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How turbulence enhances coalescence of settling particles with applications to rain in clouds S Ghosh, J Dávila, J.C.R Hunt, A Srdic, H.J.S Fernando and P.R Jonas Proc. R. Soc. A 2005 461, 3059-3088 doi: 10.1098/rspa.2005.1490

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Proc. R. Soc. A (2005) 461, 3059–3088 doi:10.1098/rspa.2005.1490 Published online 9 August 2005

How turbulence enhances coalescence of settling particles with applications to rain in clouds B Y S. G HOSH 1 , J. D A´ VILA 2 , J. C. R. H UNT 3,4 , A. S RDIC 5 , H. J. S. F ERNANDO 5 AND P. R. J ONAS 6 1

School of the Environment, University of Leeds, Leeds LS2 9JT, UK ([email protected]) 2 ´nica de Fluidos, E.S. Ingenieros, Universidad de Sevilla, Grupe de Meca Camino Descubrimientos, s/n lsla Cartuja, C.P. 41092, Spain 3 Department of Space and Climate Physics and Department of Earth Sciences, University College, London WC1E 6BT, UK 4 Delft University of Technology, 2600 AA Delft, The Netherlands 5 Environmental Fluid Dynamics Program, Department of Mechanical and Aerospace Engineering, Arizona State University, Tempe, AZ 85287-9809, USA 6 Department of Physics, UMIST, Manchester M60 1QD, UK From theoretical, numerical and experimental studies of small inertial particles with density 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 VT in a size range just below a critical radius acr is increased significantly by up to about 80%. We show that in fully developed turbulence, acr is determined by the circulation Gk of the smallest Kolmogorov micro-scale eddies, 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 acr varies approximately like n2=3 gK1=3 ðbK 1ÞK1=2 and is about 20 mm (G2 mm) for water droplets in most types of cloud. New calculations are presented to show how this phenomena causes higher collision rates between these ‘large’ droplets and those that are smaller than acr, leading to rapid growth rates of droplets above this critical radius. Calculations of the resulting 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 mm irrespective of the level of cloud turbulence is also applicable where acrw1 mm for typical sand/mud particles. This mechanism, associated with unequal droplet/particle sizes is not dependant on higher particle concentration around vortices and the results differ quantitatively 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 turbulent motions within clouds has three main stages. Firstly, condensation of saturated Received 8 April 2004 Accepted 1 April 2005

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water vapour on to nuclei causes the growth of droplets with radii generally smaller 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 smaller droplets, the larger droplets grow by collisions with the smaller ones and fall out of the cloud. The second stage is still not adequately understood or accurately modelled, in common with other processes in which the average sizes of particles and bubbles grow in turbulent flows caused by an increased rate of collision and coalescence (Jonas 1996). The developments in the understanding the structure of turbulence and the motion of particles in turbulence (Hunt et al. 1994, 2001; Da´vila & Hunt 2001) provides an opportunity to re-examine these problems. There is still some controversy as to whether this mechanism also controls the coalescence/flocculation of mud/clay particles in water. The uncertainty is partly because of the lack of experimental observations, and partly because it is still 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 reduced surface to volume ratio and as a result, even if the initial particle size spectrum is broad, subsequent growth of the particles would lead to a narrowing of the spectrum as the mean size increases, if all particles were exposed to the same supersaturation. Recently, it has become possible to measure droplets with a much higher spatial resolution. Observed droplet spectra at all levels in most water clouds are generally broader than spectra modelled on this basis. In addition, it is also observed that growth by coalescence is very slow until some droplets have reached a critical radius of ca 20 mm, whereupon in deep clouds with high values of the liquid water content, subsequent growth to drizzle size may take only a few minutes (Jonas 1996). A major concern for researchers in the field of cloud physics is to find the cause of this transition to fast growth (see figure 1). Many of the early calculations of particle growth in clouds were based on the assumption that the particle grew in stagnant air. However, observations show that most clouds are very turbulent with dissipation rates ranging from 10K4 to 10K1 m2 sK3 in cumulus clouds (Smith & Jonas 1995). In this paper, we have critically examined the role of turbulence in inducing microphysical alterations. In order to explain the broad spectrum observed, two main mechanisms have been 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 temporal fluctuations. If the effect of turbulence is not considered, the former mechanism is too slow (Mason 1952). In one of the first models for the effect of turbulence, Brunk et al. (1998) suggested that straining within Kolmogorov micro-scale eddies would accelerate the collision relative velocity of colliding droplets. Pinsky & Khain (1997a) analysed the motion of inertial particles in turbulence using a statistical model and numerical simulations to show how the centrifuging action of vortical eddies tends to concentrate particles near the periphery and thereby amplify droplet coalescence (see also Shaw 2000). However, in many microphysical models, turbulence induced fall velocity enhancements depend rather critically on the energy of the turbulence. Spectral broadening effects in cloud 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 Proc. R. Soc. A (2005)

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microscale vortices ~ 1 mm

large eddy ~ 100 m

1 km

1 km cloud droplet (10;106;1)

newly formed rain droplet (100;100;70)

aerosol particles (0.1;106;10–4) Figure 1. Schematic showing the diversity of spatial scales, vortices and particle number concentrations in cloud microphysical calculations. The numbers within brackets refer to radius in microns, number per litre and terminal velocity of droplets in cm sK1, respectively.

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 critical size (over which turbulence effects can enhance droplet coalescence) theoretically. Indeed, in this paper, we have achieved this through scaling analysis (see equation (3.5) of the present paper). In addition to the above-mentioned papers, two review papers discuss in some detail the problem of particle–turbulence interactions and the consequent implications for cloud microphysical applications. Vaillancourt & Yau (2000) have reviewed laboratory and numerical work, concluding that the majority of direct numerical simulations have not accounted for gravity and have focused on Stokes numbers close to unity where preferential concentration is found to be the most prominent. In addition, they argue that the effect of preferential concentration during diffusional growth cannot be treated as a good mechanism to explain droplet spectral broadening in adiabatic cloud cores. The other recent review paper by Shaw (2003) summarizes recent advances in this area (including the mechanism suggested by Da´vila & Hunt (2001) and its subsequent application by Ghosh & Jonas (2001)) and, in particular, they point out that the influence of fine-scale turbulence on the condensation process may be limited. Shaw also points to mechanisms of finescale intermittency, droplet number density fluctuations, entrainment and mixing in addition to the processes of collision and coalescence. From the discussions given in these two reviews, it is clear that an exploration into fluid–particle interactions that does not depend sensitively on the droplet clustering mechanisms should be explored further. Nevertheless, these and other full-scale and laboratory studies agree that the coalescence of cloud droplets is Proc. R. Soc. A (2005)

