Parameterization of Convective Boundary Layer Turbulence and Clouds
105
where the counter-gradient term is given by Kγ c . This approximation also considers
the turbulent diffusivities given by Equations 4.61 and 4.62. The counter-gradient
term obeys to
*
2
(
) ,
c
s
w i
w
a
w
z
γ
γ =
θ
′ ′
σ
(4.71)
where a ≅ 2.
Simulations with those two schemes, considering the same initial conditions of
the previous section, were made to compare its results. Figure 4.19 shows the vertical
profi les of potential temperature resulting from the three different schemes and LES
(fi fth hour average). From the inspection of Figure 4.19, it can be seen that EDMF is
the scheme showing the best agreement in all the BL sublayers.
The K-diffusion parameterization presents a typical unstable profi le. The scheme
with counter-gradient does not have this problem. However, the BL growth is rather
0.05
0.04
0.03
0.02
0.01
0.00
0.00 0.01 0.02 0.03 0.04 0.05
A
wθ /Ri
LES
w e /w *
(a)
0.05
0.04
0.03
0.02
0.01
0.00
0.00 0.01 0.02 0.03 0.04 0.05
w e /w *
A
δz /Ri
LES
(b)
0.05
0.04
0.03
0.02
0.01
0.00
0.00 0.01 0.02 0.03 0.04 0.05
LES
w ent /w *
(A
wθ +
A
δz )/Ri
(c)
FIGURE 4.18 Comparison between the fi rst-order-jump model and the LES results for the
top-entrainment rate: (a) contribution of the minimum buoyancy fl ux A wθ /Ri, (b) contribution
of the inversion thickness A δz /Ri, and (c) total (A wθ + A δz )/Ri.
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