
).
Questions on key concepts
8.1
Explain physically why warm air rises.
8.2
Explain how
θ
is defined through entropy conservation, and how that allows
it to be constant on trajectories on which both pressure and temperature vary.
8.3
Explain a
scale height
, such as that for density, physically.
8.4
Explain the meaning of a
conserved
variable.
8.5
Explain why neither the density of air nor the density of an advected
constituent is conserved in the atmosphere, but their ratio is conserved.
8.6
Show that
T
v
is the temperature of dry air having the same pressure and
density as moist air at temperature
T
.
8.7
Explain how the concept of conserved temperature is extended to include
phase change.
8.8
Explain what is meant by “thermodynamic” Reynolds terms and why they
appear.
Problems
8.1
Explain why it appears on physical grounds that a necessary condition for
the uniform mixing of a quantity of constituent added to a turbulent flow
(e.g., cream into coffee) is that the constituent be conserved.
8.2
Calculate
p
0
(z)
and
ρ
0
(z)
.
8.3
Derive the expression in
Eq. (8.13)
for the scale height for density.
8.4
Explain why the variation of kinematic viscosity with temperature should
have no effect on the dissipation rate in turbulent flow with temperature
variations.
8.5
Compare the viscous dissipation term in
Eq. (8.23)
to the temperature term
in the equation. Use the scaling
u
3
/
,
D T/Dt
˜
∼
∼
θu/
. Is the dis-
sipation term likely to be important in the boundary layer? In a supercell
thunderstorm? In a hurricane?