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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 apparent contradiction. Our approach considers an important mechanism in addition to the 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 collision efficiencies of colliding droplets in turbulence. A literature survey (including the reviews by Shaw 2003; Vaillancourt & Yau 2000) unanimously point out that the collision efficiency of cloud droplets can be increased by turbulence–particle interactions. Pinsky et al. (2000) have calculated collision efficiencies of small cloud droplets in a turbulent flow and found that the mean values of the collision efficiency and the kernel are higher in turbulent flows than in still air. This is an ongoing research area in cloud microphysical studies and featured prominently at the recent 14th International Conference in Cloud Physics in July 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 collision efficiencies and collision rates in turbulence. The results from these studies have enabled us to obtain a broad estimate of the collision kernel enhancements in turbulence. 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 new quantitative estimates for the particle motion in the typical vortices of high Re turbulence and the consequences on the droplet distribution. It is shown how this rational theory can be applied to various physical situations. Our calculations and scaling analysis establish a critical droplet size for droplet fall velocity enhancements when an ensemble of droplets interacts with a vortex. The critical size prescribes the fall speed that should be right for this amplification to be most effective. In addition, our proposed mechanism, associated with unequal droplet sizes, is not dependent on higher particle concentrations around vortices (as proposed by Falkovich et al. 2002), where the higher number concentrations ensure 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 dominant mechanisms that can affect cloud microphysical processes. In §3, we discuss the main theoretical considerations leading to our new estimates of the average settling rates for particles around vortices and the mechanism of enhanced collisions 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 microphysical simulations. We evaluate the evolution of a typical cloud droplet spectrum with and without the centrifuging action of the vortices. Finally, we evaluate raindrop spectra with and without turbulence effects, and are able to predict the existence/non existence of large raindrops without artificial adjustments for the first time. Although other mechanisms have been proposed for turbulent enhancement of rain formation, none have been able to produce a broad spectrum simply by including turbulence assisted collisions in the spectral development. In §5, we discuss the wider applications of this model to particles, droplets and bubbles in liquids. Proc. R. Soc. A (2005)

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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 TL are studied using numerical simulations, laboratory measurements and theory. Some of these earlier numerical results have shown that inertial bias causes particles to accumulate on the outside of twisted tube-like vortical structures (Jime´nez et al. 1993), with the general tendency of the particles being to disperse faster than fluid elements (e.g. Squires & Eaton 1991). Maxey & Corrsin (1986), Maxey (1987), and Wang & Maxey (1993) showed by numerical simulations that the settling rate VT of typical inertial particles with time-scale tp may be slightly larger (!20%) than the terminal velocity in still fluid VTO 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 with conclusions broadly similar to those obtained from Wang & Maxey (1993). Fevrier (2000) showed that this increase could be substantially greater for a particular range of inertial particles for which the Stokes number StZtp/TLO0 and were found to accumulate in regions of low vorticity and high strain rates (Squires & Eaton 1991). The earlier results of Wang & Maxey (1993) have shown how small-scale dynamics cause intense vorticity in turbulent flows to form at dissipation-range scales and that particles accumulate in the low vorticity regions of the flow. They do not accumulate here because the flow is faster and there are straining regions. However, they do spend an increased time at particlestagnation points. Their numerical results in homogeneous isotropic turbulence indicated that maximum preferential accumulation occurs when the inertial time-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 that particles accumulate in the low vorticity regions of the flow and are centrifuged away from the vortex cores. However, the fact that certain particles are deflected into particular zones around vortices increases the local void fraction of the particles. Could this effect further increase the fluid interaction between these particles 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 weaker effect for cloud particles than the former effect of inertial particles moving in the downflow side of the vortices. The question of the role of local concentration is also important for estimating collisions between droplets; the first mechanism we show to be the most effective is that the relative speeds of particles and therefore collision rates for different sizes are enhanced by their motion around turbulent vortices. The second is that certain sizes of particles are concentrated around these vortices, so that they encounter each other more often than in the surrounding flow and their relative acceleration is thereby affected. (b) Effects of isolated vortices on droplet settling To understand and model the average settling velocity VT of small dense particles (such as cloud droplets) descending and colliding in turbulence, we Proc. R. Soc. A (2005)

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apply the results of Da´vila & Hunt (2001) to analyse particles around an isolated vortex with circulation G (with radius Rv and maximum velocity UvZG/Rv). (Note that, generally, the acceleration in a Kolmogorov micro-scale eddy is small relative to that due to gravity (i.e. e3=4 =ðn1=4 gÞ10K1 ), but there are occasional intense vortices where this inequality does not apply). The very small particles with radius a have an acceleration dV 1 Z ðu C v TO K V Þ; dt tp

(2.1)

where v is the velocity of the particle and u is the unperturbed velocity of the fluid at the position of the particle. The settling or terminal velocity in still fluid estimated by using Stokes linear drag law is VTO xgðb K 1Þa 2 =ð9n=2Þ;

(2.2)

which the particles reach after a ‘relaxation time’ tp Z ð2=9Þðb K 1Þa2 =n;

(2.3)

where n is the kinematic viscosity of the fluid (ca 10K5 m2 sK1 in air) and b is the ratio 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 particle moves near a vortex. Numerically, the trajectories of the particles can be obtained by integrating (2.1) together with dX/dtZV given certain initial conditions (e.g. V ðX 0 ; t0 ÞZ uðX 0 ÞC V TO ). Where the ratio, u, of the terminal velocity VTO to the maximum velocity in the vortex Uv is less than about 1.0, Da´vila & Hunt (2001) show that the effect of the vortex on the settling velocity is determined by the non-dimensional ‘particle Froude number’ Fp, defined by the ratio of the stopping distance (VTOtp) of the droplet to the characteristic radius (G/VTO) of the trajectory of the droplet around the vortex: Fp Z

VTO tp : G=VTO

(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 Stokes number, which is more relevant to our analysis than the conventional definition. The Stokes number defined in Da´vila & Hunt (2001) is defined as tp/tr, where tr is the residence time of a fluid particle around the vortices. Further, if the residence time of the particles to move around the vortices is much shorter than the lifetime of the vortical structures, then particle trajectories can be calculated by considering that the flow is stationary (see Da´vila & Hunt 2001). Using the result of that paper and that of Vincent & Meneguzzi (1994), this can be 2 expressed as G=VTO / TI . TI is a time-scale of the order of the ratio of a typical large 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 (Fp/1), it passes round the vortex and the net change in VT (averaged over the whole life of the particle Proc. R. Soc. A (2005)

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near the vortex and over a range of starting positions on the scale of G/VTO or Rv, whichever is the larger) from its value VTO in still fluid is negligible (i.e. vZVT/VTOx1.0). However, when the inertia is large enough that FpwO(1), the particles are flung outwards; then, v rapidly increases to a maximum value vmax of about 2.0 (for 1.0RuR0.7, where FpxFpmax). With a small further increase in inertia, the particles ‘crash’ through the vortex and are on average slightly delayed, so that v is reduced to a minimum value vmin of about 0.7 (for u%1.0 where FpZFpminx4). For very large inertia or large u, the vortex has negligible effect on the settling rate, and vZ1.0. (These values of vmax and vmin are calculated for particles released at a level above the vortex equal to about 10 times its radius Rv, falling to an equivalent distance below the vortex.) In addition, large inertia particles have a stopping distance VTOtp greater than the distance between the eddies Dlv (see figure 2a) so that they average out the effect of individual eddies and vx1. It is important to point out that vR1.0 and Fp%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 microphysical implications. First, droplet pairs within this regime have their fall velocities enhanced when they interact with the micro-scale vortices within clouds and this can lead to increased collision and capturing among droplets that eventually yield realistic spectral distributions (see figure 6). Secondly, our analysis also supports the well-established observation that collisions between unequal droplet sizes 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 and changes in the settling velocities of particles in turbulent flows with vortices (Maxey 1987; Perkins et al. 1991; Fung 1993). However, a systematic calculation for the increase and decrease in the average settling velocity over a wide parameter range is new, as is the derivation of a new scaling in terms of (G/VTO), 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 in hydraulic flows is not relevant here where b is very large. The motions of the particles around a vortex vary greatly if their sizes lie within certain ranges. This can substantially increase the probability of collisions in a time tc between pairs of different sizes, with large and small radii a
, a^, respectively, having terminal fall speeds VTO ð
a Þ, VTO ð^ aÞ. In still air, the probability Pc is proportional to the length of a vertical collision line lco in which all the larger particles must be positioned at tZ0 if they are to collide with the smaller particle in time Dtc, where lco Z Dtc ðVTO ð
a ÞK VTO ð^ aÞÞ. As the large particle moves round a vortex (see figure 3a,b) starting at time tZ0, it collides with a small particle also released at time tZ0, if they start on a curved ‘collision line’ lc. If the large particle lies in the critical range where VT is significantly greater than VTO, then because lc is greater than lco, the probability of a collision is proportionally greater. (c) Collisions of droplet pairs around isolated vortex lines In addition to the considerations described above, some other interesting features emerge from the Da´vila & Hunt (2001) theory, which relate to droplets Proc. R. Soc. A (2005)

