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doi: 10.1098/rspa.2005.1490, 3059-30884612005Proc. R. Soc. A

S Ghosh, J Dávila, J.C.R Hunt, A Srdic, H.J.S Fernando and P.R Jonasparticles with applications to rain in cloudsHow turbulence enhances coalescence of settling

References

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How turbulence enhances coalescence of settling particles with applications to rain

in clouds

BY S. GHOSH1, J. DAVILA

2, J. C. R. HUN T3, 4, A . SRDIC

5,H. J. S. FERNANDO

5AN D P. R. JONAS

6

1School of the Environment, University of Leeds, Leeds LS2 9JT, UK ([email protected])

2Grupe de Meca ´ nica de Fluidos, E.S. Ingenieros, Universidad de Sevilla,Camino Descubrimientos, s/n lsla Cartuja, C.P. 41092, Spain

3Department of Space and Climate Physics and Department of Earth Sciences,University College, London WC1E 6BT, UK

4Delft University of Technology, 2600 AA Delft, The Netherlands 5Environmental Fluid Dynamics Program, Department of Mechanical and

Aerospace Engineering, Arizona State University, Tempe, AZ 85287-9809, USA6Department of Physics, UMIST, Manchester M60 1QD, UK

From theoretical, numerical and experimental studies of small inertial particles withdensity equal to b(O1) times that of the fluid, it is shown that such particles are‘centrifuged’ out of vortices and eddies in turbulence. Thus, in the presence of gravitational

acceleration g , their average sedimentation velocity V T in a size range just below a criticalradius a cr is increased significantly by up to about 80%. We show that in fully developedturbulence, a cr is determined by the circulation Gk of the smallest Kolmogorov micro-scaleeddies, but is approximately independent of the rate of turbulent energy dissipation e,because Gk is about equal to the kinematic viscosity n. It is shown that a cr variesapproximately like n2=3g K1=3ðbK1ÞK1=2 and is about 20 mm (G2 mm) for water droplets inmost types of cloud. New calculations are presented to show how this phenomena causeshigher collision rates between these ‘large’ droplets and those that are smaller than a cr,leading to rapid growth rates of droplets above this critical radius. Calculations of theresulting droplet size spectra in cloud turbulence are in good agreement with experimental

data. The analysis, which explains why cloud droplets can grow rapidly from 20 to 80 mmirrespective of the level of cloud turbulence is also applicable where a crw1 mm for typicalsand/mud particles. This mechanism, associated with unequal droplet/particle sizes is notdependant on higher particle concentration around vortices and the results differquantitatively and physically from theories based on this hypothesis.

Keywords: turbulence; coalescence; droplets; clouds

1. Introduction

The initiation of warm rain (where ice particles are not present) in the turbulentmotions within clouds has three main stages. Firstly, condensation of saturated

Proc. R. Soc. A (2005) 461, 3059–3088

doi:10.1098/rspa.2005.1490

Published online 9 August 2005

Received 8 April 2004Accepted 1 April 2005 3059 q 2005 The Royal Society

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water vapour on to nuclei causes the growth of droplets with radii generallysmaller than 20 mm. Secondly, there is a rapid growth of much larger droplets of about 80 mm, and thirdly, as they settle relatively faster than the smallerdroplets, the larger droplets grow by collisions with the smaller ones and fall outof the cloud. The second stage is still not adequately understood or accurately

modelled, in common with other processes in which the average sizes of particlesand bubbles grow in turbulent flows caused by an increased rate of collision andcoalescence (Jonas 1996). The developments in the understanding the structureof turbulence and the motion of particles in turbulence (Hunt et al . 1994, 2001;Davila & Hunt 2001) provides an opportunity to re-examine these problems.There is still some controversy as to whether this mechanism also controls thecoalescence/flocculation of mud/clay particles in water. The uncertainty ispartly because of the lack of experimental observations, and partly because it isstill not clear that these processes are triggered by pure collisions alone.

The growth rate of cloud particles by condensation in a supersaturated

environment decreases as the particles become larger, owing to the reducedsurface to volume ratio and as a result, even if the initial particle size spectrum isbroad, subsequent growth of the particles would lead to a narrowing of thespectrum as the mean size increases, if all particles were exposed to the samesupersaturation. Recently, it has become possible to measure droplets with amuch higher spatial resolution. Observed droplet spectra at all levels in mostwater clouds are generally broader than spectra modelled on this basis. Inaddition, it is also observed that growth by coalescence is very slow until somedroplets have reached a critical radius of ca 20 mm, whereupon in deep cloudswith high values of the liquid water content, subsequent growth to drizzle sizemay take only a few minutes (Jonas 1996). A major concern for researchers in thefield of cloud physics is to find the cause of this transition to fast growth (seefigure 1).

Many of the early calculations of particle growth in clouds were based on theassumption that the particle grew in stagnant air. However, observations showthat most clouds are very turbulent with dissipation rates ranging from 10K4 to10K1 m2 sK3 in cumulus clouds (Smith & Jonas 1995). In this paper, we havecritically examined the role of turbulence in inducing microphysical alterations.In order to explain the broad spectrum observed, two main mechanisms havebeen proposed; namely, collisions caused by differential settling velocities of particles as in the third stage, and collisions forced by the turbulent eddying

motion in the clouds caused by buoyancy forces associated with long temporalfluctuations. If the effect of turbulence is not considered, the former mechanism istoo slow (Mason 1952). In one of the first models for the effect of turbulence,Brunk et al . (1998) suggested that straining within Kolmogorov micro-scaleeddies would accelerate the collision relative velocity of colliding droplets. Pinsky& Khain (1997a ) analysed the motion of inertial particles in turbulence using astatistical model and numerical simulations to show how the centrifuging actionof vortical eddies tends to concentrate particles near the periphery and therebyamplify droplet coalescence (see also Shaw 2000). However, in manymicrophysical models, turbulence induced fall velocity enhancements depend

rather critically on the energy of the turbulence. Spectral broadening effects incloud appear whenever there are a few drops with radii of ca 20 mm. In fact,observations indicate that the collision efficiencies increase around a critical size

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of 20 mm over a wide range of turbulent energy dissipation rates (see Jonas 1996

and references therein). Thus, ideally, one should be able to derive this criticalsize (over which turbulence effects can enhance droplet coalescence) theoreti-cally. Indeed, in this paper, we have achieved this through scaling analysis (seeequation (3.5) of the present paper). In addition to the above-mentioned papers,two review papers discuss in some detail the problem of particle–turbulenceinteractions and the consequent implications for cloud microphysical appli-cations. Vaillancourt & Yau (2000) have reviewed laboratory and numericalwork, concluding that the majority of direct numerical simulations have notaccounted for gravity and have focused on Stokes numbers close to unity wherepreferential concentration is found to be the most prominent. In addition, they

argue that the effect of preferential concentration during diffusional growthcannot be treated as a good mechanism to explain droplet spectral broadening inadiabatic cloud cores. The other recent review paper by Shaw (2003) summarizesrecent advances in this area (including the mechanism suggested by Davila &Hunt (2001) and its subsequent application by Ghosh & Jonas (2001)) and, inparticular, they point out that the influence of fine-scale turbulence on thecondensation process may be limited. Shaw also points to mechanisms of fine-scale intermittency, droplet number density fluctuations, entrainment andmixing in addition to the processes of collision and coalescence. From thediscussions given in these two reviews, it is clear that an exploration into

fluid–particle interactions that does not depend sensitively on the dropletclustering mechanisms should be explored further. Nevertheless, these and otherfull-scale and laboratory studies agree that the coalescence of cloud droplets is

large eddy ~ 100 m

1 km

cloud droplet (10;106;1)

1 km

aerosol

particles

(0.1;106;10–4)

newly formed

rain droplet

(100;100;70)

microscale vortices ~ 1 mm

Figure 1. Schematic showing the diversity of spatial scales, vortices and particle numberconcentrations in cloud microphysical calculations. The numbers within brackets refer to radius inmicrons, number per litre and terminal velocity of droplets in cm sK1, respectively.

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increased by turbulence especially near the radius of about 20 mm (Jonas &Goldsmith 1972). To date, there have been no attempts to explain this apparentcontradiction. Our approach considers an important mechanism in addition tothe various other mechanisms reviewed.

