Turbulent collision-coalescence of cloud droplets: validated parameterizations of the geometric kernel, collision efficiency, Reynolds-number extrapolation, and stochastic-coalescence moment closures for cloud-scale models

Warm clouds form precipitation when droplets of roughly 10-20 μm radius cross the 'size gap' to sizes where gravitational collection is efficient; deterministic collection theory predicts crossing times far longer than the 15-20 minutes sometimes observed, and air turbulence is known to enhance collisions through the droplet radial distribution function, the radial relative velocity, and the collision efficiency. The source paper (Wang, Xue, Ayala and Grabowski, 2006, Atmospheric Research) posed the problem of turning these effects into predictive capability through five linked directions: parameterizing the radial distribution function at cloud-relevant Reynolds numbers with gravitational settling, determining the true turbulent collision efficiency, justifying the extrapolation of finite-Reynolds-number direct numerical simulations to atmospheric turbulence, closing the higher-order moments of the true stochastic coalescence equation, and integrating all of these into computationally tractable cloud-scale schemes.

Background

Warm (ice-free) clouds are turbulent suspensions of water droplets that grow first by condensation of water vapor and then by collision and coalescence, in which two colliding droplets merge into a larger one. Condensation growth slows as droplets approach roughly 10-20 μm in radius, while gravitational collection, in which larger droplets fall fast enough to sweep up smaller ones, becomes efficient only at radii of order 50 μm. Crossing this 10-50 μm 'size gap' is the rate-limiting step of warm-rain formation: deterministic collection calculations typically predict an hour or more, whereas some cumuli are observed to produce drizzle-sized particles within 15-20 minutes.

The standard predictive tool is the kinetic collection equation (KCE), a Smoluchowski-type mean-field equation for the average number of droplets in each mass bin, driven by a collection kernel $C_{jk}$ that multiplies mean droplet numbers. The KCE assumes that instantaneous droplet counts in two bins factorize into products of their means; for the finite droplet populations and narrow initial spectra of real clouds this assumption is uncontrolled. The exact description is a continuous-time Markov process (a master equation) for the joint probability of all bin populations; its first-moment equation, the 'true stochastic coalescence equation' (TSCE) derived in the source paper, governs the mean bin populations $\langle x_m \rangle$ but contains unclosed second moments $\langle x_j x_k \rangle$ whose correlations can approach magnitude unity in sparsely populated bins, invalidating the KCE factorization precisely where early droplet growth matters.

Air turbulence enters the kernel through three separable effects. First, turbulent velocity differences increase the mean radial (approach) relative velocity $\langle |w_r| \rangle$ between droplet pairs. Second, because droplets have inertia they cluster preferentially in turbulent straining and vortical regions; this preferential concentration is quantified by the radial distribution function $g_{12}$, the pair density at the collision distance normalized by its value in a spatially uniform suspension. Third, the aerodynamic disturbance flows of the two droplets and of their neighbors deflect trajectories and drain momentum from the approach; this is collected into the collision efficiency $E_{12}$, the ratio of the actual collision rate to the geometric rate for undisturbed straight trajectories. The geometric collision kernel is built from $\langle |w_r| \rangle$ and $g_{12}$, and multiplying by $E_{12}$ gives the collection kernel. The regime is set by the turbulent kinetic-energy dissipation rate $\epsilon$ (roughly 10-400 cm^2 s^-3 in convective clouds, higher in vigorous updrafts) and by the flow Reynolds number, conventionally the Taylor-microscale Reynolds number $R_\lambda$ (a standard turbulence Reynolds number built from the root-mean-square velocity fluctuation and the Taylor microscale, the effective small-scale correlation length of the turbulent velocity field), which in the atmosphere is of order 10^3-10^4, far beyond what direct numerical simulation (DNS) can reach ($R_\lambda \lesssim 10^3$). Hybrid DNS (HDNS) couples DNS of the carrier turbulence with an approximate superposition treatment of the droplets' disturbance flows so that many-body hydrodynamic interactions can be included at feasible cost.