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B

A

VTO C

G/ VTO

Rv

∆ lv

G

(b)

VT VTO 1.0

0

Fp 10–3

10–2

10–1

1

101

102

0.3

0.45

0.7

1.0

1.5

2.25

a acr

(c)

d(VTVTO) d(a /acr) 0

0.3

0.45

0.7

1.0

1.5

2.25

a acr

Figure 2. Mechanisms of particles falling near vortices: (a) typical trajectories of settling particles A, B, C moving around a vortex with strength G and radius Rv with increasing fall speed VTO. Here, Dlv is the distance between vortices; (b) average settling velocity ratio VT/VTO versus the particle Froude number Fp and particle radius ratio a/acr; (c) d(VT/VTO)/d(a/acr) versus a/acr.

Proc. R. Soc. A (2005)

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Figure 3. Collision mechanisms: (a) colliding droplets settling in still air in the absence of vortices with larger and smaller radii a
, a^. Note the vertical collision distance lco; (b) colliding droplets descending near an isolated vortex. Note the curved collision line with length lc; the large droplet collides with a smaller droplet both released at tZ0, if they start on the curved collision line; (c) variation of lc relative to lco for still fluid over a typical horizontal distance, El Z ðlc =lco K 1Þ! ðDtc V
TO Þ=ðG=V^ TO Þ for smaller particles with F^ p Z 4 and V^ TO Z 0:7, and larger particles with F
p Z 5 and V
TO Z 0:8.

Proc. R. Soc. A (2005)

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interacting within intense vortex lines. Two additional collision mechanisms are relevant to cloud microphysical calculations. Firstly, for droplets moving between line vortices, the ratio of the average collision length to the length in still air hlci/lco in a horizontal box of size Dlv can be expressed in a closed form after some algebra (see appendix A): ^ K l2 D
hlc i ðG=V^ TO Þ2 D Z1C : ^ TO 1 K l lco Dlv Dtc V

(2.5)

As before, lco is the collision distance taken by a large droplet of radius a
to collide with a small droplet of radius a^ in time Dt c in still fluid ^ TO are the settling velocities of the large ðlco Z ðV
TO K V^ TO ÞDtc Þ. V
TO and V and the small droplets in still fluid. Dlv is typically a measure of the distance between the vortices, which is comparable to the vortex diameter and ^ and D
are the dimensionless ‘drift’ integrals correspondlZ V^ TO =V
TO ðl! 1Þ, D ^ 0 and D
!0 and ing to the small and large droplets. When Fp%1, both D! 2
^ jl Dj! jDj. This results in a smaller collision distance (i.e. hlci/lco!1). This implies that in order to have the maximum collision enhancement induced by ^ !0. Equation (2.5) shows that turbulence the droplet radii must be such that D hlci/lco increases in proportion to the effective trapping radius of the vortex G=V^ TO and inversely to the assumed time between collision Dtc. Hence, we define a normalized value of the fraction of the collision length increment for the distance between the intense vortex lines given by El Z ðlc =lco K 1Þ ðDtc V
TO Þ=ðG=V^ TO Þ. By calculating how lc varies over a typical range of the initial location X0, one can estimate the average value of lc, hlci, for all particles of a particular size a
descending round a typical vortex and thence the average value of El and the probability pð
a ; a^Þ that a larger particle will collide with a smaller particle of size a^. The ratio hlci/lco is a measure of the increase in this probability compared with collisions in still fluid. The calculations show that lc has a maximum at an initial position X0M for the small particle corresponding to smaller velocities than in still fluid, and for the large particle larger velocities than in still fluid (these particles pass close to the equilibrium points described by Da´vila & Hunt 2001). Smaller particles moving in the downflow side of the vortex settle faster, which implies that lc has a minimum at X0M corresponding to particles 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 El is negative up to droplet radii smaller than 35 mm. This shows that for the most efficient droplet capture between an ensemble of colliding droplets, the maximum collision enhancement is achieved when there are some droplets whose radii exceed 35 mm. However, when the droplet radius is somewhat greater than 35 mm, El changes sign leading to collision enhancements, since the drift integral DO0 for values of FpO1 (see fig. 10 of Da´vila & Hunt 2001). For Fp!1 the effect of the vortices on the average settling velocities of droplets is smaller for the larger particles. This is because the effective area of the line vortices that modify the droplet trajectories, ðG=V^ TO Þ2 , decreases as a4. As in the theory of Falkovich (2002), it appears that collision rates are not simply related to fall speed or to concentration or probabilities of lying in the vortex core. Proc. R. Soc. A (2005)

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Figure 4a,b shows the position of droplet pairs with radii 20 and 25 mm and 5 to 10 mm, respectively. The X- and Y-axes are each normalized with the vortex radius Rv. These are calculated using (2.1) as in Da´vila & Hunt (2001) for the droplet pairs moving around a vortex with circulation GZ1.5!10K4 m2 sK1 and radius RvZ1.5 mm (to consider the effect of gravity in non-horizontal vortices, the terminal velocity should be projected on the plane perpendicular to the vortex 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 multiple collisions (figure 4b). This indicates that this vortex–particle model is consistent with the well-established observational result that the probability of collision between droplets of unequal sizes is higher than the collision probability of similarly sized droplets. Other theories have addressed the issue of droplet spectral broadening by invoking various mechanisms that depend sensitively on turbulent energy dissipation (Pinsky & Khain 1997b), on enhanced droplet concentrations around vortices (note that the higher number concentration ensures enhanced droplet collisions) as in the Falkovich (2002) study or on particle interactions with intense vortex tubes (as in the Shaw 2000 study), which are again concentrationdependent. Shaw’s scale analysis also showed that the vortex tubes were sufficiently intense and persistent so that they caused larger flux divergences in the local concentrations of cloud droplets. From our analysis, we have been able to explicitly show that the probability of collision between two droplets with widely different radii are much higher than collisions between droplets having comparable radii—a well established observational result (Pruppacher & Klett 1997). This feature is not apparent in these earlier theories. Collisions involving very small droplets (for which Fp!1 and u!1) can also occur within the vortices (see figure 2a). As explained above, most of these particles that fall towards a vortex are advected towards it and tend to be swept around the vortex by the mean streamlines and by their own inertia in the curved flow (e.g. Squires & Eaton 1991). However, as a result of other mechanisms in the interior 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. Jime´nez et al. 1993) and they surround any small particle present provided tp!tv. Then, the vortices trap the particles for a certain time tvp. In high Reynolds number turbulence tv wðe=nÞK1=2 w10K1 s in typical clouds (Pruppacher & Klett 1997). Usually, the vortices last for a longer period as they decay than during their growth phase. For a typical 10 mm droplet, tpw10K3 s, so that these droplets can certainly be trapped. The time for such low inertial particles with a fall speed VT to escape beyond the cavity region is given by tvpwRp/vp, where RpwG/VTO and 5 np wðVTO tp ÞðVTO =Rp Þ. Thence tvp wgG2 =VTO (provided Fp!1). Typically, for 10 to 20 mm particles in the atmosphere, RpwRv for micro-scale eddies and tvpTtv. We conclude that the critical scale, small inertia, particles can remain trapped in these vortices over the lifetime of the micro-scale vortices. (Although direct numerical simulations cannot describe micro-scale dynamics in turbulence with 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 collision mechanisms, which differ from those outside the vortices. The first involves Proc. R. Soc. A (2005)

