Directory UMM :Data Elmu:jurnal:A:Atmospheric Research:Vol53.Issue1-3.Mar2000:

Atmospheric Research 53 Ž2000. 131–169
www.elsevier.comrlocateratmos

Stochastic effects of cloud droplet hydrodynamic
interaction in a turbulent flow
M. Pinsky a,) , A. Khain a , M. Shapiro b
a

The Institute of Earth Sciences, The Hebrew UniÕersity of Jerusalem, GiÕat Ram, Jerusalem, Israel
Faculty of Mechanical Engineering, Technion-Israel Institute of Technology, Haifa 32000, Israel

b

Abstract
The collision efficiencies of small cloud droplets are calculated in a turbulent flow. A flow
velocity field was generated using a model of isotropic turbulence wPinsky, M.B., Khain, A.P.,
1995. A model of homogeneous isotropic turbulence flow and its application for simulation of
cloud drop tracks. Geophys. Astrophys. Fluid Dyn. 81, 33–55; Pinsky, M.B., Khain, A.P., 1996.
Simulations of drops’ fall in a homogeneous isotropic turbulence flow. Atmos. Res. 40, 223–259x.
The results indicate that in a turbulent flow the collision efficiency is a random value with high
dispersion. The collision efficiencies depend on the initial Žat the infinity. interparticle velocities

and angles of droplet approach. The mean values of the collision efficiency and the mean values
of the collision kernel in a turbulent flow exceed the values measured or calculated for still air
conditions. For instance, under turbulent intensity typical of early cumulus clouds, the mean
values of the collision efficiency for a 10 mm drop-collector are 5 to 9 times as large as the values
in the pure gravity case. In case of a 20-mm-radius drop-collector the mean collision efficiencies
are greater than the corresponding gravity values by a factor ranging from 1.2 to about 6,
depending on the size ratios of colliding droplets.
The collision efficiencies in a turbulent flow are shown to be very sensitive to relative droplet
velocities. Even a small variation Žof about 10%. of the interdrop relative velocity can result in
values of the collision efficiencies twice as large as in the calm air.
Possible effects of the stochastic nature of the collision process on the evolution of the
‘‘mean’’ cloud structure are discussed. It is shown that drop concentration inhomogeneity
accelerates the rate of droplet collisions, which are effective in the areas of the increased droplet
concentration. To illustrate the effect of the mean increase of the collision efficiency on the drop
size spectrum evolution, we solved the stochastic kinetic equation of collision. It was shown that

)

Corresponding author. Tel.: q972-2-6585822; fax: q972-2-5662581.
E-mail address: khain@vms.huji.ac.il ŽM. Pinsky..

0169-8095r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 1 6 9 - 8 0 9 5 Ž 9 9 . 0 0 0 4 8 - 4

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M. Pinsky et al.r Atmospheric Research 53 (2000) 131–169

the increase of the collision efficiency of small cloud droplets controls the broadening of the
initially narrow droplet size spectrum and accelerates the process of rain formation. The results
indicate that the increase of the collision efficiency of small cloud droplets in a turbulent flow is,
possibly, the mechanism responsible for in situ observed broadening of droplet spectra in
undiluted cores of cumulus clouds.
In numerical simulations of drop collisions within a turbulent flow, a new effect was
discovered, namely, in some cases the collision efficiencies in a turbulent flow appeared to be
equal to zero Žno collisions.. Analysis shows that in the case the ratio of droplet fall velocities is
less than a certain critical value Žwhich depends on the ratio of particle sizes. the falling particles
form a tandem, in which particles fall with equal velocities. q 2000 Elsevier Science B.V. All
rights reserved.
Keywords: Stochastical processes in clouds; Turbulence effects; Cloud microphysics; Droplet spectrum

formation

1. Introduction
1.1. Different approaches in the description of microphysical processes in clouds
When considering methods of description of microphysical processes in clouds, we
can distinguish three main approaches: deterministic, ‘‘mixed’’ and fully stochastic.
A typical example of the deterministic approach is the utilization of the equation of
continuous growth ŽPruppacher and Klett, 1997.. In this approach, the growth rate of
large drop-collector is fully determined by the concentration of smaller droplets and
terminal fall velocities of the drop-collector and small droplets. In case of equal
concentration of small droplets, initially equal drop-collectors will remain equal all
through the time period of their collision growth. Thus, the approach can hardly be used
to reproduce the formation of a wide drop size spectrum. In numerical models, in which
the equation of continuous growth is used to describe collisions of rain drops and cloud
droplets Ž‘‘bulk microphysics’’ parameterization models. the size spectra of rain drops
Žas well as of other hydrometeors. are prescribed a priori, say, in the form of the
Marshall–Palmer distribution. Thus, the approach is unfit to explain the process of size
spectra formation of cloud hydrometeors.
An example of the ‘‘mixed’’ approach is the utilization of the stochastic kinetic
equation of collision ŽPruppacher and Klett, 1997. in the so-called spectral-microphysics

models Že.g., Kogan, 1991; Khain and Sednev, 1996; Reisin et al., 1996.. In these
models size distribution functions of hydrometeors of different type are calculated
during the model integration. These models claim to reproduce the process of the
formation of drop size spectra as well as the spectra of ice particles. The utilization of
the stochastic kinetic equation of collisions means that this approach takes into account
some elements of ‘‘stochasticality’’ of collision processes. The equation permits one to
obtain comparably complicated shapes of size distribution of cloud hydrometeors.
Note, however, that the approach used in cloud models with the resolution of several
hundred meters cannot be referred to as a fully stochastic one. Because of a crude spatial

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133

Žand time. resolution, drop Žas well as ice particles. concentrations as well as the
collision kernels used in the stochastic kinetic equation actually denote values obtained
by averaging over large volumes of the order of 10 7 to 10 10 m3. A question arises,
whether microphysical processes in clouds can be described adequately under the
assumption that the concentration of cloud hydrometeors and the collision kernels are
constant within such large volumes.

