1. Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel
2. Institute of Automation and Electrometry, Novosibirsk 630090, Russia

Correspondence to: G. Falkovich¹ Correspondence and requests for materials should be addressed to G.F. (e-mail: Email: fnfal@wicc.weizmann.ac.il).

The local distribution of droplets over sizes, n(a,t,r) = n(a), changes with condensation and coalescence according to refs 1, 2 (see Table 1 for definitions of variables):

Here, v(t, r) is the droplet velocity at point r at time t, q is proportional to the supersaturation and the diffusivity of the vapour and a" = (a³ - a'³)^1/3. The collection kernel is proportional to the collision kernel, which is the product of the target area and the relative velocity v of droplets before the contact: K(a₁,a₂) (a₁ + a₂)²v. For droplets larger than couple of micrometres across, brownian motion can be neglected and the collision kernel in still air is due to gravitational settling^{1, 2}: K_g(a₁,a₂) = (a₁ + a₂)²E(a₁,a₂)|u_g(a₁) - u_g(a₂)|. When the Reynolds number of the flow around the droplet, Re_a u_ga/, is not too large and concentration is small enough (na³Re_a^-21) the settling velocity is due to the balance of gravity and friction: u_g = g with the Stokes time = (2/9)(₀/)(a²/). Here ₀, are water and air densities respectively. Hydrodynamic interaction between approaching droplets is accounted for in K_g by the collision efficiency E, for which values can be found in refs 1, 17.

Table 1: Parameters

Full table

Cloud condensation nuclei are typically of micrometre or submicrometre size and the initial stage of droplet growth is solely due to condensation: n(a,t) = af(a² - 2qt) with the function f determined by the initial distribution. An important conclusion is that while the distribution shifts to larger sizes, it keeps its small width over a² (a few micrometres squared or less). It would take many hours for condensation to grow millimetre-size raindrops, especially with the account of vapour depletion. Such growth is supposed to come from coalescence but because K_g|a₁² - a₂²| the gravitational collision rate is strongly suppressed for droplets that are close in size. With narrow local size distributions in cloud cores, rain would not start in still air for many hours. There is then the long-standing problem of the bottleneck in the transition from condensation to coalescence stage^{1, 2, 3, 4, 5, 8, 9, 10, 11, 18, 19, 20} which we discuss here.

In some cases, droplets with substantially different sizes may appear, owing to the existence of ultra-giant nuclei^{18, 19}. Another possibility that we consider here is that turbulence-induced collisions^{8, 9, 10, 11, 20} may occur. Both velocity and concentration of droplets fluctuate in a random flow. To provide meteorology with an effective computational tool, theoretical physics is expected to produce the condensation–coagulation equation (1) averaged over space. Here we derive analytically the averaged equation—that is, we obtain the effective collision kernel = K(a₁,a₂)n₁n₂/n₁n₂]—and solve it numerically to demonstrate the changes in the average distribution n(a, t) brought about by turbulence.

For the basic discussion of cloud turbulence see the reviews in refs 3–5 and the references therein. Turbulence intensity can be characterized by the energy input rate which determines the root-mean-square (r.m.s.) velocity gradient: (/)^1/2. We consider small droplets with the Stokes number St = (which characterizes mean droplet inertia) smaller than unity. If inertia is neglected, droplets follow the incompressible air flow and their concentration is uniform. Droplet motion in the air flow gradient s then provides v s(a₁ + a₂) and gives the mean collision kernel²⁰ K_t (a₁ + a₂)³. Inertia deviates droplets from the air flow, adding a contribution to v proportional to s the mean value of which is St, that is, small²⁰. Hence Saffman and Turner²⁰ concluded that only extremely energetic turbulence with > 2,000 cm² s^-3 would produce a noticeable collision kernel. We note, however, that it is rather than K that determines the mean collision rate. Inertial deviation of droplets from the air flow leads to fluctuations in droplet concentration, characterized by the factor k₁₂ = n₁n₂/n₁n₂ > 1, which may be large. Concentration fluctuations have been observed in experiments and numerical simulations^{5, 6, 7, 8, 9, 10, 11, 12} and described analytically for same-size droplets in low Reynolds flow without gravity^{13, 14}: k(a,a) = n²/n² (/a)^St2, where (³/)^1/4 is the mean correlation scale of velocity gradients.

