Validation and coupled nonlinear quantification of the tangling-clustering instability as a mechanism of rapid raindrop formation in turbulent stratified atmospheric clouds

In warm (ice-free) clouds, micron-size droplets created by condensation on cloud condensation nuclei must grow to roughly 50–100 µm to fall as rain, and observations show this happens in only 15–20 minutes — the unexplained 'condensation–coalescence bottleneck' of cloud microphysics. Elperin et al. (Phys. Rev. E 92, 013012, 2015) proposed a resolution: the tangling-clustering instability, in which turbulence tangling of the mean atmospheric temperature gradient generates small-scale temperature and pressure fluctuations that render the droplet number density linearly unstable, producing clusters with number densities enhanced by up to $n_{\max}/N\approx470$, shortening the effective collision–coalescence time by $(n_{\max}/N)^2\sim10^5$ and allowing growth from 1 µm to 40–50 µm within the observed time. The proposal rests on a mean-field theory with a depletion-limited saturation estimate, assumes unit coalescence efficiency, neglects two-way droplet–flow momentum coupling and aerosol spectral dynamics, and has never been confronted with simultaneous, sub-Kolmogorov-resolution measurements of temperature fields, turbulent heat fluxes, and droplet clustering in natural clouds. The open problem is to determine, through such measurements and through coupled nonlinear modeling with realistic microphysics, whether the tangling-clustering mechanism actually operates in natural clouds at the predicted amplitudes and quantitatively accounts for rapid raindrop formation — or to bound and refute its contribution.

Background

Warm clouds contain droplets that form by condensation of water vapor on cloud condensation nuclei (CCN, tiny aerosol particles on which vapor condenses). Droplets grown by vapor diffusion stall at radii of about 10 µm, while gravitational collision–coalescence becomes efficient only for radii of roughly 25–50 µm; the unexplained crossing of this 'condensation–coalescence bottleneck' (also called the size gap) within the observed 15–20 minutes is a long-standing open problem of cloud microphysics (reviews: Devenish et al. 2012; Grabowski & Wang 2013).

Several turbulence-based mechanisms have been studied. In inertial clustering (preferential concentration), heavy droplets with Stokes number $St=\tau_{st}/\tau_{\eta}$ — the ratio of the droplet aerodynamic response time $\tau_{st}$ to the Kolmogorov time $\tau_{\eta}$ of the smallest turbulent eddies — are centrifuged out of intense vortices and accumulate in strain regions; the Kolmogorov length scale $\ell_{\eta}$ (of order millimeters in clouds) is the size of the smallest eddies. In-situ measurements using the pair-correlation function of droplet arrival times (Lehmann et al. 2007, Tellus B) show clustering of cloud droplets even at very small Stokes numbers, but with only modest amplitude at Kolmogorov separations, and aircraft probes measure along a nearly one-dimensional path with limited spatial resolution and possible instrumental artifacts.

Elperin, Kleeorin, Liberman and Rogachevskii (2013) identified a distinct 'tangling clustering instability': in a fluid with a mean temperature gradient $\nabla T$, turbulence tangles the gradient and produces small-scale temperature fluctuations $\theta$ whose statistics are governed by the turbulent heat flux $\langle\boldsymbol{u}\theta\rangle$; for inertial particles the fluctuating pressure field then acts as an effective potential, and the particle number density becomes linearly unstable to clustering, with a growth rate of order inverse Kolmogorov times. A laboratory experiment in a stably stratified turbulent flow at Reynolds number 250 (Eidelman et al. 2010) found the two-point correlation function of particle density to be about an order of magnitude larger than for inertial clustering in isothermal turbulence — the only experimental support, from the proposing group. Elperin et al. (2015) applied the mechanism to clouds: solving the mean-field transport equation for the droplet size distribution with a standard coagulation kernel, they found that inside clusters with $n_{\max}/N$ up to the mass-loading-limited value of about 470 (the bound $n_{\max}/N<0.5\rho/\bar{\rho}_{dr}$ on their depletion-limited saturation estimate, Eq. (21)), the effective collision–coalescence time drops by $(n_{\max}/N)^2\sim10^5$, so that droplets can grow from 1 µm to 40–50 µm in 15–20 minutes, provided the relative humidity is near saturation and the droplet size exceeds the instability threshold ($d>d_{th}=0.7$ µm, with a sharp maximum of the growth rate at $d=1.75$ µm at 99–100% humidity).

The 2015 paper itself states what remains unresolved: existing atmospheric measurements never recorded the spatial temperature distributions and the vertical and horizontal turbulent heat fluxes needed to determine the tangling-clustering source function $B(\boldsymbol{R})$ (a two-point correlation of Laplacian-squared temperature fluctuations entering the instability growth rate), so 'no conclusions about tangling clustering in these experiments' are possible; resolving the predicted sharp rise of the pair-correlation function below the Kolmogorov scale requires roughly tenfold better spatial resolution than the 100–200 µm effective resolution of probe-based studies, likely via particle image velocimetry or holographic imaging. In addition, the saturation estimate is not a nonlinear, two-way-coupled solution: at finite droplet mass loading the particles modify the turbulence itself; collisions are assumed always to coalesce, whereas real outcomes (coalescence, bounce, temporary coalescence followed by separation) depend on impact parameter, size ratio, and relative velocity; and aerosol activation and evolving droplet size spectra are not coupled to the clustering dynamics.