The source paper (Wang, Xue, Ayala and Grabowski 2006) combined the TSCE derivation, Monte Carlo simulations of stochastic coalescence driven by DNS-based kinematic parameterizations, and HDNS computations of the collision efficiency, concluding that turbulence raises both the geometric collision rate and the collision efficiency, most strongly for similar-size pairs where gravitational collection is weakest. Reviews a few years later (Devenish et al. 2012; Grabowski and Wang 2013) summarized total kernel enhancements of up to about a factor of five for such pairs, and discussed these unresolved directions among the open problems of warm-cloud droplet growth.

Problem Statement

Construct a physically grounded, validated, and computationally tractable quantitative treatment of droplet collision-coalescence in turbulent cloud air, following the future-work program of Wang, Xue, Ayala and Grabowski (2006). The problem has five jointly linked components serving one common scientific objective, the quantitative prediction of cloud droplet size-distribution evolution and drizzle/rain initiation in warm clouds: (1) parameterize the pair radial distribution function $g_{12}$ and the mean radial relative velocity $\langle |w_r| \rangle$, equivalently the geometric collision kernel, for droplet pairs in the roughly 10-50 μm size range at cloud-relevant turbulence intensities with gravitational settling included, valid at or demonstrably extrapolable to atmospheric flow Reynolds numbers; (2) determine the true collision efficiency $E_{12}$, including short-range hydrodynamic and lubrication interactions in turbulent many-droplet suspensions, as a function of droplet sizes, size ratio, and dissipation rate, in a form usable in bulk and bin models; (3) provide a theoretical and/or experimental justification for extrapolating finite-Reynolds-number DNS results to atmospheric turbulence; (4) develop a systematic treatment of the higher-order moments $\langle x_j x_k \rangle$ of the stochastic coalescence process, a closure or other quantitative correction achieving the same validity, so that a deterministic mean-field collection equation remains quantitatively valid, or is explicitly corrected, in regimes where stochastic correlations and finite-number fluctuations matter; and (5) integrate components (1)-(4) into computationally tractable collision-coalescence schemes for cloud-scale models. A resolution requires an integrated treatment addressing all five components jointly; work on a single component is a partial result.