Apart from these areas of research interests that we have just discussed,

another active ongoing research area relates to the modification of collisionefficiencies of colliding droplets in turbulence. A literature survey (including thereviews by Shaw 2003; Vaillancourt & Yau 2000) unanimously point out thatthe collision efficiency of cloud droplets can be increased by turbulence–particleinteractions. Pinsky et al . (2000) have calculated collision efficiencies of smallcloud droplets in a turbulent flow and found that the mean values of thecollision efficiency and the kernel are higher in turbulent flows than in still air.This is an ongoing research area in cloud microphysical studies and featuredprominently at the recent 14th International Conference in Cloud Physics inJuly 2004. Papers by Erlick et al . (2004), Franklin et al . (2004), Pinsky et al .

(2004) and Wang et al . (2004) have presented recent estimates on collisionefficiencies and collision rates in turbulence. The results from these studies haveenabled us to obtain a broad estimate of the collision kernel enhancements inturbulence.

The objective of this paper is to apply recent research (which is reviewed in §2)on the enhanced settling of particles in turbulent flows. This leads to newquantitative estimates for the particle motion in the typical vortices of high Re turbulence and the consequences on the droplet distribution. It is shown how thisrational theory can be applied to various physical situations. Our calculationsand scaling analysis establish a critical droplet size for droplet fall velocity

enhancements when an ensemble of droplets interacts with a vortex. The criticalsize prescribes the fall speed that should be right for this amplification to be mosteffective. In addition, our proposed mechanism, associated with unequal dropletsizes, is not dependent on higher particle concentrations around vortices (asproposed by Falkovich et al . 2002), where the higher number concentrationsensure enhanced droplet collisions.

The structure of the paper is as follows. In §2, we discuss the broad issues of the interaction between particles and fluid turbulence and review the dominantmechanisms that can affect cloud microphysical processes. In §3, we discuss themain theoretical considerations leading to our new estimates of the average

settling rates for particles around vortices and the mechanism of enhancedcollisions between particles of different sizes. This is followed by applications of the theory to turbulent flows and comparison with laboratory experiments. In §4,we apply our theoretical and experimental results to cloud microphysicalsimulations. We evaluate the evolution of a typical cloud droplet spectrum withand without the centrifuging action of the vortices. Finally, we evaluate raindropspectra with and without turbulence effects, and are able to predict theexistence/non existence of large raindrops without artificial adjustments for thefirst time. Although other mechanisms have been proposed for turbulentenhancement of rain formation, none have been able to produce a broad

spectrum simply by including turbulence assisted collisions in the spectraldevelopment. In §5, we discuss the wider applications of this model to particles,droplets and bubbles in liquids.

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2. Particle movement and interactions

(a ) Review of numerical simulations

The interaction between particles and turbulent eddies with typical time-scale

T L are studied using numerical simulations, laboratory measurements andtheory. Some of these earlier numerical results have shown that inertial biascauses particles to accumulate on the outside of twisted tube-like vorticalstructures (Jimenez et al . 1993), with the general tendency of the particles beingto disperse faster than fluid elements (e.g. Squires & Eaton 1991). Maxey &Corrsin (1986), Maxey (1987), and Wang & Maxey (1993) showed by numericalsimulations that the settling rate V T of typical inertial particles with time-scaletp may be slightly larger (!20%) than the terminal velocity in still fluid V TO

because the particles tend to fall preferentially in the downward flow regions of the velocity field, which are generally formed between neighbouring regions of vorticity. In addition, some direct numerical simulation (DNS) studies by

Sundaram & Collins (1997) and Yang & Lei (1998) have also been reported withconclusions broadly similar to those obtained from Wang & Maxey (1993).Fevrier (2000) showed that this increase could be substantially greater for aparticular range of inertial particles for which the Stokes number S tZtp/T LO0and were found to accumulate in regions of low vorticity and high strain rates(Squires & Eaton 1991). The earlier results of Wang & Maxey (1993) have shownhow small-scale dynamics cause intense vorticity in turbulent flows to form atdissipation-range scales and that particles accumulate in the low vorticityregions of the flow. They do not accumulate here because the flow is faster andthere are straining regions. However, they do spend an increased time at particle-

stagnation points. Their numerical results in homogeneous isotropic turbulenceindicated that maximum preferential accumulation occurs when the inertialtime-scale of the particles are comparable to the smallest time-scales of the flow.Because of the intense vorticity at the dissipation range scales, this suggests thatparticles accumulate in the low vorticity regions of the flow and are centrifugedaway from the vortex cores. However, the fact that certain particles are deflectedinto particular zones around vortices increases the local void fraction of theparticles. Could this effect further increase the fluid interaction between theseparticles and increase their fall speed (in proportion to the local concentration)?This is the suggestion of Hainaux et al . (2000), who recently measured the fall

speed of particles in a turbulent air stream. As we shall show, this is a weakereffect for cloud particles than the former effect of inertial particles moving in thedownflow side of the vortices. The question of the role of local concentration isalso important for estimating collisions between droplets; the first mechanism weshow to be the most effective is that the relative speeds of particles and thereforecollision rates for different sizes are enhanced by their motion around turbulentvortices. The second is that certain sizes of particles are concentrated aroundthese vortices, so that they encounter each other more often than in thesurrounding flow and their relative acceleration is thereby affected.

(b) Effects of isolated vortices on droplet settling

To understand and model the average settling velocity V T of small denseparticles (such as cloud droplets) descending and colliding in turbulence, we

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apply the results of Davila & Hunt (2001) to analyse particles around an isolatedvortex with circulation G (with radius Rv and maximum velocity U vZG/Rv).(Note that, generally, the acceleration in a Kolmogorov micro-scale eddy is smallrelative to that due to gravity (i.e. e3=4=ðn1=4g Þ10K1), but there are occasionalintense vortices where this inequality does not apply). The very small particles

with radius a have an acceleration

dV

dt Z

1

tp

ðu Cv TOKV Þ; (2.1)

where v is the velocity of the particle and u is the unperturbed velocity of thefluid at the position of the particle. The settling or terminal velocity in still fluidestimated by using Stokes linear drag law is

V TOxg ðbK1Þa 2=ð9n=2Þ; (2.2)

which the particles reach after a ‘relaxation time’

tpZ ð2=9ÞðbK1Þa 2=n; (2.3)

where n is the kinematic viscosity of the fluid (ca 10K5 m2 sK1 in air) and b is theratio of the density of the particle to that of the fluid (ca 103 for cloud droplets).These quantities provide the reference scales for considering how the particlemoves near a vortex. Numerically, the trajectories of the particles can beobtained by integrating (2.1) together with dX /dt ZV given certain initialconditions (e.g. V

ðX

0; t

0ÞZu

ðX

0ÞCV

TO). Where the ratio, u , of the terminal

velocity V TO to the maximum velocity in the vortex U v is less than about 1.0,Davila & Hunt (2001) show that the effect of the vortex on the settling velocity isdetermined by the non-dimensional ‘particle Froude number’ F p, defined by theratio of the stopping distance (V TOtp) of the droplet to the characteristic radius(G/V TO) of the trajectory of the droplet around the vortex:

F pZV TOtp

G=V TO: (2.4)

It should also be noted that the new definition of the particle Froude number in

equation (2.4) enables us to also obtain an alternative definition of the Stokesnumber, which is more relevant to our analysis than the conventional definition.The Stokes number defined in Davila & Hunt (2001) is defined as tp/tr, where tr

is the residence time of a fluid particle around the vortices. Further, if theresidence time of the particles to move around the vortices is much shorter thanthe lifetime of the vortical structures, then particle trajectories can be calculatedby considering that the flow is stationary (see Davila & Hunt 2001). Using theresult of that paper and that of Vincent & Meneguzzi (1994), this can beexpressed as G=V 2TO/T I. T I is a time-scale of the order of the ratio of a typicallarge length-scale and a typical velocity-scale, and is therefore usually in the

range of 10–100 s.When the effect of the particle inertia is very small (F p/1), it passes roundthe vortex and the net change in V T (averaged over the whole life of the particle

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near the vortex and over a range of starting positions on the scale of G/V TO

or Rv, whichever is the larger) from its value V TO in still fluid is negligible (i.e.v ZV T/V TOx1.0). However, when the inertia is large enough that F pwO (1), theparticles are flung outwards; then, v rapidly increases to a maximum value v max

of about 2.0 (for 1.0Ru R0.7, where F pxF pmax). With a small further increase in

inertia, the particles ‘crash’ through the vortex and are on average slightlydelayed, so that v is reduced to a minimum value v min of about 0.7 (for u %1.0where F pZF pminx4). For very large inertia or large u , the vortex has negligibleeffect on the settling rate, and v Z1.0. (These values of v max and v min arecalculated for particles released at a level above the vortex equal to about 10times its radius Rv, falling to an equivalent distance below the vortex.) Inaddition, large inertia particles have a stopping distance V TOtp greater than thedistance between the eddies Dl v (see figure 2a ) so that they average out the effectof individual eddies and v x1. It is important to point out that v R1.0 andF p%1.0 for droplets with radii in the range of 5–10 mm, ensuring velocity

enhancements even in this size range. This has significant cloud microphysicalimplications. First, droplet pairs within this regime have their fall velocitiesenhanced when they interact with the micro-scale vortices within clouds and thiscan lead to increased collision and capturing among droplets that eventuallyyield realistic spectral distributions (see figure 6). Secondly, our analysis alsosupports the well-established observation that collisions between unequal dropletsizes are favoured over collisions between droplet pairs that have similar sizes(see figure 4a ,b).

