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where g is the acceleration due to gravity and θ 0 is the reference-level potential
temperature at the cloud-base level.
By substituting for W and taking θ as the production of temperature perturbation
at the cloud-base level by MFC, we arrive at the following expression since there is
a linear relationship between the vertical velocity perturbation W and temperature
perturbation θ (from Eqs. 1.4 and 1.6):
θ
*
(2.1)
θ
=
ln
z
=
θ
fz
.
*
k
Thus, the in-cloud vertical velocity and temperature perturbation follow the fz dis-
tribution (Fig. 1.5).
2.1.4
In-Cloud Temperature Lapse Rate Profile
The saturated adiabatic lapse rate Γ sat is expressed as
=− L
Cz
d
d
χ ,
ΓΓ
sat
p
where Γ is the dry adiabatic lapse rate, C p is the specific heat of air at constant pres-
sure, and d χ /d z is the liquid water condensed during parcel ascent along a saturated
adiabat Γ sat in a height interval d z .
In the case of cloud growth with vertical mixing, the in-cloud lapse rate Γ s can
be written as
=− L
C
d
d
q
z ,
ΓΓ
s
p
where d q , which is less than dχ, is the liquid water condensed during a parcel ascent
d z and q is less than the adiabatic LWC q a . From Eq. (2.1),
(2.2)
θ
fz
d
d
θ
θ
*
ΓΓ Γ
=−=−=−
Γ
,
s
z
r
r
where d θ is the temperature perturbation θ during parcel ascent d z . By concept, d z
is the dominant turbulent eddy radius r (Fig. 1.2).
2.1.5
Total Cloud LWC Profile
The total cloud LWC q t at any level is directly proportional to θ as given by the fol-
lowing expression:
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