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25

15

Y(t)

5

–5

–15

–25 –20 –15 –10

–5

0

5 X(t)

10

15

20

25

30

0

50 100 150 200 250 300 X(t)

(b) 250 200 150 100

Y(t)

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

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

Proc. R. Soc. A (2005)

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Table 1. Particle settling ratios outside and inside vortices showing how in different ranges of the stopping/vortex ratio Fp the effective particle settling velocity vZVT/VTO normalized on its value in still fluid, is increased or decreased for characteristic vortices (Here we focus on critical particle sizes where Fpw1, which implies that for typical Kolmogorov micro-scale eddies acrw(n/(bK1))1/2(g/RK1/4 ); that is, acr ca 20 mm in air, acr ca 1 mm in water.) v

external particlesa internal particlesb

Fp/1

Fp(1

FpT1

Fp[1

vT 1½1
E v( 1½1
I

v O1½2
E v( 1½2
I

v !1½3
E v !1½3
I

v w 1½4
E v w 1½4
I

a

Note that the effect on v of external particles is only significant if VTOtp!Dlv.bParticles can only be trapped within a vortex if tp!tv and only spend a significant time there, in relation to its passage around the vortex, if tvpTRv/VTO.

critical scale particles (FpT1), which are initially trapped and then thrown out, and in doing so, collide with very small particles (Fp/1). The second involves all the small particles (Fp!1, v!1), which are trapped and therefore collide with any much larger particles (FpT1, vO1), which crash through these vortices without being deflected. Thus, applying the two set of results to the overall distributions of droplets and vortices, we conclude that there are two ranges of the vortex–droplet parameters that need to be considered (see table 1 below). From this table, it is clear that the greatest differences of relative settling velocity 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 range close to that of the critical diameter acr, as defined by Fpw1, uw1. For the internal particles that are trapped, the largest difference occurs between large particles that cut through the vortex [4]I and the small particles that are trapped [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 are close in size to each other. These are more numerous than the larger particles and are continuously being nucleated. Although our calculations have been mainly concerned with cloud droplets with radii ca 20 mm interacting with micro-scale eddies, it is also possible that turbulence can modify the settling rates of the larger droplets interacting with even larger eddies. However, from equation (3.5) (see §3 below), we note that because the critical radius acr has a 1/6 power dependence on the vortex circulation, its value even for inertial range eddies lies in the range 20–30 mm. 3. Applications to turbulence: experiments and prediction 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 the Digimage system, they studied the effects of ‘mixing-box’ turbulence on the settling of small dense particles in water with radii varying between 22.5 and 355 mm over Proc. R. Soc. A (2005)

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a range of b between 1.4 and 8.7. In these experiments, the spectrum was only large enough for the small scales, approximate to that of high Reynolds number inertial range turbulence over a limited range of scales (Kit et al. 1997), but the Reynolds number was large enough (Rex200) so that there were active vortical motion with vortices 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, in figure 2a, the net effect of a vortex on the fall speed is only significant for particles when the characteristic distance G/VTO is less than the distance Dlv between the eddies, that is, if VT tp ! R:

(3.1)

The speeding up effect of the vortices on VT is a maximum when Fpw1, where 2 Fp Z VTO tp =G. Srdic and Fernando’s experimental results for v plotted as a 2 function of a large-scale particle Froude ratio FPL Z VTO tp =ðUL LÞ (where UL is the maximum vortex velocity with radius L) showed that v increased from 1.3 to about 1.8 when FPL was about 0.8!10K3 (G20%), and decreased to about 0.8 for FPL x3 ! 10 K 3 . As noted previously, in Fevrier’s (2000) simulations, this range of increase and decrease in v was also found. For FPL greater than 4!10K3 v was equal to 1.0. These low values of large scale FPL correspond to values of Fp of order unity for micro-scale vortices in the flow (with characteristic velocity Ukw(en)1/4 and length-scale Rkw(e/n3)K1/4). The fact that the particle motions were primarily distorted by small-scale vortices was confirmed by the measured small spikes in the frequency spectrum of the velocity of the particle at the micro time-scale lk/vk. The experiment not only confirmed the prediction of a significantly larger increase than decrease in settling velocity over a narrow parameter range, but also showed how ‘empty’ regions exist around and below the vortices from which the inertial particles have been expelled. This was expected from the theory when the ratio uL(ZVTO/UL) was between 0.2 and 1.2. Applying these results to the settling of the smallest droplets in a cloud with a radius a, it follows that v(a) can only be significantly increased if vortical eddies exist of scale R whose strength G is such that Fp(R)w1. Therefore, from equations (2.2) and (2.3) 2 GðRÞ wVTO tp w

a 6 g2 ðb K 1Þ3 : ð9=2Þ3 n3

(3.2)

Analysing the structure of turbulence provides an estimate for G(R) for the vortices formed over many scales with radius R (e.g. Sundaram & Collins 1997; Hunt 2000). At the micro-scales where RZRk, the circulation G Z Gk wUk Rk wðenÞ1=4 ðn3 =eÞ1=4 wn:

(3.3)

A key point of this theory is the observation that Gk is independent of the energy of the turbulence. Physically, this is because as e increases, the peak velocity Uk increases, but the radius Rk decreases by the same amount. For RO Rk, the circulation of eddies in the inertial range G(R) increases with their radius as GðRÞ wnðR=Rk Þ4=3 : Proc. R. Soc. A (2005)

(3.4)

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From equation (3.2), it follows that the smallest radius acr of droplets whose settling velocities are increased (by up to about 80%) is given by acr wn2=3 gK1=3 ðb K 1ÞK1=2 :

(3.5)

In air acrw20 mm, and in water, for bZ2, acrw100 mm. Note that for these particles in the micro-scale eddies VTO =Uk w

gba2 =n ðenÞ1=4

;

(3.6)

for aw10 mm. This shows why the turbulence must be intense enough and e large enough for VTO/Uk(1. Thus, for droplets with a smaller radius than 20 mm, since VT increases in proportion to a2, Fp is much less than 1.0 and therefore the particle fall speed is not on average increased (i.e. vx1). In a high Reynolds number turbulence where there is a full inertial range of vortical eddies varying in size (R) down to the Kolmogorov micro-scale, typical velocities increases with radius U(R)we1/3R1/3. We can apply the formula (3.2) to estimate the critical size of particles acr(R) that are accelerated by eddies with scale R larger than Rk; it follows that acr ðRÞ=acr ðRk Þ xðR=Rk Þ1=6 :