In other words, the question is: what is the contribution of small-scale stochastic
fluctuations of drop concentration and other parameters to cloud microphysics? The
essence of the problem is somehow similar to the role of ‘‘subgrid’’ processes in
large-scale global atmospheric circulation models. It is generally recognized that subgrid
processes Že.g., convective heating. are of crucial importance when large-scale processes
are simulated. The role of stochastic ‘‘subgrid’’ processes in cloud evolution is not
widely recognized yet because of many uncertainties.
The third approach can be referred to as a fully stochastic one. According to this
approach, drop collisions are simulated within realistic stochastic fields of velocity,
concentration, supersaturation, etc., without any parameterization. The main purpose of
this approach is to understand the effects of stochastic processes on drop growth Žsay, by
collisions. and to find ways permitting parameterization of these fine effects in case
their contribution is significant.
Numerous recent theoretical and laboratory studies Žsee below. indicate that the
impact of the small-scale stochastic effects is quite important so far as it provides an
insight into some fundamental problems of cloud physics.
1.2. Turbulence effects induced by droplet inertia
The main source of ‘‘stochasticality’’ of cloud microphysical processes is cloud
turbulence. Clouds are known to be areas of enhanced turbulence. Different aspects of
influence of cloud turbulence on cloud evolution have been discussed in a special issue

of the Atmospheric Research, Vol. 43, 1996.
We will analyze here some stochastic effects induced by the inertia of drops and ice
particles moving within a turbulent flow. The problem has attracted the attention of
investigators for a long time. About 50 years many studies claiming to reveal turbulence
effects were criticized by other authors, therefore, no consensus has been reached toward
1993, which has been stated by Beard and Ochs Ž1993. in their review. Details of the
discussion can be found in Pinsky and Khain Ž1996, 1997c., Pruppacher and Klett
Ž1997.. A new splash of interest to the effects of turbulence on cloud microphysics takes
place during a few past years.
We will mention three main mechanisms, by means of which turbulence can
potentially influence droplet collisions: Ža. change in the swept volume, Žb. formation of
concentration inhomogeneity of cloud hydrometeors and Žc. change of the drop–drop,
drop–ice and ice–ice collision efficiencies. All these turbulence effects are related to the
drop Žor ice particles. inertia, due to which their motion and interaction in a turbulent
medium is much more complicated than in calm air.
Let us consider these three mechanisms in more detail.

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1.2.1. Increase in the swept Õolume
Khain and Pinsky Ž1995. showed that due to their inertia drops deviate from the
surrounding air parcel tracks, that is, drops in a turbulent flow never fall with their
terminal fall velocities. Drops of different inertia respond differently to the wind shears
of the air flow. As a result, additional drop–drop relative velocities arise leading to the
increase of the swept volume and frequency of drop collisions. Khain and Pinsky Ž1995.
showed that, in case of a simple background flow with the wind shear, the increase of
the swept volume induced by turbulencerinertia accelerates significantly the broadening
of the drop size spectrum by collisions. Khain and Pinsky Ž1997., Pinsky and Khain
Ž1997a. generalized the study to the case of a 3-D turbulent flow. Turbulence was
assumed to be isotropic. It was shown that relative velocities between small cloud
droplets were caused mainly by flow accelerations, while larger drop responded rather to
flow shears. In these studies, the effects of the swept volume increase on the drop size
spectrum evolution were taken into account by the increase of mean-square drop–drop
relative velocities.
The increase of relative velocities and the swept volume in a turbulent flow was
shown to be especially pronounced in case of drop–ice and ice–ice collisions ŽPinsky
and Khain, 1998; Pinsky et al., 1998a.. This effect can be attributed to the following.
The increase in the swept volume in a turbulent flow is proportional to the ratio of

inertia induced relative velocities between particles to the relative velocity induced by
gravity. Ice particles having the same Žor even a larger. mass Žinertia. as compared to
water drops have significantly lower terminal fall velocities induced by gravity. Thus,
the ratio: Žinertia induced relative velocityrgravity induced relative velocity. is higher
for low-density ice particles.
1.2.2. Formation of particle concentration inhomogeneity
Pinsky and Khain Ž1996. investigated drop motion in a turbulent flow using a model
of isotropic turbulence Žsee Appendix A.. Numerous examples of drop tracks for drops
of different size demonstrate a significant influence of turbulence on different aspects of
drop motion. The tendency of drop tracks to collect along isolated paths was demonstrated. The existence of areas of two types within a turbulent flow was revealed. Due to
their inertia, drops tended to leave the areas of enhanced flow curvature. Locating
initially beyond these areas, drops tended to avoid them. At the same time drops tended
to concentrate in the areas of low curvature. Thus, the effect of drop collection and
formation of drop concentration inhomogeneity was demonstrated in turbulent flows
typical of cloud turbulence. Spatial scales of drop concentration inhomogenieties were
found to be dependent on drop mass Žinertia., so that for rain drops of a few hundred of
mm, the scales are of several meters and even tens of meters. For small cloud droplets,
these scales are of one to a few centimeters.
Earlier, Maxey Ž1987. and Wang and Maxey Ž1993. demonstrated the formation of
concentration inhomogeneity of inertial particles moving within turbulent flows. In the

last study the direct numerical simulation ŽDNS. has been used for turbulent flow
generation.
Pinsky and Khain Ž1997b. and Pinsky et al. Ž1999a. theoretically investigated some
characteristics of the cloud fine structure. Pinsky and Khain Ž1997b. showed that due to

