Do caustic-driven droplet voids and phase-change feedback operate at deep-convective-cloud Reynolds numbers ($R_\lambda \sim 10^4$)?

The review by Ravichandran and Govindarajan (Phys. Rev. Fluids 7, 110512, 2022) argued that small cloud droplets centrifuged out of intense sub-Kolmogorov vortices form caustics - multivalued droplet velocity fields that enable fast 'sling' collisions - and, by extrapolating the observed collapse of extreme-vorticity probability densities from $R_\lambda \le 1300$ to the $R_\lambda \sim 10^4$ of deep convective clouds, predicted that the volume fraction of droplet-free voids becomes order unity there, so that a significant fraction of droplets participate in caustic events and can initiate rain. The extrapolation has never been tested: direct numerical simulations resolving scales well below the Kolmogorov scale still stop at $R_\lambda \approx 1300$, no turbulence-resolving superdroplet-type scheme faithful to caustics exists, and the conjectured self-sustaining enhancement of turbulence intermittency by phase change has only recently been demonstrated at ordinary simulation scales. The open problem is to establish, by coupled Navier–Stokes + droplet + phase-change simulations at cloud-like Reynolds numbers, whether extreme-vorticity universality persists to $R_\lambda \sim 10^4$, whether the caustic void fraction is indeed $O(1)$ with bottleneck-crossing droplet growth, and whether condensation-driven turbulence enhancement operates self-sustainingly under those conditions.

Background

Warm clouds pose a long-standing puzzle known as the droplet-growth bottleneck: droplets smaller than about 10 microns grow efficiently by vapor diffusion and condensation, and drops larger than about 50–100 microns grow efficiently by gravitational collision–coalescence, but no equally efficient mechanism is known to bridge the 10–50 micron range quickly enough to explain observed warm-rain initiation (Falkovich, Fouxon and Stepanov, Nature 419, 151, 2002). Turbulence is widely believed to help. Small heavy droplets advected by turbulence are inertial particles: their motion is described, in the Stokes-drag limit, by the Maxey–Riley equation, and their inertia is measured by the Stokes number $St$, the ratio of the droplet relaxation time $\tau$ to a flow time scale. Inertial particles are centrifuged out of vortical regions and cluster in strain regions ('preferential concentration'). A stronger effect is the formation of caustics: when the droplet velocity field becomes multivalued (a fold in position–velocity space, as in ray caustics in optics), droplets at nearly the same position can have very different velocities, producing a 'sling effect' with sharply enhanced collision rates (Wilkinson and Mehlig, Europhys. Lett. 71, 186, 2005). The review of Ravichandran and Govindarajan (Phys. Rev. Fluids 7, 110512, 2022) developed a single-vortex picture of these processes: for a Lamb–Oseen (viscous) vortex, taken as a building block of small-scale turbulence, caustics form only around vortices smaller than a caustics radius $r_{cd} \simeq (\Gamma\tau)^{1/2}$, with $\Gamma$ the circulation and $\tau$ the droplet relaxation time, which selects intense vortices smaller than the Kolmogorov scale $\eta_K$ (the smallest turbulence length scale, with associated time $t_K$). Such vortices evacuate droplets, creating particle-free voids whose area is comparable to the caustics area. Turbulence intensity is measured by the Taylor-microscale Reynolds number $R_\lambda$; in a cumulonimbus (deep convective) cloud the large-scale Reynolds number is $Re \simeq 10^8$, i.e. $R_\lambda \simeq 10^4$, with turbulent dissipation $\epsilon \simeq 0.1$ m²/s³. Using the extreme-vorticity statistics of Buaria and Sreenivasan (Phys. Rev. Fluids 7, L072601, 2022) - who showed that for $R_\lambda$ from 140 to 1300 the tails of the probability density of squared vorticity $\Omega$, scaled by the fastest time scale $t_{ext}$, collapse onto one curve, reflecting the intermittency of intense small-scale vorticity - the review integrated the void fraction $f_{void} \sim (\tau/t_K)\int p(x)x^{1/2}dx$ over vortices strong enough to make caustics ($x \equiv \Omega t_K^2$ above a threshold). This gave $f_{void} \sim 10^{-10}$ at $R_\lambda = 650$, $\sim 10^{-4}$ at $R_\lambda = 1300$, and, by extrapolating the collapse of the vorticity PDF by an order of magnitude in $R_\lambda$, an $O(1)$ void fraction at $R_\lambda \sim 10^4$ - the basis of the review's 'audacious propositions' that single-vortex caustics could be important for rain initiation and that rain initiation needs $R_\lambda > 10^3$. The review itself stressed that this extrapolation is tentative: nothing short of simulations resolved well past the Kolmogorov scale at such Reynolds numbers would settle it, and the highest $R_\lambda$ for which such direct numerical simulations (DNS) of homogeneous isotropic turbulence exist is 1300. Two further ingredients sharpen the question. First, such simulations are infeasible if every droplet is tracked individually; a 'superdroplet' method - a Monte-Carlo coarse-graining in which one computational particle represents many identical physical droplets, as used for stochastic coalescence by Dziekan and Pawlowska (Atmos. Chem. Phys. 17, 13509, 2017) - that faithfully reproduces caustic (multivalued-velocity) statistics, which coarse-grained schemes can destroy, has not yet been developed. Second, cloud droplets exchange heat and vapor with the flow: after centrifuging out of vortices they grow by condensation and release latent heat locally. A two-vortex model (Ravichandran and Govindarajan, J. Fluid Mech. 832, 745, 2017) showed this can change the flow itself when the product of the droplet-inertia and phase-change Stokes numbers is large, and the review conjectured a self-sustaining cycle in which condensation at void boundaries increases the population of small, intense, caustic-producing vortices, i.e. the intermittency on which the void-fraction estimate rests - making the extrapolated prediction and the feedback hypothesis one linked question about cloud turbulence at $R_\lambda \sim 10^4$.

