Atmospheric and Climate Sciences, 2011, 1, 86-90
doi:10.4236/acs.2011.13009 Published Online July 2011 (http://www.scirp.org/journal/acs)
The Clouds Microstructure and the Rain Stimulation by
Acoustic Waves
Ovik Nalbandyan
Moscow Social-Humanitarian Institute, Moscow, Russia
Received May 10, 2011; revised June 15, 2011; accepted J une 29, 2011
Abstract
A simplified model of clouds’ microstructure dynamics under turbulent conditions is presented. The nature
of rain stimulation by acoustic waves, based on the drops injection in the region of turbulent coagulation, is
described. The conditions for effective rain stimulating are estimated.
Keywords: Cloud, Drop, Turbulence, Condensation, Coagulation, Rain Stimulation
1. Introduction
The investigations of clouds’ microstructure represent a
significant step for studying the possibilities and pros-
pects to stimulate rain and ultimately to control weather
patterns. The most commonly used methods have been
by “seeding” the cloud by various agents that causes a
large scale fracture within its structure or by direct in-
jecting of large drops of water into the cloud. However,
these methods require the use of artillery and rocket sys-
tems or flying devices which can be pretty costly. Scien-
tists have been searching for cost-effective alternative
methods.
To bring the changes in the clouds’ microstructure by
using the impact of acoustic waves were an interesting
idea from the start. Theoretical calculations have esti-
mated that the impact of intense acoustic waves upon a
cloud would initiate the coagulation of the drops which
would create a shift of the maximum in the distribution
of the drops towards larger sizes. The experiments, con-
ducted with both, overland fog and artificial fog in a
chamber have confirmed these estimations [1,2]. What
became apparent, that the impact of acoustic waves has
to be extremely intense (~140 dB) and has to last for a
long time (minutes). However, it would not be realistic to
obtain the required parameters of the acoustic waves in
the clouds while the source of sound waves is on the
ground. But the clouds in nature, comparative to an arti-
ficial fog or an overland fog may, intrinsically be “oper-
ating” under turbulent conditions. And the turbulent con-
ditions combined with acoustic waves could sufficiently
enhance the effectiveness of stimulating the rain.
There are many theoretical and empirical models de-
scribing the clouds' microstructure and its dynamics due
to the drops’ coagulation and condensational growth [1,
2]. These models represent various approaches to the
movement of the drops and their interaction. In the pre-
sent paper a simplified model of the drops’ movement,
coagulation and condensational growth under turbulent
conditions is represented. On the basis of the represented
model, the possibilities of rain stimulating by the impact
of acoustic waves under turbulent conditions are exam-
ined.
2. The Motion of Drops under Turbulent
Conditions in the Air
After the nucleation of drops in a supersaturated air the
main processes that have been involved in the formation
of the clouds microstructure are:
- Condensational growth and evaporation of drops;
- Coagulation of the drops by colliding;
- Pulverizing of the drops.
The speed of the drop relative to the air particles plays
the main role in these dynamics. Due to the air viscosity
the drop gets involved by the turbulent streams of the air.
But the density of the drops is much greater than the air
density and due to inertia of drops their trajectory deviate
from the air particles trajectory. It means that drops
move relative to the surrounding air. The relative veloc-
ity of a spherical drop in the viscous air is determined by
the following equation:
dd a
d
vv v
tt 




 (1)

O. NALBANDYAN ET AL.87
Here dis the drop velocity relative to the surround-
ing air particles,
v
2
2
9
d
d
a
r


d (2)
is the time of Stokes relaxation of the drop’s velocity in
the viscous air, a is the local velocity of the air parti-
cles at the position of the drop, d
v

and a

are the
densities of the water and the air correspondingly,

is
the kinematic viscosity of the air, d is the radius of the
drop. The air resistance, proportional to the drop's accel-
eration is neglected. The mass of the air in a hydrody-
namic boundary layer is neglected as well.
r
The pattern of the random turbulent flows is a super-
position of numerous vortices of various spatial scales
and the variance of the air particle velocity

