Parameterization of Convective Boundary Layer Turbulence and Clouds
87
∂φ
∂ φ
+ ∇ ⋅ φ +
=
∂
∂
( )
,
h
H
w
v
F
t
z
(4.32)
where v
→
H is the horizontal velocity vector and F gathers all the external forces, and
integrating φ in the A u area, dividing by A, applying the Leibnitz and the divergence
theorems leads to
∂ φ
∂
φ
+
⋅ −
φ +
=
∂
∂
∫
fronteira
1
(
) d
,
u
u u
u
f
u u
a
a w
n v v
l
a F
t
A
z
(4.33)
where
n
→ is a unit vector perpendicular to the boundary that separates the updraft from
the surroundings
v
→ is the velocity vector
v
→
f is the boundary velocity
Similarly, the continuity Equation 4.3, or Equation 4.32 assuming φ = cte, integrated
in the same horizontal section of an updraft can be written
∂
∂
+
⋅ −
+
=
∂
∂
∫
fronteira
1
(
)d
0.
u
u u
f
a
a w
n v v
l
t
A
z
(4.34)
The Equation 4.34 corresponds to the mass balance of air crossing the boundary, that
is, constitutes the net result of lateral mixing. If one defi nes E (entrainment) as the
lateral mixing associated with the air mass from the surroundings that enters in the
ascent, and D (detrainment) the term representing the process of exporting updraft
air, D − E represents the net result of both processes, therefore
− =
⋅ −
∫
fronteira
1
(
)d,
f
D E
n v v
l
A
(4.35)
where E is given by
( ) 0
1
(
)d,
f
f
n v v
E
n v v
l
A ⋅ − <
= −
⋅ −
∫
(4.36)
and D is written as
( ) 0
1
(
)d,
f
f
n v v
D
n v v
l
A ⋅ − >
=
⋅ −
∫
(4.37)
Finally, from Equation 4.34:
(
)
0.
u
u u
a
aw
D E
t
z
∂
∂
+ −
+
=
∂
∂
(4.38)
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