ATMO 551a
Fall 08
Diffusion of water vapor onto a cloud droplet
Growth of cloud droplet by diffusion involves two simultaneous processes.
1. Water vapor diffuses onto the condensation nucleus/droplet
2. Heat released by the latent heat release diffuses outward from the cloud droplet
From Fick’s second law
n
 Dv 2 n
t
(1)
n
 0  Dv 2 n
t
(2)
Assume steady state

In spherical coordinates and assuming, by symmetry, only radial dependence (no angular
dependence)

 2n 
1 d  2 d 
n 0
r
r 2 dr  dr 
 2 d 
n C
r
 dr  1

(3)
(4)
d
C
n  21
dr
r

C
 dn   r 21 dr

 1 C
n()  n(r)  C10   1
 r  r

C
n(r)  n()  1
r
(5)
(6)
(7)

We have the further constraint
that at rdrop, n = nr such that
n(rdrop )  n() 

C1
rdrop
C1  n()  n(rdrop )rdrop
such that
(8)


n(r)  n() 
rdrop
r
n()  n(r )
drop
(9)
The flux is

Fn  Dv
r
dn
 Dv drop
n()  n(rdrop )
dr
r2

1
Kursinski 11/12/08
ATMO 551a
Fall 08
Fn r  
Dv rdrop
D
n()  n(rdrop )  v n()  n(r )

r r
r
(10)
The total flux at the drop surface where r = rdrop is the flux (density) times the area of the drop:
Dv
2
 Fn,rdrop Adrop   r n()  n(rdrop )4 rdrop  Dv n()  n(rdrop )4 rdrop
(11)
drop
The change in the mass of the droplet is this number density flux times the mass per molecule
dmdrop

dt
 Dv n()  n(rdrop )4 rdropmv  Dv v ()  v (rdrop )4rdrop
(12)
where mv is the mass per molecule or per mole depending on the units of n. Clearly, for the
droplet to grow, the environmental number density of water molecules, n() , must exceed that
of thenumber density at the surface of the droplet, n(rdrop ) . If on the other hand n(rdrop ) is
greater than n() , then the cloud droplet will evaporate.

The latent heat release from the condensation of the water vapor causes the droplet to
 away from the droplet.
warm above the ambient temperature and
this extra sensible heat diffuses

dQ
(13)
 4rK s T r  T()
dt
where Ks (=  Cp Ds) is the thermal conductivity of the air in units of kg/m3 J/kg/K m2/s =
J/m/s/K.
The change in the 
temperature of the droplet is
4 3
dT
dm dQ
r L CL r  L

3
dt
dt
dt
(14)
If we assume steady state droplet temperature, then
L


dm dQ
  LDv v ()  v (rdrop )4rdrop  4rdropK T rdrop  T()
dt
dt
 ()   (r )  K
T r  T() LD
v
v
drop
(15)
(16)

The water vapor density over the droplet surface is the determined by the saturation vapor
pressure over a curved surface such that
 e ' r,T
es r  ,Trdrop 
s
rdrop
b   a 
(17)
vrdrop 

1 3 exp
 

RvTrdrop
RvTrdrop
 rdrop  rdrop 
v
drop

 

Equations (16) and (17) can be solved simultaneously numerically to determine Tdrop, vdrop and
rdrop versus time.
There is an alternative solution shown below.

2
Kursinski 11/12/08
ATMO 551a
Fall 08
Growth of the droplet size: Alternative approximate solution by Mason (1971)
The saturation vapor pressure is related to the water vapor density via the ideal gas law
es  vsRvT
vs 
 as
The logarithmic derivative is given
es
RvT
(18)
dvs des dT L dT dT





vs
es
T Rv T 2 T
(19)
where we have used the Clausius Clapeyron equation (so this equation does NOT involve the
droplet surface curvature effects). Integrating (19) from Tdrop to T and assuming T/Tdrop ~ 1
yields

T

Tdrop
dvs
vs
 ln vs T
T
drop
T
 L dT dT 
T
L 1
  
  
 ln T T
2
drop
T  Rv T Tdrop
Tdrop Rv T
T
  T 
L 1
1   T 
vs  
  
ln 




 ln 


 T

 vs  drop  Rv T Tdrop  Tdrop 


  T  L T  T   T  T 
drop
drop
vs  
 
ln 

 ln 
1 T

 T
 R  T T 
drop

 vs  drop  v  drop  
  T  L T  T  T  T 
drop
drop
vs  
 
ln 

 
 T

 T
 R  T T 
 vs  drop  v  drop   drop 

 


