Geoscience Reference
In-Depth Information
Figure 11.1 Sketches of profiles of mean quantities and their vertical fluxes in the
CBL, with its layers, heights, and parameters indicated. Left pair: Virtual potential
temperature and its flux. Right pair: Aconserved scalar and its flux. From
Deardorff
(
1979
).
From the definition of the M-O length
L
it follows that the convective velocity
scale
w
∗
,
Eq. (10.42)
,is
g
θ
0
Q
0
z
i
1
/
3
1
/
3
z
i
L
w
∗
=
0
.
7
u
∗
−
,
(11.1)
so when the boundary-layer stability parameter
z
i
/L
is very large
w
∗
is much
larger than the friction velocity
u
∗
. Thus
Deardorff
(
1970
) suggested that at
large
−
z
i
/L
a free-convection-like state emerges. Observations and numerical
simulations suggest that this state appears when
−
z
i
/L
exceeds 5-10.
This asymptotic state is called
mixed-layer similarity
.Here
m
−
−
1
=
5and
n
4, so there are two independent dimensionless quantities. The velocity scale
is
w
∗
, t
he l
ength scales are
z
and
z
i
, and the intensity scale of a conserved scalar
is
c
∗
=
=
cw
s
/w
∗
; within the mixed layer quantities nondimensionalized with these
scales are predicted to be functions only of
z/z
i
. This can be successful for velocity
statistics, as
Figure 11.2
shows. But it can fail spectacularly for scalar statistics
(Figure 11.2)
because it neglects the entrainment-induced scalar flux at the mixed-
layer top, which as pointed out by
Deardorff
(
1972a
) is an additional source of
scalar fluctuations in the upper mixed layer.
Mixed-layer similarity is now known to be incorrect in other respects. For exam-
ple, the lateral and streamwise integral scales
(Part III)
, which are equal in free
convection, can differ by as much as a factor of two with a mean horizontal wind.
As we shall discuss, it appears that other effects - e.g., vertical variation of the mean
horizontal pressure gradient (baroclinity) and
z
-dependent horizontal advection of
mean momentum - can also influence the structure of the CBL.