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135

drop inertia drop flux velocity tends to be divergent even in a non-divergent turbulent
flow. The drop flux divergence turned out to be maximal for drops of 100-mm radii.
Drops of smaller radii follow air track of non-divergent flow more closely. The
sensitivity of drops of larger size to air velocity fluctuations is lower, so that gravity
induced fall velocity dominates. Thus, the rate of concentration inhomogeneity was
found to be greatly dependent on the drop size. Inhomogeneities of drop concentration
were most pronounced for drops of 100-mm radii. However, drop flux divergence for
small cloud droplets Ž10- to 15-mm radii. was found intense enough to provide
significant drop concentration fluctuations, which can be of 20% to 40% of the mean
drop concentration. The divergence of drop flux velocity for small cloud droplets was
found to be proportional to the dissipation rate and square of droplet radii ŽPinsky et al.,

1999a.. They showed that the characteristic spatial scale of these fluctuations of droplet
concentration is equal to one or a few centimeters, hence, clouds have fine structure
Ž‘‘inch clouds’’.. The mechanisms of formation of concentration inhomogeneity of
inertial particles in different turbulent flows were studied recently by Elperin et al.,
Ž1996., Wang et al. Ž1998. and Zhou et al. Ž1998..
Note that the problem of the existence of small-scale concentration fluctuations in
clouds Žso-called preferential concentration. remains in the cloud-physics community to
be a controversial issue Žsee a comment of Grabowski and Vaillancourt, 1999 on the
paper by Shaw et al., 1998.. Recently, these centimeter-scale drop concentration
fluctuations have been retrieved from a series of drop-arrival times providing in situ
measurements using the Fast FSSP ŽPinsky et al., 1998b.. The RMS amplitude of
small-concentration fluctuation in a cumulus cloud analyzed turned out to be 30% of the
mean droplet concentration, which agreed well with the theoretical evaluations of Pinsky
et al. Ž1999a..
The process of collisions is known to be more effective in case of inhomogeneous
spatial distribution of particles Že.g., Voloshuk and Sedunov, 1975; Kasper, 1984.. Using
a simplified model of the formation of drop concentration inhomogeneity during the
process of drop collisions, Pinsky and Khain Ž1997b. demonstrated a significant
acceleration of the process of rain drop formation by small droplet collisions. Zhou et al.
Ž1998. demonstrated a significant increase of the collision rate of inertial particles using

a turbulence model based on the solution of the Navier–Stokes equation.
1.2.3. Collision efficiencies of particles
One of the most efficient ways owing to which turbulence can influence microphysical processes in clouds is its impact on the collision efficiencies of inertial particles.
Note that in all studies mentioned above the collision efficiency was taken equal to that
in the pure gravity case. In several studies the problem of hydrodynamic interaction
between droplets in a turbulent flow has been considered ŽAlmeida, 1976, 1979; Grover
and Pruppacher, 1985; Koziol and Leighton, 1996.. Almeida Ž1976, 1979., Koziol and
Leighton Ž1996. calculated the track of one drop in the vicinity of another drop under
the assumption that the relatiÕe drop Õelocity at separation distances greater than
seÕeral drop radii is equal to the difference of their still air terminal fall Õelocities.
Under this assumption, the motion of approaching droplets during their hydrodynamic
interaction is influenced by turbulent fluctuations with scales of several drop radii, i.e.,

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of the order or less than the Kolmogorov microscale Žabout 1 mm.. Corresponding
scales of turbulence vortices lie within the viscous subrange. According to the results
obtained by Koziol and Leighton Ž1996., the influence of such low energy vortices on
the relative drop motion seems to be small.
According to Khain and Pinsky Ž1997. and Pinsky and Khain Ž1997a., turbulence-induced relative velocities between drops are caused by turbulent fluctuations lying within
the inertial subrange of the spectrum. Thus, drops separated by a distance of several
radii in a turbulent flow have relative velocities that differ from those induced by pure
gravitation settling. This idea was used first by Grover and Pruppacher Ž1985., who
investigated effects of turbulence on the ability of large rain drops to collect small
particles. According to the results obtained, turbulence increases the rate at which large
cloud drops collect particles in the radius range of 0.5 to 2.5 mm significantly. However,
the authors used a simplified one-dimensional model of turbulent flow, which limits the
applicability of their results.
In a 3-D turbulent flow, the directions of drop velocities do not coincide with the
direction of their approach. This factor can lead to a significant change of drop tracks
during mutual approach and hence, to a significant change of the collision efficiency as
compared with the case of gravitational collision.
Vohl et al. Ž1996, 1999. reported the results of laboratory experiments of drop-collector growth in a wind tunnel indicating a pronounced increase of drop growth rate under
turbulent conditions.
In Pinsky et al. Ž1999b., the hydrodynamic problem of collisions of small droplets
Žwith radii less than 30 mm. in a turbulent flow was analyzed. It was shown that when
dealing with the hydrodynamic problem of drop collisions, drop-air relative velocities
formed in a turbulent flow due to drop inertia should be used as initial conditions Žat
infinity. for the calculation of mutual drop tracks, and not the difference in the still air
terminal velocities.
The collision efficiencies of cloud droplets were calculated using the superposition
method. A certain number of calculations shows that the droplet collision efficiencies in
a turbulent flow are usually larger than those in calm atmosphere. It is stressed that in a
turbulent flow the droplet collision efficiency is a random value with a significant
dispersion.
In the present study, we report some results concerning the stochastic nature of the
collision efficiency of particles in a turbulent flow. In a special section, we discuss the
possible contribution of these processes to cloud microphysics.