2
0
a
vEk

dk
l
is determined by the spatial spectral density of the ran-
dom field of rates , where is the spatial
wave number of the vortex, is the vortex size. For
atmospheric turbulence we have

Ek 2π/k
l

5/3
2π/2πv klEkCk

,


0, m
kkk,
where 0 and m are the spatial wave numbers which
correspond to the outer 0 and the inner scales of
turbulence correspondingly.
kk Lm
l
The air particles being involved with vortices of vari-
ous sizes have curvilinear trajectories of movement. So
the acceleration of the air particles in some direction may
be represented as a superposition of accelerations of
various frequencies which depend on the vortex
size

Tk

 
0sin d
aT
v
A
kkt
t


k





Let’s assume a model of dynamic turbulence. Accord-
ing to the model kinetic energy of the air particles is sta-
tionary and the particles have only tangential accelera-
tion caused be the curvilinearity of their trajectories. In
this case the magnitude of tangential acceleration equals
to the air velocity in the vortex raised to the second
power and divided by the size of the vortex, while the
frequency equals to the air velocity in the vortex divided
by the size of the vortex.

2/3
v
A
kCk

,

2/3
6π
Tv
kCk  .
In this case the quasistationary solution of Equation (1)
has the form

    
0
2/3
22
1sin cos
m
kv
dkdT
dT T
Ck
vk
ktkkt dk

 




 

. (3)
Let’s analyze the obtained solution. For small drops
the time of Stokes relaxation (2) is much less than the
characteristic time of small-scale turbulent pulsations


1
dTk
  and the amplitude of relative velocity of
the drop has the square-law dependence on the radius of
the drop
1/3
1/3 2
2
33
dvm
dvmd a
Ck
vCk r


 d (4)
For larger drop if the time of Stokes relaxation is in
the range of the turbulent pulsation’s

1
0Td
k 




Tm
k 
the dependence of the relative velocity on the
radius of the drop became linear
 
 
3/4 1/4
3/4 1/4
2π3
2
2π39
dvd
d
vd
a
vC
Cr





. (5)
Very large drop with is too inertial to
move with the air particles and its relative velocity cor-
responds to the velocity of the turbulent flows

01 dTk
 
1/3
0
6π
dv
vCk

.
3. Condensational Growth of the Drops
The clouds arise in a supersaturated air consequently to
the heterogeneous nucleation of drops. The boundary
between the water and the air is subjected to the molecu-
lar unevenness, so the kinetic coefficient of the conden-
sation q, determining the dependency of the growth rate
u on the supercooling would be quite high and we can
disregard the kinetic supercooling and; accordingly a
kinetic supersaturation. It means that the supersaturation
at the growing surface corresponds to the equilibrium
conditions.
The process of condensation gets always accompanied
by the latent heat release, and the value of the drop’s
overheating, relatively to the air is at exactly the correct
level, which is needed to transfer the latent heat of con-
densation from the growing surface of the drop to the
cloud and can be obtained by the condition of the heat
transfer

H
dd aaa
T
T
urHcK 


. (6)
Here


d
ur is the growth rate of the drop,
H
is the
specific heat of evaporation, a is the specific heat ca-
pacity of the air, a
c
K
is the thermal diffusivity of the air,
T

is the thickness of the thermal boundary layer at the
surface of the moving drop. So, the value of the steam
concentration at the growing surface S differs from
the concentration of saturation in the cloud due to
C
0
C

