94
Air Pollution and Turbulence: Modeling and Applications
temperature until the inversion region, where they converge to the values of the mean
environment. This decrease is related to the lateral mixing with the surrounding air
that penetrates in the ascendants. Globally, the vertical evolution of this excess of
the potential temperature of thermals relative to the average seems to be able to be
described by an expression not too complex. However, in the inversion region it is
clear that different processes, linked to top-entrainment and the penetration of thermals in the FA, come into play.
4.4.2 MASS-FLUX CONTRIBUTION
The contribution of MF to the total turbulent fl ux depends on the product of the
MF coeffi cient, M, proportional to the thermals vertical velocity, by the difference between the ensemble thermals’ properties and the horizontal domain average
(φ u − φ
–
). Hence, it is necessary to build an updraft model representing the properties
of the updrafts ensemble responsible for the nonlocal transport, φ u .
4.4.2.1 Updraft Model
The proposed updraft model follows the methodology of Betts (1973) to describe
cumulus convection. Consider an air parcel moving upward in the BL with lateral
mixing. From relations (Equations 4.43 and 4.48) for an updraft u, one can obtain the
equation to determine its vertical structure, given by
(
),
u
u
z
∂φ = −ε φ − φ
∂
(4.51)
where
ε is the lateral mixing rate
φ u and φ
– are, respectively, a generic property, of thermals and its horizontal
domain average
This parcel model is designed to establish updraft’s properties: the liquid water
potential temperature, θ l u ; the total specifi c humidity, q t u ; the vertical velocity, w u ;
and the BL height, z i .
Equation 4.51 needs a bottom boundary condition, that is, the initial value, to
allow its vertical integration. To the initialization of the parcel one has to estimate its
virtual potential temperature excess relatively to the surroundings, Δθ v u :
Δθ = θ
− θ
≈
θ
′ ′
( )
( )
((
) ...)
vu
vu
v
v s
z
z
f w
(4.52)
It is assumed that the excess is directly related to the surface BL variability, expressed
in function of the excess of Δθ lu and Δq tu . Considering a constant sensible heat fl ux
and knowing that
(
)
l
l s
w
u
∗
∗
θ = − ′θ ′
,
(
)
t
ts
q
wq u
∗
∗
= − ′ ′
, one can write
(
)
,
v
v s
w
w
u
b
∗ ∗
σ θ
θ = − θ ≈ σ σ
′ ′
(4.53)
where b σ is a constant, θ
σ σ
,
v
w are, respectively, the standard deviation of θ v and
of the vertical velocity, w. Siebesma and Teixeira (2000) admitted that
v
v
c θ
Δθ ≈ σ ,
© 2010 by Taylor and Francis Group, LLC
Air Pollution and Turbulence: Modeling and Applications
temperature until the inversion region, where they converge to the values of the mean
environment. This decrease is related to the lateral mixing with the surrounding air
that penetrates in the ascendants. Globally, the vertical evolution of this excess of
the potential temperature of thermals relative to the average seems to be able to be
described by an expression not too complex. However, in the inversion region it is
clear that different processes, linked to top-entrainment and the penetration of thermals in the FA, come into play.
4.4.2 MASS-FLUX CONTRIBUTION
The contribution of MF to the total turbulent fl ux depends on the product of the
MF coeffi cient, M, proportional to the thermals vertical velocity, by the difference between the ensemble thermals’ properties and the horizontal domain average
(φ u − φ
–
). Hence, it is necessary to build an updraft model representing the properties
of the updrafts ensemble responsible for the nonlocal transport, φ u .
4.4.2.1 Updraft Model
The proposed updraft model follows the methodology of Betts (1973) to describe
cumulus convection. Consider an air parcel moving upward in the BL with lateral
mixing. From relations (Equations 4.43 and 4.48) for an updraft u, one can obtain the
equation to determine its vertical structure, given by
(
),
u
u
z
∂φ = −ε φ − φ
∂
(4.51)
where
ε is the lateral mixing rate
φ u and φ
– are, respectively, a generic property, of thermals and its horizontal
domain average
This parcel model is designed to establish updraft’s properties: the liquid water
potential temperature, θ l u ; the total specifi c humidity, q t u ; the vertical velocity, w u ;
and the BL height, z i .
Equation 4.51 needs a bottom boundary condition, that is, the initial value, to
allow its vertical integration. To the initialization of the parcel one has to estimate its
virtual potential temperature excess relatively to the surroundings, Δθ v u :
Δθ = θ
− θ
≈
θ
′ ′
( )
( )
((
) ...)
vu
vu
v
v s
z
z
f w
(4.52)
It is assumed that the excess is directly related to the surface BL variability, expressed
in function of the excess of Δθ lu and Δq tu . Considering a constant sensible heat fl ux
and knowing that
(
)
l
l s
w
u
∗
∗
θ = − ′θ ′
,
(
)
t
ts
q
wq u
∗
∗
= − ′ ′
, one can write
(
)
,
v
v s
w
w
u
b
∗ ∗
σ θ
θ = − θ ≈ σ σ
′ ′
(4.53)
where b σ is a constant, θ
σ σ
,
v
w are, respectively, the standard deviation of θ v and
of the vertical velocity, w. Siebesma and Teixeira (2000) admitted that
v
v
c θ
Δθ ≈ σ ,
© 2010 by Taylor and Francis Group, LLC