2. Droplets hydrodynamic interaction in a turbulent flow
This section is based on the study of Pinsky et al. Ž1999b..
2.1. Conditions of droplet collisions in a turbulent flow
The motion of inertial droplets in a turbulent flow depends on wind shears and flow
accelerations. As it was shown by Champagne Ž1978. Žsee also, Pinsky et al., 1999a,b.,

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137

the maximum of the wind shear spectrum is reached in the transition area from the
inertial to the viscous subrange that corresponds to the length scale of about 2 cm. At a
smaller length scale of the wind shear spectrum sharply decreases due to the viscous
effect. The maximum in the spectrum of acceleration can be expected to be located near
the maximum of the shear spectrum.
The location of the maximum in a spectrum is known to be closely connected with
the correlation length. The fact that the maximum of the shear and acceleration spectra
are attained at 1 to 2 cm means that the correlation lengths of the velocity shear and the
inertial acceleration are as small as 1 to 2 cm as well. Thus, shears, as well as flow
accelerations become independent at distances greater than about 2 cm. At the same
time, at distances below 1-cm shears as well as the inertial accelerations are well
correlated. It means that within air volumes of these linear scales shears and inertial
accelerations do not experience significant variations.
While in still air drops quickly accelerate towards their terminal fall velocity, in
spatially non-uniform flows Žsheared and turbulent flows. drop inertia always leads to
the formation of additional non-zero velocity deviations from the surrounding flow.
Thus, in a turbulent flow one can see no adaptation of drop velocity to the surrounding
flow. Such adaptation does not take place because the flow itself changes in surrounding
of a moving drop and the drop experiences flow accelerations during each instance of its
motion. Thus, in a turbulent flow drops never fall with their terminal fall velocity. The
adaptation Žrelaxation. period can be defined in this case as a time period during which
initial conditions Že.g., initial drop velocities. influence drop velocity. When this
adaptation period is over, the drop velocity and its deviation from the flow velocity
depend only on the drop mass and flow parameters Žsuch as the values of the shear,
curvature of air parcel tracks, flow velocity, etc... Taking into account that the duration
of relaxation periods is short, especially for small drops, changes of drop velocity
deviations can be regarded as transitions from one adjusted state to a subsequent state. It
means that adjusted drop velocity deviations change with the characteristic time scales
of the surrounding air flow.
The time correlation scale of drop-air relative velocity can be evaluated as the
lifetime of those turbulent vortices, corresponding to the maximum shears and accelerations of the air. The lifetime for turbulent vortices t with the linear length l within the
inertial subrange can be evaluated as t f ´ y 1r3l2r3. For l of about 1 cm, t is about
0.2 s.
The interdrop relative velocity induced by the difference in drop inertia is a direct
consequence of the formation of drop-air relative velocities. Thus, interdrop relative
velocity has typical temporal and spatial scales similar to those for the velocity of a
larger drop relative to the surrounding air.
Some characteristics of relative velocity between droplets as seen from the calculations of drop motion within a turbulent flow generated by the turbulent model by Pinsky
and Khain Ž1995, 1996. are presented in Figs. 1 and 2.
Fig. 1 shows the distribution of interdrop relative velocity for 10- and 20-mm-radii
droplets normalized by the difference in their terminal fall velocity as seen from the
results of calculations using the turbulent model mentioned above. The value of
dissipation rate used in the calculations was set equal to 100 cm2 sy3 , typical of the

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M. Pinsky et al.r Atmospheric Research 53 (2000) 131–169

Fig. 1. The distribution of 10- and 20-mm-radii interdrop velocity normalized by the difference in drop
terminal velocities Žas seen from the model of turbulent flow, Pinsky and Khain, 1995..

early stage of cumulus clouds. One can see a comparably wide symmetric distribution.
The relative velocities between these comparably large cloud droplets can be by
20–30% largerrsmaller than the value in the pure gravity case.
The distribution of relative velocities is accompanied by the distribution of angles at
which drops approach ŽFig. 2.. Zero value of the angle corresponds to pure gravity case
with the drop approach in the vertical direction. One can see that in turbulent flows the
angles of drop approach can be as large as 40–508 even in the comparably low intensity
turbulence. As it will be shown below, these factors lead to a significant dispersion of
the collision efficiencies in a turbulent flow. In a turbulent flow the collision efficiency
becomes to be random value with a wide distribution.
Fig. 3 shows longitudinal Žcurve A. and lateral Žcurve B. spatial correlation functions
of x-component of relative velocity between 10 and 20 mm -radii droplets One can see
that the spatial correlation lengths are about 0.8 cm for longitudinal and 1.1 cm for
lateral spatial correlation functions, respectively.
To understand the nature of the ‘‘background’’ velocity field in which hydrodynamic
interaction between drops about to collide takes place, we will compare the characteris-