The Reynolds number of cloud turbulence is Re = uL/, where u is air velocity and L is the outer scale comparable to the cloud size. In the atmosphere, Re is large (10⁶–10⁸), that is, turbulence is intermittent and the statistics is very non-gaussian, with a substantial probability of gradients far exceeding . The role of gravity can be characterized by the ratio of the small-scale turbulent velocity to the settling velocity^{12, 21} ()^1/4/u_g; this parameter can be both larger and smaller than unity for a = 1–100 m and = 1–20 s^-1.

Here we derive the factor k₁₂ for droplets under gravity in high-Re flow. We show that contribution of large gradients can significantly increase k₁₂ compared with the low-Re case and that gravity provides for a sharp maximum of k₁₂ at a₁ = a₂. We also describe a new inertial mechanism of collisions due to rare events with large gradients (s ^-1) that produce jets of droplets initially accelerated by the air flow and then detached from it. That gives an additive contribution K_i into . We call this the 'sling effect' and show that turbulence intermittency can make K_i substantial even at small St. No realistic direct numerical simulations are possible for droplets in high-Re turbulence, so analytical derivations are indispensable. We derive (,St,Re) and show that the turbulence-induced collision rate can be substantial even for small droplets in moderate turbulence when St is small.

The field v(r, t) giving the velocity of a droplet located at r at time t satisfies the equation^{6, 22} _tv + (v)v = (u - v)/ + gz, where z is a unit vector pointing downwards. The gradients = v and s = u taken at a droplet's trajectory are related as follows: + ² = (s - )/. When ||^-1, it has a smooth evolution determined by (t) = ^t dt' exp[(t' - t)/]s(t')/. If, however, || > ^-1 then the inertial term ² dominates and may lead to an explosive evolution (t)(t₀ - t)^-1 that produces shock in v(r) and singularities in n(r).

The probability P of an explosive event is that of large and persistent gradients s. The correlation time _c(s) of the air flow gradient is given by the minimum between the turnover time |s|^-1 and the time l(s)/u_g needed for droplets to cross the region l(s) (/|s|) over which s is correlated. The gradient s that leads to || ^-1 must either exceed the threshold described by the extrapolation formula s_b = [^-2 + ^2-4]^1/2 or be larger than 1/ and occupy the region in space l(s_b)s_b/s. Because the only available data are on the single-point probability density function (PDF) P (s) we estimate the probability of explosion from below: P _1/P(||) d|| s_bsbP(|s|) d|s|, where the prefactor s_b appears because s > s_b can occur at any moment within the interval . Once a fluctuation with a negative eigenvalue _i < -^-1 occurs, the inertial term ² exceeds the driving term s_i/ and the friction term - _i/, which corresponds to a free motion of droplets along the direction of _i on a timescale of order . A negative velocity gradient means that faster droplets catch up with slower ones, creating a cubic singularity²³ in the relation between the current coordinate x(t) and the initial one y: x = y³/3l² - yt/. Here l = l(s_b) is the correlation length of || = 1/ and t is counted from the moment of singularity. Using n(x,t) = n(y)|y/x| and || = |(v/y)(y/x)| ^-1|y/x| we find the contribution of the preshocks (t < 0) into the collision rate: ||n² P^-1 dx_-⁰ dt(y/x)³n²(y)/l P^-1(l/a)^1/3ñ²[a(l/a)^2/3]. Formula ||n² assumes smoothness over the scale a, so we introduced ñ[y] coarse-grained over y, taking y a(l/a)^2/3, which corresponds to x a.