O. NALBANDYAN ET AL. 88
the curvature of the drop’s surface and the overheating of
the drop
0
00
2m
SH
d
Vd
CCC T
rRT dT

 C . (7)
Here

is the surface energy of the water-air bound-
ary, m is the molar volume of the water, is the gas
constant, is the absolute temperature.
VR
T
The steam transfer from the cloud to the growing sur-
face of the drop takes place due to the difference be-
tween the steam concentration in the cloud and at
the drop’s growing surface
eq
C
S
C

dd
eq S
aD
ur
CC
D



. (8)
Here is the steam molecular diffusion coefficient
in the air,
a
D
D

is the thickness of the diffusive boundary
layer at the surface of the moving drop.
The thickness of the boundary layers depends on the
relative velocity of the drop. The estimations have shown
that the airflow of the drops in the cloud has a laminar
character and the following expression can be used
da
Td
rK
v
 , da
Dd
rD
v
 . (9)
Substituting (7) and (8) in (6) and taking into account
(9) the expression for the drop’s growth rate can be ob-
tained

0
1
0
211
d
d
m
deq d
d
dd
aaaa d
CV
ur RTr r
v
HC
T
Dc Kr










 


. (10)
Here eq is the radius of the drop which is under the
equilibrium conditions, i.e. do not grow and do not
evaporate. It is obvious, that the condensational growth
rate depends on the drop’s velocity, relative to the air and
the atmospheric turbulence considerably intensifies the
condensational growth. However the defining factor re-
mains in the difference of the values between the drop‘s
radius and the critical radius. The value of the critical
radius is determined by the value of the supersaturation
r
00
2m
eq eqeq
V
CCCC
rRT

.
Although a size of an individual drop is minute, a
combined surface adds up to a rather large amount. For
example, in a cubic meter of the cloud with 2 g of water
capacity and the average drop size about 20 µm, the total
surface of the drops is about 0.3 m2. In the course of a
few seconds, all the steam that makes the supersaturation
condenses on the growing surface of the drops. The
spersaturation in the cloud decreases to some quasi equi-
librium level which is much less than the water volume
content of the cloud M. It means that at the stationary
conditions the growth of the larger drops occurs due to
the evaporation of smaller drops and the water content of
the cloud practically does not change.
 
2
04πd
dddd
ur Wrrr

0 (11)
and

 
d
dd
Wr Wr
ur
tr



d
. (12)
Here


d
Wr is the density of the size distribution of
the drops,

0d
dd
Wr rM

.
The Equations (11) and (12) describe the dynamics of
the size distribution of the drops at the stationary condi-
tions. Estimations give that the condensational growth
rate under the stationary conditions is very low and it is
doubtful that condensational growth can provide a fa-
vorable conditions for the rain to occur. The rate of con-
densational growth under the stationary conditions does
not rank over 1 μm/h and it would take too much time for
the rain drops to form. At the same time the condensa-
tional growth under the nonstationary conditions is very
important and efficiently changes the drops' sizes.
4. The Drops’ Coalescence
The relative motion of the drops can lead to their colli-
sions and coalescence. The disappearance of two collid-
ing drops and the appearance of the drop with the com-
mon volume lead to the change in the size distribution of
the drops. The Smoluchowski equation describing the
dynamics of the size distribution due to drops collisions
can be represented at the following form

 




1
1
1
1
12122
0
12212
33 33
33
22 12212
0
d,
d,
d,
r
r
r
Wr WrrPr rWr
t
WrrPr rWr
rP rrrWrWrr








. (10)
Here the first term corresponds to the disappearance of
the drops of radius 1 due to their collisions with the
smaller drops; the second term corresponds to the disap-
pearance of the drops of radius 1 due to their collisions
with the larger drops, and the third term corresponds to
r
r

O. NALBANDYAN ET AL.89
the appearance of the drops of radius due to colli-
sions of the smaller drops.
1
r

1
, Pr
Pr
x
The function corresponds to the space
volume that in time unit becomes free of the drops of
radius 2 due to their capture by a larger drop of radius
1. In general the determination of 2
is a very
knotty problem because it is necessary to take into ac-
counting the interaction between colliding drops. The
case of the drops movement caused by the atmospheric
turbulence is simpler because the radius of spatial corre-
lation of turbulent random field of velocities is consid-
erably greater than a distance between drops, participated
in coalescence. It means that the drops that have a possi-
bility to collide move at the same direction and the larger
drop having greater velocity runs down the smaller one.
In that simplified model a function can be
represented as