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139

Fig. 2. The distribution of angles at which drops of 10- and 20-mm-radii approach Žas seen from the model of
turbulent flow, Pinsky and Khain, 1995..

tic spatial and time scales of relative interdrop velocities induced by the drop inertia
with the characteristic scales of hydrodynamic interaction between droplets.
As it is known ŽPruppacher and Klett, 1997., drops hydrodynamic interaction begins
when drops approach at the distance of several radii of the larger drop. In this study we
deal with droplets with the radii below 30 mm. Thus, the spatial scale characterizing
hydrodynamic interaction between droplets is below 0.03 cm. A great number of
simulations of droplets motion during their hydrodynamic interaction shows that typical
hydrodynamic interaction time Žthe time during which droplets are separated by distances not exceeding several drop radii. is about 0.01 s. to 0.02 s.
Thus, in case of a non-uniform flow the difference in terminal fall velocities should
be added by the interdrop velocity induced by drop inertia. Besides, time and spatial
correlation scales of drop-air and interdrop relative velocities in a turbulent flow are
much greater than the characteristic time and spatial scale of hydrodynamic droplet
interaction.
Taking into account the differences in spatial and time scales mentioned above, the
‘‘background’’ drop-air and interdrop relative velocities within the region of hydrodynamic interaction could be considered unchangeable both in space and time. The inertia

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M. Pinsky et al.r Atmospheric Research 53 (2000) 131–169

Fig. 3. Longitudinal Žcurve A. and lateral Žcurve B. spatial correlation functions of x-component of the relative
velocity between 10- and 20-mm-radii droplets of Žsee, Pinsky et al., 1998c for more detail.. One can see that
the spatial correlation lengths are about 0.8 cm for longitudinal and 1.1 cm for lateral spatial correlation
functions, respectively.

force acts similarly to the gravity force through the process of drop hydrodynamic
interaction. All changes of drop velocities during hydrodynamic interaction take place
only due to the mutual influence of drops.
Thus, we will neglect the influence of turbulent vortices Žvelocity fluctuations. at
scales of hydrodynamic droplet interaction. Instead, we will take into account inertia-induced velocities at the periphery of drop hydrodynamic interaction zone. These velocities will be included into the boundary conditions when solving the hydrodynamic
problem of droplet interaction.
2.2. Drop motion equations and geometry of drop collisions in a turbulent flow
The equation system and approach to the calculation of the collision efficiencies are
described in Pinsky et al. Ž1999b. in detail. A short description is presented in Appendix
B.

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141

To calculate the hydrodynamic interaction between two spheres moving in the air
flow, we use here the superposition method ŽPruppacher and Klett, 1997.. According to
this method each sphere is assumed to move in a flow field induced by its counterpart
moving alone. Dating back to Langmuir Ž1948., this method has been successfully used
by many investigators Že.g., Shafrir and Gal-Chen, 1971; Lin and Lee, 1975; Schlamp et
al., 1976. for a wide range of droplet Reynolds numbers.
In Appendix B we show that drop motion equations describing their hydrodynamic
. flows can be written in the
interaction in a inhomogeneous Žfor example, turbulent™
coordinate frame moving with the background velocity Va as:

™

dV1X
dt

sy

™

dV2X
dt

sy

1

t1
1

t2

ž V™ y V™ y™u / ,

Ž 2.1 .

ž V™ y V™

Ž 2.2 .

X
1

X
1`

X
2

2

™/

X
2` y u1

™

where indices 1,2 denote the number of droplets, ViX is drop-air relative velocity of ith
drop Ž i s 1,2., t i s Vi trg ™
is the characteristic relaxation time of ith drop. This time is a
measure of drop inertia. Vi is the velocity of ith drop, ™
u1 and ™
u 2 are the perturbed
velocity fields induced by drop 1 in the point of the location of the center of drop 2, and
by drop 2 in the point of the location of the center of drop 1, respectively, so ™
u 2rt 1 is
the acceleration caused by air ™
velocity perturbations induced by the second drop.
In Eqs. Ž2.1–2.2., velocity Vi`X can be interpreted as a relative drop-air velocity of ith
droplet at sufficiently large distances where the velocity field induced by the other drop
in a drop pair can be neglected.
The system Ž2.1–2.2. is similar to a corresponding one usually used for the
calculations of the collision efficiency by the superposition method in the pure gravity
case Že.g., Lin and Lee, 1975.. The difference is that the drop velocity at infinity is not
equal to the terminal fall velocity. For example, for drop 1:

™
™
dVa
dVa
™
™
st1 gy
1` s V1,t e z y t 1

™
VX

dt

ž

dt

/

.

Ž 2.3 .

This velocity can be interpreted as a relative drop-air velocity of droplet 1 at sufficiently
large distances, where the velocity field induced by drop 2 can be neglected. V1 t is the
terminal velocity of drop 1 in calm atmosphere, ™
e z is the unit vector directed downward.
This condition is the boundary condition at infinity. Since the spatial correlation scale of
these inertia-induced velocities is 1 to 2 cm for small droplets, these velocities may be
assumed constant at the distances of drop dynamic
™ interaction. They also can be
assumed to be time-independent. The term yt 1dVardt in Eq. Ž2.3. is actually an
additional relative drop velocity induced by the drop inertia. This parameter, which was
not taken into account in earlier studies, is of crucial importance
™ since it determines the
main effects of turbulence on the drop collision efficiency. dVardt is the acceleration of
the background flow, which can be regarded as unchangeable during drop hydrodynamic
interaction.

