98
Air Pollution and Turbulence: Modeling and Applications
4.4.3 THE K-DIFFUSION CONTRIBUTION
To complete the EDMF formulation, one needs to specify the eddy diffusivities for
heat and moisture. Troen and Mahrt (1986) and Holtslag et al. (1995) proposed the
prescription of these profi les is a simple but robust way:
2
1
* 0
1
,
h
h
i
z
K
ku
z
z
− ⎛
⎞
= φ
−
⎜
⎟
⎝
⎠
(4.61)
where φ h0 is a stability function given by
1
3
0
1 39
.
h
MO
z
L
−
⎛
⎞
φ = −
⎜
⎟
⎝
⎠
(4.62)
These profi les have three important properties: (1) they obey to the similarity theory
in surface BL, (2) are nill in the inversion, and (3) its nondimensional maximum
value is (K max /w * z i ) ≅ 0.1.
4.4.4 EDMF-EMP RESULTS
The parameterizations described in the previous subsection may be called EDMFEMP (Eddy Diffusivity/Mass-Flux EMPirirical formulation scheme) due to the
empirical base of its formulation. The following results correspond to an idealized
case of a dry convective BL. This case is based on a LES model intercomparison
study (Nieuwstadt et al. 1992), and the LES results presented here were obtained
with the KNMI LES model (Cuijpers and Duynkerke 1993).
The surface force in this case corresponds to the prescription of a constant
sensible heat fl ux,
2
1
6.10 K m s
s
w
−
−
θ =
′ ′
. The surface variables are θ s = 300 K and
p s = 1000 hPa. The initial profi le of potential temperature is displayed in Figure 4.11.
In the initial state, the mean wind is negligible (u, v) = (0.01, 0)ms −1 , allowing for a
small but nonzero turbulent fl ux. Two vertical resolutions were used: 20 m (h) and
the one of the ECMWF model vertical grid (lr), having approximately 300 m in the
inner BL. The temporal evolution of potential temperature (hourly average) is presented in Figure 4.11. The BL growth and structure are clearly well captured.
The comparison of those profi les against LES, for the 5th and 10th hour, shows
the ability of the scheme to represent the vertical structure of the convective
BL. The high-resolution results are, in fact, very good. The EDMF scheme is
capable of reproducing the properties of the three BL regions: the unstable surface BL, the quasineutral mixed layer, and the slightly stable upper mixed layer.
These properties appear worst described by the coarser resolution simulation.
Nevertheless, the temporal evolution and the vertical structure remain essentially
correct (Figure 4.12).
The linear vertical profi les of the vertical fl ux of potential temperature, presented
in Figure 4.13, are also correctly captured by EDMF-EMP. The fl ux in the inversion
region, responsible for the BL growth, shows similar values to those of LES results,
© 2010 by Taylor and Francis Group, LLC
Air Pollution and Turbulence: Modeling and Applications
4.4.3 THE K-DIFFUSION CONTRIBUTION
To complete the EDMF formulation, one needs to specify the eddy diffusivities for
heat and moisture. Troen and Mahrt (1986) and Holtslag et al. (1995) proposed the
prescription of these profi les is a simple but robust way:
2
1
* 0
1
,
h
h
i
z
K
ku
z
z
− ⎛
⎞
= φ
−
⎜
⎟
⎝
⎠
(4.61)
where φ h0 is a stability function given by
1
3
0
1 39
.
h
MO
z
L
−
⎛
⎞
φ = −
⎜
⎟
⎝
⎠
(4.62)
These profi les have three important properties: (1) they obey to the similarity theory
in surface BL, (2) are nill in the inversion, and (3) its nondimensional maximum
value is (K max /w * z i ) ≅ 0.1.
4.4.4 EDMF-EMP RESULTS
The parameterizations described in the previous subsection may be called EDMFEMP (Eddy Diffusivity/Mass-Flux EMPirirical formulation scheme) due to the
empirical base of its formulation. The following results correspond to an idealized
case of a dry convective BL. This case is based on a LES model intercomparison
study (Nieuwstadt et al. 1992), and the LES results presented here were obtained
with the KNMI LES model (Cuijpers and Duynkerke 1993).
The surface force in this case corresponds to the prescription of a constant
sensible heat fl ux,
2
1
6.10 K m s
s
w
−
−
θ =
′ ′
. The surface variables are θ s = 300 K and
p s = 1000 hPa. The initial profi le of potential temperature is displayed in Figure 4.11.
In the initial state, the mean wind is negligible (u, v) = (0.01, 0)ms −1 , allowing for a
small but nonzero turbulent fl ux. Two vertical resolutions were used: 20 m (h) and
the one of the ECMWF model vertical grid (lr), having approximately 300 m in the
inner BL. The temporal evolution of potential temperature (hourly average) is presented in Figure 4.11. The BL growth and structure are clearly well captured.
The comparison of those profi les against LES, for the 5th and 10th hour, shows
the ability of the scheme to represent the vertical structure of the convective
BL. The high-resolution results are, in fact, very good. The EDMF scheme is
capable of reproducing the properties of the three BL regions: the unstable surface BL, the quasineutral mixed layer, and the slightly stable upper mixed layer.
These properties appear worst described by the coarser resolution simulation.
Nevertheless, the temporal evolution and the vertical structure remain essentially
correct (Figure 4.12).
The linear vertical profi les of the vertical fl ux of potential temperature, presented
in Figure 4.13, are also correctly captured by EDMF-EMP. The fl ux in the inversion
region, responsible for the BL growth, shows similar values to those of LES results,
© 2010 by Taylor and Francis Group, LLC