Two available parameterizations of turbulence effects on cloud-droplet collisions — the Ayala–Wang kernel and the Onishi kernel — predict qualitatively opposite Reynolds-number dependences of the enhancement of warm-rain formation. Seifert and Onishi (2016) embedded both kernels in a two-moment bulk warm-rain scheme and showed that the discrepancy propagates to large quantitative differences in simulated rain formation in shallow cumulus, while their large-eddy simulations revealed a strong, unconverged grid-resolution sensitivity implying an unparameterized contribution of resolved turbulent eddies to raindrop growth. Which kernel is realistic at the Reynolds numbers of atmospheric clouds remains undetermined: direct numerical simulations have reached only Reλ≈103Re_\lambda\approx10^3 (versus 10310^3–10410^4 in clouds), and observational tests, while confirming that turbulent coalescence materially shapes observed droplet size distributions and drizzle tails, have not discriminated between the two kernels' Reynolds-number scalings. The problem therefore remains open: as the same source concluded, it needs to be clarified which collision kernel is more realistic at cloud Reynolds numbers, while the parameterization of resolved-eddy effects on raindrop growth and the grid convergence of precipitating shallow-cumulus LES stand as closely related open modeling challenges.

Background

Warm clouds produce rain without ice through collision and coalescence of water droplets. The rate at which droplets of radii r1r_1 and r2r_2 collide per unit volume is described by a collection (collision) kernel, the product of a geometric sweeping rate, the radial distribution function (RDF, the probability of finding droplets at a given small separation relative to a random distribution — inertial droplets 'preferentially concentrate' in turbulent eddies), the droplet relative velocity, and a collision efficiency accounting for hydrodynamic interactions. Saffman and Turner (1956) gave the classical theory for turbulence effects on collision rates; its modern successors separate the effects of droplet inertia, measured by the Stokes number StSt (the ratio of the droplet aerodynamic response time to the Kolmogorov time of the smallest turbulent eddies), and of flow scale, measured by the Taylor-microscale Reynolds number ReλRe_\lambda. Reviews by Devenish et al. (2012), Grabowski and Wang (2013) and Lu et al. (2018) survey this problem of how small-scale turbulence accelerates the growth of droplets past the size range (roughly 15–40 μm radius) where purely gravitational growth is slow.

Two parameterized kernels dominate quantitative work. The Ayala–Wang kernel was derived by Ayala, Rosa and Wang (2008) from direct numerical simulation (DNS, resolving turbulence down to the Kolmogorov scale) of sedimenting droplets in isotropic turbulence, combined with theory and parameterization of the geometric collision rate and the turbulent collision efficiency of Wang et al. (2008). The Onishi kernel was derived by Onishi, Matsuda and Takahashi (2015) from a Lagrangian Cloud Simulator (LCS) that tracks individual droplets in forced isotropic turbulence, and was updated in the companion paper of Onishi and Seifert (2016) using DNS up to Reλ=1140Re_\lambda=1140. The two kernels agree at low and moderate ReλRe_\lambda but make opposite predictions at larger values: in the bulk formulation of Seifert and Onishi (2016) the turbulence enhancement of the autoconversion rate carries a factor scaling as ϵReλ1/4\epsilon Re_\lambda^{1/4} for Ayala–Wang but ϵReλ−1/8\epsilon Re_\lambda^{-1/8} for Onishi, where ϵ\epsilon is the turbulent energy dissipation rate. Atmospheric clouds have ReλRe_\lambda of order 10310^3–10410^4, with the upper end of this range beyond the reach of existing DNS, so the two kernels' extrapolations diverge precisely in the regime that matters.

Seifert and Onishi (2016) built both kernels into a two-moment bulk warm-rain scheme (a scheme predicting liquid water content and number concentration for cloud and rain categories; autoconversion is the formation of raindrops by collisions among cloud droplets, accretion is the collection of cloud droplets by falling raindrops, and self-collection denotes rain–rain collisions), generalizing the earlier Ayala–Wang-based scheme of Seifert et al. (2010). Validated against a one-dimensional bin microphysics model (which resolves the droplet size distribution explicitly), the scheme was then used in large-eddy simulations (LES) of the RICO trade-wind cumulus case (van Zanten et al. 2011). Both kernels enhanced rain formation, but with large quantitative differences (for example, only the Ayala–Wang kernel significantly lowered the trade-wind inversion, because of its faster rain formation). In addition, the LES showed strong sensitivity to horizontal grid spacing, with rain formation more rapid at finer resolution and no convergence even at 25 m; the authors attributed this to resolved larger turbulent eddies (in-cloud circulations) that lengthen raindrop residence times and augment growth, a mechanism related to the recirculation studies of Naumann and Seifert (2016), and noted that resolution sensitivity of precipitating shallow-cumulus LES had also been emphasized by Matheou et al. (2011). The authors concluded that clarifying which kernel is more realistic at cloud Reynolds numbers requires observational constraints or additional direct numerical simulations at those Reynolds numbers.

