Geoscience Reference
In-Depth Information
1) we expect the
Fourier coefficients
a
n
to vary rapidly with
z
. Since the coefficients are random
variables with zero mean, their
z
-integral will tend to zero. We expect the transition
between the
κ
n
z
At horizontal scales very small compared to
z
(i.e., at
κ
n
z
1
/z
, so the spectra of
velocity in a plane in the surface layer should behave as sketched in
Figure 10.2
.
Even though the peaks of
u
and
v
spectra occur at scales of the order of the ABL
depth
h
(corresponding to the sweeping effect of the largest convective eddies), the
peak of the
w
-spectrum is at much smaller scales - at horizontal wavenumbers of
the order of 1
/z
.
The
cospectrum of the kinematic shear stress
uw
- the density of contributions
to
uw
as a function of wavenumber in the horizontal plane
(Chapter 15)
-isat
any wavenumber proportional to the mean product of the amplitudes of the Fourier
coefficients of
u
and
w
(Part III)
. Given the wavenumber dependence of these
amplitudes
(Figure 10.2)
,
we see that this stress cospectrum must peak where the
w
-spectrum does, at
κ
1and
κ
n
z
1 regimes to occur at
κ
n
∼
1
/z
. Thus the large-scale part of the horizonta
l tu
rbulent
motions in the surface layer, that with
κ<
1
/z
, does not contribute to
uw
. These
are called
inactive motions
.
∼
Questions on key concepts
10.1 Explain the concepts underlying the “constant-flux” layer. Why is “surface-
flux” layer a better term?
10.2 Discuss how the mean-wind and stress profiles in
Figure 10.1
are inferred
from the mean-motion equations.
10.3 Explain the key concepts of dimensional analysis.
10.4 Of the other possible governing parameters in M-O similarity, which do you
feel is the most important? Explain.
10.5 Explain some of the uses of M-O similarity.
10.6 Explain some of the deficiencies of M-O similarity.
10.7 Explain the two limiting states of M-O similarity, how their physics is
simplified, and the rationale for that simplification.
10.8 Explain physically why the horizontal velocity fluctuations in
Figure 10.7
suddenly intensified shortly after 1100 but the vertical fluctuations did not.
10.9 Interpret physically the pressure-transport term in the TKE budget.
10.10 There is a Lagrangian integral time scale
(Chapter 4)
for each of the three
velocity components. How would you expect each to behave in the surface
layer? Would any be M-O similar?
10.11 Explain the notions of quasi-steadiness and local homogeneity and how they
simplify analysis of the surface layer.
