Geoscience Reference
In-Depth Information
8.6 Show that mixing ratio is a conserved variable.
8.7 Explain physically why the scalar-flux conservation equation applied to
potential temperature indicates a tendency for a negative (downward) tem-
perature flux in an isothermal planetary boundary layer, and zero flux in the
presence of the adiabatic temperature gradient.
8.8 Relate a mixing-ratio fluctuation to fluctuations in temperature and species
density. When and how can this expression be simplified?
8.9 Derive the conservation equation for the covariance of fluctuations of water
vapor mixing ratio and potential temperature. What sign is the covariance
over an evaporating surface that is warmer than the air? Over an evaporating
surface that is cooler than the air? Near the top of a convective boundary
layer that is capped by air of higher potential temperature and lower mixing
ratio?
8.10 Discuss the role of the buoyancy term in the TKE equation when the vertical
component of the temperature flux is negative. Under what physical situ-
ations does this occur? In which component energy equation is its impact
felt directly? Do you expect its impact to be felt indirectly by the other
components? By what mechanism?
8.11 Assuming that at atmospheric pressure the density of water depends only on
temperature, carry out a density expansion like
Eq. (8.8)
but for water. What
is the resulting form of the buoyancy term for water? Does warmer water
tend to rise or sink?
8.12 Can eddy-diffusion models apply to both conserved and nonconserved scalar
variables? Discuss.
8.13 Show that the Coriolis term in the TKE equation vanishes but that Coriolis
effects can transfer energy among velocity components.
8.14 Show that the Coriolis term does not affect the magnitude of the flux of a
conserved scalar but can affect its direction.
8.15 Show that
θ
e
,asdefinedby
Eq. (8.56)
, does satisfy entropy conservation in
cloud air as expressed by
Eq. (8.55)
.
8.16 Show that virtual potential temperature is a conserved variable.
8.17 Dynamically induced pressure fluctuations in turbulence have an rms value
of the order of
ρu
2
. Calculate the vertical distance over which the background
pressure in the atmospheric boundary layer changes by
ρu
2
if
u
=1ms
−
1
.
8.18 Average
Eq. (8.56)
for equivalent potential temperature. Can one express
θ
e
in
terms of average properties? Using a two-term expansion of the exponential,
try to estimate the Reynolds terms it produces when averaged. Could they
be important?
8.19 Confirm by scaling arguments that the neglected radiative covariance in
Eq.
(8.61)
is negligible.