Qualitatively, these results are well known, experimentally and theoretically,particularly the ‘centrifuging’ out of inertial particles in gas and liquid flows andchanges in the settling velocities of particles in turbulent flows with vortices(Maxey 1987; Perkins et al . 1991; Fung 1993). However, a systematic calculationfor the increase and decrease in the average settling velocity over a wideparameter range is new, as is the derivation of a new scaling in terms of (G/V TO),to replace other definitions of the Stokes number for vortices (e.g. Marcu et al .1995). The trapping of inertial particles in vortices (Toobey et al . 1977) found inhydraulic flows is not relevant here where b is very large.

The motions of the particles around a vortex vary greatly if their sizes liewithin certain ranges. This can substantially increase the probability of collisionsin a time t c between pairs of different sizes, with large and small radii a , a ,respectively, having terminal fall speeds V TO

ða

Þ, V TO

ða

Þ. In still air, the

probability P c is proportional to the length of a vertical collision line l co in whichall the larger particles must be positioned at t Z0 if they are to collide with thesmaller particle in time Dt c, where l coZDt cðV TOða ÞKV TOða ÞÞ. As the largeparticle moves round a vortex (see figure 3a ,b) starting at time t Z0, it collideswith a small particle also released at time t Z0, if they start on a curved ‘collisionline’ l c. If the large particle lies in the critical range where V T is significantlygreater than V TO, then because l c is greater than l co, the probability of a collisionis proportionally greater.

(c ) Collisions of droplet pairs around isolated vortex lines

In addition to the considerations described above, some other interestingfeatures emerge from the Davila & Hunt (2001) theory, which relate to droplets

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B A

C

V TO

Rv

∆ lv

G

acr

a10–3 10–2 10–1 101 1021

F p

V TO

V T

1.0

0

1.0 1.5 2.250.70.450.3

0

1.0 1.5 2.250.70.450.3

d(V TV TO)

d(a/acr)

acr

a

(a)

(b)

(c)

G/ V TO

Figure 2. Mechanisms of particles falling near vortices: (a ) typical trajectories of settling particlesA, B, C moving around a vortex with strength G and radius Rv with increasing fall speed V TO.Here, Dl

vis the distance between vortices; (b) average settling velocity ratio V

T/V

TOversus the

particle Froude number F p and particle radius ratio a /a cr; (c ) d(V T/V TO)/d(a /a cr) versus a /a cr.

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Figure 3. Collision mechanisms: (a ) colliding droplets settling in still air in the absence of vorticeswith larger and smaller radii a , a . Note the vertical collision distance l co; (b) colliding dropletsdescending near an isolated vortex. Note the curved collision line with length l c; the large dropletcollides with a smaller droplet both released at t Z0, if they start on the curved collision line; (c )variation of l c relative to l co for still fluid over a typical horizontal distance, E l Z

ðl c=l coK1

Þ!

ðDt c

V TOÞ=ðG=^

V TOÞ for smaller particles with^F pZ 4 and

^V TOZ 0:7, and larger particles with

F pZ 5 and V TOZ 0:8.

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interacting within intense vortex lines. Two additional collision mechanisms arerelevant to cloud microphysical calculations. Firstly, for droplets movingbetween line vortices, the ratio of the average collision length to the length instill air hl ci/l co in a horizontal box of size Dl v can be expressed in a closed formafter some algebra (see appendix A):

hl cil coZ 1C

ðG=V TOÞ2

Dl vDt cV TO

D Kl2 D

1Kl: (2.5)

As before, l co is the collision distance taken by a large droplet of radius a tocollide with a small droplet of radius a in time Dt c in still fluidðl coZð V TOKV TOÞDt cÞ. V TO and V TO are the settling velocities of the largeand the small droplets in still fluid. Dl v is typically a measure of the distancebetween the vortices, which is comparable to the vortex diameter andlZV TO= V TOðl!1Þ, D and D are the dimensionless ‘drift’ integrals correspond-

ing to the small and large droplets. When F p%1, both^

D !0 and

D !0 andjl2 D j! jD j. This results in a smaller collision distance (i.e. hl ci/l co!1). Thisimplies that in order to have the maximum collision enhancement induced byturbulence the droplet radii must be such that D !0. Equation (2.5) shows thathl ci/l co increases in proportion to the effective trapping radius of the vortexG=V TO and inversely to the assumed time between collision Dt c. Hence, we definea normalized value of the fraction of the collision length increment for thed is ta nce b et ween t he int en se v or tex l in es g iv en b y E l Zðl c=l coK1ÞðDt c V TOÞ=ðG=V TOÞ. By calculating how l c varies over a typical range of theinitial location X 0, one can estimate the average value of l c, hl ci, for all particles of a particular size a descending round a typical vortex and thence the averagevalue of E l and the probability pða ; a Þ that a larger particle will collide with asmaller particle of size a . The ratio hl ci/l co is a measure of the increase in thisprobability compared with collisions in still fluid. The calculations show that l chas a maximum at an initial position X 0M for the small particle corresponding tosmaller velocities than in still fluid, and for the large particle larger velocitiesthan in still fluid (these particles pass close to the equilibrium points described byDavila & Hunt 2001). Smaller particles moving in the downflow side of the vortexsettle faster, which implies that l c has a minimum at X 0M corresponding toparticles moving on that region. This variation is shown in figure 3c .

Considering cloud droplets settling in air with radii between 10 and 35 mm,

and with a typical vortex circulation GZ1.5!10K4 m2 sK1 (assuming G ca 10n)and radius RvZ1.5 mm, it is found that E l is negative up to droplet radii smallerthan 35 mm. This shows that for the most efficient droplet capture between anensemble of colliding droplets, the maximum collision enhancement is achievedwhen there are some droplets whose radii exceed 35 mm. However, when thedroplet radius is somewhat greater than 35 mm, E l changes sign leading tocollision enhancements, since the drift integral D O0 for values of F pO1(see fig. 10 of Davila & Hunt 2001). For F p!1 the effect of the vortices on theaverage settling velocities of droplets is smaller for the larger particles. This isbecause the effective area of the line vortices that modify the droplet trajectories,

ðG=^

V TOÞ2

, decreases as a 4

. As in the theory of Falkovich (2002), it appears thatcollision rates are not simply related to fall speed or to concentration orprobabilities of lying in the vortex core.

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Figure 4a ,b shows the position of droplet pairs with radii 20 and 25 mm and 5to 10 mm, respectively. The X - and Y -axes are each normalized with the vortexradius Rv. These are calculated using (2.1) as in Davila & Hunt (2001) for thedroplet pairs moving around a vortex with circulation GZ1.5!10K4 m2 sK1 andradius RvZ1.5 mm (to consider the effect of gravity in non-horizontal vortices,

the terminal velocity should be projected on the plane perpendicular to thevortex axis). Figure 4a shows that for the 20–25 mm droplet pair (with a 20%difference in radius), there is a very small increase in the probability of collisions,whereas for the 5 to 10 mm pair (with 100% difference), there are multiplecollisions (figure 4b). This indicates that this vortex–particle model is consistentwith the well-established observational result that the probability of collisionbetween droplets of unequal sizes is higher than the collision probability of similarly sized droplets.

Other theories have addressed the issue of droplet spectral broadening byinvoking various mechanisms that depend sensitively on turbulent energy

dissipation (Pinsky & Khain 1997b), on enhanced droplet concentrations aroundvortices (note that the higher number concentration ensures enhanced dropletcollisions) as in the Falkovich (2002) study or on particle interactions withintense vortex tubes (as in the Shaw 2000 study), which are again concentration-dependent. Shaw’s scale analysis also showed that the vortex tubes weresufficiently intense and persistent so that they caused larger flux divergences inthe local concentrations of cloud droplets. From our analysis, we have been ableto explicitly show that the probability of collision between two droplets withwidely different radii are much higher than collisions between droplets havingcomparable radii—a well established observational result (Pruppacher & Klett1997). This feature is not apparent in these earlier theories.

