Example Observations
21 November 1354 EST T = 11.7°C
Scattered altocumulus (Ac) at ~6 km
22 Nov. 0800 EST T = 8.9°C
Temperature stable for hours. Mist, fog,
overcast. Little diel cycle because of cloud
cover and condensation.
22 Nov. 1400 EST T = 17.8°C
Broken cirrus at ~7 km. Radiative heating.
Copyright © 2010 R. R. Dickerson
& Z.Q. Li
1
AOSC 620
Lecture 23
Cloud Droplet Population Growth – Warm Cloud
Precipitation (Rogers and Yau Chapt. 7)
R. Dickerson, 2010
In the last class (Lecture 22) we showed how
condensation leads to cloud droplets. How do
droplets grow and become rain drops?
Balance among:
1. Molecular diffusion of H2O
2. Heat conduction
3. Curvature (surface tension) and solute effects
(Raoult’s Law).
4. Temp. dependence of vapor pressure in CC
Equation.
Copyright © 2010 R. R. Dickerson
& Z.Q. Li
2
Final Set of Growth Equations
• Mass diffusion to the droplet
dr
D
r
=
(e ¥ - e r )
dt
rw R v T¥
dr
K
=
(Tr - T¥ )
• Conduction of latent heat away r
dt
L v rw
• Combined curvature and solute effects
• Clausius-Clapeyron equation
(1a)
(2a)
er
a
b
1

esr
r
r3
esr
 exp
es
{ ( )}
Lv
Tr – T
2
RvT
Molecular diffusion vs. eddy diffusion.
D ~ 2x10-5 m2 s-1
t = Xd2/D
Kz ~ 20 m2 s-1
t = XK2/Kz
How much travel in 1.00 min?
60 = Xd2/D → XD = (120 x 10-5)½ = 3.5 mm
→ XK = (60 x 20)½ = 350 m
• Eddy diffusion is much faster than molecular diffusion, but
eddies cannot mix water vapor toward small droplets.
Droplets move together.
How do we make it rain?
If all growth were due to condensation then there would be never
be rain. We first considered stationary droplets, now a parcel
moving along a pseudoadiabat. They we discuss how droplets
collide sometimes.
Coalescence:
Consider warm clouds, i.e., T ≥ 0ºC, no ice.
Condensation can get droplets to 20 mm.
collection
condensation
~20
Radius ↑
(mm)
Time →
Assumptions
• Isolated, spherical water droplet of mass M, radius r and
density w
• Droplet is growing by the diffusion of water vapor to the
surface
• The temperature T and water vapor density v of the
remote environment remain constant
• A steady state diffusion field is established around the
droplet so that the mass of water vapor diffusing across
any spherical surface of radius R centered on the droplet is
independent of R and time t.
Growth of a Population of Cloud Droplets
Interaction between droplets?
r ≈ 7 µm; x ≈ 1000 µm; vt ≈ 0.5 cm s-1
• The separation between droplets is so great,
relative to their sizes, that any interaction
between droplets is negligible (not counting
collisions).
• Each growing droplet experiences the same
environment. That is, droplets influence each
other only in their combined effect on the
common environment.
Qualitative Description of Condensation in a
Rising Current of Air
• As an initially unsaturated air parcel rises and expands
approximately adiabatically, the saturation ratio
S = (e / es) increases and nuclei swell.
• After the saturation level (LCL) is reached, condensation
begins to occur on the largest, most active nuclei.
Qualitative Description of Condensation in a
Rising Current of Air - cont.
• S continues to increase and more and more nuclei are
activated and begin to grow as droplets. However, the rate
of increase in S is slower than above because the growing
droplets are rapidly removing the excess water vapor from
the parcel.
• Since the large droplets remove the water vapor more
quickly than the smallest ones, the excess vapor is soon
being removed from the air as fast as it is supplied from
expansion. Then S decreases toward unity.
Growth Equations for a
Population of Nuclei
In order to calculate the rate of change of S with time, one
must consider:
• the updraft speed
• the condensation nuclei population
• the growth equations for each nuclei.
We’ll assume
1. a parcel rising with constant vertical speed dz/dt = U;
2. no mixing with the environment;
3. no collisions between droplets.
Growth Equations for a
Population of Nuclei
e
S 
e s
dS
1 de
e des

 2
dt
es dt
es dt
but
e  f1 ( p, w)
es  g (T )
T  f 2 ( z , w)
I
Growth Equations - cont.
II
Growth Equations - cont.
III
i.e., if there is condensation, w changes, and T changes
And the parcel is more nearly saturated adiabatically.
Growth Equations - cont.
Inserting III into II and then II into I
dS
1

dt es
 e

 p
e
 2
es
  dp 
1  e  dw
  U 


es  w  p dt
 w  dz 
 Lv es

2
R
T

 v
 T 
e  Lv es

 U  2 
2

z
e
R
T

w
s 
v

 T  dw


 w  z dt
but
 e   dp 
g
e g
 T 




and




 