Problem Statement

As posed by Seifert and Onishi (2016), the problem is to determine which of the two available turbulent collection kernels — the Ayala–Wang kernel or the Onishi kernel — more realistically describes droplet collision statistics at the high Reynolds numbers (ReλRe_\lambda of order 10310^3–10410^4, with the parameterization range extending to 25,000) characteristic of atmospheric clouds. The two kernels make qualitatively opposite predictions: in the bulk warm-rain formulation their autoconversion enhancement factors carry Reynolds-number exponents of approximately +1/4+1/4 (Ayala–Wang) and −1/8-1/8 (Onishi), so an answer must adjudicate (or empirically replace) these conflicting scalings under conditions representative of atmospheric clouds. The same source also noted two closely related modeling deficits as directions for future study: current two-moment bulk warm-rain schemes do not represent the augmentation of raindrop growth by resolved turbulent eddies (in-cloud circulations interacting with raindrops of Stokes number greater than 1), and large-eddy simulations of precipitating shallow convection are not grid-converged, so in such simulations the sub-grid contribution parameterized by the collision kernel cannot be separated from the resolved contribution. These deficits are related open modeling challenges that the source flagged for future work; they are not conditions attached to the open question it posed, which is the adjudication of the kernel itself.

References

  1. Seifert, A. and Onishi, R. (2016). Turbulence effects on warm-rain formation in precipitating shallow convection revisited. Atmospheric Chemistry and Physics 16, 12127–12141. doi:10.5194/acp-16-12127-2016. https://doi.org/10.5194/acp-16-12127-2016
  2. Onishi, R. and Seifert, A. (2016). Reynolds-number dependence of turbulence enhancement on collision growth. Atmospheric Chemistry and Physics 16, 12441–12455. doi:10.5194/acp-16-12441-2016. https://doi.org/10.5194/acp-16-12441-2016
  3. Ayala, O., Rosa, B., and Wang, L.-P. (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
  4. Ayala, O., Rosa, B., Wang, L.-P., and Grabowski, W. W. (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. Wang, L.-P., Ayala, O., Rosa, B., and Grabowski, W. W. (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
  6. Onishi, R., Matsuda, K., and Takahashi, K. (2015). Lagrangian tracking simulation of droplet growth in turbulence — turbulence enhancement of autoconversion rate. Journal of the Atmospheric Sciences 72, 2591–2607. doi:10.1175/JAS-D-14-0292.1. https://doi.org/10.1175/JAS-D-14-0292.1
  7. Seifert, A., Nuijens, L., and Stevens, B. (2010). Turbulence effects on warm-rain autoconversion in precipitating shallow convection. Quarterly Journal of the Royal Meteorological Society 136, 1753–1762. doi:10.1002/qj.684. https://doi.org/10.1002/qj.684
  8. Grabowski, W. W. and Wang, L.-P. (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
  9. Devenish, B. J., Bartello, P., Brenguier, J.-L., Collins, L. R., Grabowski, W. W., IJzermans, R. H. A., Malinowski, S. P., Reeks, M. W., Vassilicos, J. C., Wang, L.-P., and Warhaft, Z. (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
  10. Lu, C., Liu, Y., Niu, S., and Xue, Y. (2018). Broadening of cloud droplet size distributions and warm rain initiation associated with turbulence: an overview. Atmospheric and Oceanic Science Letters 11, 123–135. doi:10.1080/16742834.2018.1410057. https://doi.org/10.1080/16742834.2018.1410057
  11. Saffman, P. G. and Turner, J. S. (1956). On the collision of drops in turbulent clouds. Journal of Fluid Mechanics 1, 16–30. doi:10.1017/S0022112056000020. https://doi.org/10.1017/S0022112056000020