Collisions involving very small droplets (for which F p!1 and u !1) can alsooccur within the vortices (see figure 2a ). As explained above, most of theseparticles that fall towards a vortex are advected towards it and tend to be sweptaround the vortex by the mean streamlines and by their own inertia in the curvedflow (e.g. Squires & Eaton 1991). However, as a result of other mechanisms in theinterior of such vortices, a significant number of small particles may be present.This is firstly because as such vortices grow on a time-scale tv (e.g. Jimenez et al .1993) and they surround any small particle present provided tp!tv. Then, thevortices trap the particles for a certain time tvp. In high Reynolds numberturbulence tvw

ðe=n

ÞK1=2w10K1 s in typical clouds (Pruppacher & Klett 1997).

Usually, the vortices last for a longer period as they decay than during their growthphase. For a typical 10 mm droplet, tpw10K3 s, so that these droplets can certainlybe trapped. The time for such low inertial particles with a fall speed V T to escapebeyond the cavity region is given by tvpwRp/v p, where RpwG/V TO andnpwðV TOtpÞðV TO=RpÞ. Thence tvpwg G2=V 5TO (provided F p!1). Typically,for 10 to 20 mm particles in the atmosphere, RpwRv for micro-scale eddies andtvpTtv. We conclude that the critical scale, small inertia, particles can remaintrapped in these vortices over the lifetime of the micro-scale vortices. (Althoughdirect numerical simulations cannot describe micro-scale dynamics in turbulencewith a realistic spectrum, they do show low concentrations of small inertial

particles in the most intense vortices (Squires & Eaton 1991).)This trapping mechanism in the smallest eddies gives rise to two collisionmechanisms, which differ from those outside the vortices. The first involves

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250

200

150

100

50

0

–50

–100

–150

–200

–200 –150 –100 –50 0 50 100 150 200 250 300–250

X (t )

X (t )

Y ( t )

Y ( t )

–20–25

–15

–5

5

15

25

–15 –10 –5 0 5 10 15 20 25 30

(a)

(b)

Figure 4. (a ) Colliding droplet pairs with radii 20 and 25 mm descending near an isolated vortexwith circulation GZ1.5!10K4 m2 sK1 and radius RvZ1.5 mm calculated from the Davila & Hunt(2001) theory. Note the absence of collisions. (b) Colliding droplet pairs with radii 5 and 10 mmdescending near an isolated vortex with circulation GZ1.5!10K4 m2 sK1 and radius R

vZ1.5 mm

calculated from the Davila & Hunt (2001) theory. Note the presence of multiple collisions.

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critical scale particles (F pT1), which are initially trapped and then thrown out,and in doing so, collide with very small particles (F p/1). The second involvesall the small particles (F p!1, v !1), which are trapped and therefore collide withany much larger particles (F pT1, v O1), which crash through these vorticeswithout being deflected.

Thus, applying the two set of results to the overall distributions of dropletsand vortices, we conclude that there are two ranges of the vortex–dropletparameters that need to be considered (see table 1 below).

From this table, it is clear that the greatest differences of relative settlingvelocity occur between particles in categories [2]E, [3]E in the table; that is,

for the external particles that lie outside the vortices and in a size rangeclose to that of the critical diameter a cr, as defined by F pw1, u w1. For theinternal particles that are trapped, the largest difference occurs between largeparticles that cut through the vortex [4]I and the small particles that aretrapped [2]I.

From the point of view of collisions, the difference in the former category of external particles is more significant because it applies to small particles that areclose in size to each other. These are more numerous than the larger particles andare continuously being nucleated.

Although our calculations have been mainly concerned with cloud dropletswith radii ca 20 mm interacting with micro-scale eddies, it is also possible thatturbulence can modify the settling rates of the larger droplets interacting witheven larger eddies. However, from equation (3.5) (see §3 below), we note thatbecause the critical radius a cr has a 1/6 power dependence on the vortexcirculation, its value even for inertial range eddies lies in the range 20–30 mm.

3. Applications to turbulence: experiments andprediction of critical particle sizes

The previous theoretical concepts for particles near vortices are now compared

with Srdic and Fernando’s (Srdic 1998) laboratory measurements. Using theDigimage system, they studied the effects of ‘mixing-box’ turbulence on the settlingof small dense particles in water with radii varying between 22.5 and 355 mm over

Table 1. Particle settling ratios outside and inside vortices showing how in different ranges of the stopping/vortex ratio F p the effective particle settling velocity v ZV T/V TO normalized on its value in still fluid, is increased or decreased for characteristic vortices

(Here we focus on critical particle sizes where F pw1, which implies that for typical Kolmogorovmicro-scale eddies a crw(n/(bK1))1/2(g /Rv

K1/4); that is, a cr ca 20 mm in air, a cr ca 1 mm in water.)

F p/1 F p(1 F pT1 F p[1

external particlesav T1½1E v O1½2E v !1½3E v w 1½4E

internal particlesbv (1½1I v (1½2I v !1½3I v w 1½4I

aNote that the effect on v of external particles is only significant if V TOtp!Dl v.bParticles can onlybe trapped within a vortex if tp!tv and only spend a significant time there, in relation to itspassage around the vortex, if tvpTRv/V TO.

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a range of b between 1.4 and 8.7. In these experiments, the spectrum was only largeenough for the small scales, approximate to that of high Reynolds number inertialrange turbulence over a limited range of scales (Kit et al . 1997), but the Reynoldsnumber was large enough (Re x200) so that there were active vortical motion withvortices formed having diameters with typical magnitude R and circulation G(R).

These were observed to be separated by distances of the order of R. Thus, infigure 2a , the net effect of a vortex on the fall speed is only significant for particleswhen the characteristic distance G/V TO is less than the distance Dl v between theeddies, that is, if

V Ttp!R: (3.1)

The speeding up effect of the vortices on V T is a maximum when F pw1, whereF pZV 2TOtp=G. Srdic and Fernando’s experimental results for v plotted as afunction of a large-scale particle Froude ratio F PL

ZV 2TOtp=ðU LLÞ (where U L is themaximum vortex velocity with radius L) showed that v increased from 1.3 to about1.8 when F PL was about 0.8!10K3 (G20%), and decreased to about 0.8 forF PLx3! 10K 3. As noted previously, in Fevrier’s (2000) simulations, this range of

increase and decrease in v was also found. For F PLgreater than 4!10K3 v was equal

to 1.0. These low values of large scale F PLcorrespond to values of F p of order unity

for micro-scale vortices in the flow (with characteristic velocity U kw(en)1/4 andlength-scale Rkw(e/n3)K1/4). The fact that the particle motions were primarilydistorted by small-scale vortices was confirmed by the measured small spikes in thefrequency spectrum of the velocity of the particle at the micro time-scale l k/v k. Theexperiment not only confirmed the prediction of a significantly larger increase thandecrease in settling velocity over a narrow parameter range, but also showed how

‘empty’ regions exist around and below the vortices from which the inertialparticles have been expelled. This was expected from the theory when the ratiou L(ZV TO/U L) was between 0.2 and 1.2.

Applying these results to the settling of the smallest droplets in a cloud with aradius a , it follows that v (a ) can only be significantly increased if vortical eddiesexist of scale R whose strength G is such that F p(R)w1. Therefore, fromequations (2.2) and (2.3)

GðRÞwV 2TOtpwa 6g 2ðbK1Þ3

ð9=2

Þ3n3

: (3.2)

Analysing the structure of turbulence provides an estimate for G(R) for thevortices formed over many scales with radius R (e.g. Sundaram & Collins 1997;Hunt 2000). At the micro-scales where RZRk, the circulation

GZGkwU kRkwðenÞ1=4ðn3=eÞ1=4wn: (3.3)

A key point of this theory is the observation that Gk is independent of theenergy of the turbulence. Physically, this is because as e increases, the peakvelocity U k increases, but the radius Rk decreases by the same amount. For RORk, the circulation of eddies in the inertial range G(R) increases with their

radius as

GðRÞwnðR=RkÞ4=3: (3.4)

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From equation (3.2), it follows that the smallest radius a cr of droplets whosesettling velocities are increased (by up to about 80%) is given by

a crwn2=3g K1=3ðbK1ÞK1=2: (3.5)

In air a crw

20 mm, and in water, for bZ

2, a crw

100 mm. Note that for theseparticles in the micro-scale eddies

V TO=U kwg ba 2=n

ðenÞ1=4; (3.6)

for a w10 mm. This shows why the turbulence must be intense enough and elarge enough for V TO/U k(1. Thus, for droplets with a smaller radius than20 mm, since V T increases in proportion to a 2, F p is much less than 1.0 andtherefore the particle fall speed is not on average increased (i.e. v x1). In a highReynolds number turbulence where there is a full inertial range of vortical

eddies varying in size (R) down to the Kolmogorov micro-scale, typicalvelocities increases with radius U (R)we

1/3R1/3. We can apply the formula (3.2)to estimate the critical size of particles a cr(R) that are accelerated by eddieswith scale R larger than Rk; it follows that

a crðRÞ=a crðRkÞxðR=RkÞ1=6: (3.7)

This shows that even the eddies that are 100 times larger than the micro-scaleeddies also tend to accelerate and concentrate particles with diameters in therange 20–30 mm. This is why the result (3.5) is quite robust.