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M. Pinsky et al.r Atmospheric Research 53 (2000) 131–169

™

The term Ž ™
g y dVard t . does not change during the process of drop hydrodynamic
interaction and can be interpreted as ‘‘modified gravity’’. In case illustrated in Fig. 1,
mean deviations of ‘‘modified gravity’’ from pure gravity are about 20 %, but in some
cases they can exceed 40%.
The geometry of droplet approach corresponding to the droplet motion equations is as
follows. We consider the droplet motion in the coordinate frame  xX , yX , zX 4 moving with
the velocity of the background flow. The frame is oriented in such a way that the z-axis
should be directed anti-parallel to the relative velocity.
Fig. 4a,b illustrates the definition of the collision efficiency in case Ža. of a turbulent
flow and Žb. calm air, respectively. We consider
drop motion in the frame  xX , yX , zX 4
™
moving with the background flow velocity
. The™frame is oriented in such a way that
™X Vaand
X
X
4 plane should contain vectors V1`
the  xX , z™
V2`
, and, correspondingly, the relative
™
™
X
X
X
velocity V12` s V1` y V2` . This approach permits one to reduce the three-dimensional
drop interaction problem to a symmetric geometry Žquasi two-dimensional. problem.1
In the turbulent case, xX-components of relative drop velocities at the infinity are
non-zero but equal to each other, as a result of the choice of the  xX , yX , zX 4 coordinate
frame.
The zX-axis can have any direction depending on the direction of relative velocity
™
V12` . It is important that droplets approach does not coincide with the directions of drop
velocities at infinity.
™
In case of still air the vector V12` is the difference in the droplets terminal velocities.
In a turbulent flow, the drop-air relative velocities are oriented at angles a 1 and a 2
relative to their approaching direction ŽFig. 4a.. These angles are determined
the
™X y™uby and
directions
of
drop
velocities
with
respect
to
the
surrounding
air
velocities:
V
2
™X y™u . The  xX , zX 4 plane is a symmetry plane for the flow fields induced1`by moving
V2`
1
droplets. This symmetry arises due to the fact that the velocity fields induced by the
moving droplets are axisymmetric Žsee Appendix B..
Droplets initially located in the plane  xX , zX 4 will remain within this plane during
drop approach. If one of the droplets does not initially lie in this plane, its track will not
lie in the plane either. In this case there exists a point symmetric to the initial position of
the drop with respect to this plane. A drop put at the point will have a track symmetric
to the track of the first drop with respect to the plane. In other words, droplet motion
during their hydrodynamic interaction appears to have a mirror symmetry with respect to
the  xX , zX 4 plane.
In the pure gravity case the geometry of drop interaction is axisymmetric with respect
to zX axis ŽFig. 4b.. One can see that the determination of the collision cross-sections
and the collision efficiencies in a turbulent flow can be defined in a manner similar to
the case of pure gravity collisions. Fig. 4c,d illustrates the definition of the collision
efficiency in a turbulent flow and calm air, respectively. In pure gravity case grazing
tracks are axisymmetric with respect to the zX-axis ŽFig. 4d.. In a turbulent flow segment
X c Žextension of the collision cross-section in the xX-direction. is located asymmetrically
with respect to the zX-axis ŽFig. 4c..

1

™X

™X

Strictly speaking, one can always define a plane containing two vectors V1` and V2` by their parallel
transfer.

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143

Fig. 4. Illustration of the definition of the collision efficiency in case Ža. of a turbulent flow and Žb. calm air,
X X X
respectively. The  x , y , z 4 coordinate frame moving with respect to the immovable  x, y, z 4 frame with the
™
™X
X
velocity of the background flow Va and with z axis directed anti-parrallel to relative velocity V12` . We define
™X
X X
X
X
the  x , z 4 plane Žthe x axis is perpendicular to the z axis. in such a way that it contains both vectors V1` and
™
X
X
V2` . While in the pure gravity case grazing tracks are axisymmetric with respect to the z axis Žb., one can
X
show that in a turbulent flow segment Žthe extension of the collision cross-section in x -direction. will be
X
located asymmetrically with respect to the z axis Ža. Žsee Pinsky et al, 1999b for more detail.. The definition
of the collision efficiency in Žc . a turbulent flow and Žd. calm air. While in the pure gravity case grazing
X
tracks are axisymmetric with respect to the z axis Žd., one can see that in a turbulent flow segment X c
X
Žextension of the collision area in x -direction. is located asymmetrically with respect to the zX axis Žc..

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The grazing tracks form the effective collision cross-section of area Sc on the plane
perpendicular to the direction of interdrop relative velocity at infinity. In case of a
turbulent flow the effective collision cross-sections are not circular and can be shifted
with respect to the zX-axis, remaining, however, symmetric with respect to the  xX , zX 4
plane.
The collision efficiency in a turbulent flow Et is calculated as the ratio of Sc to the
geometric cross-section p Ž a1 q a 2 . 2 :
Et s Scrp Ž a1 q a2 .

2

Ž 2.4 .

where a1 and a 2 are the radii of interacting droplets. The collision efficiency for pure
gravity collisions Eg is determined in a similar way, but allowing for the axisymmetric
geometry it can be calculated as:
2

Eg s Yc2r4 Ž a1 q a2 . .

Ž 2.5 .
X

In this case the collision cross-section is a circle with the center located on the z -axis.
The linear collision efficiency in  xX , zX 4 and  yX , zX 4 planes can be defined as
E x s Xcr 2 Ž a1 q a 2 .

