In the early stage of a cloud, droplets are generally sufficiently small (r < 10 µm) that gravitational settling and coagulation is relatively unimportant. We want to determine the microphysical
properties of the parcel as it rises. To do this properly actually requires a numerical model. The
basic physics we need to consider is that heat and energy in the parcel are conserved
• Q = χ + w = const. (Conservation of mass)
• hm = cp T + gz + Lv w = const. (Conservation of energy)
So as the parcel rises we have
dQ
dχ dw
=
+
=0
dt
dt
dt
dhm
dT
dz
dw
= cp
+ g + Lv
=0
dt
dt
dt
dt
Also recognizing that
w=
²Ses
p
the Clausius-Clapeyron equation and
dχ
= 4πrD (S − 1) ws N
dt
where N depends on the Kohler Curve, the aerosol population, and the Saturation ratio. Based
on all this information we can derive equations for the rate of change of χ (or r ), S, N and T
within the air parcel as it rises in the cloud. The basic outline to all this is in Rogers and Yau.
The main thing to recognize is that as a parcel rises the Saturation ratio in the cloud depends on
two factors. The Saturation ratio is increased by cooling during ascent since as ws decreases more
vapor is made available for condensation. On the other hand, as time progresses, the Saturation
ratio decreases due to condensation. Eventually the parcel reaches an equilibrium. This equation
can be expressed as
dS
=P −C
dt
where P is the production term from cooling and C is the condensation term due to condensation.
More specifically
dS
dχ
= Q1 W − Q2
dt
dt
where W is the updraft velocity and
"
#
"
#
1 ²Lv g
g
Q1 =
−
T Rd cp T
Rd
²L2v
Rd T
+
Q2 = ρ
²es
pT cp
Values are shown below
And the solutions to the equations of droplet growth are plotted below
1
Figure 1: Values of Q1 and Q2
Figure 2: Numerical model results in Rogers and Yau
2