Problem Statement

The problem, as posed by Ravichandran and Govindarajan (Physical Review Fluids 7, 110512, 2022), is to establish whether the single-vortex caustics picture of inertial droplets in turbulence survives to the Reynolds numbers of deep convective clouds, where it would transform rain initiation. Specifically, for a dilute suspension of small cloud droplets (Stokesian Maxey–Riley dynamics with droplet relaxation time $\tau$) in turbulent flow at Taylor-microscale Reynolds number up to $R_\lambda \sim 10^4$: (i) determine whether the collapse of the tails of the extreme-vorticity probability density, established only for $R_\lambda \le 1300$, persists another order of magnitude in $R_\lambda$; (ii) determine, correspondingly, whether the volume fraction of droplet-free voids created by centrifugal evacuation around intense sub-Kolmogorov vortices becomes $O(1)$, as the extrapolated estimate predicts, so that a significant fraction of droplets participate in caustic (sling) events capable of bridging the droplet-growth bottleneck and initiating rain (the review's proposition that rain initiation needs $R_\lambda > 10^3$); and (iii) determine whether phase change makes this regime self-sustaining - droplets centrifuged out of vortices growing by condensation and releasing latent heat at void boundaries, generating small-scale turbulence and a larger population of intense caustic-producing vortices. Because three-way coupled direct numerical simulation - Navier–Stokes turbulence resolved well below the Kolmogorov scale simultaneously with droplet dynamics (including caustics) and the thermodynamics of phase change - is required and is currently infeasible droplet-by-droplet at these Reynolds numbers, whether any coarse-grained representation of the droplet population preserves caustic (multivalued-velocity) statistics is unresolved.

The sub-questions are posed over the range of $R_\lambda$ that can be attained, and any statement about $R_\lambda \sim 10^4$ is an extrapolation from that range whose standing turns on whether a coarse-grained Lagrangian representation of the droplet population reproduces the caustic (multivalued-velocity) statistics of droplet-resolved direct numerical simulations of the same flow in a regime where both are feasible. Sub-question (i) is a claim about the flow's extreme-vorticity statistics alone, whereas (ii) and (iii) are claims about the droplet response and its feedback, so each is independently refutable; resolving any one of them, or establishing that a caustics-faithful coarse-grained method cannot be validated, is a genuine partial result rather than a resolution of the problem as a whole.

