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for θ and c we can derive, using the approach in Eqs. (1 1.27 ) a nd (11.28) ( Problem
11.16) , the following expressions for the profiles of θc and θ 2 :
R θ R c f t +
f b ,R θ =
0 wc 0
w 2
1
0
wc 1
θc
=
(R θ +
R c )f tb +
,R c =
wc 0 ,
(11.31)
(wθ 0 ) 2
w 2
R θ f t +
2 R θ f tb + f b .
θ 2
=
(11.32)
R θ , i.e., unless the top/
bottom flux ratio R is the same for the s cala r c and θ , the buoyant production term
in the dimensionless budgets of and wc will not be the same. The nature of the
CBL makes R θ negative, but R c can be positive, as for water vapor in a CBL over
an evaporating surface with drier air in the capping inversion. Thus the two flux
budgets, and the two eddy diffusivities, are apt to differ.
Equations (11.31) and (11.32) show that unless R c
=
11.3.2 Generalizations of K-closure for scalars
We have seen that in top-down diffusion (where the flux of c -stuff is nonzero at the
top of the mixed layer and zero at the bottom) K appears to be well behaved, but
in bottom-up diffusion K is significantly larger and has a midlayer singularity.
This latter phenomenon was characterized earl ier by Deardorff ( 1966 ), for exam-
ple, as “the existence of upward heat flux, wθ > 0, with vanishing or counter
(positive) potential temperature gradient, ∂/∂z
0.” He cited several studies in
which this had been observed, beginning with the Great Plains Turbulence Field
Program in 1957. It is also evident in high-resolution LES ( Moeng and Wyngaard ,
1989 ). Here the mean-gradient production term in the temperature variance budget
(10.44) is negligible or even a slight loss term; the gain is by turbulent transport of
variance from below.
Because the simplicity of K -closure is attractive in numerical modeling, this
evidence of its misbehavior stimulated early efforts to develop improved versions.
The simplest is the modified form suggested by Deardorff ( 1972b ),
K
γ θ ,
=−
∂z
(11.33)
with a si mpl e m odel of the conservation equation (10.48) used to estimate
γ θ as 2 0 w 2 . Holtslag and Moeng ( 1991 ) obtained another estimate through a
different model of that equation.
 
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