Geography Reference
In-Depth Information
δm of entrained environmental air, which has an amount of the arbitrary variable
given by δmA
env
. The change in the value of A within the cloud, δA
cld
, is then
given by the mass balance relationship
δmA
env
+
DA
cld
Dt
S
mδt (9.48)
where
DA
cld
Dt
S
designates the rate of change of A
cld
due to sources and
sinks unrelated to entrainment. Dividing through by δt in (9.48), neglecting the
second-order term, and rearranging yields
(m
+
δm)(A
cld
+
δA
cld
)
=
mA
cld
+
DA
cld
Dt
δA
cld
δt
1
m
δm
δt
=
S
−
(A
cld
−
A
env
)
(9.49)
Noting that in the time increment δt an ascending parcel rises a distance δz
wδt,
where w is the speed of ascent of the parcel, we can eliminate δt in (9.49) to
obtain an equation for the vertical dependence of A
cld
in a continuously entraining
convective cell:
=
DA
cld
Dt
w
dA
cld
dz
=
S
−
wλ (A
cld
−
A
env
)
(9.50)
where we have defined the
entrainment rate
; λ
≡
d ln m/dz.
ln θ
e
and noting that θ
e
is conserved in the absence of entrain-
ment, (9.50) yields
d ln θ
e
dz
Letting A
cld
=
λ
(ln θ
e
)
cld
−
(ln θ
e
)
env
=−
λ
L
c
c
p
T
ln
T
cld
T
env
cld
≈−
(q
s
−
q
env
)
+
(9.51)
where (9.40) is used to obtain the latter form of the right-hand side. Thus an
entraining convective cell is less buoyant than a nonentraining cell. Letting A
cld
=
w in (9.50), applying (9.45), and neglecting the pressure contribution to buoyancy,
we find that the height dependence of kinetic energy per unit mass is given by
w
2
2
g
T
cld
−
d
dz
T
env
T
env
λw
2
=
−
(9.52)
An entraining cell will undergo less acceleration than a nonentraining cell not
only because the buoyancy is reduced, but also because of the drag exerted by
mass entrainment.
Equations (9.51) and (9.52), together with suitable relations for the cloud mois-
ture variables, can be used to determine the vertical profile of cloud variables. Such