References

1. S. Ravichandran and R. Govindarajan, 'Waltz of tiny droplets and the flow they live in', Physical Review Fluids 7, 110512 (2022). DOI 10.1103/PhysRevFluids.7.110512, https://doi.org/10.1103/PhysRevFluids.7.110512 (preprint arXiv:2207.00540)
2. D. Buaria and K. R. Sreenivasan, 'Intermittency of turbulent velocity and scalar fields using three-dimensional local averaging', Physical Review Fluids 7, L072601 (2022). DOI 10.1103/PhysRevFluids.7.L072601, https://doi.org/10.1103/PhysRevFluids.7.L072601
3. D. Buaria and A. Pumir, 'Universality of extreme events in turbulent flows', Physical Review Fluids 10, L042601 (2025). DOI 10.1103/PhysRevFluids.10.L042601, https://doi.org/10.1103/PhysRevFluids.10.L042601 (preprint arXiv:2412.09801)
4. N. N. Makwana, B. Kumar, Y. P. M. Sethuraman and R. Govindarajan, 'Spatially Inhomogeneous Phase Change, and Its Effect on Cloud Turbulence', Geophysical Research Letters 52, e2025GL117378 (2025). DOI 10.1029/2025GL117378, https://doi.org/10.1029/2025GL117378
5. C. Rajarshi and R. Govindarajan, 'Caustics of finitely dense inertial particles', Physical Review Fluids 11, 074304 (2026). DOI 10.1103/fwkk-s3w4, https://doi.org/10.1103/fwkk-s3w4 (preprint arXiv:2601.14179)
6. L. A. Codispoti, D. W. Meyer and P. Jenny, 'Polydisperse collision kernels in droplet-laden turbulence with implications for rain formation', Physical Review Fluids 11, 064601 (2026). DOI 10.1103/kxxm-ghty, https://doi.org/10.1103/kxxm-ghty (preprint arXiv:2507.15326)
7. T. Oujia, K. Matsuda and K. Schneider, 'Divergence and convergence of inertial particles in high-Reynolds-number turbulence', Journal of Fluid Mechanics 905, A14 (2020). DOI 10.1017/jfm.2020.672, https://doi.org/10.1017/jfm.2020.672 (preprint arXiv:2005.00525)
8. S. Li, A. D. Bragg and G. Katul, 'Reduced Sediment Settling in Turbulent Flows Due To Basset History and Virtual Mass Effects', Geophysical Research Letters 50, e2023GL105810 (2023). DOI 10.1029/2023GL105810, https://doi.org/10.1029/2023GL105810
9. S. Olivieri, F. Picano, G. Sardina, D. Iudicone and L. Brandt, 'The effect of the Basset history force on particle clustering in homogeneous and isotropic turbulence', Physics of Fluids 26, 041704 (2014). DOI 10.1063/1.4871480, https://doi.org/10.1063/1.4871480
10. P. Dziekan and H. Pawlowska, 'Stochastic coalescence in Lagrangian cloud microphysics', Atmospheric Chemistry and Physics 17, 13509–13520 (2017). DOI 10.5194/acp-17-13509-2017, https://doi.org/10.5194/acp-17-13509-2017
11. S. Ganga Prasath, V. Vasan and R. Govindarajan, 'Accurate solution method for the Maxey–Riley equation, and the effects of Basset history', Journal of Fluid Mechanics 868, 428–460 (2019). DOI 10.1017/jfm.2019.194, https://doi.org/10.1017/jfm.2019.194
12. G. Falkovich, A. Fouxon and M. G. Stepanov, 'Acceleration of rain initiation by cloud turbulence', Nature 419, 151–154 (2002). DOI 10.1038/nature00983, https://doi.org/10.1038/nature00983
13. M. Wilkinson and B. Mehlig, 'Caustics in turbulent aerosols', Europhysics Letters 71, 186–192 (2005). DOI 10.1209/epl/i2004-10532-7, https://doi.org/10.1209/epl/i2004-10532-7
14. S. Ravichandran and R. Govindarajan, 'Vortex-dipole collapse induced by droplet inertia and phase change', Journal of Fluid Mechanics 832, 745–776 (2017). DOI 10.1017/jfm.2017.677, https://doi.org/10.1017/jfm.2017.677
15. S. Ravichandran, E. Meiburg and R. Govindarajan, 'Mammatus cloud formation by settling and evaporation', Journal of Fluid Mechanics 899, A27 (2020). DOI 10.1017/jfm.2020.439, https://doi.org/10.1017/jfm.2020.439
16. S. Ravichandran and R. Govindarajan, 'Instability driven by settling and evaporation in a shear flow: A model for asperitas clouds', Physical Review Fluids 7, 010501 (2022). DOI 10.1103/PhysRevFluids.7.010501, https://doi.org/10.1103/PhysRevFluids.7.010501