86
Air Pollution and Turbulence: Modeling and Applications
1971; Betts 1973; Yanai et al. 1973). Siebesma et al. (2004) showed, through LES
simulations, that in the BOMEX case, corresponding to a shallow cumulus BL above
the ocean, more than 80% of the total fl ux can be attributed to the MF term. Similar
results have been used to justify the approximation:
(
)(
)
(
) ,
u
u
u
e
u
e
w
a w w
M
′ϕ′ ≅
−
ϕ − ϕ ≡
ϕ − ϕ
(4.28)
where M is the MF coeffi cient associated with the updrafts: M = a u (w u − w e ). This
result is fundamental in this approach, stating that the vertical fl ux of a property is
proportional to the difference between the mean value of that property in the updraft
regions and its mean value in the surrounding areas. Neglecting the fi rst two terms
of Equation 4.27 corresponds to consider a u << 1 and that the turbulence in the surrounding environment contributes scarcely to the total fl ux. In the same context, it is
acceptable to consider w e ≅ 0, resulting in
.
u u
M a w
≅
(4.29)
The validity of Equation 4.28 was extensively studied for different types of BL. It
is important to mention the work of Wyngaard and Moeng (1992), where analytical
and LES simulation results were compared. These authors divide the atmosphere in
regions of updrafts and downdrafts (d) and assume that w and φ obey a Gaussian
PDF, allowing to write
(
),
ud w
u
d
w
b
φ = σ φ − φ
′ ′
(4.30)
where
=
π
=
( 2 4) 0.627
ud
b
σ w is the vertical velocity standard deviation
Admitting that vertical velocity also obeys to a Gaussian distribution, = σ
π
(
2 )
w
M
,
this equation can be written as a function of
(
),
u
u
d
w
M
φ = ν
φ − φ
′ ′
(4.31)
where ν u = (2π/4) ≈ 1.57, implying that in Equation 4.27, for a Gaussian PDF, approximately 60% (≈1/ν u ) of the total fl ux is described by the MF term. Note that in
Equation 4.28 ν u is implicitly equal to 1.
Equation 4.28 constitutes the fundamental expression of the MF approach and
is presented as an alternative way to parameterize the turbulent fl uxes and to close
the Reynolds system of Equations 4.3 through 4.6. For that, one needs to know the
profi les of M, φ u , and φ e . The computation of those profi les is not trivial. A moving
air parcel in the BL is not an isolated system; whenever it ascends or descends it
exchanges properties with the surrounding air modifying its own properties in the
mixing process.
The evolution of the variable φ of an updraft horizontal section, with area a u , that
is, φ u , has to refl ect the lateral mixing process. Following Siebesma (1996), starting
from the prognostic equation of φ:
© 2010 by Taylor and Francis Group, LLC
Air Pollution and Turbulence: Modeling and Applications
1971; Betts 1973; Yanai et al. 1973). Siebesma et al. (2004) showed, through LES
simulations, that in the BOMEX case, corresponding to a shallow cumulus BL above
the ocean, more than 80% of the total fl ux can be attributed to the MF term. Similar
results have been used to justify the approximation:
(
)(
)
(
) ,
u
u
u
e
u
e
w
a w w
M
′ϕ′ ≅
−
ϕ − ϕ ≡
ϕ − ϕ
(4.28)
where M is the MF coeffi cient associated with the updrafts: M = a u (w u − w e ). This
result is fundamental in this approach, stating that the vertical fl ux of a property is
proportional to the difference between the mean value of that property in the updraft
regions and its mean value in the surrounding areas. Neglecting the fi rst two terms
of Equation 4.27 corresponds to consider a u << 1 and that the turbulence in the surrounding environment contributes scarcely to the total fl ux. In the same context, it is
acceptable to consider w e ≅ 0, resulting in
.
u u
M a w
≅
(4.29)
The validity of Equation 4.28 was extensively studied for different types of BL. It
is important to mention the work of Wyngaard and Moeng (1992), where analytical
and LES simulation results were compared. These authors divide the atmosphere in
regions of updrafts and downdrafts (d) and assume that w and φ obey a Gaussian
PDF, allowing to write
(
),
ud w
u
d
w
b
φ = σ φ − φ
′ ′
(4.30)
where
=
π
=
( 2 4) 0.627
ud
b
σ w is the vertical velocity standard deviation
Admitting that vertical velocity also obeys to a Gaussian distribution, = σ
π
(
2 )
w
M
,
this equation can be written as a function of
(
),
u
u
d
w
M
φ = ν
φ − φ
′ ′
(4.31)
where ν u = (2π/4) ≈ 1.57, implying that in Equation 4.27, for a Gaussian PDF, approximately 60% (≈1/ν u ) of the total fl ux is described by the MF term. Note that in
Equation 4.28 ν u is implicitly equal to 1.
Equation 4.28 constitutes the fundamental expression of the MF approach and
is presented as an alternative way to parameterize the turbulent fl uxes and to close
the Reynolds system of Equations 4.3 through 4.6. For that, one needs to know the
profi les of M, φ u , and φ e . The computation of those profi les is not trivial. A moving
air parcel in the BL is not an isolated system; whenever it ascends or descends it
exchanges properties with the surrounding air modifying its own properties in the
mixing process.
The evolution of the variable φ of an updraft horizontal section, with area a u , that
is, φ u , has to refl ect the lateral mixing process. Following Siebesma (1996), starting
from the prognostic equation of φ:
© 2010 by Taylor and Francis Group, LLC