L
1 
vs_ env
ln

T

T





  env drop R T T

 vs _ drop 
 v drop env Tdrop 
(20)

 Tenv  Tdrop  L

vs _ env   vs _ drop

1





vs _ drop
Tenv
RvTenv 


(21)

 
dmdrop
L 
1
dQ 
L 
L
vs _ env   vs _ drop
 1
 1







vs _ drop

  RvTenv 4rK sTenv dt  RvTenv 4rK sTenv dt
(22)


Subbing from (13) and then (15)
(22) shows the fractional difference between saturation vapor densities of the environment minus
that at the droplet for flat surfaces. Note that (22) would actually cause the droplet to evaporate
because
Tenv < Tdrop.

3
Kursinski 11/12/08
ATMO 551a
Fall 08
We also have (12) that includes the actual vapor pressure of the environment which is
supersaturated and the actual saturation vapor density over the curved droplet surface. From (12),
we can write

v _ env
()  v (rdrop )
v (rdrop )

1
dmdrop
Dv 4rdropv (rdrop )
dt
(23)
This should be rewritten to include the curvature and solution effects on the droplet surface
which means its saturation vapor pressure depends on the size of the droplet according to (9)
from the droplet
 activation notes.
 b   2   b  a 
es ' r,T 
vs ' r,T 
(24)

 1 3 exp
 1 3 exp 
 r  r 
es r  ,T  vs r  ,T   r  rRv L T  
Therefore (23) can be rewritten as



 b  a 
 v _ env ()   v _ drop (r )1 3 exp 
dmdrop
 r  r 
1


 b  a 
Dv 4 rdropv (rdrop ) dt
v _ drop (r )1 3 exp 
 r  r 


 b   a 
 v _ env ()1 3 exp   v _ drop (r )
dmdrop
 r   r 
1



 v _ drop (r )
Dv 4 rdrop v (rdrop ) dt

v _ drop
() f (r)  v _ drop (r )
v _ drop (r )


1
dmdrop
Dv 4rdropv (rdrop )
dt
(26)
 b   a   a b 
f (r)  1 3 exp  1  3 
 r   r   r r 
where
(25)
(27)

Subtracting (22) from (26) and assuming v _ drop (r ) =vs_drop,

v _ env ()
 f (r)  vs _ env   
vs _ drop

(28)

drdrop
L  L 
2
 1
L 4rdrop

dt
 RvT 4rK sT 
(29)
Dv 4rdropvs _ drop
v _ env () f (r)  vs _ env   
vs _ drop

L  L dmdrop
 1


 RvT 4rK sT  dt
1
1
Dv 4rdropvs _ drop

Now we must rearrange (29) to solve for drdrop/dt. So more algebra…
v _ env () f (r)  vs _ env   

vs _ drop

drdrop
L  L 
 1
L rdrop

dt
Dv vs _ drop  RvT K sT 
v _ env () f (r)  vs _ env   


vs _ drop
1

L L L  drdrop
 1
rdrop

dt
Dves (rdrop )  RvT K sT 
L RvT
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Kursinski 11/12/08
ATMO 551a
Fall 08
v _ env () f (r)  vs _ env 
rdrop
drdrop
dt

 vs _ drop
(30)
 L
 L L   L Rv T 
1
 



Rv T K sT Dv es (rdrop ) 
We need to do something with the numerator…
v () f (r)  vs _ env  vs _ env v _ env () f (r)  vs _ env  vs _ env

rdrop
drdrop
dt

vs _ drop
vs _ env

 L
L L  L RvT 
1
 


Rv T K sT Dv es (rdrop )
vs _ env
vs _ drop
 L
 L   R T 
1 L   L v 

RvT K sT Dv es (rdrop ) 
Now we substitute the supersaturation, S = e/es = v/vs , and get

rdrop
So we now need to consider
drdrop
dt
vs _ env
S

f (r) 1
 vs _ env
 vs _ drop
(31)
 L
 L L   L Rv T 
1
 



Rv T K sT Dv es (rdrop ) 
vs _ drop which is the saturation vapor density in the
environment divided
 by the saturation vapor density at the surface of the droplet. There are two
issues. First the curvature of the droplet surface means its saturation vapor pressure depends on
the size of the droplet according to (9) from the droplet activation notes.

 b   2   b  a 
es ' r,T 
(32)
 1 3 exp
 1 3 exp 
 r  r 
es r  ,T   r  rRv L T  
Second, the temperature at the droplet is higher than that of the environment so the saturation
vapor pressure for a flat surface at the droplet is higher than that of the environment.