Cp
R ' T
 z  w
 p  w  dz 
(remember lapse rate and scale height)
e

e
since w  
, dw  de  2 dp  0
p
p
p
 e
so, 
 p

e
dp
pg
 
; and
  g  
p
dz
R ' T
w
Growth Equations - cont.
1
es
e
 2
es
 e  g 
 e   dp 

U

  U  
 p  w  dz 
 es  RT 
 Le
 v s
 R T2
 v
 T 
 Lg


e
v
 



U


 e  R C T 2
 z  w
 s  v p 


U


Growth Equations - cont.
 e
1  e 
p
 

 
es  w  p es  es
e  Lv es
es2  RvT2

 p

 e
2
 T 


 Lv

  S 
 w  z
T pC p



   RT
  S 

 e




The First Law  Lv dw  c p dT  dp
Lv
 T 
therefore, 
 
 w  p c p
and
dw
d

( i.e.,   liquid water content )
dt
dt



Growth Equations – final
The change in Saturation ratio, S, is Production minus
Condensation (loss).
dS
 P  C , or
dt
dS
d
 Q1U  Q2
dt
dt
 Lv g

g

Q1  

2
 R' C T

R
'
T

 v p 

 L2
R ' T
v

Q2   

 T p Cp
e

R&Y
Eq. 7.23




Eq. 7.24
Equation for Liquid Water Content
where ni is the number of condensation nuclei of size i
per unit volume, and ri is the radius of a droplet
formed on the ith nucleus.
Liquid Water Content - cont.
d
d  M 
1 dM 






dt
dt   a 
 a dt
thus
w
d
2 d ri

 ni 4 ri
dt
a i
dt
where

eri 


S 

esr 
d ri


ri

dt

eri

C

C
2
 1
esr





Growth Equations
d
dS
 SQ1U  SQ2
dt
dt
d  w
2 d ri

 ni 4 ri
dt
dt  a i
eri
a b
1  3
ri ri
esr
Lv d
g
dT
 U 
C p dt
Cp
dt

eri 
 S  
esr 
d ri


ri
dt 
eri 
 C1  C2 
esr 

Q1 is increase in S due to adiabatic cooling;
Q2 is the decrease due to condensation.
Growth Calculations
• Thus, the analysis yields a set of simultaneous equations.
• Given initial conditions of T, p, w, U and ni, we would like
to solve for
S as a function of time (height in a cloud)
ri as a function of time/height
Results of Calculation of droplet spectrum
Initial Conditions
U - 15 cm s-1
moderate concentration
of nuclei at cloud base
Results
Fig. 7.3 from Rogers (1989)
• All droplets begin to
grow as they ascend
• S* reaches a max ~ 10m
above cloud base
• Size distribution is very
narrow
• Largest droplets descend
Results of Calculations - cont.
from Rogers (1989
Initial conditions
Population of NaCl nuclei with Nc = 650•S*0.7 cm-3
U = 2 m s-1 (solid) and 0.5 m s-1 (dashed)
The variable im/Ms is
just the molar mixing
ratios. The droplets
formed from small
CCN fail to activate;
they evaporate back
into haze. The From
Wallace & Hobbs
Overall Results
(what does our cloud look like?)
• Sharp rise and gentle settling of supersaturation.
• S* reaches its peak within 100 m of cloud base. Thus the
cloud droplet concentration is determined at low levels in the
cloud.
• The maximum of S* increases with U as does the number of
nuclei activated.
• Rapid increase of droplet concentration to a steady value
reached at point of maximum supersaturation
• Narrow droplet size distribution.
Vertical variations
of cloud droplet
sizes and liquid
water density for
low-level stratiform
clouds compiled
from various in-situ
measurements.
Note the general
linear increasing
trends!
After Miles et al. (JAS, 2000 JAN)
Vertical variations of
cloud number
concentration. Note
the difference
between continental
and marine clouds.
Marine clouds has
much smaller # of
droplets which does
not change much
with height. The
opposite is the case
for continental
clouds. ?
After Miles et al. (JAS, 2000 JAN)
Problems with Diffusion Growth Theory
• Kinetic effects - The heat mass and momentum transfer
equations used are valid only for drops much larger than
the molecular mean free path of water vapor in air
(~0.06mm). In this case, the field of water vapor and
temperature may be regarded as continua so that the
Maxwell continuum approximation
can be used to
m
determine the transfer of water and heat.
• When kinetic effects are included, the size reached by a
drop is smaller at a given time than our model. The kinetic
corrections result in a relatively broader spectrum of sizes.
Ventilation effects - When a drop is large
enough to fall through the air with a
significant speed, the vapor field becomes
distorted from the spherical symmetry used in
our model. These effects are negligibly small
for cloud droplets, although they can become
more significant for precipitation.
Problems - cont.
• Nonstationary growth - The vapor and temperature fields are
not steady because the surface of the droplet is expanding or
contracting. However, calculations show that molecular
diffusion quickly establishes a field around the droplet that
corresponds to the steady state solution.
• Unsteady updraft - Calculations show the results are relatively
insensitive to the updraft structure because there is a strong
correlation between Smax and U.
• Statistical effects - Mixing and sedimentation prevent a
population of droplets from staying together indefinitely
(Rogers, 1989). Some drops may grow faster than others
because they experience higher than average S’s. Not yet
accepted conclusively.