  12. van Zanten, M. C., Stevens, B., Nuijens, L., Siebesma, A. P., Ackerman, A. S., Burnet, F., Cheng, A., Couvreux, F., Jiang, H., Khairoutdinov, M., Kogan, Y., Lewellen, D. C., Mechem, D., Nakamura, K., Noda, A., Shipway, B. J., Slawinska, J., Wang, S., and Wyszogrodzki, A. (2011). Controls on precipitation and cloudiness in simulations of trade-wind cumulus as observed during RICO. Journal of Advances in Modeling Earth Systems 3, M06001. doi:10.1029/2011MS000056. https://doi.org/10.1029/2011MS000056
  13. Naumann, A. K. and Seifert, A. (2016). Recirculation and growth of raindrops in simulated shallow cumulus. Journal of Advances in Modeling Earth Systems 8, 520–537. doi:10.1002/2016MS000631. https://doi.org/10.1002/2016MS000631
  14. Matheou, G., Chung, D., Nuijens, L., Stevens, B., and Teixeira, J. (2011). On the fidelity of large-eddy simulation of shallow precipitating cumulus convection. Monthly Weather Review 139, 2918–2939. doi:10.1175/2011MWR3599.1. https://doi.org/10.1175/2011MWR3599.1
  15. Li, X.-Y., Brandenburg, A., Svensson, G., Haugen, N. E. L., Mehlig, B., and Rogachevskii, I. (2018). Effect of turbulence on collisional growth of cloud droplets. Journal of the Atmospheric Sciences 75, 3469–3487. doi:10.1175/JAS-D-18-0081.1. https://doi.org/10.1175/JAS-D-18-0081.1
  16. Chen, S., Yau, M. K., Bartello, P., and Xue, L. (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
  17. Saw, E.-W. and Meng, X. (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
  18. Ababaei, A., Rosa, B., Pozorski, J., and Wang, L.-P. (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
  19. Michel, A., Ababaei, A., and Rosa, B. (2023). The influence of gravity and mass loading on the coalescence of aerodynamically interacting droplets in homogeneous isotropic turbulence. Journal of the Atmospheric Sciences 80, 2207–2220. doi:10.1175/JAS-D-22-0267.1. https://doi.org/10.1175/JAS-D-22-0267.1
  20. Codispoti, L. A., Meyer, D. W., and Jenny, P. (2026). Polydisperse collision kernels in droplet-laden turbulence with implications for rain formation. Physical Review Fluids 11, 064601. doi:10.1103/kxxm-ghty. https://doi.org/10.1103/kxxm-ghty
  21. Witte, M. K., Chuang, P. Y., Ayala, O., Wang, L.-P., and Feingold, G. (2019). Comparison of observed and simulated drop size distributions from large-eddy simulations with bin microphysics. Monthly Weather Review 147, 477–493. doi:10.1175/MWR-D-18-0242.1. https://doi.org/10.1175/MWR-D-18-0242.1
  22. Borque, P., Varble, A., and Hardin, J. (2022). Peak rain rate sensitivity to observed cloud condensation nuclei and turbulence in continental warm shallow clouds during CACTI. Journal of Geophysical Research: Atmospheres 127, e2022JD036864. doi:10.1029/2022JD036864. https://doi.org/10.1029/2022JD036864
  23. Chandrakar, K. K., Morrison, H., Grabowski, W. W., and Lawson, R. P. (2024). Are turbulence effects on droplet collision–coalescence a key to understanding observed rain formation in clouds? Proceedings of the National Academy of Sciences 121, e2319664121. doi:10.1073/pnas.2319664121. https://doi.org/10.1073/pnas.2319664121
  24. Chandrakar, K. K. and Morrison, H. (2025). Dampening of the precipitation response to aerosol pollution from turbulence in cumulus clouds. Geophysical Research Letters 52, e2025GL118693. doi:10.1029/2025GL118693. https://doi.org/10.1029/2025GL118693
  25. Thiede, B., Larsen, M. L., Nordsiek, F., Schlenczek, O., Kim, Y., Bodenschatz, E., and Bagheri, G. (2026). Highly localized droplet clustering in shallow cumulus clouds. Proceedings of the National Academy of Sciences 123, e2602976123. doi:10.1073/pnas.2602976123. https://doi.org/10.1073/pnas.2602976123