We also have to consider the ratio V T

(a cr

(R))/U (R) for the critical size of particles for eddies in this range. The typical eddy velocity U (R)w(e1/3R1/3)increases in proportion to R1/3, and from (2.2), (3.7) (V T(a cr)) increases as a 2 orR1/3. It follows that the ratio V T(a cr(R))/U (R) is approximately constant so thatif (3.6) is satisfied the basic criterion for the ‘acceleration’ effect is satisfied for allparticles in this range.

When particles reach 80–100 mm, their fall speed becomes comparable withthat of the energy containing eddies in the turbulence so that U (R) reaches itsmaximum value. Then, V T/U (R) increases and the acceleration effect ceases.

These calculations are consistent with the experiments of Srdic and Fernandoand earlier measurements of enhanced settling rates. In fact, experimental

evidence of much enhanced sedimentation in turbulence (by even more than80%) was observed by Nielsen (1992). When the void fraction is greater thanthe typical cloud value of about 10K5 (or the mass loading greater than 10K2),the tendency of inertial particles in the critical range to be concentrated(around vortices) causes them to settle even faster (Hainaux et al . 2000).(As discussed in §2, we conclude this is not the relevant mechanism for clouddroplet formation.)

In order to justify the applicability of the Davila & Hunt (2001) calculations tocloud microphysics, some further fluid mechanical details need to be elaborated.This is now discussed. First, it is important to note that the vortex representation

in the Davila & Hunt formulation can indeed be applicable to clouds. The theory isnot only valid for horizontal vortices (see eqn (6.3) of that paper). Although thecalculation of the drift integral in the Davila & Hunt paper is for horizontal

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vortices, one can work with the projection of the terminal velocity on the planeperpendicular to the vortex axis for any other orientation. We have focused on theeffect of cylindrical vortices because only in regions of strong vorticity can thetrajectories of heavy particles change so that the average velocity is altered. This isbased on the fact that slow accelerations lead to small Stokes–Froude numbers for

the particles and therefore the average velocity is close to that in still fluid.Moreover, as pointed out by Maxey (1987) and others, only regions of highvorticity can create nearly empty regions and deviations in particle trajectories.We ensured that the inter-vortex distance was comparable to the diameters of thevortices. We have considered only values of vortex circulation of the order of thekinematic viscosity, the typical values found in DNS, although Jimenez et al .(1993) suggest that there may be a weak dependence on the Reynolds numberbased on the Taylor micro-scale ðGf ffiffiffiffiffiffiffiffi

Re lp Þ.

The methodology of this calculation is not based on a precise statement thatturbulence consists exclusively of vortices. It is an approximate physical

calculation in which one isolates a mechanism that has a strong macroscopiceffect and estimates its significance in relation to overall data. This has alwaysbeen the approach in cloud physics studies, and the earlier results have not beenin agreement with data because the collision/settling mechanism was not themost significant mechanism. The sign and magnitude of the effects of themechanism do not depend sensitively on the precise distribution and orientationof vortices.

The laboratory experiments of homogeneous turbulence of Srdic and Fernando(Re of about 100; Srdic 1998) show through visualization how particles of acertain size are deflected/excluded from random vortices in the flow. More

significantly, they show how the mean fall speed rises (by 80%) and falls (by20%) with turbulence relative to still fluid by an amount that approximatelycorresponds to the Davila & Hunt (2001) calculation for a typical distribution of single vortices. The separations of the vortices are comparable with the diameterof the vortices.

For very high Reynolds number turbulence, there is still no detailed dataavailable; except laboratory visualizations (e.g. by Douady et al . 1991) certainlyshow these structures exist and with a distribution in space (a factor of 10 radiibetween them) quite comparable with the hypothesized spacing in the Davilaand Hunt model.

There is some evidence from the experiments and numerical simulations of Perkins et al . (1991) that the increased fall speed is also found in inhomogeneousshear flows. They found that in a horizontal turbulent air jet, small inertiaparticles, where V Twu o, were found to fall significantly faster than in still air,but for the heavier particles, where V TOu o, there was no such effect. Thisexplains the discrepancy presented in that paper between the experimentalresults and the predictions of their stochastic simulation model which did notaccount for the faster settling of particles (which is normally assumed in suchmodels).

Finally, we wish to point out that the Davila and Hunt mechanism has now

been referred to in recent work concerned with rain enhancements in turbulenceas in the Falvovich (2002) paper published in Nature and also in the review paperby Shaw (2003).

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4. Droplets in cloud turbulence

(a ) Collision rates

The collision rate C ða ; a Þ between larger droplets (collector droplets) of radius a

and smaller droplets (collected droplets) of radius^

a is determined by the velocitydifference DV Tða ; a Þ between the droplets and the efficiency E with which theycollide. The velocity enhancements are size dependent, as was shown by Davila &Hunt (2001), and hence the relative velocity DV Tða ; a Þ between droplet pairs isvery often different from a conventional relative velocity using still air fallvelocities. Although we have seen in figure 3b,c how the collision process iscomplex around a vortex, an estimate for the collision rate allowing for itsincrease in these vortical flows is

C ða ; a ÞZ jDV Tða ; a ÞjE ða ; a Þ; (4.1)

where E ða ;^a Þ (O0) is the collision efficiency. We used the stochastic collectionequation (SCE) solver employing the flux method developed by Bott (1998).

From various sensitivity studies, Bott (1998) showed that the flux methodremains numerically stable for different choices of the grid mesh and theintegration time-step. The hydrodynamic collection kernel that is used in theSCE solver is simply the collision rate given by equation (4.1) multiplied bypða C a Þ2 (Pruppacher & Klett 1997). In the original SCE solver developed byBott (1998), the collision efficiencies were taken from Long (1974), although Bott(1998) used other collision efficiencies (e.g. Davis 1972; Hall 1980) for variousnumerical experiments with satisfactory results. While evaluating the hydro-dynamic kernels, we used look-up tables for specifying the collision efficiencies,which varied with every pair of the collector and the collected drop radii. In ourpaper, since we are mainly concerned with relatively small droplets (where thecollector drop radius a !40 mm) the data are taken from Davis (1972) and Jonas(1972). Figure 5a shows the kernels with and without the effects of enhancedsedimentation; the solid lines correspond to the case with the enhancedsedimentation and the dashed lines to pure gravitational settling. In order tosingle out the sedimentation effects exclusively, we did not enhance thehydrodynamic kernels in this figure; that is, the same collision efficiencies wereused for both set of curves shown in figure 5a . Figure 5a shows that with theeffects of turbulence, which increases DV T for certain pairs of droplet radii (see

figure 2b), the hydrodynamic kernel values can be up to about 15–35% higher,because of the enhanced sedimentation alone. This increase is evident for dropletradii pairs up to 16 mm. For droplet pairs approaching 40 mm, there is even areversal of the contours. This is because with larger droplet pairs, the fall velocityenhancements are proportionately smaller. As a result, the velocity differencebetween the droplet pairs is actually smaller than the velocity difference withoutturbulence effects, and this causes a reversal of the contours. However, it isunrealistic to use still air collision efficiencies when one considers settling of droplets in turbulence, because the settling rates are enhanced in turbulence, andthe collision efficiencies are also expected to increase, causing an overall increase

in the hydrodynamic kernel. This feature is implicit in the Davila & Hunt (2001)theory, which shows that with increasing settling rates, the collision efficienciesare expected to be higher and can indeed be enhanced by at least 50–100%.