After the shock (t > 0), folds appear in the map y(x) in the region |x| < 2l(t/)^3/2/3: for every x value there are three y values which correspond to the three groups of droplets that came from different places and have different velocities. Nearby droplets from the same group have v a and contribute ||n² P^-1(l/a)^1/2ñ²[(la)^1/2]. However, this contribution and that of preshocks are both less than that given by collisions of droplets from different groups. Groups coming from afar appear because droplets are shot out of curved streamlines with too high a centrifugal acceleration, an effect known to anyone who has used a sling to throw stones. That is why we call this the 'sling effect'. Droplets separated at the beginning of the free motion by a distance l have v l/ and provide for the inertial collision kernel:

Subscripts denote different droplet sizes, (a₁) = ₁, P₁ = P(|| > 1/₁). Because the velocities and the concentrations of the different groups are uncorrelated, K_i(a₁,a₂)n₁n₂ = K_i(a₁,a₂)n₁n₂. The collision efficiency E' in equation (2) can be expressed via E taken for the effective sizes, giving the same v. We thus obtain E' 0.93–0.98 in the interval 15–100 m for collinear velocities (noncollinearity further increases E'; ref. 3), so with our accuracy, E' 1. We note that K_i(a, a) is larger than the contribution due to droplets from the same group by the factor (l/a)ñ²[l]/ñ²[(la)^1/2]. It is shown below that ñ²[r]r^- with < 1 so that K_i indeed dominates. The ratio K_i/a³ PSt^-1(l/a) has the smallness of P compensated by two large factors and can exceed unity even at small St. Most importantly, K_i(a,a)0.

We now describe concentration fluctuations and derive k₁₂. Because of inertia, the divergence of v is nonvanishing⁶, albeit small: tr = - exp[(t' - t)/]tr²(t') dt'|| at ||1. Negative tr² corresponds to elliptic flows (vortices) which act as centrifuges decreasing n. Droplets concentrate in hyperbolic regions (between the vortices) where tr² > 0 > tr. Clusters of droplets are created with sizes not exceeding , as follows from theory^{13, 14} and as seen in numerics²⁴ and observations²⁵. We note in passing that as clusters do not exceed the fluctuations of droplet concentration do not produce significant fluctuations in vapour concentration (because the vapour diffusivity is comparable to the air viscosity) so that droplet distribution over sizes cannot be significantly broadened during the condensation stage; see also ref. 26. Clustering can be readily understood: a compressible flow with lagrangian chaos creates a fractal concentration, the so-called Sinai–Ruelle–Bowen measure¹⁶. The moments of the fractal measure behave as powers of the scale ratio: ñ[r] n(/r)⁽⁾, where () is convex and (0) = (1) = 0. As '(0) is negative¹³ (it is proportional to the sum of the backward-in-time Lyapunov exponents of the v-flow), then (2) > 0. Droplets of different sizes have additional relative velocity |₁ - ₂|(g + ²) that stops clustering at r |₁ - ₂|(g + ²)/_d. We thus find:

We distinguish a₁ from a₂ only in |₁ - ₂| giving the sharpest dependence. The exponent is described by equation (6), derived in the Methods section below. For sufficiently small droplets (St < 1 and > 1) and not very high Re, we have St²F₃, where F₃ ^-3|s|³P(s) ds is a growing function of Re that describes how turbulence intermittency amplifies the effect of small droplet inertia. At low Re and St < 1, does not depend on , so the only dependence k₁₁() can come from shock contribution and has to be weak (logarithmic), which agrees with numerics¹². At large Re, both and k₁₂ have a maximum at St ².

We now write the effective collision kernel for small heavy particles in turbulence:

To compare with numerics done for low Re without gravity^{9, 10} we use equations (2), (3), (4) and (6) with P exp(- St^-2) and l St^1/2. Analytics and numerics agree well, showing fast growth of with St at St < 1 and a (broad) maximum at a₁ = a₂ (refs 9, 10). As St approaches unity, k₁₂1 and K_i(a₁ + a₂)³, which explains the observation⁹ that contributions of both preferential concentration and relative velocity are important. Gravity suppresses K_i increasing s_b at ² < St. It also makes k₁₂ a sharp function of |a₁ - a₂| so that the gravitational collision rate is only weakly enhanced by preferential concentration, because k₁₂ > 1 where K_g 0. At small St, we can also neglect the turbulence-induced increase of the vertical flux¹².