  26. Shaw, R. A., Chandrakar, K. K., Krueger, S. K., Ren, Y., Yang, F., and Yeom, J. M. (2026). Unifying observations, simulations, and theory for drizzle size distribution tails. Geophysical Research Letters 53, e2026GL125405. doi:10.1029/2026GL125405. https://doi.org/10.1029/2026GL125405
  27. Jin, H.-G., Lee, H., and Baik, J.-J. (2022). Large-eddy simulations of drizzling shallow cumuli using a turbulence-aware autoconversion parametrization. Quarterly Journal of the Royal Meteorological Society 148, 3885–3900. doi:10.1002/qj.4395. https://doi.org/10.1002/qj.4395
  28. Noh, Y., Oh, D., Hoffmann, F., and Raasch, S. (2018). A cloud microphysics parameterization for shallow cumulus clouds based on Lagrangian cloud model simulations. Journal of the Atmospheric Sciences 75, 4031–4047. doi:10.1175/JAS-D-18-0080.1. https://doi.org/10.1175/JAS-D-18-0080.1
  29. Oh, D. and Noh, Y. (2022). Roles of drop size distribution and turbulence in autoconversion based on Lagrangian cloud model simulations. Journal of Geophysical Research: Atmospheres 127, e2022JD036495. doi:10.1029/2022JD036495. https://doi.org/10.1029/2022JD036495
  30. Igel, A. L., Morrison, H., Santos, S. P., and van Lier-Walqui, M. (2022). Limitations of separate cloud and rain categories in parameterizing collision–coalescence for bulk microphysics schemes. Journal of Advances in Modeling Earth Systems 14, e2022MS003039. doi:10.1029/2022MS003039. https://doi.org/10.1029/2022MS003039
  31. Liu, Y., Yau, M.-K., Shima, S.-i., Lu, C., and Chen, S. (2023). Parameterization and explicit modeling of cloud microphysics: approaches, challenges, and future directions. Advances in Atmospheric Sciences 40, 747–790. doi:10.1007/s00376-022-2077-3. https://doi.org/10.1007/s00376-022-2077-3
  32. Liu, Y. (2025). Impact of turbulence on the autoconversion rate from cloud droplets to raindrops. Science China Earth Sciences 68, 2949–2956. doi:10.1007/s11430-025-1633-1. https://doi.org/10.1007/s11430-025-1633-1
  33. Stanford, M. W., Fridlind, A. M., Ackerman, A. S., van Diedenhoven, B., Xiao, Q., Wang, J., Matsui, T., Hernandez-Deckers, D., and Lawson, P. (2025). Warm-phase microphysical evolution in large-eddy simulations of tropical cumulus congestus: evaluating drop size distribution evolution using polarimetry retrievals, in situ measurements, and a thermal-based framework. Atmospheric Chemistry and Physics 25, 11199–11231. doi:10.5194/acp-25-11199-2025. https://doi.org/10.5194/acp-25-11199-2025
  34. Lim, J.-S., Noh, Y., Lee, H., and Hoffmann, F. (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
  35. Sato, Y., Shima, S.-i., and Tomita, H. (2018). Numerical convergence of shallow convection cloud field simulations: comparison between double-moment Eulerian and particle-based Lagrangian microphysics coupled to the same dynamical core. Journal of Advances in Modeling Earth Systems 10, 1495–1512. doi:10.1029/2018MS001285. https://doi.org/10.1029/2018MS001285
  36. Lamaakel, A., Venters, E., Teixeira, J., and Matheou, G. (2023). Computational domain size effects on large-eddy simulations of precipitating shallow cumulus convection. Atmosphere 14, 1186. doi:10.3390/atmos14071186. https://doi.org/10.3390/atmos14071186
  37. Krueger, S. K. and Kerstein, A. R. (2018). An economical model for simulating turbulence enhancement of droplet collisions and coalescence. Journal of Advances in Modeling Earth Systems 10, 1858–1881. doi:10.1029/2017MS001240. https://doi.org/10.1029/2017MS001240
  38. Wang, A., Jiang, P., Burrows, S. M., Glienke, S., Ovchinnikov, M., and Mahfouz, N. (2026). Inverse mapping of the collision kernel and wall flux scaling in a tall convection-cloud chamber using local sensors and knowledge-informed deep learning. Journal of Advances in Modeling Earth Systems 18, e2025MS005463. doi:10.1029/2025MS005463. https://doi.org/10.1029/2025MS005463