Ž 2.6a .

E y s Ycr 2 Ž a1 q a 2 . ,

Ž 2.6b .

and

respectively. As it was mentioned above, segment Yc is located symmetrically relative to
the  xX , zX 4 plane. In calm air E x s E y s Ycrw2Ž a1 q a 2 .x. In a turbulent flow E x is
usually not equal to E y .

3. Results of calculation of the collision efficiencies in a turbulent flow
In the calculations the dissipation rate of turbulent kinetic energy was taken equal to
100 cm2 sy3 , typical of early cumulus clouds.
The calculation of the collision efficiency was carried out in two steps. The first step
consisted in the calculation of drop-air and interdrop relative velocities between two
closely separated droplets in a turbulent flow. The following algorithm was used.
Applying the equation of the drop motion Žsee Khain and Pinsky, 1995., tracks of drops
of different masses in a turbulent velocity field were calculated Žwithout taking into
account drop interaction.. The turbulent flow velocity field was generated by the model
described in detail by Pinsky and Khain Ž1995, 1996. Žsee Appendix A for more details..
The model permits one to generate different realization of the random velocity field
components with given latitudinal and lateral correlation functions and a spatial structure, which obeys the properties of isotropic turbulence both in the viscous and inertial
subranges ŽMonin and Yaglom, 1975.. The model generates turbulent velocity fields
within a wide range of spatial scales from about 5 mm to 50 m.
The drop-air relative velocities and the interdrop relative velocity were calculated in
points of droplet track intersection. These velocities were calculated in different points
randomly chosen within a turbulent flow for each pair of drop sizes in order to take into

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account the stochastic nature of drop collisions. As a result, a set of the drop-air and
interdrop relative velocities Žwithout hydrodynamic interaction. was calculated for each
drop pair.
These velocities were used as boundary conditions at the second step for solving the
hydrodynamic problem of drop interaction. Varying the initial location of one of the
smaller droplets, the grazing trajectories and the effective collision cross-section were
determined. Then, the collision efficiencies were calculated. Equations of drop motion
were solved using the fifth order Runge–Kutta method with automatic precision control
and automatic choice of integration time step ŽPress et al., 1992..
Fig. 5 presents some examples of the collision cross-sections for drop pairs containing 10- and 20-mm-radii droplets in different points of a turbulent flow Žshaded areas..
Small open circles mark the collision cross-sections in calm air Žpure gravity case..
Large open circles are geometric cross-sections. The collision efficiencies Et and Eg
calculated for the cases of the turbulent flow and the pure gravity setting, respectively,
are presented as well.
Significant variations of the collision efficiency in the turbulent flow can be seen.
Thus, the collision efficiencies are sensitive both to relative interdrop velocity at the
infinity and the angle of drop tracks intersection.
Fig. 6a,b presents the mean values of the collision efficiencies in a turbulent flow
Žcurve A. and in calm air Žcurve B. as a function of the ratio of smaller droplet and
droplet-collector radii. The drop-collector radius was 10 mm in Fig. 6a and 20 mm in
Fig. 6b.
Fig. 7 shows a scattering diagram indicating the dependence of the collision
efficiency Žnormalized with respect to the pure gravity value. on the interdrop relative
velocity Žnormalized to the difference of drop terminal fall velocities induced by gravity.
Fig. 8 presents the distribution of the collision efficiencies of 10- and 20-mm-radii
droplets normalized with respect to the pure gravity value.

Fig. 5. Examples of the collision cross-sections for drop pairs containing 10- and 20-mm-radii droplets in
different points of a turbulent flow Žshaded areas.. The collision cross-sections in calm air Žpure gravity case.
are marked by small open circles. Large open circles are geometric cross-sections. The collision efficiencies Et
and Eg calculated for the cases of the turbulent flow and the pure gravity setting, respectively, are presented as
well. Significant variations of the collision efficiency in the turbulent flow can be seen. Hence, the collision
efficiencies are sensitive to the relative interdrop velocity at the infinity and to the angle of drop tracks
intersection.

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147

Fig. 7. A scattering diagram indicating the dependence of the collision efficiency Žnormalized to the pure
gravity value. on the interdrop relative velocity Žnormalized with respect to the difference of drop terminal fall
velocities induced by gravity.

Several conclusions can be drawn from the figures.
Ža. The mean values of the collision efficiencies in case of a turbulent flow are higher
than in the pure gravity case. The effect of turbulence increases with the decrease of the
ratio of the radii of smaller droplets to drop-collector. The increase of the mean values
of the collision efficiency in case of a 10-mm drop-collector significantly greater than in
case of a 20-mm-radius drop-collector: it can be by factor of 5 to 9 as large as the
corresponding values in the pure gravity case ŽFig. 6a.. In case of 20-mm-radius
drop-collector the mean collision efficiency is greater than corresponding gravity values
by factor ranged from 1.2 to about 6, depending on the ratios of colliding droplet radii
ŽFig. 6b..

Fig. 6. Mean values of collision efficiency as a function of the ratio of the radius of smaller droplet and the
radius of the drop collector in a turbulent flow Žcurve A. and in calm air Žcurve B.. In Ža. the radius of the
drop collector is 10 mm, in Žb. the radius of the drop-collector is equal to 20 mm.

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Fig. 8. A distribution of the collision efficiencies of 10- and 20 mm-radii droplets normalized with respect to
the pure gravity value.

