Parameterization of Convective Boundary Layer Turbulence and Clouds
85
approach is suitable for the representation of the transport of active chemical tracers
(Chatfi eld and Brost 1987).
Originally, the MF approach was based on the decomposition of the atmosphere
in regions of ascending motion, in the interior of clouds, and regions in its environment. But, if it is also intended to describe the clear sky BL, this criterion obviously
needs to be adjusted. A more general criterion of decomposition may use the signal
of vertical velocity, not forgetting that more restrictive criteria could be applied.
Therefore, the MF approach consists in the decomposition of the atmosphere in
updraft regions (u) and the surrounding environment (e). Considering an horizontal
domain of the atmosphere with the area A = L x L y , A u is the area with upward motion
and A e is the surrounding area (A = A u + A e ), and one can compute the mean value of a
variable in the domain, φ
–
(z), and in each of the complementary regions, respectively
φ
– u (z) and φ
– e (z):
( )
( )
( )
φ ≡
φ
φ
=φ ≡
φ
φ
=φ ≡
φ
∫ ∫
∫∫
∫∫
0 0
1
( , , ) d d ,
1
( , , ) d d ,
1
( , , ) d
.
y
x L
L
u
u
u
u
e
e
e
e
z
xyz x y
A
z
x y z xy
A
z
xy z x d y
A
(4.24)
Normalizing the areas, with a u = (A u /A) as the fraction of the domain covered by
updrafts, the mean φ for all the domain is given by,
(1
) .
u u
u
e
a
a
φ = φ + − φ
(4.25)
The turbulent vertical fl uxes for the domain and for the partial domains can be easily
deduced using the Reynolds averaging, yielding
φ = φ − φ
′ ′
φ = φ − φ
′ ′
φ = φ − φ
′ ′
,
,
,
u
u
u u
e
e
e e
w
w w
w
w
w
w
w
w
(4.26)
and knowing that w – = a u w u + (1 − a u )w e and
(1
)
u
e
u
u
w
a w
a w
φ =
φ + −
φ , and Equation
4.25, the following relations between fl uxes are obtained:
(1
)
(
)(
).
u
e
u
u
u
u
u
e
w
a w
a w
a w w
φ =
φ + −
φ +
−
φ − φ
′ ′
′ ′
′ ′
(4.27)
The third term of the r.h.s. expresses the updraft contribution to the vertical turbulent transport of φ, and is the MF term. In the context of shallow convection that
is considered the more signifi cant component of the turbulent transport (Ooyama
© 2010 by Taylor and Francis Group, LLC
85
approach is suitable for the representation of the transport of active chemical tracers
(Chatfi eld and Brost 1987).
Originally, the MF approach was based on the decomposition of the atmosphere
in regions of ascending motion, in the interior of clouds, and regions in its environment. But, if it is also intended to describe the clear sky BL, this criterion obviously
needs to be adjusted. A more general criterion of decomposition may use the signal
of vertical velocity, not forgetting that more restrictive criteria could be applied.
Therefore, the MF approach consists in the decomposition of the atmosphere in
updraft regions (u) and the surrounding environment (e). Considering an horizontal
domain of the atmosphere with the area A = L x L y , A u is the area with upward motion
and A e is the surrounding area (A = A u + A e ), and one can compute the mean value of a
variable in the domain, φ
–
(z), and in each of the complementary regions, respectively
φ
– u (z) and φ
– e (z):
( )
( )
( )
φ ≡
φ
φ
=φ ≡
φ
φ
=φ ≡
φ
∫ ∫
∫∫
∫∫
0 0
1
( , , ) d d ,
1
( , , ) d d ,
1
( , , ) d
.
y
x L
L
u
u
u
u
e
e
e
e
z
xyz x y
A
z
x y z xy
A
z
xy z x d y
A
(4.24)
Normalizing the areas, with a u = (A u /A) as the fraction of the domain covered by
updrafts, the mean φ for all the domain is given by,
(1
) .
u u
u
e
a
a
φ = φ + − φ
(4.25)
The turbulent vertical fl uxes for the domain and for the partial domains can be easily
deduced using the Reynolds averaging, yielding
φ = φ − φ
′ ′
φ = φ − φ
′ ′
φ = φ − φ
′ ′
,
,
,
u
u
u u
e
e
e e
w
w w
w
w
w
w
w
w
(4.26)
and knowing that w – = a u w u + (1 − a u )w e and
(1
)
u
e
u
u
w
a w
a w
φ =
φ + −
φ , and Equation
4.25, the following relations between fl uxes are obtained:
(1
)
(
)(
).
u
e
u
u
u
u
u
e
w
a w
a w
a w w
φ =
φ + −
φ +
−
φ − φ
′ ′
′ ′
′ ′
(4.27)
The third term of the r.h.s. expresses the updraft contribution to the vertical turbulent transport of φ, and is the MF term. In the context of shallow convection that
is considered the more signifi cant component of the turbulent transport (Ooyama
© 2010 by Taylor and Francis Group, LLC