 LT  T 
drop
env
es r  ,Tenv   es r  ,Tenv 
(33)
es r  ,Tdrop  1
R
T
T

v drop env 


Combining the last two equations
 LT  T  b  a 
 b  a 
drop
env

1 3 exp  (34)
es ' rdrop,Tdrop  es r ,Tdrop 1 3 exp  es r ,Tenv 1
 r  r 
R
T
T

v drop env 

 r  r 
Also we are dealing with water vapor densities rather than vapor pressures so there is an
additional conversion required

vs _ env evs _ env Rv Tdrop evs _ env Tdrop


vs _ drop evs _ drop Rv Tenv evs _ drop Tenv
(35)
Combining these last two equations yields

5
Kursinski 11/12/08
ATMO 551a
Fall 08
vs _ env evs _ env Tdrop


vs _ drop evs _ drop Tenv 
1
Tdrop
LTdrop  Tenv  b  a  Tenv
1
1 3 exp 
R
T
T


v
drop
env

 r  r 


(36)
vs _ env Tdrop  LTdrop  Tenv  b   a 
1
1 3 exp 

vs _ drop Tenv 
R
T
T

v
drop
env

 r   r 
(37)
vs _ env  Tdrop  Tenv LTdrop  Tenv  a b 
 1

  3 
vs _ drop 
T
R
T
T
r r 
env
v
drop
env


(38)
 vs _ env  Tdrop  Tenv 
L  a b 
 1
1

 R T 
  r 3 
 vs _ drop 
Tenv



v drop  r

(39)

Now let’s try to understand the size of the non-unity terms. From (16) and the definition of K,
we can write

LDv
LDv
(40)
T rdrop  T() 
v ()  v (rdrop )

 ()  v (rdrop )
K
C p Ds v


The two diffusivities, the sensible heat, Ds, and diffusion of water vapor in air, Dv, are similar.
The diffusion of sensible heat is the result of the diffusion of N2 and O2 molecules through other
N2 and
 O2 molecules. The diffusion of water vapor is a bit different because it is the diffusion of
water vapor molecules through N2 and O2 molecules. We can guess that the collisional
crosssection is larger for water molecules which will decrease the diffusivity but the mass of
water vapor is smaller which will increase the diffusivity. So we can argue crudely that Ds and
Dv, are similar so their ratio is about 1. The supersaturation is perhaps a percent of the saturation
value. Near the surface, a rough approximation is v/a ~ 1%. So v/a ~ 1e-4. L/Cp ~
2.5e6/1e3 K = 2.5e3 K. So the temperature difference for near surface conditions is of the order
of 2.5e3 K 1e-4 = 0.25 K. At colder temperatures, the temperature difference will be still smaller
because less water vapor is available in the air and therefore less latent heating to drive a
temperature difference. Therefore the T difference term in (39) is approximately
Tdrop  Tenv 
L  0.25



~  250 20  0.02
Tenv 
 RvTdrop 
This is comparable to the –a/r term for typical supersaturations. So

 vs _ env
 1  
 vs _ drop
(41)
where  can reach a few percent near the surface and is less at colder temperatures at higher
altitudes.
Now we return to the other
 term in the numerator of (31), S f (r) 1.
S
f (r) 1 
v  b   a 
  a b 
a b
1 3 exp 1  v 1  3 1  S 1  3

vs  r   r 
vs  r r 
r r
(42)


6
Kursinski 11/12/08
ATMO 551a
Fall 08
Combining (41) and (42), the numerator of (31) becomes
S
f (r) 1

vs _ env 
a b 
a b 
 S 1  3 1    S 1  3 

vs _ drop 
r r 
r r 
(43)
Plugging (43) into (31) yields when the supersaturations are not too large


a b 
S 1  3 
dr

r r 
rdrop drop 
 L
 L   R T 
dt
1 L   L v 

Rv T K sT Dv es (rdrop ) 
(44)
According to Rogers and Yau, this equation estimates the growth rate of cloud droplets quite
accurately.

Cloud droplet growth by diffusion
1000
Series1
Radius (microns)
100
10
1
10
100
1000
10000
100000
0.1
time (sec)
7
Kursinski 11/12/08