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–

–

40

30

20

10

0 10 20 30 40radius of capturing drops (µm)

r a d i u s o f c a p t u r e d d r o p s

( µ m )

1 × 1

0 – 1 2 1

× 1 0

– 1 3

1 × 1 0 – 1 0

1 × 1 0

– 1 5

1 ×

1 0 – 1 0

1 × 1 0

– 1 0

1 × 1 0

– 1 0

1 × 1 0

– 1 3

1 × 1 0

– 1 7

1 × 1 0 – 1 3

1× 10 – 10

1×10 –151×10

–131×10 –12

1 × 1 0

– 1 2

1 ×

1 0

– 1 5

1 ×

1 0

– 1 7

1 × 1 0 – 1 2

1 × 1 0

– 1 3

1 × 1 0

– 1 3

1 ×1 0

–15 1 × 1 0

– 1 7

1 – 1 0

– 1 5

(a)

40

30

20

10

0

r a d i u s o f c a p t u r e d d r o p s ( µ m )

1 × 1 0

– 1 2

1 × 1 0

– 1 3

1 × 1 0

– 1 5

1 × 1 0 – 1 2

1 × 1 0 – 1 3

1 ×

1 0

– 1 5

1 ×

1 0

– 1 7

1 × 1 0 – 1 0

1 × 1 0 – 1 2

1 × 1 0 – 1 3

1 × 1 0 – 1

5

1 × 1 0 – 7

51×10– 10

1 × 1 0 – 1 0

1 × 1 0

– 1 0

1 × 1 0

– 1 5

1 × 1 0

– 1 2

1 × 1 0

– 1 3

1 × 1 0

– 1 7

1×10– 131×10– 121×10– 17

10 20 30 40radius of capturing drops (µm)

(b)

Figure 5. (a ) Collision kernel (pða C a Þ2E ða ; a ÞjDV Tða ; a Þj) (m3 sK1) with (solid lines) and without(dotted lines) turbulence induced enhancements of the differences DV T in the settling velocities.In order to single out the sedimentation effects, the collision efficiencies E ða ; a Þ are not enhanced.

a and a are the radii of the larger capturing drops and the captured collected drops, respectively.(b) Collision kernel (pa C a Þ2E ða ; a ÞjDV Tða ; a Þj) (m3 sK1) with (solid lines) and without (dottedlines) turbulence induced enhancements of the differences DV T in the settling velocities. In orderto include turbulence effects on the sedimentation as well as on the collision efficiencies E ða ; a Þ,the turbulent kernel is enhanced by 50%. a and a are the radii of the larger capturing drops andthe captured collected drops, respectively. (c ) Mass distribution after 15 min without (dottedline) and with (dashed line) the effects of turbulence induced velocity enhancements. The solidline corresponds to the initial profile represented by a gamma distribution function with a

prescribed mean radius (8 mm) and cloud water content (2.75 g mK3). Note that a bimodalspectrum is obtained only with the faster settling rates shown in (b).

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From Davila & Hunt (2001) one obtains progressively decreasing velocityenhancements with increasing droplet sizes. For example, the enhancementsdecrease from 90 to 18% as the droplet radii increase from 11 to 20 mm. For therange of droplet sizes considered in this study, the velocity amplification effect isthe highest for the smallest droplets. With increasing droplet radii, the effectprogressively decreases; this is consistent with our discussions in §§2 and 3. Usingthe Davila & Hunt (2001) theory and the analysis of §3, our calculations areextended over a wide range of F p values. The computations depend sensitively onthe droplet size as the Davila & Hunt (2001) paper implies, because the velocityamplification ratio v increases in proportion to (1KaD ), where D is the averagevalue of the ‘drift integral’ (which is essentially a measure of the differencebetween the vertical settling distances with and without the vortex for particles

starting at a fixed point and falling for a fixed period of time) for different valuesof F p and V TO appearing in the flow. Here, a is the effective volume fractionoccupied by the vortices, so that awððG=V TOÞ=Dl vÞ2. D becomes more negativewith decreasing values of V TO, which varies as a 2 for small cloud droplets; thus,the V TO values become progressively smaller with smaller a and this causeslarger negative values of D and larger amplification. Therefore, for theparameters relevant to this study, the velocity amplification effect fades withincreasing drop radii and becomes extremely small for a w40 mm.

In the recent study by Franklin et al . (2004), the authors showed thatincreases in the collision kernels in turbulence can sometimes be larger by a

factor of 3. The Davila & Hunt (2001) analysis also implicitly indicates that thecollision efficiencies can indeed be enhanced by up to 100%. We have performedsome sensitivity studies and found that with an increase of 50% in the collision

5

4

3

2

1

0 10 100 1000

m a s s d i s t r i b u t i o n ( g m – 3 / d l n ( r a d i

u s ) )

radius (µm)

(c)

Figure 5. (continued .)Figure 5. (Continued.)

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efficiency, the kernels with turbulence enhancements are always higher than thestill air kernels for all droplet pairs (see figure 5b). As expected, with even higherincreases in the collision efficiencies (not shown here for want of space) thedifferences between the turbulent and the still air contours are even greater.Although, there are two identical halves in figure 5a ,b, owing to the symmetrical

kernels, only one half of the contours are considered, and the Bott (1998) codeensures that there is no double counting. In order to study the impact of theseenhanced sedimentation rates, we applied them first to an idealized massdistribution (shown as the solid line in figure 5c ) and used the SCE solver tostudy the spectral evolution with time. The initial mass distribution correspondsto a total cloud water content of 2.75 g mK3 and a mean radius of 8 mm. In figure5c , we have also shown the mass distribution after 15 min without (dotted line)and with (dashed line) the effects of turbulence induced velocity enhancements.Note that a bimodal spectrum is obtained only with the faster settling ratesshown in figure 5b. It is well known that when a bimodal spectrum develops, the

resulting collision-induced second mode has the propensity to rapidly initiaterain formation.

(b) Droplet spectra in cumulus cloud

Because clouds consist of finite volumes of particles and water vapour movingunsteadily, mainly up and down, the distribution or ‘spectrum’ of droplet sizes,and thereby the formation of rain, have to be calculated as time dependantprocesses. The development of the spectrum caused by collisions after the initialcondensational growth was calculated from the SCE (Pruppacher & Klett 1997)using the Bott (1998) code, which accounts for the fact that not all droplets of agiven size grow at the same rate, since a small fraction of drops experience aparticularly favourable sequence of collisions and grow much more rapidly thanother drops.

Next, we show results from our model simulations where a perturbed gammadistribution was used to create the initial distribution shown in figure 6.The total liquid water content is 3.33 g mK3 and the mean droplet radius isca 7 mm. The initial mass distribution and a subsequent distribution are shown infigure 6 where we have considered collisions between droplets over a time periodof 20 min. In these calculations, the fall velocity enhancements were calculatedusing the Davila & Hunt (2001) mechanism described earlier. The collision

kernels were enhanced by 50% (the solid lines in figure 5b) for simulating theturbulent case. During a time span of 20 min, the small droplets with radii of ca 10 mm can recirculate about four times within cloud eddies which typicallyhave length-scales of ca 50 m and circulation velocities of ca 1.0 msK1. Details of the calculation procedure are given in appendix B. Further details of the fluxmethod for the numerical solution of the SCE can be found from Bott (1998).

We compared our model simulations with some observed data where acollision-induced spectrum was observed. It must be pointed out that the presentcalculations include only the process of spectral broadening owing to dropletcollisions while ignoring all other dynamical effects. Thus, for the sake of

consistency, we chose observed data points with radii greater than 15 mm tocompare with model runs; these larger drops are expected to have grown fromcollisions. For a definitive model simulation vis-a-vis observations, one would

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need to use these calculations in a cloud model with detailed dynamical andmicrophysical processes. This will form the basis of a later study. For themoment, we have aimed to determine the effect of these higher collision rates onthe large end tail of a cumulus cloud spectrum. Figure 6 incorporates someobservations reported by Mason & Jonas (1974), where they computed the meanof two spectra observed by Warner (1969a ,b) near the top of a cumulus cloud

1.4 km deep. There are two peaks in the observed distribution—one centredaround 17 mm and a second around 24 mm radius. With the turbulence-enhancedcalculations, we obtain, as expected, a broader distribution that agreesreasonably well with the observed data points; the second peak centred atca 24 mm is well captured. The simulation with the enhanced fall rates indicatesthe presence of coalescence induced peaks for radii larger than 30 mm. Withoutthese enhanced collision rates, there is no second peak. The important point isthat within a time span of 20 min, we obtain a broader spectrum with theenhanced collision rates than a conventional run. Without these enhanced fallrates, the simulation would have to be extended for a longer time period to fall in

the range of the observed values.These simulations suggest that even with small increase in the collision ratesðC ða ; a ÞÞ, because the droplet number concentration N (d ) decays exponentially

5

4

3

2

1

01 10 100

m

a s s d i s t r i b u t i o n ( g m –

3 / d l n ( r a d i u

s ) )

radius (µm)

Figure 6. Mass distribution in a cumulus cloud. The solid line represents an initial distribution witha total liquid water content of 3.33 g mK3 and a mean droplet radius of 7 mm. The broaderdistribution obtained with the faster settling rates (dashed line) agree better with the observationsas compared with the case with normal settling rates (dotted line). Both correspond to a simulation

time of 20 min. With the normal settling rates a longer time-interval is necessary to have the samespread as for the case with the faster settling rates. The observed data points ( ) are from Mason &Jonas (1974) based on cloud top measurements for a cumulus cloud 1.4 km deep.