References

1. Lian-Ping Wang, Yan Xue, Orlando Ayala, Wojciech W. Grabowski (2006), 'Effects of stochastic coalescence and air turbulence on the size distribution of cloud droplets', Atmospheric Research 82(1-2):416-432, doi:10.1016/j.atmosres.2005.12.011, https://doi.org/10.1016/j.atmosres.2005.12.011 (source paper)
2. Lian-Ping Wang, Orlando Ayala, Scott E. Kasprzak, Wojciech W. Grabowski (2005), 'Theoretical Formulation of Collision Rate and Collision Efficiency of Hydrodynamically Interacting Cloud Droplets in Turbulent Atmosphere', Journal of the Atmospheric Sciences 62:2433-2450, doi:10.1175/jas3492.1, https://doi.org/10.1175/jas3492.1
3. Orlando Ayala, Wojciech W. Grabowski, Lian-Ping Wang (2007), 'A hybrid approach for simulating turbulent collisions of hydrodynamically-interacting particles', Journal of Computational Physics 225:51-73, doi:10.1016/j.jcp.2006.11.016, https://doi.org/10.1016/j.jcp.2006.11.016
4. Orlando M. Ayala, Bogdan Rosa, Lian-Ping Wang, Wojciech W. Grabowski (2008), 'Effects of turbulence on the geometric collision rate of sedimenting droplets. Part 1. Results from direct numerical simulation', New Journal of Physics 10:075015, doi:10.1088/1367-2630/10/7/075015, https://doi.org/10.1088/1367-2630/10/7/075015
5. Orlando M. Ayala, Bogdan Rosa, Lian-Ping Wang (2008), 'Effects of turbulence on the geometric collision rate of sedimenting droplets. Part 2. Theory and parameterization', New Journal of Physics 10:075016, doi:10.1088/1367-2630/10/7/075016, https://doi.org/10.1088/1367-2630/10/7/075016
6. Lian-Ping Wang, Orlando M. Ayala, Bogdan Rosa, Wojciech W. Grabowski (2008), 'Turbulent collision efficiency of heavy particles relevant to cloud droplets', New Journal of Physics 10:075013, doi:10.1088/1367-2630/10/7/075013, https://doi.org/10.1088/1367-2630/10/7/075013
7. Yan Xue, Lian-Ping Wang, Wojciech W. Grabowski (2008), 'Growth of Cloud Droplets by Turbulent Collision-Coalescence', Journal of the Atmospheric Sciences 65:331-356, doi:10.1175/2007JAS2406.1, https://doi.org/10.1175/2007JAS2406.1
8. M. B. Pinsky, A. P. Khain, M. Shapiro (2007), 'Collisions of Cloud Droplets in a Turbulent Flow. Part IV: Droplet Hydrodynamic Interaction', Journal of the Atmospheric Sciences 64:2462-2482, doi:10.1175/jas3952.1, https://doi.org/10.1175/jas3952.1
9. A. Khain, M. Pinsky, T. Elperin, N. Kleeorin, I. Rogachevskii, A. Kostinski (2007), 'Critical comments to results of investigations of drop collisions in turbulent clouds', Atmospheric Research 86:1-20, doi:10.1016/j.atmosres.2007.05.003, https://doi.org/10.1016/j.atmosres.2007.05.003
10. Bogdan Rosa, Hossein Parishani, Orlando Ayala, Wojciech W. Grabowski, Lian-Ping Wang (2013), 'Kinematic and dynamic collision statistics of cloud droplets from high-resolution simulations', New Journal of Physics 15:045032, doi:10.1088/1367-2630/15/4/045032, https://doi.org/10.1088/1367-2630/15/4/045032
11. Sisi Chen, Peter Bartello, M. K. Yau, P. A. Vaillancourt, Kevin Zwijsen (2016), 'Cloud Droplet Collisions in Turbulent Environment: Collision Statistics and Parameterization', Journal of the Atmospheric Sciences 73:621-636, doi:10.1175/jas-d-15-0203.1, https://doi.org/10.1175/jas-d-15-0203.1
12. Sisi Chen, M. K. Yau, Peter Bartello (2018), 'Turbulence Effects of Collision Efficiency and Broadening of Droplet Size Distribution in Cumulus Clouds', Journal of the Atmospheric Sciences 75:203-217, doi:10.1175/jas-d-17-0123.1, https://doi.org/10.1175/jas-d-17-0123.1
13. Sisi Chen, Man-Kong Yau, Peter Bartello, Lulin Xue (2018), 'Bridging the condensation-collision size gap: a direct numerical simulation of continuous droplet growth in turbulent clouds', Atmospheric Chemistry and Physics 18:7251-7262, doi:10.5194/acp-18-7251-2018, https://doi.org/10.5194/acp-18-7251-2018
14. A. Ababaei, B. Rosa, J. Pozorski, Lian-Ping Wang (2021), 'On the effect of lubrication forces on the collision statistics of cloud droplets in homogeneous isotropic turbulence', Journal of Fluid Mechanics 918:A22, doi:10.1017/jfm.2021.229, https://doi.org/10.1017/jfm.2021.229
15. Ahmad Ababaei, Bogdan Rosa (2023), 'Collision efficiency of cloud droplets in quiescent air considering lubrication interactions, mobility of interfaces, and noncontinuum molecular effects', Physical Review Fluids 8:014102, doi:10.1103/physrevfluids.8.014102, https://doi.org/10.1103/physrevfluids.8.014102