(3.7)

This shows that even the eddies that are 100 times larger than the micro-scale eddies also tend to accelerate and concentrate particles with diameters in the range 20–30 mm. This is why the result (3.5) is quite robust. We also have to consider the ratio VT(acr(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) (VT(acr)) increases as a2 or R1/3. It follows that the ratio VT(acr(R))/U(R) is approximately constant so that if (3.6) is satisfied the basic criterion for the ‘acceleration’ effect is satisfied for all particles in this range. When particles reach 80–100 mm, their fall speed becomes comparable with that of the energy containing eddies in the turbulence so that U(R) reaches its maximum value. Then, VT/U(R) increases and the acceleration effect ceases. These calculations are consistent with the experiments of Srdic and Fernando and earlier measurements of enhanced settling rates. In fact, experimental evidence of much enhanced sedimentation in turbulence (by even more than 80%) was observed by Nielsen (1992). When the void fraction is greater than the 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 cloud droplet formation.) ´vila & Hunt (2001) calculations to In order to justify the applicability of the Da cloud 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 Da´vila & Hunt formulation can indeed be applicable to clouds. The theory is not only valid for horizontal vortices (see eqn (6.3) of that paper). Although the calculation of the drift integral in the Da´vila & Hunt paper is for horizontal Proc. R. Soc. A (2005)

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vortices, one can work with the projection of the terminal velocity on the plane perpendicular to the vortex axis for any other orientation. We have focused on the effect of cylindrical vortices because only in regions of strong vorticity can the trajectories of heavy particles change so that the average velocity is altered. This is based 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 high vorticity can create nearly empty regions and deviations in particle trajectories. We ensured that the inter-vortex distance was comparable to the diameters of the vortices. We have considered only values of vortex circulation of the order of the kinematic viscosity, the typical values found in DNS, although Jime´nez et al. (1993) suggest that there may be a weak pffiffiffiffiffiffiffiffi dependence on the Reynolds number based on the Taylor micro-scale ðGf Rel Þ. The methodology of this calculation is not based on a precise statement that turbulence consists exclusively of vortices. It is an approximate physical calculation in which one isolates a mechanism that has a strong macroscopic effect and estimates its significance in relation to overall data. This has always been the approach in cloud physics studies, and the earlier results have not been in agreement with data because the collision/settling mechanism was not the most significant mechanism. The sign and magnitude of the effects of the mechanism do not depend sensitively on the precise distribution and orientation of vortices. The laboratory experiments of homogeneous turbulence of Srdic and Fernando (Re of about 100; Srdic 1998) show through visualization how particles of a certain size are deflected/excluded from random vortices in the flow. More significantly, they show how the mean fall speed rises (by 80%) and falls (by 20%) with turbulence relative to still fluid by an amount that approximately corresponds to the Da´vila & Hunt (2001) calculation for a typical distribution of single vortices. The separations of the vortices are comparable with the diameter of the vortices. For very high Reynolds number turbulence, there is still no detailed data available; except laboratory visualizations (e.g. by Douady et al. 1991) certainly show these structures exist and with a distribution in space (a factor of 10 radii between them) quite comparable with the hypothesized spacing in the Davila and 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 inhomogeneous shear flows. They found that in a horizontal turbulent air jet, small inertia particles, where VTwuo, were found to fall significantly faster than in still air, but for the heavier particles, where VTOuo, there was no such effect. This explains the discrepancy presented in that paper between the experimental results and the predictions of their stochastic simulation model which did not account for the faster settling of particles (which is normally assumed in such models). Finally, we wish to point out that the Da´vila and Hunt mechanism has now been referred to in recent work concerned with rain enhancements in turbulence as in the Falvovich (2002) paper published in Nature and also in the review paper by Shaw (2003). Proc. R. Soc. A (2005)

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4. Droplets in cloud turbulence (a) Collision rates The collision rate C ð
a ; a^Þ between larger droplets (collector droplets) of radius a
and smaller droplets (collected droplets) of radius a^ is determined by the velocity difference DVT ð
a ; a^Þ between the droplets and the efficiency E with which they collide. The velocity enhancements are size dependent, as was shown by Da´vila & Hunt (2001), and hence the relative velocity DVT ð
a ; a^Þ between droplet pairs is very often different from a conventional relative velocity using still air fall velocities. Although we have seen in figure 3b,c how the collision process is complex around a vortex, an estimate for the collision rate allowing for its increase in these vortical flows is C ð
a ; a^Þ Z jDVT ð
a ; a^ÞjEð
a ; a^Þ;

(4.1)

where Eð
a ; a^Þ (O0) is the collision efficiency. We used the stochastic collection equation (SCE) solver employing the flux method developed by Bott (1998). From various sensitivity studies, Bott (1998) showed that the flux method remains numerically stable for different choices of the grid mesh and the integration time-step. The hydrodynamic collection kernel that is used in the SCE solver is simply the collision rate given by equation (4.1) multiplied by pð
a C a^Þ2 (Pruppacher & Klett 1997). In the original SCE solver developed by Bott (1998), the collision efficiencies were taken from Long (1974), although Bott (1998) used other collision efficiencies (e.g. Davis 1972; Hall 1980) for various numerical experiments with satisfactory results. While evaluating the hydrodynamic 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 our paper, since we are mainly concerned with relatively small droplets (where the collector 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 enhanced sedimentation; the solid lines correspond to the case with the enhanced sedimentation and the dashed lines to pure gravitational settling. In order to single out the sedimentation effects exclusively, we did not enhance the hydrodynamic kernels in this figure; that is, the same collision efficiencies were used for both set of curves shown in figure 5a. Figure 5a shows that with the effects of turbulence, which increases DVT 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 droplet radii pairs up to 16 mm. For droplet pairs approaching 40 mm, there is even a reversal of the contours. This is because with larger droplet pairs, the fall velocity enhancements are proportionately smaller. As a result, the velocity difference between the droplet pairs is actually smaller than the velocity difference without turbulence effects, and this causes a reversal of the contours. However, it is unrealistic to use still air collision efficiencies when one considers settling of droplets in turbulence, because the settling rates are enhanced in turbulence, and the collision efficiencies are also expected to increase, causing an overall increase ´vila & Hunt (2001) in the hydrodynamic kernel. This feature is implicit in the Da theory, which shows that with increasing settling rates, the collision efficiencies are expected to be higher and can indeed be enhanced by at least 50–100%. Proc. R. Soc. A (2005)

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3

–1 –100 0

1×10–121×10–1

30

0 1 1×10–11× 1 × 1

1×

10 –

713 –10 –1–

10

010 10 ×11× 1×

3

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10

12

1×

–

10 –

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10 1×3–15 1 – 0 10 ×11– –1 1×

20

–17 –15 1×10 1×10

10 0 –13

radius of captured drops (µm)

(a) 40

7

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1×

–

1×10 –10

1×1

–1

–12 –15 –13 1×10 1×101×10

–15

1×10

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20 30 radius of capturing drops (µm)

1×10–13 1×10–12

0 1× 1×11×1 13 1–7

– 010 ×11×

1×

10 –

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2

–1

0 –13

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1×10–10 –12 1×10–13 1×10 1×10–17

1×1

55 –0–17 0 1 1 × 11×

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103 –1 1×0–1×10 1 ×1

1×

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1× 10 –1 0

30

2 –51 –10 –1001

–17 1×1 1×100–15

radius of captured drops (µm)

(b) 40

40

10

20 30 radius of capturing drops (µm)

40

Figure 5. (a) Collision kernel (pð
a C a^Þ2 Eð
a ; a^ÞjDVT ð
a ; a^Þj) (m3 sK1) with (solid lines) and without (dotted lines) turbulence induced enhancements of the differences DVT in the settling velocities. In order to single out the sedimentation effects, the collision efficiencies Eð
a; 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 (p
a C a^Þ2 Eð
a; a^ÞjDVT ð
a; a^Þj) (m3 sK1) with (solid lines) and without (dotted lines) turbulence induced enhancements of the differences DVT in the settling velocities. In order to include turbulence effects on the sedimentation as well as on the collision efficiencies Eð
a; a^Þ, the turbulent kernel is enhanced by 50%. a
and a^ are the radii of the larger capturing drops and the captured collected drops, respectively. (c) Mass distribution after 15 min without (dotted line) and with (dashed line) the effects of turbulence induced velocity enhancements. The solid line 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 bimodal spectrum is obtained only with the faster settling rates shown in (b).

