Geoscience Reference
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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.
 
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