Geoscience Reference
In-Depth Information
9.10 Discuss the determination of eddy diffusivity for momentum in the ABL in
the context of required averaging times.
9.11 Explain the time lag between sunrise and the appearance of positive surface
temperature flux
(
Figure 9.5
)
and between sunset and negative flux.
9.12 In view of conservation of energy, the rate of loss of TKE to buoyant destruc-
tion must reappear as a source term somewhere. Rewrite the buoyancy term
in the equation of vertical motion in terms of density fluctuations and use
the vertical integral of the continuity equation times height to show where it
emerges. Interpret this physically.
9.13 What is the physical meaning of the first term on the right side of
Eq. (9.4)
?
Show that it integrates to zero over the ABL.
9.14 Consider flow in a horizontally homogeneous, quasi-steady ABL capped by
a strong inversion at height
z
i
such that the parameter
fz
i
/u
∗
1. Scale
the two mean horizontal momentum equations and show that they reduce to
that for turbulent channel flow. Where does the adjustment to geostrophic
flow occur in this limit? What is the change in mean flow direction in this
adjustment?
9.15 Write an expression for a turbulence Rossby number, the ratio of typical iner-
tial and Coriolis forces on energy-containing eddies. Estimate its magnitude
in the ABL.
9.16 Derive
Eqs. (9.29)
and
(9.30)
from
Eq. (9.28)
.
9.17 Show that in the ABL the Coriolis terms in the turbulence second-
moment budgets
(8.70)
,
(8.71)
tend to be small compared to the leading
terms.
9.18 Assume the flow over the earth's surface were laminar, not turbulent. Use the
surface energy balance to explain why at mid latitudes during clear summer
weather, shallow water on the surface could boil during the day and freeze
at night.
9.19 Turbulent channel flow
(
Figure 3.1
)
with the upper and lower walls moving
in opposite directions at speed
U
w
is called
turbulent Couette flow
.Itis
simulated in the laboratory with counter-rotating concentric cylinders of
diameter
R
and
R
+
δR
, with
δR
R
; this allows the use of cartesian
coordinates.
(a) Using the streamwise periodicity of this flow, show that the mean
streamwise pressure gradient is zero.
(b) Solve the streamwise mean momentum equation for the total stress
profile.
(c) Using an eddy-viscosity (
K
) closure, show that the mean velocity profile
depends sensitively on the
K
profile.
(d) Sketch a
K
profile that you feel gives a realistic mean velocity profile.
(e) Discuss the implications for
K
closure.