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with increasing droplet size beyond the peak at a w7 mm, there is a large increasein the capture of the small droplets with radii of the order of 10 mm. This leads tothe subsequent maxima in the droplet size distribution, for droplets of 17 and24 mm radius. Without the enhancement, the latter is absent and the former isless pronounced. For the typical cumulus-like air parcel in a cloud of depth h

with velocity w , the period of its movement ðwh =w w250 m=0:2 m sK 1Þ is about20 min. The computations and observations agree somewhat better when theamplified fall velocities are accounted for, compared with the calculations basedonly on still air fall velocities. In the latter case, even after 20 min of simulation,the mass and consequently the number concentration corresponding to dropswith radii greater than 20 mm is very small. By contrast, when the number of thelarger cloud drops begin to grow exponentially by differential settling velocitiesand by turbulence-enhanced collisions, the increase in mass (and consequentlyN (d )) makes the difference between rain and no rain!

In §4.3 therefore we shall examine how the turbulence assisted fall velocity

amplifications can lead to a more accurate estimation of raindrop spectra instratocumulus clouds.

(c ) Droplet spectra in Stratocumulus cloud

It is well recognized today by meteorologists that even shallow layers of warmstratocumulus clouds are capable of producing drizzle that reaches the ground.However, as Mason (1952) first pointed out, the production of precipitation-sizedparticles by shallow layers of cloud is incompatible with simple models of dropgrowth (invoking only the effects of condensation and coalescence). Thecalculated growth rates were far too slow, because a drop would fall out of thecloud long before attaining the size necessary to survive the fall to the ground.However, Mason recognized that cloud turbulence could have an effect simply byextending the residence time of the droplets within the cloud. This paperprovides a new approach for quantifying the effect, and a partial verificationusing new observation techniques. Measurements of turbulence and drop sizespectra can now be made on instrumented aircraft and these measurementsconfirm that turbulent diffusion is potentially important in determining thevertical distribution of even quite large drops with radii of ca 100 mm, sinceupdraughts exceeding the terminal velocities of these large drops of ca 1 m sK1

are quite often observed. In a recent theoretical paper Ghosh & Jonas (2001)

derived some analytical expressions for the growth of drizzle drops in turbulentclouds. Their estimates of the velocity amplification effects were based on theDavila & Hunt (2001) results and their calculations showed that it was necessaryto include the dependence of the radii of the smaller captured drop in thecollection growth equation in addition to the turbulence effects. The results fromthis study were more consistent with observations than those of earlier theories(e.g. Baker 1993) which neglected these effects.

Here, we consider the evolution of drizzle and apply it to anotherstratocumulus related case study, which is based on observations of anextensive, horizontally uniform stratocumulus cloud over the North Sea on 22

July 1982. The mean depth of the cloud was ca 450 m, the cloud ‘auto-conversion’ rate (i.e. the rate at which cloud liquid water is partitioned as rainwater) was 3.2!10K9 kg mK3 sK1, the maximum horizontally averaged cloud

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liquid water content was 0.6 g mK3 and drizzle was observed below the clouddown to the lowest flight level (90 m above the sea level). The numerical modeldeveloped by Nicholls (1987) describes the growth of precipitation-sized drops ina warm stratocumulus cloud and combines the effects of stochastic turbulentdiffusions with explicit microphysical calculations. Further details of both the

observations and the model can be found in Nicholls (1987). The most significantfact that emerged from this study is that there was a considerable improvementof model predictions when the effect of air turbulence was considered withvertical r.m.s. velocity fluctuations (swZ0.36 m sK1) as well as the Lagrangianintegral time-scale (T LZ360 s). Nicholls found that the distribution of mainlythe larger drops changed with sw. The concentration of droplets with radiismaller than 20 mm responds rapidly to supersaturation and are controlled bycondensation and evaporation, and these are only minor variations with sw.However, even with the inclusion of turbulent air velocity fluctuations, theNicholls model still substantially under-estimated the number densities of the

larger drops. Nevertheless, this study was a significant improvement over earliersimpler models that ignored turbulence effects altogether. It showed for the firsttime that steady-state concentrations of precipitation sized drops are found to beincreased by some orders of magnitude when realistic levels of turbulence areincluded compared with an identical situation where swZ0. This arises, asNicholls pointed out, because a few particles have a relatively unlikely (butfinite) chance of encountering a significantly higher than average proportion of updraughts. This leads to enhanced growth rates by extending their lifetimeswithin cloud and in some cases by recycling drops upwards through regions of higher liquid water content. This explains why even shallow layers of warmcloud can produce significant amount of drizzle.

The main limitation of the Nicholls model is that he assumed that althoughthe cloud droplets are moved up and down by the turbulent updraughts anddowndraughts, their fall velocities are still equal to the classical still air values.Baker (1993) proposed an analytic version of Nicholls’ model. The non-localturbulence closure was replaced with a stochastic diffusion equation for aturbulent plume of sedimenting raindrops. In addition, Baker (1993) specified aproduction rate for the smallest raindrops, and let the diffusing drops grow byaccretion with a time constant determined by the liquid water content and thesize-dependent fall speed. This enables one to calculate an equilibrium raindropsize distribution as a function of height within the cloud.

Inthispaper,weadoptedtheBaker (1993) model, but included thedependence of theradii of thesmaller captured droplets in thecollection equation,using theDavila& Hunt (2001) results to calculate the new turbulence-enhanced fall speeds. Infigure 7, we show the equilibrium raindrop spectrum at a distance of 95 m above thecloud base. We also show results from Nicholls (1987) as described in their‘standard’ run. The observations are shown for droplet radii that are greater than40 mm where the two-dimensional probe measurements are expected to be free fromany counting errors. From this figure, we find that the Nicholls model under-estimates the number of large drops. In the next stage, we included the effectsof turbulence as by using parameters specified by Nicholls (1987), that is,

swZ0.36 m sK1

and T LZ360 s and used it in the equilibrium rain spectrum model(for details, see Baker 1993; Ghosh & Jonas 2001). Although this second case(marked Baker 1993in figure7) is an improvement over the case whenswZ0,itdoes

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not yield the right number concentrations of the larger drops. In order to match theobservational results, Nicholls (1987) had to artificially alter the spectral shape toslightly larger radii. When we used turbulence-enhanced collision rates using theDavila & Hunt (2001) formalism (marked ‘this calc.’ in figure 7), we find that theresultant spectrum matched the observations well and without any artificialadjustments. It yields drop number concentrations of ca 100 cmK3 for drop sizes of ca 200 mm as is observed.

5. Wider implications

The analysis presented in this paper has shown how turbulent eddies amplify thefall velocity of cloud droplets in the range 10–40 mm and thereby increasescollision kernels in the initial range of particle sizes and then leads to animproved prediction of cloud and raindrop spectra. The theoretical and scalinganalysis, supported by matching laboratory experiments and numericalsimulations for settling velocities, have provided convincing results todemonstrate the effectiveness of this centrifuging action for the first time. Inaddition, when these amplified velocities are accounted for, the predicted cloud

and raindrop spectra agree well with observations. A bimodal spectrum is easilyproduced—even with empirical elements, previous calculations could not achievethat straightforwardly.

106

104

102

100

10–2

10–4

10–6

10–8

c o n c .

( c m – 3

m

m – 1 )

40 50 60 70 80 90 100 200

radius ( m m)

obs

this calc.

Baker (1993)Nicholls (1987)

Figure 7. Raindroplet spectra for a drizzling stratocumulus cloud with and without the effects of turbulence assisted fall velocity enhancements. The observational points (solid squares) are fromNicholls (1987).

