104
Air Pollution and Turbulence: Modeling and Applications
Sullivan et al. (1998) used LES simulations to address the problems of topentrainment and the structure of different CBLs, characterized by distinct Richardson
numbers, Ri. From this study, it is relevant to emphasize two conclusions for the
interval 13.6 ≤ Ri ≤ 43.8: top-entrainment is mostly due to thermals and the normalized top-entrainment rate (w ent /w * ) varies inversely with Ri. Dividing Equation
4.68 by w * , simplifying the expression, and defi ning
(
) (
)
i
w
vz
vs
A
w
w
θ = − θ
θ
′ ′
′ ′ and
ˆ
( /(
) ) ( / )
z
v s
A
z w
t
δ = δ
θ
∂θ ∂
′ ′
, one gets
1 (
) .
ent
w
z
w
A
A
w
Ri
θ
δ
∗
=
+
(4.69)
Sullivan et al. (1998) considering (
)
ˆ
i
h
z
t
t
∂θ ∂ = ∂ θ ∂ verifi ed the consistency of
this model, stressing the need for two contributions, A wθ and A δz for A Ri . A wθ is related
to the contribution of the top buoyancy fl ux and A δz is associated with the inversion
thickness.
Similarly, using results from LES experiments, one may diagnose the ratio
w ent /w * , and the terms A wθ /Ri (Figure 4.18a) and A δz /Ri (Figure 4.18b) separately, and
their sum (Figure 4.18c), verifying the validity of Equation 4.69. Figures 4.18a and
b show that both ratios are greater than w ent /w * , emphasizing the insuffi ciency of the
simpler approach, supported in the representation of the top-entrainment by the ratio
between minimum buoyancy fl ux and the surface fl ux. Therefore, top-entrainment
parameterizations should include a nonzero inversion thickness.
The requirement of the two previous contributions for the top-entrainment rate
allows establishing a parallelism with the need of the two contributions, K-diffusion
and MF, for the total vertical fl ux in the region of the inversion. Actually, the MF
contribution is preponderant in the inversion region of fi nite thickness, where
(φ u − φ
–
) < 0, taking into account the effect of the penetration of thermals into the freeatmosphere. If the inversion thickness was zero, then A Ri = A wθ = cons and the approximation Equation 4.66 would be suffi cient to parameterize the top-entrainment. In
this validation experiment, like in Sullivan et al. (1998), the term A δz contributes
more than A wθ to the value of A Ri , suggesting that top-entrainment is not simply controlled by surface fl uxes, but requires the representation of the more vigorous mixing
at the inversion done by the overshooting of thermals.
4.4.4.2 Comparison with Other Approaches
The previous results illustrate the potential of the EDMF-EMP scheme. However, it is
important to compare it with other schemes, namely, K-diffusion (Holtslag 1998) and
K-diffusion with counter-gradient term (Holtslag and Moeng 1991). The fi rst scheme
corresponds to consider the turbulent diffusivities described by the Expressions 4.61
and 4.62. The second parameterization is modifi ed with the introduction of a counter-gradient term:
∂φ
φ = −
+ γ
′ ′
∂
,
c
w
K
K
z
(4.70)
© 2010 by Taylor and Francis Group, LLC
Air Pollution and Turbulence: Modeling and Applications
Sullivan et al. (1998) used LES simulations to address the problems of topentrainment and the structure of different CBLs, characterized by distinct Richardson
numbers, Ri. From this study, it is relevant to emphasize two conclusions for the
interval 13.6 ≤ Ri ≤ 43.8: top-entrainment is mostly due to thermals and the normalized top-entrainment rate (w ent /w * ) varies inversely with Ri. Dividing Equation
4.68 by w * , simplifying the expression, and defi ning
(
) (
)
i
w
vz
vs
A
w
w
θ = − θ
θ
′ ′
′ ′ and
ˆ
( /(
) ) ( / )
z
v s
A
z w
t
δ = δ
θ
∂θ ∂
′ ′
, one gets
1 (
) .
ent
w
z
w
A
A
w
Ri
θ
δ
∗
=
+
(4.69)
Sullivan et al. (1998) considering (
)
ˆ
i
h
z
t
t
∂θ ∂ = ∂ θ ∂ verifi ed the consistency of
this model, stressing the need for two contributions, A wθ and A δz for A Ri . A wθ is related
to the contribution of the top buoyancy fl ux and A δz is associated with the inversion
thickness.
Similarly, using results from LES experiments, one may diagnose the ratio
w ent /w * , and the terms A wθ /Ri (Figure 4.18a) and A δz /Ri (Figure 4.18b) separately, and
their sum (Figure 4.18c), verifying the validity of Equation 4.69. Figures 4.18a and
b show that both ratios are greater than w ent /w * , emphasizing the insuffi ciency of the
simpler approach, supported in the representation of the top-entrainment by the ratio
between minimum buoyancy fl ux and the surface fl ux. Therefore, top-entrainment
parameterizations should include a nonzero inversion thickness.
The requirement of the two previous contributions for the top-entrainment rate
allows establishing a parallelism with the need of the two contributions, K-diffusion
and MF, for the total vertical fl ux in the region of the inversion. Actually, the MF
contribution is preponderant in the inversion region of fi nite thickness, where
(φ u − φ
–
) < 0, taking into account the effect of the penetration of thermals into the freeatmosphere. If the inversion thickness was zero, then A Ri = A wθ = cons and the approximation Equation 4.66 would be suffi cient to parameterize the top-entrainment. In
this validation experiment, like in Sullivan et al. (1998), the term A δz contributes
more than A wθ to the value of A Ri , suggesting that top-entrainment is not simply controlled by surface fl uxes, but requires the representation of the more vigorous mixing
at the inversion done by the overshooting of thermals.
4.4.4.2 Comparison with Other Approaches
The previous results illustrate the potential of the EDMF-EMP scheme. However, it is
important to compare it with other schemes, namely, K-diffusion (Holtslag 1998) and
K-diffusion with counter-gradient term (Holtslag and Moeng 1991). The fi rst scheme
corresponds to consider the turbulent diffusivities described by the Expressions 4.61
and 4.62. The second parameterization is modifi ed with the introduction of a counter-gradient term:
∂φ
φ = −
+ γ
′ ′
∂
,
c
w
K
K
z
(4.70)
© 2010 by Taylor and Francis Group, LLC