Proc. R. Soc. A (2005)

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(c)

5

mass distribution (g m–3/dln (radius))

Coalescence of particles in rain clouds

4

3

2

1

0

10

100 radius (µm)

1000

Figure5.5. (Continued.) (continued.) Figure

´vila & Hunt (2001) one obtains progressively decreasing velocity From Da enhancements with increasing droplet sizes. For example, the enhancements decrease from 90 to 18% as the droplet radii increase from 11 to 20 mm. For the range of droplet sizes considered in this study, the velocity amplification effect is the highest for the smallest droplets. With increasing droplet radii, the effect progressively decreases; this is consistent with our discussions in §§2 and 3. Using the Da´vila & Hunt (2001) theory and the analysis of §3, our calculations are extended over a wide range of Fp values. The computations depend sensitively on ´vila & Hunt (2001) paper implies, because the velocity the droplet size as the Da amplification ratio v increases in proportion to (1KaD), where D is the average value of the ‘drift integral’ (which is essentially a measure of the difference between 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 values of Fp and VTO appearing in the flow. Here, a is the effective volume fraction occupied by the vortices, so that a wððG=VTO Þ=Dlv Þ2 . D becomes more negative with decreasing values of VTO, which varies as a2 for small cloud droplets; thus, the VTO values become progressively smaller with smaller a and this causes larger negative values of D and larger amplification. Therefore, for the parameters relevant to this study, the velocity amplification effect fades with increasing drop radii and becomes extremely small for aw40 mm. In the recent study by Franklin et al. (2004), the authors showed that increases in the collision kernels in turbulence can sometimes be larger by a ´vila & Hunt (2001) analysis also implicitly indicates that the factor of 3. The Da collision efficiencies can indeed be enhanced by up to 100%. We have performed some sensitivity studies and found that with an increase of 50% in the collision Proc. R. Soc. A (2005)

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efficiency, the kernels with turbulence enhancements are always higher than the still air kernels for all droplet pairs (see figure 5b). As expected, with even higher increases in the collision efficiencies (not shown here for want of space) the differences 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) code ensures that there is no double counting. In order to study the impact of these enhanced sedimentation rates, we applied them first to an idealized mass distribution (shown as the solid line in figure 5c) and used the SCE solver to study the spectral evolution with time. The initial mass distribution corresponds to a total cloud water content of 2.75 g mK3 and a mean radius of 8 mm. In figure 5c, 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 rates shown in figure 5b. It is well known that when a bimodal spectrum develops, the resulting collision-induced second mode has the propensity to rapidly initiate rain formation. (b) Droplet spectra in cumulus cloud Because clouds consist of finite volumes of particles and water vapour moving unsteadily, mainly up and down, the distribution or ‘spectrum’ of droplet sizes, and thereby the formation of rain, have to be calculated as time dependant processes. The development of the spectrum caused by collisions after the initial condensational growth was calculated from the SCE (Pruppacher & Klett 1997) using the Bott (1998) code, which accounts for the fact that not all droplets of a given size grow at the same rate, since a small fraction of drops experience a particularly favourable sequence of collisions and grow much more rapidly than other drops. Next, we show results from our model simulations where a perturbed gamma distribution 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 is ca 7 mm. The initial mass distribution and a subsequent distribution are shown in figure 6 where we have considered collisions between droplets over a time period of 20 min. In these calculations, the fall velocity enhancements were calculated using the Da´vila & Hunt (2001) mechanism described earlier. The collision kernels were enhanced by 50% (the solid lines in figure 5b) for simulating the turbulent 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 typically have 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 flux method for the numerical solution of the SCE can be found from Bott (1998). We compared our model simulations with some observed data where a collision-induced spectrum was observed. It must be pointed out that the present calculations include only the process of spectral broadening owing to droplet collisions while ignoring all other dynamical effects. Thus, for the sake of consistency, we chose observed data points with radii greater than 15 mm to compare with model runs; these larger drops are expected to have grown from collisions. For a definitive model simulation vis-a`-vis observations, one would Proc. R. Soc. A (2005)

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mass distribution (g m–3/dln (radius))

5

4

3

2

1

0

1

10 radius (µm)

100

Figure 6. Mass distribution in a cumulus cloud. The solid line represents an initial distribution with a total liquid water content of 3.33 g mK3 and a mean droplet radius of 7 mm. The broader distribution obtained with the faster settling rates (dashed line) agree better with the observations as 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 same spread 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.

need to use these calculations in a cloud model with detailed dynamical and microphysical processes. This will form the basis of a later study. For the moment, we have aimed to determine the effect of these higher collision rates on the large end tail of a cumulus cloud spectrum. Figure 6 incorporates some observations reported by Mason & Jonas (1974), where they computed the mean of 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 centred around 17 mm and a second around 24 mm radius. With the turbulence-enhanced calculations, we obtain, as expected, a broader distribution that agrees reasonably well with the observed data points; the second peak centred at ca 24 mm is well captured. The simulation with the enhanced fall rates indicates the presence of coalescence induced peaks for radii larger than 30 mm. Without these enhanced collision rates, there is no second peak. The important point is that within a time span of 20 min, we obtain a broader spectrum with the enhanced collision rates than a conventional run. Without these enhanced fall rates, 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 rates ðC ð
a ; a^ÞÞ, because the droplet number concentration N(d ) decays exponentially Proc. R. Soc. A (2005)