12
, Prr


r
rr

1
, r

2

2
1212ma
, πPr rvvR , (14)
Here 1 and 2 are the velocities of the larger and
the smaller drops correspondingly, max is the maximal
transversal distance between trajectories of drops which
could have a collision. If a hydrodynamic interaction
between the drops is neglected then max1 2.
However the moving drop in a real viscous air is sur-
rounded by a laminar hydrodynamic boundary layer with
thickness
v vR
Rrr
1
1
1
r
v

 .
The opposite airs particles and drops have to flow
round the moving drop. The tangential shift of the iner-
tialess drop equals
12 1a
Srr2

  ,
where 2

is the hydrodynamic boundary layer thick-
ness of the opposite drop. Taking into account inertia of
the opposite drop and assuming for the simplicity the
circular motion of air round the larger drop we have the
less value of its tangential shift



12 1 21
2
12
π
1exp2
1
d
rr v
Sr
vv r
  







 

,
where


21 2
12 1 2
π
2
vv
rr





 ,
2

is the time of Stokes relaxation of the opposite drop.
Then the maximal transverse distance between trajecto-
ries of drops which could have a collision equals:



max1 2
12 1 21
12 2
12
π
1exp2
1
d
RrrS
rr v
rr r
vvr
 

 

 
 
 


(15)
Thus the possibility of collision depends on the drops’
sizes and velocities. The velocities of the drops depend
on the intensity of turbulent mixing and for any intensity
of turbulence; there exists a minimal radius for the drop
that allows it to absorb smaller drops. For example, for
the following parameters of turbulence L0 = 200 m, Lm =
10 mm and the root mean square velocity of the turbulent
flows a
v = 3 m/s, the low borderline of the diapason for
the turbulent coalescence c equals 43 µm, while at ra
v
= 5 m/s we have c = 32 µm. and at ra v = 10 m/s - c
= 20 µm. If the drops in the cloud are distributed in the
range of diapason of the radiuses, smaller than c, the
coalescence does not take place, the cloud remain stable
and the drops have continue to grow slowly through the
condensational mechanism. The rate of growth in such a
cloud could have pick up for example, with the changed
temperature or increased intensity of turbulence.
r
r
The rain condition can be caused also by injecting the
cloud with drops of the radiuses substantially larger than
c. The required number of injected drops that can intake
all the water content of the cloud is about 200 in a cubic
meter. If the radius of the injected drop is significantly
greater than the average radius of water drops in the
cloud its exponential growth would have occurred.
r
5. The Rain Stimulation by Acoustic Waves
A propagating sound wave evokes a wavy motion of air
particles.
2π
sin
S
aas S
Pt
vc





. (16)
Here S is the sound wave’s pressure, S P

is the
wave period of sound,
s
c is the sound speed. Due to air
viscosity the drops have a wavy motion as well. The am-
plitude of relative velocity of the drop is
22
Sd
das ds
P
vc




 (17)
Awakened by sound wave, the movement of the drops
intensifies the heat and the mass transfer and according
to (11) increases the condensational growth rate. How-
ever, the effect of sound wave could be even more in-
dispensable for the coagulation of the drops. The mecha-
nism that is involved in the coalescence under the impact
of acoustic wave is actually similar to the mechanism
that is involved in the coalescence under the turbulent
conditions. For example, the effectiveness of the coales-