For practical applications to high-Re flows, and K_i have to be evaluated with P(s) determined experimentally. To make an estimate from below, we numerically evaluate equations (2), (3), (4) and (6) for moderate turbulence with Re 10⁶, taking P(s) from ref. 27. Figure 1 shows the effective collision kernel normalized by 8a³. The normalized kernel has a maximum at St ² which corresponds to the balance between inertia and gravity when the Stokes time is the universal value (/g²)^1/3. Droplets of such size take time to fall through the vortex with a turnover time . The effect of centrifugal force is less both for smaller droplets (which are less inertial) and for larger ones (which spend less time inside the vortex). This is to be contrasted with the maximum at St 1 in low-Reynolds numerics without gravity^{9, 10}. How inertia and gravity influence droplet settling is discussed in refs 12, 21, 28; see also refs 29, 30 on the role of turbulence.

Figure 1: Normalized effective collection kernel for equal-size droplets at Re 10⁶ according to equations (2), (3), (4) and (6).

From bottom to top, = 10, 15 and 20 s^-1.

High resolution image and legend (29K)

We see that the interplay between gravity and turbulence intermittency makes inertial enhancement of the turbulence-induced collision rate significant only in the restricted interval of droplet sizes that depends on the air density (between 20 and 60  for = 10^-3 g cm^-3). The condensation–coagulation bottleneck is expected precisely in this interval, so turbulence must be able to alleviate it. To illustrate the effect, we solve space-averaged equation (1) numerically with the mean-field collection kernels K_g + K_t (dashed line in Fig. 2) and with (solid line in Fig. 2) for = 20 s^-1, = 6 mm and q = 5 10^-9 cm² s^-1. We took 50 droplets per cm³ with initial sizes in the interval 2–3 m. Even for such relatively low level of turbulence and small size of droplets, the difference in coalescence-produced secondary peaks is apparent after only 10 min. The main peak is at a = 25 m. The number density of coalescence-produced droplets is 1.06 cm^-3 with the newly found versus 0.64 cm^-3 with the old mean-field values.

Figure 2: Distribution over sizes after 10 min.

The dashed line is the solution of equation (1) with the mean-field collision kernel and the solid line is the solution of equation (1) with equation (4).

High resolution image and legend (24K)

We thus conclude that turbulence-induced inertial effects can substantially accelerate the transition from condensation to coalescence stage in the interval of few tens of micrometres. Our results are valid for a low concentration of small droplets and not very energetic turbulence, conditions compatible with the data for most clouds^{1, 2, 4, 5}, so we believe that equations (2), (3), (4) and (6) provide for an effective tool in rain prediction.