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with increasing droplet size beyond the peak at aw7 mm, there is a large increase in the capture of the small droplets with radii of the order of 10 mm. This leads to the subsequent maxima in the droplet size distribution, for droplets of 17 and 24 mm radius. Without the enhancement, the latter is absent and the former is less pronounced. For the typical cumulus-like air parcel in a cloud of depth h  the period of its movement ðw h=w  w 250 m=0:2 m s K 1 Þ is about with velocity w, 20 min. The computations and observations agree somewhat better when the amplified fall velocities are accounted for, compared with the calculations based only on still air fall velocities. In the latter case, even after 20 min of simulation, the mass and consequently the number concentration corresponding to drops with radii greater than 20 mm is very small. By contrast, when the number of the larger cloud drops begin to grow exponentially by differential settling velocities and by turbulence-enhanced collisions, the increase in mass (and consequently N(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 in stratocumulus clouds. (c) Droplet spectra in Stratocumulus cloud It is well recognized today by meteorologists that even shallow layers of warm stratocumulus clouds are capable of producing drizzle that reaches the ground. However, as Mason (1952) first pointed out, the production of precipitation-sized particles by shallow layers of cloud is incompatible with simple models of drop growth (invoking only the effects of condensation and coalescence). The calculated growth rates were far too slow, because a drop would fall out of the cloud long before attaining the size necessary to survive the fall to the ground. However, Mason recognized that cloud turbulence could have an effect simply by extending the residence time of the droplets within the cloud. This paper provides a new approach for quantifying the effect, and a partial verification using new observation techniques. Measurements of turbulence and drop size spectra can now be made on instrumented aircraft and these measurements confirm that turbulent diffusion is potentially important in determining the vertical distribution of even quite large drops with radii of ca 100 mm, since updraughts 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 turbulent clouds. Their estimates of the velocity amplification effects were based on the Da´vila & Hunt (2001) results and their calculations showed that it was necessary to include the dependence of the radii of the smaller captured drop in the collection growth equation in addition to the turbulence effects. The results from this 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 another stratocumulus related case study, which is based on observations of an extensive, horizontally uniform stratocumulus cloud over the North Sea on 22 July 1982. The mean depth of the cloud was ca 450 m, the cloud ‘autoconversion’ rate (i.e. the rate at which cloud liquid water is partitioned as rain water) was 3.2!10K9 kg mK3 sK1, the maximum horizontally averaged cloud Proc. R. Soc. A (2005)

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liquid water content was 0.6 g mK3 and drizzle was observed below the cloud down to the lowest flight level (90 m above the sea level). The numerical model developed by Nicholls (1987) describes the growth of precipitation-sized drops in a warm stratocumulus cloud and combines the effects of stochastic turbulent diffusions with explicit microphysical calculations. Further details of both the observations and the model can be found in Nicholls (1987). The most significant fact that emerged from this study is that there was a considerable improvement of model predictions when the effect of air turbulence was considered with vertical r.m.s. velocity fluctuations (swZ0.36 m sK1) as well as the Lagrangian integral time-scale (TLZ360 s). Nicholls found that the distribution of mainly the larger drops changed with sw. The concentration of droplets with radii smaller than 20 mm responds rapidly to supersaturation and are controlled by condensation and evaporation, and these are only minor variations with sw. However, even with the inclusion of turbulent air velocity fluctuations, the Nicholls model still substantially under-estimated the number densities of the larger drops. Nevertheless, this study was a significant improvement over earlier simpler models that ignored turbulence effects altogether. It showed for the first time that steady-state concentrations of precipitation sized drops are found to be increased by some orders of magnitude when realistic levels of turbulence are included compared with an identical situation where swZ0. This arises, as Nicholls pointed out, because a few particles have a relatively unlikely (but finite) chance of encountering a significantly higher than average proportion of updraughts. This leads to enhanced growth rates by extending their lifetimes within cloud and in some cases by recycling drops upwards through regions of higher liquid water content. This explains why even shallow layers of warm cloud can produce significant amount of drizzle. The main limitation of the Nicholls model is that he assumed that although the cloud droplets are moved up and down by the turbulent updraughts and downdraughts, their fall velocities are still equal to the classical still air values. Baker (1993) proposed an analytic version of Nicholls’ model. The non-local turbulence closure was replaced with a stochastic diffusion equation for a turbulent plume of sedimenting raindrops. In addition, Baker (1993) specified a production rate for the smallest raindrops, and let the diffusing drops grow by accretion with a time constant determined by the liquid water content and the size-dependent fall speed. This enables one to calculate an equilibrium raindrop size distribution as a function of height within the cloud. In this paper, we adopted the Baker (1993) model, but included the dependence of ´vila the radii of the smaller captured droplets in the collection equation, using the Da & Hunt (2001) results to calculate the new turbulence-enhanced fall speeds. In figure 7, we show the equilibrium raindrop spectrum at a distance of 95 m above the cloud base. We also show results from Nicholls (1987) as described in their ‘standard’ run. The observations are shown for droplet radii that are greater than 40 mm where the two-dimensional probe measurements are expected to be free from any counting errors. From this figure, we find that the Nicholls model underestimates the number of large drops. In the next stage, we included the effects of turbulence as by using parameters specified by Nicholls (1987), that is, swZ0.36 m sK1 and TLZ360 s and used it in the equilibrium rain spectrum model (for details, see Baker 1993; Ghosh & Jonas 2001). Although this second case (marked Baker 1993 in figure 7) is an improvement over the case when swZ0, it does Proc. R. Soc. A (2005)

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conc. (cm–3 m m–1)

102 100 10–2 10–4

obs this calc. Baker (1993) Nicholls (1987)

10–6 10–8 40

50

60

70

80 90 100 radius ( m m)

200

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 from Nicholls (1987).

not yield the right number concentrations of the larger drops. In order to match the observational results, Nicholls (1987) had to artificially alter the spectral shape to slightly larger radii. When we used turbulence-enhanced collision rates using the Da´vila & Hunt (2001) formalism (marked ‘this calc.’ in figure 7), we find that the resultant spectrum matched the observations well and without any artificial adjustments. 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 the fall velocity of cloud droplets in the range 10–40 mm and thereby increases collision kernels in the initial range of particle sizes and then leads to an improved prediction of cloud and raindrop spectra. The theoretical and scaling analysis, supported by matching laboratory experiments and numerical simulations for settling velocities, have provided convincing results to demonstrate the effectiveness of this centrifuging action for the first time. In addition, when these amplified velocities are accounted for, the predicted cloud and raindrop spectra agree well with observations. A bimodal spectrum is easily produced—even with empirical elements, previous calculations could not achieve that straightforwardly. Proc. R. Soc. A (2005)

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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 by Ghosh et al. (2000) and Rosenfeld (2000) concerning reduced growth of raindrop when there is an excess of nucleation particles in urban areas. These studies imply that in situations where cloud droplets grow in polluted air masses with a very large number of nucleating particles, the resulting cloud droplets have reduced sizes. This is because a great number of particles start competing with each other for a limited amount of the available water vapour. As the droplet sizes are reduced, the collision rates also fall, and this eventually leads to precipitation suppression. Similarly, cloud seeding experiments afford another application in this context. Seeding is usually achieved with larger sized particles so that the nucleated particles can rapidly grow by collisions to yield precipitation-sized drops. Our results suggest that if seeding experiments can be conducted in a turbulent air mass, then the particles will actually fall faster than their still air fall velocities. This implies that seeding with smaller sizes now can also induce precipitation. This happens because, although the particles have modest sizes, in turbulence, they fall faster and produce the same effect as larger particles. The mechanism proposed in our study and its associated critical particle size of 20 mm radii shows that if too few droplets of this size are nucleated, by comparison with the smaller sizes, then the enhanced collision rate and droplet growth will not occur. In addition, the bimodal ‘tail’ in the size spectrum probably has applications in other environmental and industrial processes involving sedimenting, coalescing and flocculating particles in turbulent flows. These calculations can also be extended to calculate the settling rates of atmospheric aerosols and particulate matter. For example, it is now known that heterogeneous processing on polar stratospheric cloud particles (PSCs; in particular the larger type 2 PSCs) is crucial to the correct quantification of stratospheric denitrification and heterogeneous ozone depletion. It is also expected that the estimated fall velocities of these type 2 PSCs (typically with radii of ca 10 mm) would be higher than their still air values when one accounts for the turbulence within the stratospheric vortex. This faster settling would possibly lead to greater denitrification of the stratosphere, which would eventually lead to larger heterogeneously processed ozone depletion. In forthcoming work, the effects of including this mechanism in computational models for cloud processes will be tested—especially its interaction between ice crystals, aerosol particles and droplets. To date, current climate models, including those at the UK Meteorological Office, continue to use still air fall velocities for cloud droplets. It is expected that by using turbulenceassisted fall velocities in climate models, one can obtain a better precipitation characterization and forecast. In addition, recent research suggests that turbulence effects on droplet condensational growth can also be important (see Celani et al. 2005 and references therein). The contribution from this latter effect may also complement the well-established turbulence effects on settling and coalescence. In general, the implications for improving numerical models for weather and climate predictions are also being considered. Our future work will be aimed at formulating the above results in closed-formed parametric results so that they can be easily incorporated into large-scale climateprediction models. Proc. R. Soc. A (2005)