16. Ahmad Ababaei, Antoine Michel, Bogdan Rosa (2024), 'Collision Statistics of Droplets in Turbulence Considering Lubrication Interactions, Mobility of Interfaces, and Non-continuum Molecular Effects', Flow, Turbulence and Combustion 112:33-59, doi:10.1007/s10494-023-00450-1, https://doi.org/10.1007/s10494-023-00450-1
17. R. V. Kearney, G. P. Bewley (2022), 'Relative accelerations characterize the hydrodynamic interaction of cloud droplets', arXiv:2204.07688, https://arxiv.org/abs/2204.07688
18. B. J. Devenish, P. Bartello, J.-L. Brenguier, L. R. Collins, W. W. Grabowski, R. H. A. IJzermans, S. P. Malinowski, M. W. Reeks, J. C. Vassilicos, L.-P. Wang, Z. Warhaft (2012), 'Droplet growth in warm turbulent clouds', Quarterly Journal of the Royal Meteorological Society 138:1401-1429, doi:10.1002/qj.1897, https://doi.org/10.1002/qj.1897
19. Wojciech W. Grabowski, Lian-Ping Wang (2013), 'Growth of Cloud Droplets in a Turbulent Environment', Annual Review of Fluid Mechanics 45:293-324, doi:10.1146/annurev-fluid-011212-140750, https://doi.org/10.1146/annurev-fluid-011212-140750
20. L. Alfonso, G. B. Raga, D. Baumgardner (2008), 'The validity of the kinetic collection equation revisited', Atmospheric Chemistry and Physics 8:969-982, doi:10.5194/acp-8-969-2008, https://doi.org/10.5194/acp-8-969-2008
21. L. Alfonso (2015), 'An algorithm for the numerical solution of the multivariate master equation for stochastic coalescence', Atmospheric Chemistry and Physics 15:12315-12326, doi:10.5194/acp-15-12315-2015, https://doi.org/10.5194/acp-15-12315-2015
22. S. Shima, K. Kusano, A. Kawano, T. Sugiyama, S. Kawahara (2009), 'The super-droplet method for the numerical simulation of clouds and precipitation: a particle-based and probabilistic microphysics model coupled with a non-hydrostatic model', Quarterly Journal of the Royal Meteorological Society 135:1307-1320, doi:10.1002/qj.441, https://doi.org/10.1002/qj.441
23. Piotr Dziekan, Hanna Pawlowska (2017), 'Stochastic coalescence in Lagrangian cloud microphysics', Atmospheric Chemistry and Physics 17:13509-13520, doi:10.5194/acp-17-13509-2017, https://doi.org/10.5194/acp-17-13509-2017
24. Piotr Zmijewski, Piotr Dziekan, Hanna Pawlowska (2024), 'Modeling collision-coalescence in particle microphysics: numerical convergence of mean and variance of precipitation in cloud simulations using the University of Warsaw Lagrangian Cloud Model (UWLCM) 2.1', Geoscientific Model Development 17:759-780, doi:10.5194/gmd-17-759-2024, https://doi.org/10.5194/gmd-17-759-2024
25. Jambajamts Lkhamjav, Hyunho Lee, Ye-Lim Jeon, Jong-Jin Baik (2017), 'Examination of an improved quasi-stochastic model for the collisional growth of drops', Journal of Geophysical Research: Atmospheres 122:1713-1724, doi:10.1002/2016JD025904, https://doi.org/10.1002/2016JD025904
26. Simon Unterstrasser, Fabian Hoffmann, Marion Lerch (2020), 'Collisional growth in a particle-based cloud microphysical model: insights from column model simulations using LCM1D (v1.0)', Geoscientific Model Development 13:5119-5145, doi:10.5194/gmd-13-5119-2020, https://doi.org/10.5194/gmd-13-5119-2020
27. Ewe-Wei Saw, Xiaohui Meng (2022), 'Intricate relations among particle collision, relative motion and clustering in turbulent clouds: computational observation and theory', Atmospheric Chemistry and Physics 22:3779-3788, doi:10.5194/acp-22-3779-2022, https://doi.org/10.5194/acp-22-3779-2022
28. Jung-Sub Lim, Yign Noh, Hyunho Lee, Fabian Hoffmann (2025), 'The critical number and size of precipitation embryos to accelerate warm rain initiation', Atmospheric Chemistry and Physics 25:5313-5329, doi:10.5194/acp-25-5313-2025, https://doi.org/10.5194/acp-25-5313-2025
29. Yangze Ren, Kamal Kant Chandrakar, Fan Yang, Raymond A. Shaw (2026), 'Evaluating the Collision-Coalescence Process in Idealized Cloud Convection Using Large-Eddy Simulations With Lagrangian Microphysics', Geophysical Research Letters 53:e2025GL118873, doi:10.1029/2025gl118873, https://doi.org/10.1029/2025gl118873
30. Jun-Ichi Yano, Marta Waclawczyk, Maarten Ambaum (2026), 'Stochastic Collection Equation: Invariant Evolution of the Cloud Droplet-Size Distribution', EGUsphere preprint, doi:10.5194/egusphere-2026-3530, https://doi.org/10.5194/egusphere-2026-3530
31. Itzhak Fouxon, Seulgi Lee, Changhoon Lee (2022), 'Intermittency and collisions of fast sedimenting droplets in turbulence', Physical Review Fluids 7:124303, doi:10.1103/physrevfluids.7.124303, https://doi.org/10.1103/physrevfluids.7.124303