O. NALBANDYAN ET AL. 90
cence under turbulence with parameters Cv = 0.265
m4/3·s–2 and Lm =10 mm can be matched up by the sound
wave with S = 180 Pa and frequency 17 Hz. The coa-
lescence of the drops under the effect of acoustic wave
can be described by the same Equations (13-15), as it
were under the turbulent conditions, however, the value
of the drops’ velocities have to correspond to (17).
P
Figure1. The change of the density of the size distribution of
the drops


d
δWr under the impact of acoustic wave.
It is evident, the best result in the coalescence can be
achieved if the period of the sound equals to the time of
Stokes velocity relaxation of the larger of the two coa-
lescing drops. That would secure the maximum relative
velocity for the larger drop, provide the minimum thick-
ness for its boundary hydrodynamic layer and ensure the
maximum contrast of velocities between the larger and
the smaller drops. Note, that the turbulence frequency of
17 Hz is the optimum for the drops with the radius 67
µm. For the smaller drops the effectiveness of turbulent
coalescence decreases (proportional to ) and acoustic
waves with the correspondent frequency would be more
effective.
2
r
120 Pa the number of the newly formed in a cubic meter
drops came to 10000 and at = 140 Pa came to
13500.
S
P
Thus the stimulating effect of acoustic wave results in
producing not actually the rain drops, but the drops
which are good and ready to coalesce under the turbulent
conditions. The impact of acoustic waves becomes espe-
cially effective with a near-rain cloud. It means that the
drops are distributed in a close proximity to the border-
line of the diapason for a successful turbulent coales-
cence and a single coalescence would be sufficient.
As it were in the case of the coalescence under the ef-
fect of turbulence, for any intensity of the sound wave,
there exists a minimum radius for the drop that is able to
consume a smaller drop. Or, in other words, for a drop of
any size, there exists a threshold point of intensity for the
sound wave which would have just ensured a consump-
tion of smaller drops. For example, for the drops with the
radius 30, the threshold point of intensity for the sound
wave has to be 50 Pa with the frequency 120 Hz, for the
drops with the radius 25 μm we have 60 Pa with the fre-
quency 175 Hz and for the drops with the radius 20 μm
we have 80 Pa with the frequency 265 Hz.
The main concern in the realization of the acoustic
wave rain stimulation is the actual delivery of the right
intensity sound wave to the cloud. For example, the in-
tensity of 100 Pa with the radius of the acoustic spot of
200 m corresponds to the acoustic power of 3 MW. That
power could not very likely be achieved by the means of
electro-mechanical translators. However, a certain per-
spectives have become visible with a connection to the
sound wave’s generation by means of the fuel gas explo-
sion. For example, the energy of a single impulse of in-
dicated intensity and the duration correspondent to nec-
essary frequency is about 30 kJ , which corresponds to
mechanical energy, released from the explosion of 2g of
propane.
The impact of sound wave is especially effective for
the sizes of the drops, which are unable to coalesce under
turbulent conditions, in other words for the drops, which
are in the range for the stable cloud. In Figure 1 the
change of the density of the size distribution of the drops
(109 m–4) under the impact of acoustic wave is
represented. In the represented case the water capacity is
2 g/m3, the radiuses of the drops were initially distributed
in the range from 20 µm to 30 µm. Acoustic wave has
the intensity S = 100 Pa, frequency is 130 Hz, the im-
pact duration is 1s. It can be seen the disappearance of
the drops in the ranges from 27 to 29 µm and from 24 to
25 µm. Their coalescence leads to formation of the drops
with the radius around 32 µm. A total number of ap-
peared single coalesced drops is about 6800 in a cubic
meter. If the size of appeared drops is in the range of an
effective coalescence under turbulence they could repre-
sent the base for the follow up rain. Note that at =

d
Wr 
P
S
P
6. Acknowledgements
The author wants to acknowledge Dr. A.Vardanyan initi-
ated the present investigation, Dr. I.Chunchuzov for nu-
merous discussions and Dr. O.Krymova for the interest
to the present investigation and the technical support.
7. References
[1] L. G. Kachurin, “Fizicheskie Osnovi Vozdeistvia Na
atmjcfernie Processi,” Gidrometeoizdat, Leningrad, 1990.
[2] T. V. Tulaikova, A. V. Mischenko and S. R. Amirova,
“Akusticheskie Dojdi,” Fizmatkniga, Moscow, 2010.