Žb. The distribution of the collision efficiencies is rather wide. Comparably small
changes of the interdrop relative velocity can lead to a significant increaseror decrease
of the collision efficiency. The significant dispersion of the collision efficiency for the
same absolute values of the interdrop velocities can be attributed to the fact that the
collision efficiency largely depends not only on the magnitude of the interdrop relative
velocity, but on the angles of droplet approach as well.
Žc. In some cases drops are unable to collide Žcollision efficiencies are equal to zero.
in spite of the fact that at the periphery of the area of drop hydrodynamic interaction
drop approach takes place Žin Fig. 7, these cases are marked by symbols ‘‘zero’’..
Calculations show that the percent of cases with zero collision efficiencies varies from
0% to 20% of the total number of cases. The percent of these cases depends of the radii
of interacting droplets.
The results show that the increase of the collision efficiency in a turbulent flow
depends on the intensity of turbulence and on sizes of colliding droplets. We interpret
the latter as follows. The smallest droplets Žbelow about 5 mm in radii. follow air stream
lines. Thus, turbulence hardly influences the collision efficiency in case a drop pair
contains droplets, whose radii are smaller than 5 mm.

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149

The collision efficiencies in the pure gravity case grow rapidly with the drop size. For
example, the collision efficiency of drops with radii 30 and 10 mm is as large as 0.5
ŽPruppacher and Klett, 1997.. As the values of the collision efficiency are limited to 1,
an additional increase of the collision efficiency due to turbulence cannot be significant.
At the same time, the collision efficiency of small droplets of different radii Žsay, 3
and 10 mm; 4 and 20 mm. is about 0.02, so that a significant potential towards an
increase of the collision efficiency can be supposed to be real. As it was shown by
Pinsky et al. Ž1999a., small droplets tend to deviate from air stream lines in phase. Thus,
the larger the difference in the droplet size Žinertia., the higher must be relative velocity
induced by drop inertia and the higher must be the increase of the collision efficiency in
a turbulent flow ŽFig. 6..
Effects of conclusions Ža. and Žb. on drop collisions will be discussed below in
Section 4.
The mechanism leading to no collisions is discussed in Appendix C. Here we
summarize the conclusions concerning the mechanism. Analysis shows that in the case
the ratio of the droplets fall velocities is less than a certain critical value Žwhich depends
on the ratio of particle masses., the interacting droplets form a tandem, in which droplets
fall with the same velocities. The fall velocity of the tandem is greater than the velocity
of a greater particle. In the case of two nearly equal particles Ždrops. the fall velocity of
the tandem exceeds the velocity of each particle in the case of independent fall by about
50%. The increase of the fall velocity of the drop tandems increases their probability to
collect other droplets. This possible effect requires special analysis. The formation of
drop tandems can increase the probability of triple collisions.
The physical mechanism of the formation of such tandems involves in the interaction
of velocity fields induced by the droplets during their fall. On the one hand, the
difference in the terminal velocities due to gravity and the additional relative velocity
stemming from the drop inertia Žin case additional velocity increases the total interdrop
relative velocity. favors the drop collision. On the other hand, the lowermost drop
experiences acceleration by the flow induced by the upper one Žby the collector. which
favors droplet repulsion. This influence increases during the particle approach. In cases
of a comparably small ratio between droplet velocities at the infinity Žbeyond the zone
of hydrodynamic interaction., both particles reach the same velocity at a certain
separation distance and continue falling as a tandem.

4. Stochastic nature of drop collisions and its importance for cloud microphysics
In this section, we discuss the effects of stochastic processes on cloud microphysics.
The main result of the paper is that in a turbulent flow interdrop relative velocities and
collision efficiencies are random values with a wide dispersion. Hence, the collision
kernels determining the probability of drop collisions are random values with a wide
dispersion. If we take into account the fact that drop concentration fluctuations in a
turbulent flow are determined by turbulent flow fluctuations, which are also random, all
processes of drop–drop Žas well as drop–ice and ice–ice. collisions should be stochas-

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tic. Characteristic time and spatial scales of variations of all these microphysical values
are determined by the properties of atmospheric turbulence and droplet size. For small
cloud droplets a typical time scale is about 0.5 s and a typical spatial scale is one to few
centimeters.
Thus, the process of collisions in a turbulent flow is highly inhomogeneous in space
and in time. The question arises, what might be the effects of these inhomogeneities on
the process of droplet collisions and other microphysical cloud properties.
Let us start the discussion with the fluctuations of droplet concentration. The droplet
concentration inhomogeneity can by caused by many mechanisms. As it was shown in
Section 1, the combined effect of turbulence and drop inertia can lead to the formation
of drop concentration inhomogeneity. The spatial scales of these concentration fluctuations depend on the drop size. For small cloud droplets these scales equal one to few
centimeters. The RMS amplitude of these fluctuations can amount to 30–40% of the
mean concentration ŽPinsky et al. 1999a.. Pinsky et al. Ž1998b. found in a typical
cumulus cloud the RMS amplitude of droplet fluctuation concentration to be 30% of the
mean concentration. The probability of droplet collisions increases in these small
volumes of enhanced concentration. Let us evaluate a potential effect of the droplet
concentration fluctuations Žunder assumption that these fluctuations do exist in clouds..
As follows from the stochastic kinetic equation of collisions, the growth rate
Žfrequency of drop collisions. is proportional to the multiplication of their concentrations
n1 and n 2 . Thus, the frequency of collisions increases in the areas of enhanced droplet
concentration by a factor of 1.7 to 2. To evaluate the effect of