References

1. Pruppacher, H. & Klett, J. Microphysics of Clouds and Precipitation (Kluwer, Dordrecht, 1998)
2. Seinfeld, J. & Pandis, S. Atmospheric Chemistry and Physics (Wiley, New York, 1998)
3. Pinsky, M., Khain, A. & Shapiro, M. Stochastic effects of cloud droplet hydrodynamic interaction in a turbulent flow. Atmos. Res. 53, 131–169 (2000) | Article | ISI |
4. Jonas, P. Turbulence and cloud microphysics. Atmos. Res. 40, 283–306 (1996) | Article | ISI | ChemPort |
5. Vaillancourt, P. A. & Yau, M. K. Review of particle-turbulence interactions and consequences for cloud physics. Bull. Am. Met. Soc. 81, 285–298 (2000) | Article | ISI |
6. Maxey, M. R. The gravitational settling of aerosol particles in homogeneous turbulence and random flow field. J. Fluid Mech. 174, 441–465 (1987) | ISI |
7. Squires, K. & Eaton, J. Measurements of particle dispersion from direct numerical simulations of isotropic turbulence. J. Fluid Mech. 226, 1–35 (1991) | ISI | ChemPort |
8. Pinsky, M. & Khain, A. Turbulence effects on droplet growth and size distribution in clouds—a review. J. Aerosol Sci. 28, 1177–1214 (1997) | Article | ISI | ChemPort |
9. Sundaram, S. & Collins, L. Collision statistics in an isotropic particle-laden turbulent suspension. J. Fluid Mech. 335, 75–109 (1997) | Article | ISI | ChemPort |
10. Zhou, Y., Wexler, A. & Wang, L.-P. Modelling turbulent collision of bidisperse inertial particles. J. Fluid Mech. 433, 77–104 (2001) | ISI | ChemPort |
11. Reade, W. & Collins, L. Effect of preferential concentration on turbulent collision rates. Phys. Fluids 12, 2530–2540 (2000) | Article | ISI | ChemPort |
12. Wang, L. P. & Maxey, M. Settling velocity and concentration distribution of heavy particles in homogeneous isotropic turbulence. J. Fluid Mech. 256, 27–68 (1993) | ISI | ChemPort |
13. Balkovsky, E., Falkovich, G. & Fouxon, A. Intermittent distribution of inertial particles in turbulent flows. Phys. Rev. Lett. 86, 2790–2793 (2001) | Article | PubMed | ISI | ChemPort |
14. Brumfiel, G. How raindrops form. Phys. Rev. Focus 7, story 14 (22 March 2001), http://focus.aps.org/v7/st14.html.
15. Shraiman, B. & Siggia, E. Scalar turbulence. Nature 405, 639–646 (2000) | Article | PubMed | ISI | ChemPort |
16. Falkovich, G., Gawedzki, K. & Vergassola, M. Particles and fields in fluid turbulence. Rev. Mod. Phys. 73, 913–975 (2001) | Article | ISI |
17. Pinsky, M., Khain, A. & Shapiro, M. Collision efficiency of drops in a wide range of Reynolds numbers. J. Atmos. Sci. 58, 742–766 (2001) | Article | ISI |
18. Johnson, D. B. The role of giant and ultragiant aerosol particles in warm rain initiation. J. Atmos. Sci. 39, 448–460 (1982) | Article | ISI |
19. Levin, Z., Wurzler, S. & Reisin, T. Modification of mineral dust particles by cloud processing and subsequent effects on drop size distribution. J. Geophys. Res. 105, 4501–4512 (2000) | Article | ISI |
20. Saffman, P. & Turner, J. On the collision of drops in turbulent clouds. J. Fluid Mech. 1, 16–30 (1956) | ISI |
21. Raju, N. & Meiburg, N. The accumulation and dispersion of heavy particles in forced two-dimensional mixing layers. Phys. Fluids 7, 1241–1264 (1995) | Article | ISI | ChemPort |
22. Maxey, M. R. & Riley, J. J. Equation of motion for a small rigid sphere in a nonuniform flow. Phys. Fluids 26, 883–889 (1983) | Article | ISI |
23. Vekshtein, G. Physics of Continuous Media: A Collection of Problems with Solutions 93–94 (Adam Hilger, Bristol, 1992)
24. Grits, B., Pinsky, M. & Khain, A. Formation of small-scale droplet concentration inhomogeneity in a turbulent flow as seen from experiments with an isotropic turbulence model. Proc. 13th Int. Conf. on Clouds and Precipitation Vol. 1, 138–141 (Am. Met. Soc., Reno, 2000).
25. Kostinski, A. & Shaw, R. Scale-dependent droplet clustering in turbulent clouds. J. Fluid Mech. 434, 389–398 (2001) | Article | ISI | ChemPort |
26. Brenguier, J.-L. & Chaumat, L. Droplet spectra broadening in cumulus clouds. J. Atmos. Sci. 58, 628–641 (1999) | Article |
27. Belin, F., Maurer, J., Tabeling, P. & Willaime, H. Velocity gradient distribution in fully developed turbulence: an experimental study. Phys. Fluids 9, 3843–3850 (1997) | Article | ISI | ChemPort |
28. Davila, J. & Hunt, J. C. R. Settling of small particles near vortices and in turbulence. J. Fluid Mech. 440, 117–145 (2001) | Article | ISI | ChemPort |
29. Crowe, C. T., Chung, J. N. & Troutt, T. R. Particulate Two Phase Flow Ch. 18 (ed. Roce, M. C.) 626 (Butterworth-Heinemann, Oxford, 1993)
30. Marble, F. E. Dynamics of dusty gases. Annu. Rev. Fluid Mech. 2, 397–461 (1970) | Article | ISI |

Acknowledgements

We thank A. Khain, V. Lebedev and M. Pinsky for discussions, and the Minerva and Israel Science Foundations for support.

Competing interests statement

The authors declare no competing financial interests.