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This paper also has wider geophysical as well as meteorological applications.Our collision mechanism is consistent with the recent observational study byGhosh et al . (2000) and Rosenfeld (2000) concerning reduced growth of raindropwhen there is an excess of nucleation particles in urban areas. These studiesimply that in situations where cloud droplets grow in polluted air masses with a

very large number of nucleating particles, the resulting cloud droplets havereduced sizes. This is because a great number of particles start competing witheach other for a limited amount of the available water vapour. As the dropletsizes are reduced, the collision rates also fall, and this eventually leads toprecipitation suppression. Similarly, cloud seeding experiments afford anotherapplication in this context. Seeding is usually achieved with larger sized particlesso that the nucleated particles can rapidly grow by collisions to yieldprecipitation-sized drops. Our results suggest that if seeding experiments canbe conducted in a turbulent air mass, then the particles will actually fall fasterthan their still air fall velocities. This implies that seeding with smaller sizes now

can also induce precipitation. This happens because, although the particles havemodest sizes, in turbulence, they fall faster and produce the same effect as largerparticles. The mechanism proposed in our study and its associated criticalparticle size of 20 mm radii shows that if too few droplets of this size arenucleated, by comparison with the smaller sizes, then the enhanced collision rateand droplet growth will not occur. In addition, the bimodal ‘tail’ in the sizespectrum probably has applications in other environmental and industrialprocesses involving sedimenting, coalescing and flocculating particles inturbulent flows.

These calculations can also be extended to calculate the settling rates of atmospheric aerosols and particulate matter. For example, it is now known thatheterogeneous processing on polar stratospheric cloud particles (PSCs; inparticular the larger type 2 PSCs) is crucial to the correct quantification of stratospheric denitrification and heterogeneous ozone depletion. It is alsoexpected that the estimated fall velocities of these type 2 PSCs (typically withradii of ca 10 mm) would be higher than their still air values when one accountsfor the turbulence within the stratospheric vortex. This faster settling wouldpossibly lead to greater denitrification of the stratosphere, which wouldeventually lead to larger heterogeneously processed ozone depletion.

In forthcoming work, the effects of including this mechanism in compu-tational models for cloud processes will be tested—especially its interaction

between ice crystals, aerosol particles and droplets. To date, current climatemodels, including those at the UK Meteorological Office, continue to use stillair fall velocities for cloud droplets. It is expected that by using turbulence-assisted fall velocities in climate models, one can obtain a better precipitationcharacterization and forecast. In addition, recent research suggests thatturbulence effects on droplet condensational growth can also be important (seeCelani et al . 2005 and references therein). The contribution from this lattereffect may also complement the well-established turbulence effects on settlingand coalescence. In general, the implications for improving numerical modelsfor weather and climate predictions are also being considered. Our future work

will be aimed at formulating the above results in closed-formed parametricresults so that they can be easily incorporated into large-scale climate-prediction models.

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We are grateful to the sponsors for their support of this research: E.C. (S.G.); Spanish Ministry of Science and Technology (J.D.); Isaac Newton Trust (S.G., J.C.R.H.); NSF EnvironmentalGeochemistry and Biogeochemistry (J.C.R.H., A.S., H.J.S.F.); NERC grant to Centre for PolarObservation and Modelling (U.C.L., J.C.R.H.). We have greatly benefited from conversations withT. Choularton, Rob Wood, Yan Yin, J.-L. Brenguier, J. Fung, M.R. Maxey and A.P. Khain. We

are grateful to A. Bott for the SCE solver.

Appendix A. Collision length of small heavy particles settling aroundline vortices

A solid particle released at a level Y ZY 0 far above a vortex of circulation G, afterfalling a time Dt c[2Y 0/V TO will be at

Y ZY 0KV TODt cCðG=V TOÞDhðX 0Þ; (A1)

where Dh(X 0) is the dimensionless differential settling length with respect tosettling in still fluid, a function of the initial horizontal position X 0 (see Davila &Hunt 2001). If DhO0, then the particle settles more slowly. If a small particle(with terminal velocity V TO) collides at a fixed level Y 1 with a larger particle(with terminal velocity V TO), then the collision length of particle pairs is

l cZ l coK ½ðG= V TOÞDhKðG=V TOÞDhZ l coKðG= V TOÞðlDhKDhÞ; (A2)

where l coZð V TOK V TOÞDt cZ V TODt cð1KlÞ with lZV TO= V TOð!1Þ. Hencethe normalized fractional increase in collision length is

E l Zl cl co

K1 V TODt c

G=V TO

ZDhKlDh

1Kl: (A3)

This is the formulation used to obtain figure 3c , where we have plotted E l versusthe initial horizontal position of the particles X 0 for critical small particles withF pZ 4 and V TOZ 0:7, and larger particles with F pZ 5 and V TOZ 0:8.Because the objective of these calculations is the average value of l c over all thehorizontal initial positions, we have not taken into account that the initialhorizontal positions of the larger and smaller particles may be different in orderto have a collision at the fixed level Y 1. From (A3) using the definition of the

drift integral (i.e. the average settling length around the vortex)

D Z

ð N

KN

DhðX 0ÞdX 0

G=V TO; (A4)

equation (2.5) can be obtained. In addition, using (A1), the average settlingvelocity of particles moving around vortices results in

V TZV TOð1KaD Þ; (A5)

where the effective volume fraction occupied by the vortices aZ

ððG=V TO

Þ=Dl v

Þ2.

For a given value of the droplet radius, we first calculate the fall velocity V TO,the particle Froude number F p, the volume fraction a. Then, from (A 4), the driftintegral is evaluated.

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Appendix B. Time evolution of the cloud droplet spectra

The purpose of this appendix is to provide a brief outline of the computationalprocedure that was adopted to evaluate the time evolution of the cloud dropletspectra. In the droplet collection model, all condensation and mixing with the

surroundings were neglected. The droplets were assumed to grow only bycoalescing with each other as they fell. Depending on their sizes, the actual fallvelocities were generally higher than their still air values and this added valuewas precisely estimated using the Davila & Hunt (2001) formalism described inthe paper. For drops with radii less than 30 mm, the still air terminal velocityV TO was calculated as

V TOi Z ka 2i ; (B1)

where k w1.18!108 mK1 sK1 (Rogers & Yau 1994) and a i is the droplet radius.Davila and Hunt’s (2001) calculations depend significantly on the droplet

radius and significant variations are observed in the particle Froude number F p,the radius of the droplet trajectory around the vortex Rtraj as well as the Stokesnumber S t with increasing drop radii. This is shown in table 2 below.

Using the values listed in table 2, we find that the velocity amplification dropsfrom ca 90% for a drop with 11 mm radius to about ca 18% for a 20 mm radius.

Prediction of growth times for precipitation-sized drops also includes thestochastic nature of the collection growth. Because raindrop concentrations aretypically 105–106 times smaller than cloud drop concentrations, one would expectthat the fate of the ‘favoured’ small fraction of drops that happen to grow muchfaster than the average rate would be quite important in the overall process of

precipitation development (Pruppacher & Klett 1997). Our calculations haveborne out this expectation as is briefly described below.

Let the number of drops with radii between a i and a i Cda be N i per unitvolume, and let the probability that a drop of size j will encounter a collision withone of size i in unit time be P ij (with i O j ).

From the above prescription, it follows that the number of drops of size i coalescing with drops of size j in a single time-step Dt , is

DN ij ZP ij N i N j Dt ; (B2)

per unit time. However, the probability P ij that a drop of size i can collide withone of size j in unit time depends on their collection efficiency E ða i ; a j Þ and theirrelative fall speeds V Ti and V T j . This implies that

P ij ZE ða i ; a j Þpða i C a j Þ2ðV Ti KV T j Þ: (B3)

Once the values of E ða i ; a j Þ, V Ti , V T j were estimated, the appropriate values of P ij were calculated. For the control runs (i.e. the effect of turbulence neglectedaltogether), only thestill air terminal velocity of droplets (equation (B 1)) were usedwhile evaluating P ij . The number of drops lost in a particular class and the sizerange that the resulting larger drops covered were also estimated. Finally, the new

values of the droplet concentrations were updated in each size class and the outlinedprocedure repeated for every pair of drop sizes. The history of the droplet spectrumwas followed for 20 min. The SCE solver that is used is described in Bott (1998) and

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uses an accurate flux method, which ensures accurate mass conversion. The massaveraging process consists of a two-step procedure. In the first step, the massdistribution of drops with mass x 0 that have been newly formed in a collision processis entirely added to grid box k of the numerical grid mesh with x k %x 0%x k C1. In thesecond step, a certain fraction of the water mass in grid box k is transported to k C1.This transport is achieved by means of an advection procedure. Further details of the flux method can be obtained from Bott (1998).

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20 0.48000!10K1 0.75233!10K1 0.31250!10K2 0.73469 0.24414!10K1

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