Problem Statement

Determine whether the tangling-clustering instability of small cloud droplets in temperature-stratified atmospheric turbulence operates in natural clouds at the amplitudes predicted by Elperin et al. (2015), and whether it quantitatively accounts for the observed rapid growth of droplets into raindrops — or, alternatively, establish bounds that exclude it as a resolution of the condensation–coalescence bottleneck. Two complementary components support this determination:

(a) Observational determination of the tangling-clustering source and amplitude in natural clouds: simultaneous measurements of spatial temperature distributions, vertical and horizontal turbulent heat fluxes $\langle\boldsymbol{u}\theta\rangle$, two-point temperature correlation functions, and droplet concentration fields (e.g., the radial distribution function, i.e., the ratio of the local droplet number density at a given separation to the volume-averaged mean, or equivalently the pair-correlation function of droplet positions) at spatial resolutions at or below the Kolmogorov length scale, sufficient to determine $B(\boldsymbol{R})$ directly and to compare the observed clustering amplitude and its scale dependence with the theory's predictions (growth rates of order ten inverse Kolmogorov times; saturation at $n_{\max}/N$ up to the mass-loading-limited value of order a few hundred; a sharp rise of clustering below the Kolmogorov scale), with controls against instrumental artifacts, finite-sample effects, large-scale inhomogeneity from entrainment and mixing, and ordinary inertial clustering.

(b) Coupled nonlinear quantification: extension of the theory beyond its mean-field, linear-saturation form — including two-way momentum coupling between droplets and turbulence at finite mass loading, realistic collision outcomes (non-unity coalescence efficiency, bounce, breakup) in place of the assumed always-coalesce rule, and coupling to aerosol activation and evolving droplet size spectra — to determine the mechanism's net effect on the time for droplets to grow from about 1 µm to raindrop sizes under realistic cloud conditions, including the sensitivity of the predicted 15–20 minute growth times to each neglected effect.

Completion means the mechanism's contribution to rapid warm-rain formation is quantitatively established, bounded, or refuted: either the evidence confirms the predicted stratification-driven, sub-Kolmogorov clustering signature and growth times consistent with observation, or the combined evidence shows the clustering amplitudes or growth rates to be too small to matter.

References

1. Elperin T, Kleeorin N, Krasovitov B, Kulmala M, Liberman M, Rogachevskii I, Zilitinkevich S (2015). Acceleration of raindrop formation due to the tangling-clustering instability in a turbulent stratified atmosphere. Physical Review E 92, 013012. doi:10.1103/PhysRevE.92.013012, https://doi.org/10.1103/PhysRevE.92.013012 (preprint arXiv:1312.3250)
2. Elperin T, Kleeorin N, Liberman MA, Rogachevskii I (2013). Tangling clustering instability for small particles in temperature stratified turbulence. Physics of Fluids 25, 085104. doi:10.1063/1.4816643, https://doi.org/10.1063/1.4816643
3. Eidelman A, Elperin T, Kleeorin N, Melnik B, Rogachevskii I (2010). Tangling clustering of inertial particles in stably stratified turbulence. Physical Review E 81, 056313. doi:10.1103/PhysRevE.81.056313, https://doi.org/10.1103/PhysRevE.81.056313
4. Lehmann K, Siebert H, Wendisch M, Shaw RA (2007). Evidence for inertial droplet clustering in weakly turbulent clouds. Tellus B 59, 57. doi:10.1111/j.1600-0889.2006.00229.x, https://doi.org/10.1111/j.1600-0889.2006.00229.x
5. Devenish BJ, Bartello P, Brenguier J-L, Collins LR, Grabowski WW, IJzermans RHA, Malinowski SP, Reeks MW, Vassilicos JC, Wang L-P, 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
6. Grabowski WW, 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
7. Beals MA, Fugal JP, Shaw RA, Lu J, Spuler SM, Stith JL (2015). Holographic measurements of inhomogeneous cloud mixing at the centimeter scale. Science 350, 87–90. doi:10.1126/science.aab0751, https://doi.org/10.1126/science.aab0751
8. Dodson DS, Small Griswold JD (2019). Droplet inhomogeneity in shallow cumuli: the effects of in-cloud location and aerosol number concentration. Atmospheric Chemistry and Physics 19, 7297–7317. doi:10.5194/acp-19-7297-2019, https://doi.org/10.5194/acp-19-7297-2019
9. Li X-Y, Brandenburg A, Haugen NE, Svensson G (2017). Eulerian and Lagrangian approaches to multidimensional condensation and collection. Journal of Advances in Modeling Earth Systems 9, 1116–1137. doi:10.1002/2017MS000930, https://doi.org/10.1002/2017MS000930
10. Chen S, Yau M-K, Bartello P, 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
11. Madival D (2019). A model study of the effect of clustering on the last stage of drizzle formation. Quarterly Journal of the Royal Meteorological Society 145, 3790–3800. doi:10.1002/qj.3659, https://doi.org/10.1002/qj.3659
12. Srinivas T, Tomar G (2025). Weakly nonlinear analysis of particle-laden Rayleigh–Bénard convection. Journal of Fluid Mechanics 1025. doi:10.1017/jfm.2025.10956, https://doi.org/10.1017/jfm.2025.10956
13. Srinivas T, Tomar G (2025). A generalized correction scheme for two-way coupled particle-laden Euler–Lagrange simulations. Computers & Fluids 290, 106554. doi:10.1016/j.compfluid.2025.106554, https://doi.org/10.1016/j.compfluid.2025.106554
14. Chandrakar KK, Morrison H, Grabowski WW, Lawson RP (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
15. Reartes C, Mininni PD (2023). Dynamical regimes and clustering of small neutrally buoyant inertial particles in stably stratified turbulence. Physical Review Fluids 8, 054501. doi:10.1103/PhysRevFluids.8.054501, https://doi.org/10.1103/PhysRevFluids.8.054501