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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 Environmental Geochemistry and Biogeochemistry (J.C.R.H., A.S., H.J.S.F.); NERC grant to Centre for Polar Observation and Modelling (U.C.L., J.C.R.H.). We have greatly benefited from conversations with T. 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 around line vortices A solid particle released at a level YZY0 far above a vortex of circulation G, after falling a time Dtc[2Y0/VTO will be at Y Z Y0 K VTO Dtc C ðG=VTO ÞDhðX0 Þ;

(A 1)

where Dh(X0) is the dimensionless differential settling length with respect to settling in still fluid, a function of the initial horizontal position X0 (see Da´vila & Hunt 2001). If DhO0, then the particle settles more slowly. If a small particle (with terminal velocity V^ TO ) collides at a fixed level Y1 with a larger particle (with terminal velocity V
TO ), then the collision length of particle pairs is h K ðG=V^ TO ÞD^ h
Z lco K ðG=V
TO ÞðlD
h K D^ hÞ; lc Z lco K ½ðG=V
TO ÞD


(A 2)

^ TO ÞDtc Z V
TO Dtc ð1K lÞ with lZ V^ TO =V
TO ð!1Þ. Hence where lco Z ðV
TO K V the normalized fractional increase in collision length is  
V TO Dtc lc D^ h K lD
h El Z K1 Z : (A 3) lco 1 Kl G=V^ TO This is the formulation used to obtain figure 3c, where we have plotted El versus the initial horizontal position of the particles X0 for critical small particles with F^ p Z 4 and V^ TO Z 0:7, and larger particles with F
p Z 5 and V
TO Z 0:8. Because the objective of these calculations is the average value of lc over all the horizontal initial positions, we have not taken into account that the initial horizontal positions of the larger and smaller particles may be different in order to have a collision at the fixed level Y1. From (A 3) using the definition of the drift integral (i.e. the average settling length around the vortex) ðN dX0 DZ DhðX0 Þ ; (A 4) G=V KN TO equation (2.5) can be obtained. In addition, using (A 1), the average settling velocity of particles moving around vortices results in VT Z VTO ð1 K aDÞ;

(A 5)

where the effective volume fraction occupied by the vortices aZ ððG=VTO Þ=Dlv Þ2 . For a given value of the droplet radius, we first calculate the fall velocity VTO, the particle Froude number Fp, the volume fraction a. Then, from (A 4), the drift integral is evaluated. Proc. R. Soc. A (2005)

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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 computational procedure that was adopted to evaluate the time evolution of the cloud droplet spectra. In the droplet collection model, all condensation and mixing with the surroundings were neglected. The droplets were assumed to grow only by coalescing with each other as they fell. Depending on their sizes, the actual fall velocities were generally higher than their still air values and this added value was precisely estimated using the Da´vila & Hunt (2001) formalism described in the paper. For drops with radii less than 30 mm, the still air terminal velocity VTO was calculated as VTOi Z kai2 ;

(B 1)

where kw1.18!108 mK1 sK1 (Rogers & Yau 1994) and ai is the droplet radius. Da´vila and Hunt’s (2001) calculations depend significantly on the droplet radius and significant variations are observed in the particle Froude number Fp, the radius of the droplet trajectory around the vortex Rtraj as well as the Stokes number St with increasing drop radii. This is shown in table 2 below. Using the values listed in table 2, we find that the velocity amplification drops from 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 the stochastic nature of the collection growth. Because raindrop concentrations are typically 105–106 times smaller than cloud drop concentrations, one would expect that the fate of the ‘favoured’ small fraction of drops that happen to grow much faster than the average rate would be quite important in the overall process of precipitation development (Pruppacher & Klett 1997). Our calculations have borne out this expectation as is briefly described below. Let the number of drops with radii between a
i and a
i C d
a be Ni per unit volume, and let the probability that a drop of size j will encounter a collision with one of size i in unit time be Pij (with iOj). 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 DNij Z Pij Ni Nj Dt;

(B 2)

per unit time. However, the probability Pij that a drop of size i can collide with one of size j in unit time depends on their collection efficiency Eð
a i ; a^j Þ and their relative fall speeds VTi and VTj. This implies that a i ; a^j Þpð
a i C a^j Þ2 ðVTi K VTj Þ: Pij Z Eð


(B 3)

Once the values of Eð
a i ; a^j Þ, VTi, VTj were estimated, the appropriate values of Pij were calculated. For the control runs (i.e. the effect of turbulence neglected altogether), only the still air terminal velocity of droplets (equation (B 1)) were used while evaluating Pij . The number of drops lost in a particular class and the size range 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 outlined procedure repeated for every pair of drop sizes. The history of the droplet spectrum was followed for 20 min. The SCE solver that is used is described in Bott (1998) and Proc. R. Soc. A (2005)

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S. Ghosh and others Table 2. Size dependence of model parameters

(Fp is the particle Froude number; Rtraj, the radius of the droplet trajectory around the vortex; St, the Stokes number; a is the drop radius; VTO, the still air fall velocity; and a is the volume fraction.) a (mm)

VTO (m sK1)

Fp

Rtraj (m)

St

a

11 12 13 14 15 16 17 18 19 20

0.14520!10K1 0.17280!10K1 0.20280!10K1 0.23520!10K1 0.27000!10K1 0.30720!10K1 0.34680!10K1 0.38880!10K1 0.43320!10K1 0.48000!10K1

0.20825!10K2 0.35101!10K2 0.56740!10K2 0.88510!10K2 0.13390!10K1 0.19722!10K1 0.28374!10K1 0.39982!10K1 0.55303!10K1 0.75233!10K1

0.10331!10K1 0.86806!10K2 0.73964!10K2 0.63776!10K2 0.55556!10K2 0.48828!10K2 0.43253!10K2 0.38580!10K2 0.34626!10K2 0.31250!10K2

0.22224 0.26449 0.31041 0.36000 0.41327 0.47020 0.53082 0.59510 0.66306 0.73469

0.26680 0.18838 0.13677 0.10168 0.77160!10K1 0.59605!10K1 0.46770!10K1 0.37211!10K1 0.29974!10K1 0.24414!10K1

uses an accurate flux method, which ensures accurate mass conversion. The mass averaging process consists of a two-step procedure. In the first step, the mass distribution of drops with mass x 0 that have been newly formed in a collision process is entirely added to grid box k of the numerical grid mesh with xk%x 0 %xkC1. In the second step, a certain fraction of the water mass in grid box k is transported to kC1. 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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